Handbook of Wind Energy Aerodynamics 3030313085, 9783030313081

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Bernhard Stoevesandt Gerard Schepers Peter Fuglsang Yuping Sun Editors

Handbook of Wind Energy Aerodynamics

Handbook of Wind Energy Aerodynamics

Bernhard Stoevesandt • Gerard Schepers • Peter Fuglsang • Yuping Sun Editors

Handbook of Wind Energy Aerodynamics With 678 Figures and 33 Tables

Editors Bernhard Stoevesandt Fraunhofer Institute for Wind Energy Systems Oldenburg, Germany

Gerard Schepers TNO Energy Transition Netherlands Organisation for Applied Sci, Petten The Netherlands Hanze University of Applied Science, Zernikelaan, Groningen, The Netherlands

Peter Fuglsang Siemens Gamesa Renewable Energy Brande, Denmark

Yuping Sun Goldwind Americas Chicago, IL,USA

ISBN 978-3-030-31306-7 ISBN 978-3-030-31307-4 (eBook) https://doi.org/10.1007/978-3-030-31307-4 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The purpose of a handbook is to cover the state of the art of a current topic – in this case the topic of wind energy aerodynamics – in a broader sense. When I was asked to create this handbook, it became instantly clear that this book should not be written by a single author – or even a small group of authors. Wind energy is still a rather young technology. To cover the relevant topics in this book, it was necessary to have a broader knowledge on the emerging technology and its issues. Since wind and its interaction with the turbine is, very complex and constantly changing, we have to face the fact that there is no correct way to deal with the aerodynamics directly, but several approaches. Wind itself is complex, and it is again influenced by turbines. As we really wanted a high-quality handbook, it became clear: We need the experts of the very specific fields to contribute. Now, experts are always very busy people. Contributing to such a book is nice, but might not be on top of the agenda in an expert’s life. So, in the end, it took us, the editors, a lot longer than expected to collect all contributions. Nevertheless, in the end, we are quite proud of the outcome. This handbook hopefully helps those who develop in wind energy in the future. By this we hope to foster the further development of wind energy itself. We are very thankful to Divya Nithyanandam, Swati Meherishi, Andrew Spencer, and the rest of the Springer Nature team for their patience and support to make all this possible. We would also like to thank all authors for their effort and patience throughout this very long but unique project. Further, we thank all reviewers who made this book possible and ensured its high quality. I do believe it was worth the effort! Oldenburg, Germany January, 2022

Bernhard Stoevesandt

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Contents

Volume 1 1

The Issue of Aerodynamics in Wind Energy . . . . . . . . . . . . . . . . . . . . Bernhard Stoevesandt, Gerard Schepers, Yuping Sun, and Peter Fuglsang

1

Part I Aerodynamic Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2

Economic Aspects of Wind Turbine Aerodynamics . . . . . . . . . . . . . . Panagiotis Chaviaropoulos

19

3

The Actuator Disc Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A. M. van Kuik

47

4

Airfoil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Bak

95

5

Rotor Blade Design, Number of Blades, Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giorgos Sieros

6

Blade Design with Passive Flow Control Technologies . . . . . . . . . . . Álvaro González-Salcedo, Alessandro Croce, Carlos Arce León, Christian Navid Nayeri, Daniel Baldacchino, Kisorthman Vimalakanthan, and Thanasis Barlas

Part II

123 151

Aerodynamic Modeling Techniques . . . . . . . . . . . . . . . . . . . . . .

203

7

History of Aerodynamic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerard J. W. van Bussel

205

8

Interacting Boundary Layer Methods and Applications . . . . . . . . . . Hüseyin Özdemir

253

9

CFD Simulations for Airfoil Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nilay Sezer-Uzol, O˘guz Uzol, and Ezgi Orbay-Akcengiz

303

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Contents

10

Dynamic Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricardo Santos Pereira

331

11

Thick Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francesco Grasso

353

12

The Effect of Add-Ons on Wind Turbine Blades . . . . . . . . . . . . . . . . Kristian Godsk

375

13

Pragmatic Models: BEM with Engineering Add-Ons . . . . . . . . . . . . Gerard Schepers

393

14

CFD for Wind Turbine Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . Elia Daniele

437

15

Aeroelastic Simulations Based on High-Fidelity CFD and CSD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Sayed, P. Bucher G. Guma, T. Lutz, and R. Wüchner

16

Aeroelastic Stability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jessica G. Holierhoek

Part III

17

Experimental Approaches to Wind Turbine Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Wind Tunnel Wall Corrections for Two-Dimensional Testing up to Large Angles of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. A. Timmer

491 559

599

601

18

Examples of Wind Tunnels for Testing Wind Turbine Airfoils . . . . . Özlem Ceyhan Yilmaz

631

19

Wind Tunnel Rotor Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koen Boorsma

659

20

3D Wind Tunnel Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alberto Zasso, Alessandro Fontanella, and Marco Belloli

687

21

Corrections and Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alois Peter Schaffarczyk

705

22

Doppler Lidar Inflow Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . Marijn Floris van Dooren

717

23

Load Measurements on Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . Malte Fredebohm and Nora Denecke

751

24

Surface Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scott Schreck

763

Contents

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Volume 2 Part IV

Aerodynamics and Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . .

803

25

Introduction to Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joachim Peinke, Matthias Wächter, and Raúl Bayoán Cal

805

26

Turbulent Inflow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Ehrich

833

27

Wind Shear and Wind Veer Effects on Wind Turbines . . . . . . . . . . . Julie K. Lundquist

859

28

Turbulence of Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ingrid Neunaber

881

Part V

Wind Farm Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

913

29

Wake Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Ivanell

915

30

Industrial Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonas Schmidt and Lukas Vollmer

927

31

Wake Meandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gunner Chr. Larsen

955

32

CFD-Type Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 Björn Witha

33

Wind Farm Cluster Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Martin Dörenkämper and Gerald Steinfeld

34

Wind Tunnel Testing of Wind Turbines and Farms . . . . . . . . . . . . . . 1077 Carlo L. Bottasso and Filippo Campagnolo

35

Wake Measurements with Lidar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127 Julia Gottschall

36

SAR Observations of Offshore Windfarm Wakes . . . . . . . . . . . . . . . 1145 Johannes Schulz-Stellenfleth and Bughsin Djath

37

Met Mast Measurements of Wind Turbine Wakes . . . . . . . . . . . . . . . 1179 J. W. Wagenaar

38

Aerodynamics of Wake Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 Jennifer King, Paul Fleming, Luis Martinez, Chris Bay, and Matt Churchfield

39

Optimizing Wind Farm Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 Xiao-Yu Tang

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Contents

Part VI Alternative Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251 40

Kites for Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 Paul Williams and Evgeniy Pechenik

41

Vertical-Axis Wind Turbine Aerodynamics . . . . . . . . . . . . . . . . . . . . . 1317 Delphine De Tavernier, Carlos Ferreira, and Anders Goude

Part VII

Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363

42

Wind Turbine Aerodynamic Noise Sources . . . . . . . . . . . . . . . . . . . . . 1365 Franck Bertagnolio and Andreas Fischer

43

Wind Turbine Noise Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 Matthew Cand, Andrew Bullmore, and Timothy Van Renterghem

44

Measuring and Analyzing Wind Turbine Noise . . . . . . . . . . . . . . . . . 1427 Bo Søndergaard, Tomas R. Hansen, and Stefan Oerlemans

45

Wind Turbine Noise Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447 Stefan Oerlemans

46

Direct Prediction of Flow Noise Around Airfoils Using an Adaptive Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . . . 1463 M. Grondeau and R. Deiterding

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487

About the Editors

Bernhard Stoevesandt is heading the department Aerodynamics CFD and Stochastic Dynamics at Fraunhofer Institute of Wind Energy and Energy System Technologies based in Oldenburg. He studied physics at the University of Bremen (Dipl. Phys., 2000). Afterwards, he worked as a freelancer in the field of X-ray lenses. In 2002, he joined the University of Oldenburg in the field of wind energy power forecasting. In 2004, he started his PhD at the university on turbulence in wind energy aerodynamics, finishing in 2010. In 2011, he joined Fraunhofer to soon become the head of the department. He has participated and continues to participate in the IEA Wind Task 29 and 31 projects focusing on wind fields and aerodynamics. A main interest of the work lies in the development of advanced techniques for aerodynamic and site simulations for wind turbines. The main developments have been in the field of numerical flow simulations for wind turbines.

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About the Editors

Gerard Schepers graduated in 1986 from the Faculty of Aerospace Engineering at TUDelft in the Netherlands. Since then, he has been working at the Energy Research Center of the Netherlands ECN (now TNO Energy Transition) in the field of wind energy. His main expertise is aerodynamics (and related areas), where he is active as researcher and coordinator of several large national and international projects. In 2012, he completed a PhD at TUDelft, which was based on the experiences gained in the field of aerodynamic engineering methods since the start of his career. Since 2018, he has also been working as Professor of Wind Energy at Hanze University of Applied Sciences, Groningen, the Netherlands, 2 days a week. Peter Fuglsang is head of blades R&D in the offshore business unit of Siemens Gamesa Renewable Energy. He graduated in 1994 as a mechanical engineer from Aalborg University, Denmark. From 1994 to 2004, he worked at Risø National Lab as a researcher in the fields of wind turbine aerodynamics, aeroelasticity, aeroacoustics, numerical optimization, cost modelling, wind tunnel testing, and wind turbine blade design. He has been the project manager of National and European collaborative research projects and involved in consultancy in wind turbine blade design and wind tunnel testing, supervision of PhD students, and teaching at the Technical University of Denmark. In 2004, he joined LM Wind Power A/S as manager of the aerodynamics team doing wind turbine blade design, airfoil design, and passive and active flow control, being responsible for strategy and innovation in the area. He was responsible for establishing the LM Wind Tunnel. In 2011, he joined Siemens Wind Power A/S and currently works as the head of blades R&D in the Siemens Gamesa Renewable Energy offshore business unit having the full responsibility for design, prototyping, certification, and validation of new blades and upgrades for Siemens Gamesa offshore wind turbines. He is an examiner in the Danish engineering education system. He has been opponent for several PhD defenses and has more than 10 peer-reviewed articles, numerous conference presentations and papers, as well as more than 20 granted patents.

About the Editors

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Yuping Sun graduated in 1994 with a PhD from the University of British Columbia, specializing in turbulent flow and heat transfer simulation and testing. His main achievements include proving of convergence theorem for finite analytic method, developing fractional finite analytic method for numerical solution of NavierStokes equations, developing modified turbulence nearwall model and then verifying through simulation, and testing for gas turbine blade film cooling. His work in aerospace and wind energy includes business jet external configuration design optimization, wind turbine blade airfoil design optimization for efficiency and roughness insensitivity, and various wind turbine blade optimization theories and applications. Over the years, he has published more than 10 journal papers, many more conference papers, and a few patents for wind turbines. He joined Vestas in 2009 as a principal engineer and led various projects involving wind turbine blade/airfoil aerodynamic design analysis. Currently, he works at Goldwind Americas as chief scientist responsible for wind turbine blade technology R&D and new blade development. He is one of the Chinese National Aerodynamic experts, recognized by Chinese central government.

Contributors

Carlos Arce León LM Wind Power, Schiphol, The Netherlands Christian Bak DTU Wind Energy, Roskilde, Denmark Daniel Baldacchino Delft, The Netherlands Thanasis Barlas DTU Wind Energy, Roskilde, Denmark Chris Bay National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, USA Raúl Bayoán Cal Department of Mechanical and Materials Engineering, Portland State University, Portland, OR, USA Marco Belloli Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy Franck Bertagnolio DTU Wind Energy, Roskilde, Denmark Koen Boorsma TNO Energy Transition Petten, The Hague, The Netherlands Carlo L. Bottasso Technical University of Munich (TUM), Garching b. München, Germany P. Bucher Structural Analysis, Technical University of Munich (TUM), Munich, Germany Andrew Bullmore Hoare Lea, Bristol, UK Filippo Campagnolo Technical University of Munich (TUM), München, Germany

Garching b.

Matthew Cand Hoare Lea, Bristol, UK Özlem Ceyhan Yilmaz Sirris, Heverlee, Belgium Panagiotis Chaviaropoulos iWind Renewables P.C., Athens, Greece Matt Churchfield National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, USA Alessandro Croce Department of Aerospace Science and Technology, Politecnico di Milano, Milano, Italy xv

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Contributors

Elia Daniele TPI Composites Germany GmbH, Berlin, Germany R. Deiterding Aerodynamics and Flight Mechanics Research Group, University of Southampton, Boldrewood Innovation Campus, Southampton, UK Nora Denecke Fraunhofer IWES, Bremerhaven, Germany Delphine De Tavernier Department of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Bughsin Djath Helmholtz-Zentrum Hereon, Institute of Coastal Systems – Analysis and Modeling, Geesthacht, Germany Martin Dörenkämper Fraunhofer Institute for Wind Energy Systems (IWES), Oldenburg, Germany Sebastian Ehrich Department of Physics, University of Oldenburg, Oldenburg, Germany Carlos Ferreira Department of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Andreas Fischer DTU Wind Energy, Roskilde, Denmark Paul Fleming National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, USA Alessandro Fontanella Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy Malte Fredebohm Fraunhofer IWES, Bremerhaven, Germany Peter Fuglsang Siemens Gamesa Renewable Energy, Brande, Denmark Kristian Godsk Vestas Wind System, Ry, Denmark Álvaro González-Salcedo Wind Energy Department, CENER, Sarriguren, Spain Julia Gottschall Fraunhofer Institute for Wind Energy Systems IWES, Bremerhaven, Germany Anders Goude Department of Electrical Engineering, Uppsala University, Uppsala, Sweden Francesco Grasso Aerodynamics and Acoustics, Vestas Blades Technology UK LTD, Newport, UK M. Grondeau Aerodynamics and Flight Mechanics Research Group, University of Southampton, Boldrewood Innovation Campus, Southampton, UK G. Guma Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany Tomas R. Hansen Siemens Gamesa Renewable Energy, Brande, Denmark

Contributors

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Jessica G. Holierhoek JEHO BV, Rotterdam, The Netherlands Stefan Ivanell Section of Wind Energy, Department of Earth Sciences, Uppsala University, Campus Gotland, Visby, Sweden Jennifer King National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, USA Gunner Chr. Larsen DTU Wind Energy, DTU, Risø Campus, Roskilde, Denmark Julie K. Lundquist ATOC, University of Colorado Boulder, Boulder, CO, USA T. Lutz Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany Luis Martinez National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, USA Christian Navid Nayeri Hermann-Föttinger Institute, Technische Universität Berlin, Berlin, Germany Ingrid Neunaber LHEEA (CNRS) – École Centrale de Nantes, Nantes, France Institute of Physics and ForWind – University of Oldenburg, Oldenburg, Germany Stefan Oerlemans Siemens Gamesa Renewable Energy, Brande, Denmark Ezgi Orbay-Akcengiz Department of Aerospace Engineering, Middle East Technical University (METU), METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey Hüseyin Özdemir TNO Energy Transition, Petten, The Netherlands Evgeniy Pechenik Ampyx Power, The Hague, The Netherlands Joachim Peinke ForWind, Institute of Physics, University of Oldenburg, Oldenburg, Germany Ricardo Santos Pereira Faculty of Aerospace, Delft University of Technology, Delft, The Netherlands M. Sayed MesH Engineering GmbH, Stuttgart, Germany Alois Peter Schaffarczyk Mechanical Enginering, Kiel University of Applied Sciences, Kiel, Germany Gerard Schepers TNO Energy Transition, Netherlands Organisation for Applied Sci, Petten, The Netherlands Hanze University of Applied Science, Zernikelaan, Groningen, The Netherlands Jonas Schmidt Aerodynamics, CFD and Stochastic Dynamics, Fraunhofer IWES, Oldenburg, Germany

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Contributors

Scott Schreck Siemens Gamesa Renewable Energy, Conceptual Blade Design Department (SGRE OF TE TD BL CBD AT), Boulder, CO, USA Johannes Schulz-Stellenfleth Helmholtz-Zentrum Hereon, Institute of Coastal Systems – Analysis and Modeling, Geesthacht, Germany Nilay Sezer-Uzol Department of Aerospace Engineering, Computational Aerodynamics Lab, METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey Giorgos Sieros iWind Renewables, Gerakas, Greece Bo Søndergaard SWECO, Aarhus, Denmark Gerald Steinfeld ForWind, Carl von Ossietzky University Oldenburg, Oldenburg, Germany Bernhard Stoevesandt Fraunhofer Institute for Wind Energy Systems, Oldenburg, Germany Xiao-Yu Tang College of Control Science and Engineering, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, China W. A. Timmer Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands O˘guz Uzol Department of Aerospace Engineering, METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey Gerard J. W. van Bussel Delft University of Technology, Delft, The Netherlands Marijn Floris van Dooren Research Group Wind Energy Systems, ForWind – University of Oldenburg, Oldenburg, Germany G. A. M. van Kuik Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Timothy Van Renterghem Ghent University, Ghent, Belgium Kisorthman Vimalakanthan Wind Energy Department, ECN>TNO, Petten, The Netherlands Lukas Vollmer Aerodynamics, CFD and Stochastic Dynamics, Fraunhofer IWES, Oldenburg, Germany Matthias Wächter ForWind, Institute of Physics, University of Oldenburg, Oldenburg, Germany J. W. Wagenaar TNO Wind Energy, Petten, The Netherlands Paul Williams Ampyx Power, The Hague, The Netherlands Björn Witha ForWind, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany Energy & Meteo Systems GmbH, Oldenburg, Germany

Contributors

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R. Wüchner Institut für Statik und Dynamik, Technische Universität Braunschweig, Braunschweig, Germany Yuping Sun Goldwind Americas, Chicago, IL, USA Alberto Zasso Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy

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The Issue of Aerodynamics in Wind Energy Bernhard Stoevesandt, Gerard Schepers, Yuping Sun, and Peter Fuglsang

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring Wind Turbine Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind and Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Turbine or Wind Farm Aerodynamics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustics of Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 5 8 10 11 14 15 16 16

B. Stoevesandt () Fraunhofer Institute for Wind Energy Systems, Oldenburg, Germany e-mail: [email protected] G. Schepers TNO Energy Transition, Netherlands Organisation for Applied Sci, Petten, The Netherlands Hanze University of Applied Science, Zernikelaan, Groningen, The Netherlands e-mail: [email protected] Y. Sun Goldwind Americas, Chicago, IL, USA e-mail: [email protected] P. Fuglsang Siemens Gamesa Renewable Energy, Brande, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_76

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Abstract

Wind turbine aerodynamics remains to be a central field in the development of wind turbines. It comprises several aspects, and methods to be understood really make the best use of it in today’s wind turbine and wind farm development. This chapter gives an overview of all aspects in wind energy aerodynamics covered within this book and the reasons why these aspects are covered. This way it already gives an overview for developers on all the issues wind turbine might face, when dealing with topics related to wind turbine aerodynamics. Keywords

Wind · Energy · Aerodynamics · Wakes · Acoustics · Blade Design

Introduction Wind turbines are machines driven by aerodynamics. The aerodynamics describe the interaction of the wind with the wind turbines and what this results in. This makes the aerodynamics and its related topics a central field for wind turbine design and research. The second striking issue of wind energy is the dependency on the outer conditions. While in most other fields of engineering, the outer system driving conditions are to some extend controlled, this is not true for the wind conditions. This makes wind energy aerodynamics a challenging and still extending field, we as editors hope all our authors are still excited about. Even, if the general principles of the aerodynamics of wind turbines have already been covered by other fields, such as helicopter aerodynamics, the aerodynamics of airplanes or ship propellers, the issue of the ambient conditions and the pure size of the rotors necessary for efficient extraction of electrical energy have made wind energy aerodynamics a very unique field. While helicopters and propellers of airplanes also have similar rotational features and problems evolving from this characteristic, they operate under different conditions: • For helicopters or airplanes it makes sense to limit the size of the rotor, for wind turbines a larger rotor area means a larger covered area from which the energy can be extracted from the wind. • Due to the limit in size of the rotor, rotor blades are also limited in costs. Thus, it might be efficient to use more rotor blades. In wind energy, the costs of the blades make up a decisive cost of the full turbines. To reduce the number of blades is therefore advantageous. • Although noise is also an issue for helicopters and airplanes, it is a question of compatibility for onshore wind turbines. Helicopters and airplanes come and go, while wind turbines are fixed to the ground. Their noise level is much stricter. This leads to the situation that onshore turbines have a very limited tip speed ratio.

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• To achieve a similar efficiency, the dimensioning of the blades need to be quite different for wind turbines. This again leads to several issues with the approaches applied in traditional aeronautics. • As additional complication, the ambient condition of the turbulent, sheared flow has a far larger impact on the aerodynamics. • Calculation time is a much more important constraint in wind turbine technology than it is in other areas. This is due to the fact that a wind turbine needs to survive 20 years without the option of taking it apart for maintenance. So the statistics of the wind over 20 years should be simulated in turbine design. This is done by performing a huge number of load calculations, which tend to be quite special. Although a lot of the original researchers and engineers in wind energy came from traditional aeronautics, today, wind turbine aerodynamics has become a field of its own. As a result wind turbine aerodynamics has generated a lot of interest in research and industry. This has led, for example, to one of the most continuous IEA Wind research task, which focuses on wind energy aerodynamics with a very broad field of participants (Schepers et al. 2012; Boorsma et al. 2018).

Blade Design About until the year 2000 the aerodynamic design of rotor blades was done more or less straight forward as described in many textbooks on wind energy or wind energy aerodynamics (Gasch and Twele 2011; Burton et al. 2001; Hansen 2015; Schaffarczyk 2020). A lot of the approaches described have been developed over the preceding century in aeronautics, wind energy, or hydrology. Up to this time wind turbine blades have been constructed in a rigid form, strongly based on a rotating, yet two-dimensional concept, based on a simplified actuator disc theory. Since it was already known, that this was not suitable for the actual physics of the flow, some corrections for three-dimensional effects have already been added at this time. Since the rotors were smaller, the simplifications of a homogeneous inflow condition were not too far off the real conditions in most cases. And for more difficult load cases it was possible to also define approaches which were sufficiently good. This, however, changed rapidly, when wind energy started to become an energy source competing with other energy sources. The industry grew; competition and new materials enabled new approaches to and new sizes of rotor blades. There has now been a discussion for decades on the possible limit of the size of the horizontal axis wind turbine concept of today (de Vries 2008). Until now, this discussion has remained theoretical and the sizes of the rotors keep growing. On the other side this brings ever new challenges in the design of the blades. These challenges are mostly a mixture of aerodynamic, structural, and control topics. The aim has always been to reduce the weight of the blades in relation the blade length. On the other side the larger rotors pose a challenge in the structure due to the necessary twist of the blade. The large rotor diameters face a stronger impact of a

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nonhomogeneous inflow. Shear and veer play an increasing role after the rotor sizes have doubled in the 20 years after 2000. Additionally, there is a necessary focus on the manufacturing cost, which leads to compromises concerning the aerodynamic shape. All of this has led to a steep increase in aerodynamic research and development for wind turbines. To cover the most important and promising developments in depth, we have asked specialists of the specific topics to contribute to this book. One thing that has not changed over the whole period of discussions in wind energy aerodynamics are the basic principles of physics. One of the basic rules, which still apply, is the so-called Betz law, defining the maximum of possible energy extracted within an plane which is passed by the wind. The theory, which is based on an assumed disc, which is actuating the flow in a specific way, is still the base of all aerodynamics for wind turbines. It gives all people active in wind energy a physical base on a simple assumption on which things could be possible, and which are out of the physical bounds. Still, there is a discussion on this theory. It has been improved and developed in recent years. Since it is the base of everything else, the state of this discussion is the  Chap. 3, “The Actuator Disc Concept” of this book and in a way the conclusion of the great contribution to wind turbine aerodynamics of Gijs van Kuik. This chapter provides already a deep understanding of wind turbine aerodynamics as such. The most striking thing on the development is that the main driver for the development is not within a technical field: Since wind energy has become a competitive form of energy production, economics plays a crucial role in the development of wind turbine blades. The design of the blades of today has to undergo several economic considerations, as discussed by Panagiotis Chaviaropoulos in his chapter on the economic aspects of blade design  Chap. 2, “Economic Aspects of Wind Turbine Aerodynamics”. A blade design might differ considerably, depending on the targeted market, even if the rated power or targeted rotor diameter might be the same between two blades. Nevertheless, the pure size of the blades of today poses questions to the blade designers. Using aerodynamics to cope with the challenges is a very attractive approach. If it is possible to use aerodynamic mechanisms to cope with the all of the mentioned problems and aspects in turbine design, a lot of issues in the design of the blade structure, tower, generator, or control are strongly reduced. If it is possible to change the aerodynamic properties to fulfill all the needs for the complete turbine design, blade designers would be happy. Giorgos Siros describes therefore the general approach to blade design today in  Chap. 5, “Rotor Blade Design, Number of Blades, Performance Characteristics”. Of course blade design is not purely driven by aerodynamics. In contrary: It is the design driver of the aerodynamics. The design defines the need for specific aerodynamic properties. The question on how to obtain these properties can be approached in two ways. The traditional and most preferable way is the design of better airfoils, which fulfill all the desired needs. How this is currently done is described by Christian Bak in  Chap. 4, “Airfoil Design”. Coping with the aerodynamic issues by optimal airfoils, with the right properties is nice, since they only form the blade – nothing additional needs to be considered. Thus airfoil optimization for the right aerodynamic properties is always the first choice in blade design.

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In recent years the challenges of blade design have increased so much that most blade manufacturers have opted to use additional aerodynamic devices on the blades to influence the aerodynamic or aeroacoustic characteristics of the blade. Most of the time these devices are not an integral part of the blade, but added after the base design of the blade to influence the characteristics in the desired manner. Thus these devices are often called “add-ons.” There is a broad variety of such devices for blades. Most of them are passive, which means the device is without any active control mechanism, which would add to the complexity of turbine control and could be prone to failures. There are at least in research also some devices discussed comprising control mechanisms. Such devices add options for the developers in blade design, when they have to consider their aerodynamics. For  Chap. 6, “Blade Design with Passive Flow Control Technologies” Álvaro Gonzáles-Salcedo gathered a team of experts to write about the options blade designers have today and which effects they can expect, when choosing the specific one of the options. In the end, there have been several design drivers for the development of blade design. Some of them have been technical, since new developments made new design possible. Some of them are of social manner – first of all there is the economic aspect. Nevertheless, noise of the wind turbines, especially in case of onshore turbines, has become one of the main sources of social concerns about wind energy. The aeroacoustic noise of the blades is currently still the main source for wind turbine noise. It is inherently connected to the aerodynamic properties of the blades and functions as a constraint in the blade design, if noise has to be considered. This is the reason why we dedicated a whole section with several chapters to this topic in the last section of this book.

Model Issues Today’s design of the blades would not have been possible if the methods and models of the design would have stayed the same over the last decades. This is not the case. Even if the base of the design is still very similar, in detail, the standard models have made large steps of improvements. However, it remains challenging to decide when to use what kind of model for which purpose. This often needs a lot of experience and background knowledge: which model is capable to provide what type of information under which accuracy. Error bars in wind turbine modeling are still rare. Thus, the specialists using the models have to know the strengths and limits of their models to come up with a reliable idea of what a useful information is, they gain by their calculation. For this reason the second section is dedicated to the relevant aspects of aerodynamic modeling for wind turbines. To give an understanding on how the different models evolved Gerard van Bussel gives an overview in  Chap. 7, “History of Aerodynamic Modelling” on the historical development of the aerodynamic modeling for wind turbines. Even if there have been ancient wind turbines constructed before, the systematic modeling of wind turbines first became possible in the nineteenth century. With the development of the Navier-Stokes equations, which describe the full flow physics, a complete

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description would even have been possible – if the momentum equation would have an analytical solution. However, the question if there even exists a solution remains a millennium question of mathematics and is not answered. This is why engineering model approximations have taken over in aerodynamics over the time as described in this chapter, many of them still in use today. A lot of these models are still based on the aerodynamic characteristics of twodimensional aerodynamically formed blade sections called airfoils. Such airfoils are assumed to be homogeneous in the spanwise direction. The advantage of this approach is that the aerodynamic properties of the airfoils can be more easily gained in wind tunnel experiments. Since such experiments are expensive, numerical models for such airfoils are of large importance. The most used and very convenient way to do this is to use viscous-inviscid boundary layer methods to calculate the aerodynamics. Hüseyin Özdemir has described the method and its applications in  Chap. 8, “Interacting Boundary Layer Methods and Applications”. While the viscous-inviscid method is quick and useful for many application, still it is a simplification of the full flow physics in the Navier-Stokes equations. In the last 30 years the more direct approach to solve these equations numerically, with some modeling for the turbulence has strongly evolved as an alternative. These socalled computational fluid dynamics (CFD) methods have broadly been used also for airfoil aerodynamics computations, even though they are more computationally expensive. Nilay Sezer-Uzol, O˘guz Uzol and Ezgi Orbay-Akcengiz have compiled a chapter on the use of CFD for airfoil simulations. The pitfalls and strengths of the approach are discussed especially showing the importance of turbulence to the accuracy of the results. While in other field of aeronautics the given methods would already be sufficient, their use in wind energy often needs additional know-how for their application. This is mainly due to the special complexity wind turbine blades face because of their structure or the unsteady environmental conditions. The latter might lead in a 2D view to the unpleasant phenomenon of dynamic stall. This phenomenon is also known in helicopter aerodynamics. However, Ricardo Santos Pereira discusses dynamic stall in  Chap. 10, “Dynamic Stall” from the point of view of wind turbines. The other very wind turbine specific problem in modeling airfoil aerodynamics correctly is the structurally conditioned matter of very thick airfoils necessary in the blade design. In the end, toward the hub, all blades have to converge to a cylindrical shape. The cylinder is optimal for structural reasons and is required due to the interface to the blade bearing. In terms of the aerodynamics, it is, however, more or less already a bluff body. The modeling and simulations of such thick airfoils thus remains a challenge, since flow separation plays a crucial role in the aerodynamics. Francesco Grasso describes in  Chap. 11, “Thick Sections” how to approach this issue best. Another one of these issues is the use of aerodynamic devices as described already in the blade design section. Codes relying on 2D aerodynamic information will need a correction for such add-ons, although the flow at most of such add-on devices is not two dimensional at all. This has been taken up by Kristian Godsk in

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 Chap. 12, “The Effect of Add-Ons on Wind Turbine Blades” on the modeling of

add-ons for wind turbines. All of the 2D airfoil information is then needed and most used in the calculations of the aerodynamics in codes for the load calculations of wind turbines. Such mostly aeroelastic codes are the base of all wind turbine designs. They are meant to provide information on the durability of wind turbines according to the complete set of load cases corresponding to the full life time as described by the IEC standard 61400-1 (IEC 2019). The codes providing such information need to be fast and yet reliable. Most of them are based on the blade element momentum theory, which has undergone several improvements or extensions by other methods as Gerard Schepers describes in  Chap. 13, “Pragmatic Models: BEM with Engineering Add-Ons”. So in the end even the old approach of the blade element momentum theory has not remained the same over the last decades. With the use of CFD, on the other hand, the natural question comes up: Why such 2D-based models could not be replaced? CFD offers in general the option to simulate the full flow and aerodynamics as a three-dimensional phenomenon. While in the year 2000 this was rarely done, it is a method applied frequently today in research and industry. However, CFD has its obstacles. This is why Elia Daniele described the applicability of CFD to wind turbine blades and full rotor aerodynamics in  Chap. 14, “CFD for Wind Turbine Simulations”. He shows the challenges and advantages of the method by discussing the approaches and results by several published works of different groups. By this an impression can be gained on the advantage, but also the pitfalls of CFD for wind turbine aerodynamics, which still remains a computationally expensive task. Even more computational expensive becomes the method, when even the flexibility of the wind turbines is taken into account. As mentioned, since the era of the wind turbines around the year 2000, the structures of wind turbine blades have become more and more flexible. This has large advantages in the aerodynamics and loads for the wind turbines. However, the flow over the deflected blade is again diverging from the originally assumed two-dimensional structure. In this case, CFD is of course a wonderful option to gain as much information as possible around the blade. On the other side poses the movement of the structure another challenge to the CFD. To describe the methods and their success Mohamed Sayed has had the lead of a group of researchers from the University of Stuttgart and TU-Munich in  Chap. 15, “Aeroelastic Simulations Based on High-Fidelity CFD and CSD Models”. In this chapter the methods of fluid structure interaction (FSI) in a CFD framework are discussed in detail as the current state of the art. This could be the end of the story of aerodynamic modeling, if there was not the issue of the computational effort connected to full CFD-FSI simulations would not be so prohibitively large. The flexibility of the turbines necessitates investigations on the vibrations and aeroelastic stability of wind turbines. Such computations need often long time series data or direct spectral analysis codes. To compute this for all possible load cases with CFD-FSI based methods is far out of reach. This is why Jessica Holierhoek has added  Chap. 16, “Aeroelastic Stability Models” on current approaches of aeroelastic stability models.

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Measuring Wind Turbine Aerodynamics Now, after so much progress in the modeling of wind turbine aerodynamics, why do people still do measurements? This is a question that could be asked from people who are not familiar with the struggles involved with modeling. Even fully resolved CFD simulations of wind turbines still have to model the turbulence to a certain account. However, in terms of the aerodynamics, this account is decisive: The turbulence and the transition to turbulence can have a major impact on the pressure distribution on the blades. This is even more the case, if flow separation is involved. The separation point on the blade decides strongly on the pressure, thus the aerodynamic characteristics in the specific situation. Other approaches then fully resolved CFD simulations contain more model portions. This might not necessarily lead to worse results, but the reliability of the models remains limited. Engineers using wind turbine load models based on aerodynamic airfoil characteristics often try to overcome this by using airfoil measurements from wind tunnels instead of purely modeled data. This has the advantage of a hopefully reproducible result in controlled conditions. Until today most airfoils before being used in wind turbine design actually are being tested under wind tunnel conditions. The issue with such wind tunnel airfoil experiments is that wind tunnel differs a lot from another. Airfoils are usually tested in closed test sections. Even by this constraint, the closed test section wind tunnels differ in decisive form: The size of the cross section, inflow wind speeds, or turbulence intensity are some of the aspects that need to be taken care of. One main issue for wind turbine applications is, that it is desired, that the Reynolds number is as high as possible, since the Reynolds numbers at today’s wind turbine blades easily reach the order of 107 . On the other side, the Mach number is to be kept low, since stronger compressibility effects are prone to noise generation and thus avoided in wind turbine design. Ozlem Ceyhan Yilmaz gives in  Chap. 18, “Examples of Wind Tunnels for Testing Wind Turbine Airfoils” an overview on wind tunnels and their approaches to airfoil measurements, keeping in mind that even if a bigger wind tunnel could be considered better, the costs of such experiments also increase with the capabilities of most wind tunnels. Even though it might sound simple to measure the aerodynamic characteristics of the airfoils, in detail it is again quite complex. In the end wind speed and pressure or even accumulated forces might be measured in the wind tunnel. However, already from the Navier-Stokes equations it can be estimated by their hyperbolic characteristics that every obstacle in the experiment will have an influence on the result. A closed test section with walls might be already such an obstacle, which needs to be considered in the interpretation of the results. This phenomenon called blockage is the topic of Nando Timmer who gives in  Chap. 17, “Wind Tunnel Wall Corrections for Two-Dimensional Testing up to Large Angles of Attack” an insight on how to deal with this issue in the experiments. While there is a very straightforward practical interest in measurements of airfoil aerodynamics, a lot of modeling problems arise due to the rotational movement of the wind turbine. Thus it is tempting to also do experiments with full wind turbine

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rotors in wind tunnels. The limit in size of the wind tunnels introduces severe scaling effect issues. There just isn’t a wind tunnel close to the size of commercial wind turbines of today. The scaled model wind turbines in wind tunnel experiments thus need to be well designed to improve the knowledge on the wind turbines. Such experiments might pose as validation cases for simulations, since they are run under controlled conditions, or might be used to answer specific questions for problem only found on rotational systems. Due to the scaling such experiments pose a difficult task. Koen Boorsma has provided an introduction to this task in  Chap. 19, “Wind Tunnel Rotor Measurements”. One of the main issues for wind tunnel experiments with full turbines is the design of the actual scaled turbine, which is still supposed to feature as many characteristics of the large commercial wind turbines as possible. Although it might not be possible to reach the same Reynolds numbers, at least some characteristic features of the large turbines can be imitated by the models. This can even go as far as to floating turbines like Alberto Zasso, Alessandro Fontanella, and Marco Belloni discuss in  Chap. 20, “3D Wind Tunnel Experiments”. On the other side, wind tunnel experiments with full rotors face the issue of the influence of the turbine on the flow conditions itself as well as other issues, which cause uncertainties. This is a special field discussed by Peter Schaffarczyk in  Chap. 21, “Corrections and Uncertainties”. This shows even in controlled conditions measuring wind turbine aerodynamics remains a challenge. This is even more true when field measurements are taken into account. In the field, the issue of the scale of the turbines might not play a crucial role anymore. Any type of turbine, if accessible, could be considered for aerodynamic measurements. One of the main issues in this case is the question, under which condition the actual measurement is taking place? The incoming wind is most likely not homogeneous or even laminar, but sheared, maybe veered, highly unsteady, and turbulent. This will have effects on the aerodynamics. Thus it is desirable to measure the wind conditions as good as possible. Marijn van Dooren describes in  Chap. 22, “Doppler Lidar Inflow Measurements” how this can be achieved by lidar measurements. Another indicator for the aerodynamics of the wind turbine, which is also not so prone to errors, is to measure the loads on the turbine. The loads on turbine and blades give information on integrated forces on the blades, which helps to overcome the uncertainties of the wind field to some extent. The averaged results can already help gain an estimation on the aerodynamics. How this is done is described by Malte Fredebohm and Nora Denecke in  Chap. 23, “Load Measurements on Wind Turbines”. For many aspects in wind turbine aerodynamics, however, such integrated information is not enough. To gain real insight into the actual dynamic aspects of the local aerodynamics on wind turbine blades, pressure measurements are needed. These measurement techniques are difficult to accomplish, since these neither need wholes in the blades or additional fixation techniques for the pressure sensors. Both is tricky: Wholes n the blades are unpopular with wind turbine owners and lead to an exchange of the blades in the long run. Additional fixation techniques might

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influence the flow field, which is to be measured. Thus such measurements are rare. Scott Schreck is one of the few persons with a long-term experience in such measurements. He describes in  Chap. 24, “Surface Pressure Measurements” the approaches for such measurements and some of the experiments that took place over the last decades.

Wind and Aerodynamics Result from issues with the wind turbine aerodynamics and a lot of the issues of wind turbine loads result of issues with the wind. While the turbine and its behavior can be to a certain extend understood and analyzed, this can hardly be said about wind. Wind is turbulent, unsteady, and differs depending on a lot of factors like latitude, thermal stratification, site-specific surrounding, or the existence and shape of weather systems or phenomenon (Landberg 2015). For the aerodynamics of the wind turbines, it is crucial to understand and describe the wind field, as it drives the turbines and the turbines extract energy from it. Generally, the wind is driven on large scales by pressure differences between different latitudes on the globe (Emeis 2018). These differences lead, driven by the coriolis force, to weather systems further away north and south of the equator. All of this happens in altitudes far away from ground-based wind energy generation. These operate currently in heights below 300 m most of the time, but not necessarily always, within the planetary boundary layer (PBL) (Emeis 2018). This boundary layer is characterized by a turbulent sheared flow. Sometimes the turbines even face severe veer conditions. The larger the rotors become, the bigger this problem develops to be. Julie Lundquist has investigated this topic and summarized the findings in  Chap. 27, “Wind Shear and Wind Veer Effects on Wind Turbines” to give a comprehensive overview on shear and veer for wind turbines. The shear and veer within the wind, but also the interaction of the wind with all kind of obstacles it encounters, leads to the generation of turbulence. Of the momentum equation of the Navier-Stokes equations, turbulence can be considered to be generated by the nonlinear terms (Pope 2000). For the calculation of the flow, this is not preferable, but unavoidable. Basically, this means that the flow can only be described by statistical approaches, and specific characteristics – or by fargoing stochastic approach as Joachim Peinke, Matthias Wächter and Raúl Bayoán Cal explain in  Chap. 25, “Introduction to Turbulence” on the characteristics of turbulence and their description. For all kinds of aerodynamic simulations, however, it becomes more and more important to use turbulent inflow. There are standard methods to do this, as specified in the IEC-Standard 61400-1-4 (IEC 2019), but the mentioned models pose also a reduced description of real wind characteristics. A further issue arises, when the models are to be applied in CFD simulations: The CFD field will transport the turbulence according to the mathematics within the used approach. This might change and distort the turbulence properties. Sebastian Ehrich has gathered an overview over current methods on turbulent inflow generation in

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 Chap. 26, “Turbulent Inflow Models”. He has made it clear that this field will

remain to be a field of further research. One additional aspect of turbulence is specific for wind energy: Wind turbines generate turbulence, which is in wind farms part of the inflow of other turbines, which again modify the turbulence. Since the turbulence generated at an object is close to the object still characterized by the object, the characteristics of the turbulence in the wakes of wind turbines is of importance in wind energy. A description for such turbulence is given in  Chap. 28, “Turbulence of Wakes” by Ingrid Neunaber. As she describes, still this field remains open for a lot of research.

Wind Turbine or Wind Farm Aerodynamics? The turbulence of wakes is even more relevant, since wind energy is most of the time not anymore generated by single wind turbines, but in wind farms. It is more efficient to plan the construction of a complete wind farm, than to place single turbines everywhere in the landscape. The latter would most likely not contribute to a favorable popular opinion as well. Most of the time, the space to place wind turbines is also limited. So the question is in the end: How to gain the largest energy yield from a certain area? This also changed the focus of wind turbine design. Today, it might not be preferable to gain the most energy from one turbine. If the induction of a single turbine is reduced, it might be possible to place additional turbines closer to another in the area of the envisioned site. This broadens the perspective of the aerodynamics of wind energy of today. The aerodynamics of complete wind farms come to play a more and more important role in wind energy. Due to this, more aspects have to be considered in the aerodynamics, due to site conditions. On the other side the scales to be simulated or modeled increase again considerably. This poses new challenges to the developers, who use additional approaches to cope with them. The most used and straightforward approach to consider the effect of wind turbines on the flow in wind farms is to use a more or less simplified model. There are numerous models. Most of them describe the development of wake deficits. Especially models used for industrial applications to calculate for example wind farm designs in a quick, yet accurate manner need to generalize by using few aerodynamic parameters of the single wind turbine to come up with the best wind field in the wakes possible. A broad collection of such models has been gathered and discussed by Jonas Schmidt in  Chap. 30, “Industrial Wake Models” to give an idea on what these models do and what they are capable of achieving. One issue of such models is that they need to average and sometimes homogenize the flow characteristics behind the turbines. However, the closer the wake flow is still at the turbine, the less homogeneous it is. This is why the flow field in wakes is often differentiated between the near wake and the far wake, where a more homogeneous wake deficit can be considered. The near wake on the other side needs to be described in a different way, since it might still be characterized by tip and root vortex. Stefan Ivanell gives in  Chap. 29, “Wake Structures” a detailed insight into the near wake characteristics.

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On the other side, in far wakes also the flow is not as steady and homogeneous as the industrial models might suggest: Due to the inflow turbulence, the flow in the far wake of the turbines tends to move around. This effect is called meandering. The effect is quite important to the specific loads of the turbines, since they might be influenced by the short-term changes in the flow field and not just the average field. To cope with this, models for wake meandering have been developed. Gunner Larsen gives an insight into the phenomenon, its physics, the mathematical description, and the approach to model it in  Chap. 31, “Wake Meandering”. Another approach to the wake modeling is again the use of CFD methods. Although it might theoretically be possible to simulate the fully resolved wind turbine with its wake in CFD, this is still prohibitively computational expensive. For wind farm simulations, this is for most applicants out of scope. Thus other techniques have been developed for the CFD framework to generate the effects of wind turbine wakes in the CFD flows. The methods and their applications are described in  Chap. 32, “CFD-Type Wake Models” by Björn Witha to provide a comprehensive overview. In recent years, as wind farms increase in size, further aspects have become point of consideration in the discussion of wakes: If the farms are so large, how do the wakes of the farms behave, which in a way merge to one big farm wake like structure after to farm? How far do the influence the surrounding? How can this be modeled or simulated? Again to investigate this, different approaches need to be taken, since the scales to be taken into account increase considerably. These are the questions Martin Dörenkämper and Gerald Steinfeld discuss in  Chap. 33, “Wind Farm Cluster Wakes”. Now, one might ask, how about the validity of all these models? This is the most important question to all the models. It brings back the issue of measurements. While the measurements of single wind turbines have been difficult already, the scales of wakes, multiple wakes, or even wind farms pose another hard challenge. Again the approaches to tackle it are to some extend similar. The most reproducible and therefore accurately measurable approach is to use wind tunnel experiments. However, the wind tunnels in use for such experiments do have to mimic the conditions of the wind in the field with a sheared inflow. Also, the experiments need to be able to measure the wakes several rotor diameters behind the actual rotor to give a good indication of the wakes behind the rotor. This leads to a situation in which either the turbine models in the wind tunnel need to be extremely small, or the wind tunnels are extremely large. Wind turbines to be used for such purposes need to somehow have meaningful characteristics to be comparable to real field turbines. In  Chap. 34, “Wind Tunnel Testing of Wind Turbines and Farms” Carlo Bottasso describes how this can be approached and successfully managed in wind tunnel experiments. It is also possible to measure wakes in field experiments, although the conditions are far more difficult. The main issue is usually the lack of information on the flow field. The wind turbine itself can be used as one measuring point: Power and wind measurements can be used to gain an estimate of the flow at the turbine. If, by chance, there is a remaining met mast near that turbine, this data can be used to gather information on the shape of the wake of the turbine at a specific distance.

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However, this is not trivial. There is a time lag between turbine and the mast, the incoming flow to the turbine is hardly ever one directional and depending on the direction, and there is also some distortion in the signal of the met mast. How this is nevertheless done is discussed by Jan Willem Wagenaar in  Chap. 37, “Met Mast Measurements of Wind Turbine Wakes”, where he gives insight into this technique. A technique delivering more information than a met mast, which is fixed at a certain position at specific height, is the use of lidars to measure the wake. While vertical profiling lidars can be used in a similar way as met masts, scanning lidars are able to gain spatial information in the wake of wind turbines, thus generating a more comprehensive view on the flow field behind running wind turbines. However, the use of such systems is not easy, since again there are constraints to the technology. Julia Gottschall explains in  Chap. 35, “Wake Measurements with Lidar” how lidars can be and are used for the analysis of wind turbine and sometimes even wind farm wakes. Looking at larger scales of complete large wind farms wake, it becomes hard to use lidars. Under some conditions wind farm wake can extend to several tens of kilometers. At such an extension, lidar measurements are out of range for an accurate detection of the wind with a reasonable resolution. In recent years, however, satellite synthetic aperture radar (SAR) measurements have become an option to determine the wind velocities on the surface of the sea. Thus SAR measurements are an option for offshore wind farm wake detection. Again this technology has its advantages and drawbacks. Nevertheless, it is amazing to which resolution wind speeds at the sea surface can be detected today from satellites, as Johannes Schulz-Stellenfleth and Bughsin Djath state in  Chap. 36, “SAR Observations of Offshore Windfarm Wakes”. Knowing the wakes and the form of wakes now leads to several applications. The most straightforward application is the wind farm optimization. Currently there are two ways of optimization being practically used: The optimization of wind farm layouts and the optimization of the wind farm control, either by induction or by steering the wakes. As optimization processes are often computational expensive, thus take a longer period of time, their approaches to wakes need to be rather fast – yet accurate – to actually gain optimal results. The optimization of wind farm layouts is even a whole process as Xiao-Yu Tang explicates in  Chap. 39, “Optimizing Wind Farm Layouts”. Here the accuracy of the wake aerodynamics is crucial for the final result. This is even more true in case of wind farm control optimization. Generally, there are two different ways to optimize the control of the wind farm for a specific target: The optimization of the induction is just controlling the induction of the turbine, and by doing so the intensity of the wakes. This might lead to an increase in yield or a decrease in loads. The other method is the attempt to steer the wakes by intentional yawing of the turbines. By doing so, the wakes are deflected from the center line of the rotor. This can be used to “steer” the wake around the position of the downwind placed turbine. The method is not as easy as it sounds, and Jennifer King discusses how it works and what the obstacles are in  Chap. 36, “Aerodynamics of Wake Steering”.

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Alternative Concepts “But couldn’t we do it all completely different and gain more?” This is a question most people at least in research have to listen to quite often. Why does it have to be this horizontal axis upwind turbine with three blades? This is certainly a valid question. One answer could be: “Because it works.” This concept has proven its reliability now for several tens of thousands of turbines. Nevertheless, this does not mean it is the best choice within the development of wind energy technology. Although not all ideas appearing can be considered as valid or reasonably promising, some at least appear to make sense. The basic first criterion to determine if a concept is to some point reasonable is the pure discussion of energy conservation and momentum conservation: 1. You cannot take more energy out of the wind, than is in the wind. 2. The energy of the wind is basically kinetic energy. If you take all energy out of the wind, the air will be at rest and block all other air approaching the wind energy source. Thus you can never take all energy out of the wind. 3. You can only take energy out of the wind according to the wind energy generating device affected area of the wind. By looking at these principles it becomes clear that for large amounts of energy, wind energy systems need to be large. They need to be at places where there is a lot of wind. This does not mean the horizontal upwind turbine concept is the only one. We present two of these concepts here, even though in general there are more. Vertical axis wind turbines (VAWT) are an example of a different concept. They are even commercially available at smaller scales currently. In the EU-funded “DeepWind” project, investigations have been made for use of VAWT at a scale of large multi-megawatt turbines on floating foundations. The idea behind it is striking: Since the axis of rotation is from bottom to top, it would be easy to place the generator on the bottom, thus having a much lower center of gravity. This would reduce the material costs for the floaters a lot. Also, VAWT do not need any yaw mechanism, they are less vulnerable to turbulence or wind direction changes. The state of the research is presented by Delphine de Tavernier, Carlos Ferreira Simao, and Anders Goude in  Chap. 41, “Vertical-Axis Wind Turbine Aerodynamics” to give an idea, on this so far always promising principle. A newer approach is presented by Paul Williams and Evgeniy Pechenik in  Chap. 40, “Kites for Wind Energy”, based on airborne extraction methods. The idea is also striking: In higher altitudes, the wind is not slowed down like the PBL near the ground. Thus, there is less turbulence and more energy in the flow. Why not extract the energy at the source, instead of using a lot of landscape on the ground with large towers? This idea is so striking that even though there are some research group working on it, it is mainly driven by commercial companies. There exist several concepts as shown in the chapter: Kites on ropes or autonomously flying, drone like systems. The idea is fascinating, although so far it has not been made into a commercially available state.

1 The Issue of Aerodynamics in Wind Energy

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Acoustics of Wind Turbines In the end, the aerodynamics of the wind turbine has a directly related side effect, which is often disliked, but still of large importance: The aeroacoustic noise generation. Wind turbines tend to generate noise, which is a source of large public debate, worries, and even health issues are in discussion. The main source of the noise of wind turbines of today, if they operate under regular conditions, is the noise generated by the airflow on the blades – the aeroacoustic noise. Aerodynamics and the acoustics are directly linked to each another. Today, for most blade designs, the acoustics has to be taken into account as a constraint to the aerodynamics. Therefore, aeroacoustics have to be considered in blade design and knowledge on acoustics is crucial for all wind turbine blade designers and aerodynamic specialists. First of all, it is important to understand how noise is generated and where it can be generated at the turbines. Franck Bertagnolio and Andreas Fischer discuss in  Chap. 42, “Wind Turbine Aerodynamic Noise Sources” the sources of noise in general and the specific sources within the operation of wind turbines. They give a comprehensive overview on the different aspects and how they are analyzed by several methods. Although sound itself is physically the same phenomenon, the way it gets to be differs and thus the methods for calculation differ. While the generation of noise at wind turbines is something better to be avoided, the most important aspect of the noise is its perception. While the noise emission from the turbine can be influenced by turbine design, the emission at the perceiving party strongly depends on the environment. In terms of the acceptance of the turbine and in some countries even the permission of the turbine site, the emission is the important factor. In  Chap. 43, “Wind Turbine Noise Propagation” Matthew Cand, Andrew Bullmore, and Timothy Van Renterghem elaborate on the factors influencing the emission of noise from wind turbine sites and present the various methods to calculate it with their pros and cons. In the end, noise models are nice, but the real noise of the turbines counts. This is why the measurement of wind turbine noise is still central topic. This sounds more easy than it actually is: Noise might be emitted in a directional manner. Once the measurement outside is setup, one might recognize that the turbine is not only a source of noise. So, how can noise measurement be reliably done? Which techniques are available to gain what kind of information? Bo Søndergaard, Tomas Hansen, and Stefan Oerlemans have brought together the information in  Chap. 44, “Measuring and Analyzing Wind Turbine Noise” on acoustic measurements for wind turbines. One of the important topics for blade designers is the detailed analysis of aeroacoustics of airfoils. While trailing edge noise might be modeled, this task is much harder for other sources of noise on the airfoils. It might get worse, once flow devices are involved. The most accurate method to do this is to resolve the turbulent flow around airfoils by large eddy simulations (LES). However, such simulations are enormously computationally expensive. This is even more true if whole blade sections need to be taken into account – like the blade tip. One promising method to reduce the computational costs in the future is the use of Lattice Boltzmann methods (LBM) in the fluid dynamics. This semi-statistical

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approach to fluid mechanics allows linear equations, which are computationally very fast. Even better, they approach the flow topic in a resolution of the scale of LES. Thus the approach is presented by Mikaël Grondeau and Ralf Deiterding in  Chap. 46, “Direct Prediction of Flow Noise Around Airfoils Using an Adaptive Lattice Boltzmann Method” with example calculations.

Conclusion Taking all of this together, we can say, we find the aerodynamics of wind turbines still an exciting field. Even though aerodynamics themselves have been a topic for a long period, the aspects of wind energy aerodynamics remain special. A lot of challenges remain and future developments will be on the way. For wind energy this field remains of central importance: Without the aerodynamics, no wind energy system will function. Since the global climate crisis necessitates the expansion of wind energy as one of the main renewable energy source, we will see its expansion. The developments in aerodynamics can help to minimize the impact of this expansion and its costs. We hope we contribute by this work to the dissemination of the necessary knowledge for the aerodynamic developments in the future.

References Boorsma K, Schepers J, Gomez-Iradi S, Herraez I, Lutz T, Weihing P, Oggiano L, Pirrung G, Madsen H, Shen W et al (2018) Final report of iea wind task 29 mexnext (phase 3). Wind Energy 2017:2016 Burton T, Sharpe D, Jenkins N, Bossanyi E (2001) Wind energy handbook, 1st edn. Wiley, Chichester Emeis S (2018) Wind energy meteorology: atmospheric physics for wind power generation. Springer Gasch R, Twele J (2011) Wind power plants: fundamentals, design, construction and operation. Springer Science & Business Media Hansen MO (2015) Aerodynamics of wind turbines. Routledge IEC E (2019) International Standard IEC 61400-1, Wind turbines Part 1: Design Requirements. Tech. rep., International Electrotechnical Commission Landberg L (2015) Meteorology for wind energy: an introduction. Wiley Pope S (2000) Turbulent flows. Cambridge University Press Schaffarczyk AP (2020) Introduction to wind turbine aerodynamics. Springer Nature Schepers J, Boorsma K, Cho T, Gomez-Iradi S, Schaffarczyk P, Jeromin A, Lutz T, Meister K, Stoevesandt B, Schreck S et al (2012) Final report of iea task 29, mexnet (phase 1): analysis of mexico wind tunnel measurements. IEA de Vries E (2008) Is there a limit to wind turbine size? Sun Wind Energy 1:146–149

Part I Aerodynamic Blade Design

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Economic Aspects of Wind Turbine Aerodynamics Panagiotis Chaviaropoulos

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE as a Design Driver for Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE Definition and Target Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE Contributors (AEP, CAPEX, OPEX) Onshore/Offshore. Turbine (Subsystems Level), Balance of Plant, and Operation and Maintenance Parts . . . . . . . . . . . LCoE Evaluation Using Cost Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Critical Role of Increasing AEP in LCoE Reduction, Especially Offshore . . . . . . . . . . LCoE Evolution with Technology Maturity, Learning Curve, and Short-/Mid-/Long-Term Target Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE and Turbine Size and Architecture (Emphasis on Rotors) . . . . . . . . . . . . . . . . . . . . . . . . Turbine Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE, Upscaling, and Optimum Turbine Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tendencies in Onshore/Offshore Rotor Designs. Impact of Larger and Lower Power Densities, Cpmax, and Lower Induction Designs . . . . . . . . . . . . . . . . . . . . . . Exploring Passive and Active Load Alleviation Techniques to Increase Rotor Size and AEP or Reduce CAPEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Aerodynamics and LCoE Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Aerodynamics and AEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Aerodynamics and CAPEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LCoE Reduction Potential Through Innovative and Advanced Aerodynamic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 21 22 23 25 25 27 27 29 33 35 36 36 37 39 44 44

P. Chaviaropoulos () iWind Renewables P.C., Athens, Greece e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_1

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Abstract

Wind turbines are aerodynamically driven machines. The energy produced is chiefly associated with the efficiency of their rotors to convert the kinetic energy of wind to mechanical power. Still, mechanical power must be transformed to electrical through a drivetrain, and the whole system should be kept in place through a support structure. The selection of the proper architecture and the design of these three main subsystems (rotor, drive train, support structure) allow for much freedom, but at the end of the day, it is driven by economical aspects. The optimal turbine design for a given site (onshore/offshore) with known external conditions is the one which can produce electricity in the lowest possible cost, usually expressed through a metric called levelized cost of electricity (LCoE). We acknowledge the fact that since a few years, wind farms are subjected to variable market price mechanisms, and the value of the produced electricity is depending on the market specifics, possibly leading to a different optimum than the one suggested by minimum LCoE. As, however, LCoE remains a pure metric for technology assessment, the goal of this chapter is to make the connection between LCoE and technological selections and design aspects, with focus on rotor aerodynamics. Mastering this connection allows for better understanding the critical areas where the emphasis should be placed for improving cost-efficiency of wind turbine designs. Keywords

LCoE reduction · Wind turbine design · Turbine architecture · upscaling · Rotor aerodynamics

Introduction The goal of this chapter is to make the connection between LCoE and technological selections and design aspects, with focus on rotor aerodynamics. Most of the information presented below has been acquired through the author’s participation and contribution to two recently concluded FP7 European research projects, addressing the design of 10–20 MW offshore wind turbines. The first, INNWIND.EU (http:// www.innwind.eu/), investigated and demonstrated innovative designs that have the potential to significant reduce the levelized cost of offshore wind energy for 10–20 MW ffshore wind turbines, while the second, AVATAR (http://www.eeraavatar.eu/), closed knowledge gaps and developed and validated high-fidelity aerodynamic and aeroelastic design tools for such multi-MW turbines. The chapter elaborates on four main themes: • • • •

LCoE as a design driver for wind turbines LCoE and turbine size and architecture Rotor aerodynamics and LCoE reduction LCoE reduction potential through innovative and advanced aerodynamic design

2 Economic Aspects of Wind Turbine Aerodynamics

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LCoE as a Design Driver for Wind Turbines LCoE Definition and Target Values The levelized cost of electricity (LCoE) has been adopted during the last decade as an overarching Key Performance Indicator for monitoring the evolution of wind energy technology. LCoE represents the sum of all costs over the lifetime of a given wind project, discounted to the present time and levelized based on annual energy production. LCoE is expressed in Euro per megawatt-hour (e/MWh). LCoE can be calculated with several methods or approaches to represent several differing perspectives. A simplified version for LCoE is used here, the details of which are presented in Key Performance Indicators for the European (2013) by EWII (European Wind Industrial Initiative) along with the assumptions and parameters needed to establish its reference values (2011) for both onshore and offshore wind energy. The system in which the reference cases are to be evaluated must be clearly defined by determining which aspects are within the system boundaries and which are not. The usual assumptions are that permitting costs, connection from the wind farm substation to the external grid, civil works outside the wind farm, financing costs, overheads, and decommissioning costs are outside our system boundary. Other parameters necessary for the calculation of the LCoE, apart from the capital expenditure (CAPEX) and the operation expenditure (O & M costs), are the balancing costs, capacity factor (CF) of the wind farm, project lifetime, and real discount rate. Capacity factor is directly related to the annual energy production (AEP). CAPEX is often split into two parts, one addressing the turbine itself (C in Fig. 1) and another for the balance of plant (BoP) where the (costly, in particular for offshore) foundation system is accounted for. Already in 2011 EWII set a target of 20% LCoE reduction in the period 2010– 2020 for both onshore and offshore wind. This had serious implications in European Wind Energy Research and Development, where the successful research proposals had to clearly demonstrate in the sequel of their impact on LCoE reduction. In the

Fig. 1 Definition of LCOE following Key Performance Indicators for the European (2013)

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USA, DOE through NREL had already implemented such a strategy one decade earlier through the successful WindPACT program (https://www.osti.gov/biblio/ 785133). In Europe, a consistent implementation of innovation-driven offshore wind LCoE reduction came through the INNWIND.EU FP7project (2012–2017) (http:// www.innwind.eu/).

LCoE Contributors (AEP, CAPEX, OPEX) Onshore/Offshore. Turbine (Subsystems Level), Balance of Plant, and Operation and Maintenance Parts Figures 2 and 3 present the percentage contribution of CAPEX and OPEX as well as their inner split in turbine and BoP sub-components for onshore and offshore wind, respectively. For a typical deepwater (35–50 m) bottom-mounted offshore wind plant, the following contributions to LCoE are expected: CAPEX turbine 25%, CAPEX BoP 35%, and OPEX 40%. The relevant shares for a typical onshore wind farm are 50%, 20%, and 30%. Note that in the offshore case, the rotor’s share in LCoE is only 6.2% instead of 11% onshore. This implies that using a larger more expensive rotor which will substantially increase the annual energy production but keep the same support structure might be beneficial for the cost of wind energy offshore. Note also that the largest contribution to offshore LCoE is that of the foundation system (15%). The above demonstrates that LCoE reduction strategies in onshore and offshore are not

Turbine

Rotor

CAPEX

ONSHORE

Nacelle systems

Electrics & control

OPEX

BoP

Tower Other Infrastructure

Transportation Assembly & Instal Other Labour Parts Operation Equipment Facilities

Rotor lock Blades Hub Gearbox Generator Rotor brak e Nacelle cover Nacelle structure Couplings Shaft Yaw system Bearings Pitch system Variable speed system

Roads & civil work s Electrics & grid conn Foundations Transportation Assembly & installation Financial & legal

0.0050 0.0911 0.0125 0.0844 0.0332 0.0075 0.0125 0.0170 0.0050 0.0150 0.0150 0.0150 0.0321 0.0484

0.0176 0.0607 0.0538

0.1086

0.50

0.70

0.2047

0.0805 0.0787 0.0275 0.1321

0.0291 0.0058 0.0330 0.1320 0.1050 0.0360 0.0150 0.0120

0.20

0.30

0.30

Fig. 2 Subcomponents’ % contribution to LCoE. Onshore wind, 3 MW turbine size. (From http:// www.innwind.eu/publications/deliverable-reports Del 1.22)

2 Economic Aspects of Wind Turbine Aerodynamics

OFFSHORE

Turbine

CAPEX

BoP

OPEX

Rotor

Rotor lock Blades Hub Nacelle systems Gearbox Generator Rotor brak e Nacelle cover Nacelle structure Couplings Shaft Yaw system Bearings Electrics & control Pitch system Variable speed system Tower Other Foundation system Offshore transportation and installation Offshore electrical I&C

O&M Offshore

23 0.0000 0.0585 0.0036 0.0340 0.0185 0.0035 0.0036 0.0074 0.0000 0.0050 0.0033 0.0032 0.0070 0.0132

0.0621

0.26

0.61

0.0785

0.0202 0.0693 0.0342

0.1545

0.1545

0.1054

0.1054

0.0913

0.0913

0.35

0.39

0.39

Fig. 3 Subcomponents’ % contribution to LCoE. Offshore wind, 5 MW turbine size. (From http:// www.innwind.eu/publications/deliverable-reports Del 1.22)

necessarily the same from the point of view of the wind turbine. Using an upscaled and marinized onshore turbine at offshore wind farms is probably not the best option, and optimal solutions should seek for offshore-specific turbine architectures.

LCoE Evaluation Using Cost Models Any quantitative decision on the optimal turbine size for a specific onshore or offshore site which also accounts for the technology evolution and the potential that innovation adds to LCoE reduction should be based on a proper cost model integrated to the LCoE calculator. Mass-based cost models are used in most cases; see, for instance, Fingersh et al. (2006) for the WindPACT and (http:// www.innwind.eu/publications/deliverable-reports) for the INNWIND.EU model. Both models are developed at the sub-components level; they are based on key turbine design parameters (rated power, diameter, hub-height, rated torque, etc.), and operating conditions (wind class, etc.), are intrinsically considering turbine scaling relations and may also account of variations in raw materials pricing, inflation, and currency fluctuations so that cost data from different periods and markets can be synchronized. The input/output section of the INNWIND.EU cost model applied to the project’s 10 MW reference wind turbine is shown in Fig. 4. Panels in blue font are input data, while black and red fonts present results. The input panel has a menu for selecting the desired subsystem configuration, including blades, drivetrain, and

Tower Model

O Offshore Support Structure

1

1

Total Capacity (MW) Wake Losses (%) Electrical Losses (%) Availability Losses (%)

WIND FARM DATA 9.20 2.00

0.507

0.830

Mean Annual Wind Speed (m/s) Weibull shape factor k (-) O&M Class 2b or 2c

SITE CONDITIONS

Turbine Capacity Factor

Drive Train Efficiency @ paral load (10%)

Omega (rad/s) RPM max Rated Torque (kNm) Rotor swept area (m2) Rotor Cp_max (-) Drive Train Efficiency @ full load (100%)

INTERMEDIATE TURBINE RESULTS 1.01 9.66 10520 24885 0.478 0.940

RESULTS

BoP Price/Cost Multiplier

WT Price/Cost of components

€/£ (2012) $/€ (2012)

OTHER DATA

LCOE (€/MWh)

O&M Direct Costs (€/MWh)

CAPEX (M€2012/MW)

BoP Cost (M€2012/MW)

Turbine Cost (M€2012/MW)

WF Capacity Factor

Fig. 4 The I/O section of the INNWIND.EU cost model. (From http://www.innwind.eu/publications/deliverable-reports Del 1.23)

500 9.0% 2.0% 6.0%

Reliability Surcharge (%)

Drive Train Model

2

0%

Blade Model

Power (kW) Diameter (m) Max Tip Speed (m/s) Hub height (m) Rated speed (m/s) Design speed (m/s)

TURBINE INPUT PARAMETERS

1

10000 178.0 90.0 119.0 11.4 10.0

1.250 1.320 1.400 1.000

0.425 1.372 1.695 3.066 34.81 98.56

24 P. Chaviaropoulos

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support structure. Turbine and wind farm capacity factors are critical results along with CAPEX and OPEX figures (per turbine) and, finally, LCoE.

The Critical Role of Increasing AEP in LCoE Reduction, Especially Offshore Annual energy production and, therefore, wind turbine and wind farm capacity factors are of paramount importance in determining LCoE values. This is a straightforward conclusion from the LCoE definition, where AEP is the sole entry of the denominator of the defining relation. The numerator, on the other hand, includes several entries including the individual cost components of turbine and BoP CAPEX and OPEX and the cost of financing as expressed through FCR. Figure 5 is quite instructive for evaluating the sensitivity of LCoE to CAPEX and CF. It is seen that in the CAPEX area of offshore wind interest, increasing CF from 0.40 to 0.45 leads to a reduction of LCoE by 10%. This is equivalent to the reduction resulting from cutting the CAPEX down by 500 e/kW, which is quite much. The above remarks combined with the findings of Figs. 2 and 3 are critical in decision-making regarding optimal onshore and offshore turbine architectures. Cost reduction becomes mostly important for those subsystems that are AEP irrelevant, such as support structures for given hub height. For subsystems affecting AEP, primarily the rotor and consequently also the drivetrain through its partial and full load efficiency, proper selections come as a trade-off between the percentage contribution of the subsystem to LCoE and its influence on AEP. In offshore wind, for instance, where the turbine CAPEX contribution to LCoE is smaller than onshore due to the expensive support structure, more elaborate and costlier rotor and drivetrain designs, increasing AEP, are well justified in the battle for LCoE reduction.

LCoE Evolution with Technology Maturity, Learning Curve, and Short-/Mid-/Long-Term Target Values EWII target for a 20% Research Technology Development and Innovation-driven cost reduction in onshore and offshore wind is well in line with the target that the wind energy market has also set. A comprehensive analysis for generation costs of renewables in 2017 is given by IRENA in IRENA (2018). The report provides indicative LCoE trends through learning curves for the global weighted average of LCoE of different renewable energy technologies in terms of time and cumulative deployment in MW. For onshore wind the averaged LCoE for 2016 is reported as 70 $/MWh, and its projection for 2020 is 50 $/MWh. For offshore wind the corresponding values are 150 $/MWh and 85 $/MWh. In the German wind auctions, the average surcharge awarded arrived at 47.3 e/MWh in the first onshore wind tender for 2018, won by 83 projects totalling 709 MW (https://renewablesnow.com/news/germany-opens-bids-in-new-670-mw-

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Fig. 5 LCoE dependence on CAPEX and Capacity Factor for fixed OPEX. (From http://www. innwind.eu/publications/deliverable-reports Del 1.22)

onshore-wind-tender-604159/, 2018). The early 2018 onshore wind tender in France came with a total of 22 projects, totalling 508 megawatts, being awarded at an average winning price of e65 e/MWh (https://cleantechnica.com/2018/03/ 05/costs-fall-latest-french-onshore-wind-tender/, 2018). There are several publications and press releases by major offshore wind farm developers and operators such as Ørsted (Renewable energy 2013) and E.ON (http://www.windpoweroffshore.com/2013/02/21/eon_focuses_on_95mw_goal/#.U ZtdAbVmh8E) stating that they aim to cut the cost of wind energy in the North Sea to less than 90 e/MWh by 2020 compared with the 160 e/MWh of 2012. To get this 60% cost reduction, Ørsted has plans to radically increase the size of the offshore turbines it will install, from 3 to 4 megawatts in 2012 to 8 to 10 MW in 2016 through

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2020. In a recent publication, Siemens unveiled a new turbine platform in the next 5 years, hinting at a 10 MW+ turbine as it targets e80/MWh by 2025 (http://www. windpowermonthly.com/article/1399841/siemens-teases-10mw+-turbine). Also, in 2018 GE started the development of its IEC Class IB Haliade-X 12 MW wind turbine, with a rotor diameter of 220 m and an intended capacity factor of 63% in North Sea wind conditions https://www.ge.com/renewableenergy/wind-energy/ offshore-wind/haliade-x-offshore-turbine. German offshore wind auction, earlier in 2017, resulted in 1.49 GW of offshore wind, 1.38 GW of which was subsidy free (https://seekingalpha.com/article/40631 37-offshore-wind-subsidy-free-dong-energy-enbw-submit-zero-subsidy-bids-1_38gw, 2017.). These bids have an extended delivery through 2024. Issues underpinning subsidy-free energy bids are bigger turbines (13–15 MW), wind farm scale, location (wind speed >10 m/sec), extended lifetime (from 25 to 30 years), and grid connection provided by TSO and thus not included. Bloomberg New Energy Finance (BNEF) anticipated bids in this round of about 55 e/MWh. It’s not just in Germany where the costs of offshore wind power are falling. The UK and Netherlands have both seen record low bids during the past years. In September 2017, developers led by Ørsted won bids to develop wind farms in British waters for as little as 57.50 pounds/MWh (https://www.bloomberg. com/news/articles/2017-04-13/germany-gets-bids-for-first-subsidy-free-offshore-wi nd-farms, 2017). The first subsidy free or market prices exposed offshore wind farms will be built in the Netherlands by Vattenfall in 2021.

LCoE and Turbine Size and Architecture (Emphasis on Rotors) Turbine Architectures Setting the range of interest to 2.5–6.0 MW turbines for onshore and 10–20 MW turbines for offshore, the design platforms are classified, following http://www. innwind.eu/ to three categories: evolutionary, new, and revolutionary. Evolutionary architectures are considered those based on technologies (at both the system and component levels) with TRL levels near to market, i.e., TRL6 and greater. The radically new platforms are engaging technologies that still need lab testing and proof of concept for acceptance by the industry, while revolutionary denotes those design platforms with present TRL level 3. In the present document, we shall mainly focus on the evolutionary architectures and on their rotors. In the evolutionary class, we include the three-bladed, upwind, (individual) pitch-variable speed turbines with high-, medium-, or lowspeed gearbox, tubular steel tower linked to a monopile, or jacket substructure when offshore. For evolutionary architectures LCoE reduction is expected through a combination of upscaling and innovative design at the components and system level. Apart of upscaling wind turbine size seeking for its optimum at a given

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site, we shall address here three innovations in rotor aerodynamic design that can significantly reduce LCoE, (i) the low induction rotor, (ii) the aeroelastically tailored rotor, and (iii) the use of trailing edge flaps. We have selected these three innovations from many others as they have been commonly considered in the recently concluded INNWIND.EU and AVATAR (http://www.eera-avatar.eu/) FP7 projects. INNWIND.EU developed these innovative concepts for 10–20 MW size offshore wind turbines, while AVATAR developed the aeroelastic tools for analyzing their impact on performance and loads. Although developed for large offshore turbines, all three concepts are also applicable to smaller onshore machines; however the economic impact can then be completely different. In the class of new architectures, we include non-conventional two-bladed upwind or downwind offshore designs. Although such designs are conventional for onshore wind, they present some challenges when it comes to offshore. The reason is the difficulty to prevent resonance of the offshore support structure with the waves’ frequency and at the same time with the 1-P, 2-P, and 4-P rotor excitations (which are narrower banded than the three-bladed 1-P, 3-P, and 6-P). If resonance is not avoided, the turbine will suffer from higher fatigue loads in the wind speed range where such excitation takes place. The 3-P excitation can be alleviated through an exclusion zone in the variable speed controller, which however compromises the power performance of the turbine and does not totally prevent the problem. Given that Cp_max of a two-bladed turbine is already slightly lower than a three-bladed and the high dependence of LCoE to AEP, such a compromise is to be avoided. Two-bladed offshore designs are only cost-effective when combined with special offshore substructure designs, such as the semi-floater developed in INNWIND.EU. From the upscaling and aerodynamic innovative design point of view, the elements that will be presented for the evolutionary designs are also applicable here. In revolutionary architectures INNWIND.EU classified a 20 MW floating multirotor system and a 10 MW floating vertical-axis wind turbine (VAWT). Multirotor technology has a long history, and the multirotor concept persists in a variety of modern innovative systems. The multirotor concept of having many rotors on a single support structure avoids the upscaling disadvantages of the unit turbine at the expense of a more complicated and costlier tower structure and facilitates the benefits of large unit capacity (potentially much larger than will be economically feasible for the single turbine) at a single location. Interestingly, during April 2016, VESTAS announced the erection, testing, and validation of a 900 kW MRS concept turbine (4/225 kW) which has been installed at the RISOE DTU Wind Campus. About VAWTs, their main advantage is their lower position of the center of gravity, which, in the case of a floating system, may have implications on the design and cost of the support structure. Also, in favor to VAWT is that there is no need for a yawing mechanism which simplifies the design of the support structure. Most of the available information on VAWT comes from old studies which are regarded as being outdated, especially at multi-MW scale. Recent research on VAWTs include the 5 MW DeepWind (http://www.deepwind.eu/) and the (unfinished) Nenuphar VERTIWIND (http://www.nenuphar-wind.com/en/) projects. Thanks to their advantages stated above, VAWTs may enjoy smaller CAPEX values per installed MW than

2 Economic Aspects of Wind Turbine Aerodynamics

29

HAWTs. However, their aerodynamic performance as expressed through Cp_max is considerably less. This can be only compensated by increasing the swept area of the rotor and therefore the mass and CAPEX. It is still unclear whether their AEP/CAPEX/OPEX trade-off will render VAWTs competitive to HAWTs in terms of LCoE. The scaling rules discussion that will follow is also applicable, in first principles, to multirotors and VAWTs as regards the turbine itself. Scaling rules for floaters are not fully mastered yet due to the multiplicity of floater design concepts and materials (steel, concrete). We shall therefore avoid any further reference to that. The aerodynamic innovations we are addressing are also applicable to multirotors.

LCoE, Upscaling, and Optimum Turbine Size Upscaling and AEP Even classical upscaling has a positive effect on the capacity factor of an onshore or offshore wind farm. In the onshore case, this is mainly due to the increased hub height which, in most of the cases, leads to higher winds. Exceptions are highly complex terrain sites where extreme speedups come along with reverse wind shear where larger turbines with higher towers lose their advantage. Larger turbines also allow the cost-effective use of higher-fixed-cost control and protection systems, which can also improve AEP. For large offshore wind farms, the effect was studied in the UPWIND Project (Politis and Chaviaropoulos 2008) where the (aerodynamic) wind farm capacity factor was calculated as a function of the WT rated power. The mean wind speed distribution used at the hub height of all designs was a Rayleigh distribution with mean 10 m/s, while the wind rose was assumed uniform direction-wise. Two wind farm sizes were considered, with 500 and 1000 MW installed capacity. The spacing of the turbines was 7D X 7D, leading to similar offshore area requirements for all turbine sizes. The wind farm aerodynamic capacity factor (the production of all turbines including the wake effects) increases with the size of the single turbine. Going from 5 to 10 MW, we have a nearly linear increase of almost 2.5 percentage units, with an additional increase of 2.5 units from 10 to 20 MW. This effect is attributed to the reduction of wake effects due to the smaller number of turbines in the wind farm when the rated power of the individuals increases. Such a capacity factor improvement is not related to better wind resource available at larger distances from the shore (and deeper waters) or going at larger hub heights. These are additional factors that might further increase the farm capacity and are (indirectly) linked to the single turbine size. Upscaling and CAPEX In classical upscaling we assume that the scaling exponent for CAPEX is λC = 3 for the turbine and its main sub-components (Chaviaropoulos 2007; Sieros et al. 2012) and λC = 2 for the BoP part. Namely, the turbine CAPEX scales up with s 3 where s is the linear scale factor (defined with the assumption that rated power scales up

30

P. Chaviaropoulos

with s 2 ). For the turbine part, we have shown in the referred papers that the weight of the main components (blades, low-speed shaft, gearbox, tower) scales up with λW = 3 + ε (ε δ. The azimuthal velocity w in the wake depends only on . Inside the core, w depends on the assumed characteristics of the core. The force field rotates with angular velocity . A Joukowsky rotor is characterized by the same vortex at the axis which continues

γϕ

γϕ γx , γr

n

γr

s U∞

Ω

Γaxis

U∞

Fig. 5 The circulation distribution for the Joukowsky actuator disc with swirl, left, and the Froude disc without swirl, right

60

G. A. M. van Kuik

Fig. 6 The load on bound radial vorticity of a wind turbine rotor blade. Sign conventions are shown in Fig. 3. The number of blades is B, and axis = −BB

cross section C thickness ε

L Γ

Γ U∞

r

Ω

x ϕ

in a constant blade bound circulation, as shown in Fig. 6 for a one-bladed rotor. For a rotor with B blades axis = −BB . The disc: The non-conservative force field of a steady actuator disc converts power as given by (17): 

 f · vdV =

P = V

H vn,S dS

(31)

S

with volume V having surface S shown in Fig. 7. At the cross section with the far wake vn,S = u1 . At the cross section with the stream tube far upstream, the velocity is undisturbed U∞ . For the part of V outside the stream tube, H = H∞ and  vn,S dS = 0, so the expression for the converted power becomes:  P =

H u1 d A1 − H∞ U∞ A∞ ,

(32)

A1

where A∞ is the cross section of the stream tube far upstream and A1 the same far downstream. For the Joukowsky disc, the azimuthal velocity is: w=

 2π r

⎫ ⎪ for r ≥ δ ⎪ ⎪ ⎬

⎪ ⎪  r  ⎪ C for r < δ ⎭ = 2π δ δ

(33)

3 The Actuator Disc Concept

61 S

RS

U∞ U∞

s u1

ud

U∞ A∞

x

Ad

A1

Fig. 7 Half-sphere with surface S as control volume for the momentum balance, crossing the stream tube of an actuator disc far upstream with undisturbed flow and far downstream with a fully developed wake. Only half of the cross section is displayed

where C(1) = 1 and C(0) = 0 but otherwise unspecified. As an example, with C(r/δ) = r/δ, the vortex core is modelled as a Rankine vortex. With (24) it follows that: 1  (H − H∞ ) = ρ 2π

⎫ ⎪ for r ≥ δ ⎪ ⎪ ⎬

⎪ ⎪  r  r  ⎪ = C for r < δ ⎭ 2π δ δ

(34)

Outside the core H is constant, so the power becomes: R1 P = ρ

u1 r1 dr1 = ρ

 ud R 2 2

for a J-disc, δ → 0,

(35)

0

where conservation of mass U∞ A∞ = u1 A1 = ud Ad is used to express the last term at the right-hand side of (32) in far wake properties and to convert the integral from plane A1 to the disc area, with ud being the disc averaged axial velocity. For r < δ, integration of u1 (H − H∞ ) across the vortex core cross section and letting δ → 0 shows that the contribution of the vortex core to P vanishes. This is checked by modelling C as a series development in (r/δ)n , showing that each term yields a zero contribution after integration across the core cross section. With the introduction of the non-dimensional vortex strength q = /(2π RU∞ ) and tip speed ratio λ = R/U∞ , (34) becomes for r ≥ δ:

62

G. A. M. van Kuik

H = qλ 2 ρU∞

for

r ≥ δ.

(36)

Herewith the dimensionless power coefficient Cp becomes, with Pdisc = −P as explained in the text box Sign Conventions at page 4: Cp =

Pdisc 1 3 2 2 ρU∞ π R

= −2qλ

ud for a J-disc, δ → 0. U∞

(37)

The thrust T is obtained by integration of (26) on the disc area. With (33) the contribution by the region outside the vortex core in the limit δ → 0 becomes: R T |δ≤r≤R =ρπ

(2r 2 w − w 2 r) dr = ρ

 R  2 2 R −ρ ln for a J-disc, δ→0 2 4π δ

δ

(38) In the same way, the contribution to the thrust by the vortex core cross section is  2 found to be a constant, like −ρ π4 ( 2π ) in case of the Rankine distribution C(r/δ) = r/δ. For δ → 0 these constants vanish compared to the singular term in (38), so the thrust is: T =ρ

 2 ρ R − π 2 2

 2π

2 ln

2 R for a J-disc, δ → 0. δ

(39)

The first term at the right-hand side is the thrust converting power. The second term is the thrust due to the swirl-induced pressure gradient. In dimensionless notation, the thrust coefficient becomes: ⎫ Tdisc ⎪ ⎪ CT = ⎪ 1 ⎪ 2 2 ⎪ 2 ρU∞ π R ⎪ ⎪ ⎪ ⎪ ⎬ J-disc, δ → 0 (40) CT , H = −2λq ⎪ ⎪ ⎪ ⎪

2 ⎪ ⎪ ⎪ R ⎪ 2 ⎪ = q ln CT , w ⎭ δ which is independent from the choice of the vortex core model. For δ → 0, CT , w → ∞. The consequences of this will be discussed at the end of this section. H is proportional to , so with H kept constant, w vanishes like −1 as well as γd and  when  → ∞. The disc without swirl is the Froude actuator disc; see Fig. 5. The rotor with B blades: The converted power P is torque Q times , so with the azimuthal component of (30):

3 The Actuator Disc Concept

63

R  P = Q = −Bρ

uωr rdrdC.

(41)

δ C

In (30) the limit C → 0 was included, leading to  = −BB , yielding:

 C

uωr dC = uB B . As before

R P = ρ

uB rdr for a J-rotor, δ → 0,

(42)

uB for a J-rotor, δ → 0. U∞

(43)

δ

so the power coefficient becomes: Cp = −2λq

The thrust at the rotor is defined by the axial component of (30). With (5) and with  = −BB : R T = −ρ

wrot,B dr = ρ

 2 R − ρ 2

δ

R wB dr for a J-rotor, δ → 0.

(44)

δ

In the wake the azimuthally averaged value w¯ = /(2π r), but in the rotor plane, it is half this value: wx=0 = /(4π r). The azimuthal distribution of w will be approximately uniform for low values of r/R as the induction by the root vortex dominates. However, for larger r/R values, the tip vortices will add a harmonic distribution. With the actuator line and lifting line calculations of which the axial velocity is shown in Fig. 2, the order of magnitude of the approximation wB = w¯ is estimated: the deviation in CT is δ, so the result for CT is, with again CT = CT , H + CT , w : CT

=

CT , H

=

CT , w

= ≈

Trotor 1 2 2 2 ρU∞ π R

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

−2λq R J-rotor, δ → 0. wB ⎪ dr ⎪ 4q ⎪ ⎪ U∞ R ⎪ ⎪ ⎪ δ ⎪ ⎪   ⎭ 2 R 2 q ln δ

(45)

64

G. A. M. van Kuik

2 π rdr), where L is the The local thrust coefficient Ct is defined as BdLx /(ρU∞ x axial component of (30):

Ct = −2λq + 2q

wB R , for a J-rotor, r ≥ δ, U∞ r

(46)

or, with Ct = Ct, H + Ct, w : Ct, H Ct, w

=

−2λq wB R = 2q ∞ r

U R 2 2 ≈ q r

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

J-rotor, δ → 0.

(47)

The expressions for the rotor and disc are identical, apart from the approximation in (45) and (47) for the rotor thrust component CT , w . For the disc as well as rotor, the conversion of power by the force field is expressed in the increase of the amount of wake swirl. CT , w becomes ∞ for δ → 0 as long as q 2 is non-zero. This is the case for finite  when H or qλ remains constant in this limit. With δ R the singular term is positive, so it adds up to CT , H for a wind turbine rotor. For practical conditions δ mimics the root cut-out radius, which is the radius below which the nacelle and blade root connection occupy the space. A practical value is 0.15R. For λ > 7 we find that CT , w ≤ 0.02CT , H , so this contribution to the thrust may be ignored. In section “Analysis of Joukowsky Actuator Disc Flows” CT , H will be used as parameter defining flow states, together with λ. The unknown in the power coefficient CP is u¯ d for the disc and u¯ B for the rotor. The determination of u¯ d is the topic of the following sections.

Analysis of Froude Actuator Disc Flows This actuator disc momentum theory for Froude discs, so discs without swirl, is sometimes called “one-dimensional” as only the axial momentum balance is included. It is a special case of Joukowsky discs as it results from the limit  → ∞. However, in accordance with its place in the history of rotor aerodynamics, it is treated first as an independent topic. The theory gives the average value of the axial velocity at the disc, not the velocity distribution. Modern computational approaches are able to provide flow details like the shape and strength of the vortex tube that separates the wake from the outer flow. Using a CFD solver for the Navier-Stokes equations, the velocity field for propeller as well as wind turbine flows states was first published by Sørensen et al. (1998). The method to find details of the actuator disc flow used in this chapter is based on the inviscid method of Øye (1990). The method is described in van Kuik and Lignarolo (2016) and van Kuik (2018a).

3 The Actuator Disc Concept

65

The Momentum Balance The general expression for the momentum balance is given by Batchelor (1970, p. 138). For inviscid flow the balance in x-direction drawn on a volume V enclosed by a surface S is: 

 ex · en pdS = ρ

T − S

u(u − U∞ )dS

(48)

S

with p being the pressure acting at the boundary S. When applied to the stream tube passing through the disc, the pressure integral becomes zero, as will be shown. Section “Momentum Theory Including Conservative Forces” treats the balance of an annulus instead of the entire stream tube, where the annulus-based pressure integral will be shown to be non-zero. Usually the stream tube passing through the actuator is used as the control volume V . Several proofs have been published that the pressure acting at the stream tube boundary does not contribute to the momentum balance, e.g., by Thoma (1925). Here another control volume is used. Figure 7 shows the control volume bounded by a sphere with radius RS , with the center of the sphere coinciding with the center of the actuator disc, and with RS → ∞. The advantage of this control volume is that only the flow conditions at infinite distance need to be known, not at the vortex sheet itself. Furthermore this control volume can be used for the flow induced by a static disc (U∞ = 0) which does not have a stream tube extending upstream. Outside the wake of the actuator disc at a large distance from the origin, the flow can be considered as a summation of a parallel flow and a source flow. Analogous to Batchelor (1970, p. 351), momentum and pressure terms in (48) at the sphere S but outside the wake vanish for RS → ∞. This is because the summation of undisturbed U∞ and source-induced velocities v source gives rise to momentum 2 , u 2 flux and pressure terms containing U∞ source U∞ , vsource U∞ , and |v|source . The source velocity vanishes like RS−2 by continuity of mass, so the |v|2source term does not contribute after integration on S for RS → ∞. The mixed terms containing usource U∞ , vsource U∞ do not vanish for increasing RS but do not contribute after integration on S, due to the symmetry of the source flow with respect to the plane 2 . What remains for x = 0. The same holds for the constant term containing U∞ RS → ∞ are the contributions by the disc itself and the momentum transport at stream tube cross sections A∞ far upstream and A1 far downstream. The pressure acting at these cross sections is undisturbed, p∞ , so the pressure integral in (48) vanishes and the momentum balance becomes:  T = ρ u1 (u1 − U∞ )dA1 . (49) A1

The same result is obtained when we use the stream tube as control volume and assume that the pressure (p − p∞ ) at the stream tube boundary does not result

66

G. A. M. van Kuik

in an axial force acting on the control volume. In other words, the momentum balance using the sphere as control volume confirms this assumption, so it may be considered as an indirect proof that the stream tube pressure does not contribute.

Momentum Theory Without Conservative Forces The Bernoulli equation applied to the upstream and downstream part of a streamline (the dashed line in Fig. 7) can be coupled by the pressure jump p, giving the energy balance: p =

1 2 ). ρ(u21 − U∞ 2

(50)

As p is uniform for Froude discs; also the velocity in the wake u1 is uniform. With T = pAd the momentum balance (49) becomes: pAd = ρu1 (u1 − U∞ )A1 = ρ u¯ d (u1 − U∞ )Ad ,

(51)

where mass conservation ud Ad = u1 A1 is used and where ud is the velocity averaged on the disc area. Elimination of p from (50) and (51) gives the famous result, first obtained by Froude (1889): ud =

1 (u1 + U∞ ). 2

(52)

The converted power P = pud Ad , so in dimensionless form the power coefficient is, with Pdisc = −P : 1 Cp = 1 =− 3 2 2 ρU∞ Ad Pdisc



u1 U∞



2 −2

 u1 +1 . U∞

(53)

.

(54)

The thrust coefficient follows by (50): CT =

Tdisc 1 2 2 ρU∞ Ad

=1−

u1 U∞

2

Expressed in ud both coefficients become:

ud Cp = −4 U∞ and:

2

 ud −1 U∞

(55)

3 The Actuator Disc Concept

67

ud CT = −4 U∞

 ud −1 . U∞

(56)

Elimination of ud from (55) and (56) gives: Cp =

   1 C T 1 + 1 − CT , 2

(57)

which is the solid line in Figs. 1 and 9. Differentiation of (55) to ud to find the coefficient for maximum power extraction gives: Cp,max =

ud 16 2 8 for = , CT = , 27 U∞ 3 9

(58)

which was obtained by Joukowsky (1920) and Betz (1920), for which reason it is called the Betz-Joukowsky maximum; see Okulov and van Kuik (2012).

Momentum Theory Including Conservative Forces In the previous section, the momentum balance is applied to the entire stream tube, with the non-conservative disc load Tnon−cons = pAd as the only load entering this balance. Now it is assumed that  conservative loads are present in case the pressure integral in (48): Tcons = − S ex · en pdS = 0. Herewith the momentum balance becomes:  Tnon−cons + Tcons = ρ u(u − U∞ )dS (59) S

with: ⎫ Tnon−cons = ⎪ ⎬  pA Tcons = − ex en pdS. ⎪ ⎭

(60)

S

The energy equation (50) is unaffected by Tcons so the combination of (59) with (51) gives:

ud =

Tcons Tnon−cons

 U∞ + u1 . +1 2

(61)

Expression (55) for the power coefficient becomes:

Cp =

Tcons Tnon−cons

 + 1 Cp,Tcons =0

(62)

68

G. A. M. van Kuik

and for the thrust coefficient: CT = CTcons + CTnon−cons

(63)

Equations (61) and (62) have first been derived by van Holten (1981) for discs or rotors placed in a shroud or ring wing or with tip vanes. The lift on the additional device contributes to the momentum balance but does not convert energy, so is conservative. The average axial velocity at the disc is then given by (61). When both trust components have the same sign, the average velocity increases and so does the power coefficient. Sørensen et al. (2015, section 3.4) presents a survey of recent publications on the so-called diffuser-augmented wind turbines and discusses the associated momentum theory in detail. This theory is outside the scope of the present chapter. Equation (61) has also been derived by Sørensen and Mikkelsen (2001) and is included in Sørensen and van Kuik (2011a) although they, as well as van Holten (1981), did not use the classification cons and non − cons. Sørensen and Mikkelsen (2001) derived (61) for the momentum theory applied to a stream annulus instead of stream tube. A stream annulus is a part of the stream tube, e.g., the volume bounded by the streamline shown as a dashed line in Fig. 7 or the volume between two such streamlines passing the disc at radii r and r + r as shown in Fig. 8. The evaluation of (61) for an annulus as control volume will be done in section “The Momentum Balance Per Annulus”.

Numerical Assessment of Actuator Disc Performance In order to supplement the results of the momentum theory with flow details like the velocity and pressure distributions at the disc and in the wake, van Kuik and Lignarolo (2016) developed a numerical potential flow code which calculates the position and strength of the wake boundary for a prescribed uniform pressure jump p. With the wake vorticity known, all flow details can be calculated.

p

Fig. 8 The annulus as control volume for the momentum balance, including the contribution of the pressure at the surface of the annulus

3 The Actuator Disc Concept

69

0.7

Cp

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

CT Fig. 9 Comparison of momentum theory (–) and computed Cp as a function of CT . The data displayed by an open square do not have the required accuracy of maximal 0.3% deviation from momentum theory

Figure 9 shows the efficiency for wind turbine discs. The results of Froude’s actuator disc momentum theory are reproduced accurately: for CT = 8/9, R/R1 deviates 0 at s = 0. Using (65) this implies that vs increases immediately after s = 0 reaching a maximum value at a small distance behind the disc, whereafter vs decreases until vs,1 is reached in the far wake. The strength of the vortex sheet seems to exhibit a singular behavior at the disc leading edge as shown in Fig. 11. For s → 0 the strength γ → −∞, but as discussed in van Kuik (2018b), it is not possible to draw qualitative conclusions with respect to the singular behavior. For all flow cases, the shape of the vortex sheet close to its leading edge is somewhat curved but does not show a particular behavior. The slope of the vortex sheet at x = 0 is always less than 90◦ , so the sheet does not turn upwind of x = 0. For CT = 0.998 the slope is 65◦ ; for CT = 8/9 it is 46◦ .

Pressure at the Axis Figure 12 shows the pressure distribution at the axis for CT = 8/9. The pressure jump across the disc is not symmetric: |(p − p∞ )|upstream = |(p − p∞ )|downstream . A symmetric jump would require, by the Bernoulli equation, that at the axis of the disc 1 1 2 2 2 ρ(ur=0 − U∞ )x=0 = 2 p, leading to ur=0 /U∞ = 0.745. This differs from the calculated value 0.685 shown in Fig. 13. Apart from this numerical disagreement, there is no argument found in the momentum theory why the pressure jump should be symmetric.

-0.8

/| 1| -0.9

‐1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s/R

Fig. 11 The distribution of the vortex sheet strength γ (s) as a function of the distance s/R measured along the sheet, for CT = 8/9

72

G. A. M. van Kuik 0.6

p - p∞ |Δp|

0.4

0.2

0 5

4

3

2

1

0

1

0.2

2

3

4

5

x/R

0.4

0.6

Fig. 12 The pressure distribution at the disc axis for CT = 8/9

The Velocity Distribution at the Disc The velocity components at the disc are presented in Fig. 13, showing that |v| is constant (with a deviation of ±0.2%) for CT = 8/9. The same behavior is found for all other values of CT . To explain this constant velocity, the radial component of the equation of motion (1) is helpful: ρv

∂v ∂p =− . ∂s ∂r

(66)

As the disc force field is only axial, f is absent in (66). When following a streamline passing the disc and observing the increase or decrease of v, it is clear that ∂v/∂s > 0 when travelling from upstream infinity to the disc as the induction by the wake vorticity increases. After having passed the disc, the wake vorticity in between the disc and the downstream position of observation induces a negative v, contributing ∂v/∂s < 0 for increasing s. The vortex tube downstream of this position of observation remains semi-infinite for increasing s. Consequently the radial induction by the downstream tube remains constant, apart from the effect of varying strength and radial position of the expanding part of the vortex tube. When the effect of these variations is negligible, the result is that downstream of the disc ∂v/∂s < 0 with ∂v/∂s > 0 upstream of the disc, so at the disc ∂v/∂s = 0. Then, by (66), the pressure at the disc is uniform and by Bernoulli’s law |v d | = uniform. This is confirmed by the isobar pattern shown in Fig. 10 and the velocity shown in Fig. 13. Apparently, the effect of the varying strength and expansion of the first part of the wake boundary does not jeopardize the line of arguments used above, for wind turbine actuator discs (in van Kuik (2018a) discs representing propellers were discussed: now the absolute velocity at the disc is not uniform). The conclusion is that the absolute velocity at the disc is uniform, while the axial velocity is non-uniform. These results differ from the results of vortex models without wake expansion. By neglecting wake expansion, analytical treatments become into reach; see Branlard (2017) for a comprehensive treatment of this topic. These vortex models reproduce the result of momentum theory that the

1.5

u / U∞

0 2

Fig. 13 The velocity components at x = 0 for CT = 8/9

0

1

0

0.5

0.2

0.2

0

0.4

0.4

0.8

1

0.6

|v| / U∞

0.6

0.8

1

0.5

1

r/R

1.5

0

0.2

0.4

0.6

0.8

1

0 2

v / U∞

0.5

1

1.5

2

3 The Actuator Disc Concept 73

74

G. A. M. van Kuik

averaged induced velocity at the disc is the average of the velocities far up- and downstream. However, the present analysis reveals an essential difference between the two approaches: • For the semi-infinite, straight vortex tube: ud is uniform; |v d | is non-uniform, • For the actuator disc with wake expansion: ud is non-uniform; |v d | is uniform. As long as expansion may be neglected, the vortex tube offers elegant analytical treatments providing physical insights. However, modern wind turbines operate at thrust coefficients CT > 0.5 at which the expansion is significant, so the nonuniformity in ud has to be taken into account.

The Momentum Balance Per Annulus Applying the momentum balance to an annulus can only be done with the volume of the annulus as control volume, by which (48) becomes:  pAd,ann −

 ex · en pdSann = ρ

ann

u1 (u1 − U∞ )dA1,ann ,

(67)

A1

where Sann is the surface of the annulus, Ad,ann and A1,ann the cross sections of the annulus with the disc and far wake, and en theunit vector normal to Sann . With Tnon cons = Td = pAd,ann and Tcons = Tann = − ex · en pdSann , (61) becomes:

ud =

 Tann U∞ + u1 . +1 Td 2

(68)

Tann is known only after flow and pressure field calculations. In the Blade Element Momentum theory, the implicit assumption is made that Tann = 0 by which the results of the actuator disc theory are assumed to be valid per annulus. Consequently, each annulus is considered to be independent of the other annuli. It is known that this assumption is invalid, as shown theoretically by Goorjian (1972) and numerically by Sørensen and Mikkelsen (2001), but the consequences of this assumption were assumed to be modest. n Here the ratio Tann /Td has been calculated for the annuli defined by  = 10 d with n from 0 to 10 for flow state CT = 8/9. The pressure integral is calculated with x/R = ±21 as up- and downstream limits. Figure 14 shows the calculated distribution of ud , the calculated average value in the respective annulus, the result of the annulus momentum theory (68) with calculated Tann /Td , and the disc averaged value (U∞ + u1 )/2. The results show a very good match of the calculated average per annulus and the momentum theory value, except close to the disc edge where the steep change of ud requires a finer resolution of annuli to capture the distribution accurately.

3 The Actuator Disc Concept Fig. 14 ud /U∞ at the disc for CT = 8/9: the calculated distribution, the calculated average per annulus, the result from the momentum balance per annulus, and the average. The two annuli lines coincide except in the outboard annulus

75 0.7

u U∞ 0.6

calculated distribution calculated average on annulus annulus momentum theory

0.5

disc average

0.4 0

0.2

0.4

0.6

0.8

r/R

1

One of the remarkable results is that in the center of the disc, u is higher than (U∞ + u1 )/2, so for an actuator disc, the local power coefficient exceeds the BetzJoukowsky limit. Sørensen and Mikkelsen (2001) have done the same analysis by viscous CFD calculations, with approximately the same result. Similar distributions of the axial velocity have been calculated by several others, e.g., Madsen (1996), Crawford (2006), Madsen et al. (2007, 2010), and Mikkelsen et al. (2009). The authors suggest several mechanisms to explain the non-uniformity of the induction, but the conclusion is that this is due to the pressure at the annuli, acting as a conservative contribution to the momentum balance.

An Engineering Model for the Velocity at the Disc With the distribution of ud calculated for all CT values shown in Fig. 9, a surface fit to ud has been made, showing the non-uniformity as defined by the ratio Tann /Td in (68). This is presented as a distribution function G(r, CT ). Surface fitting gives the following engineering approximation: ⎫ G(r, C T )0.5 CD1 ρ (V − ωR)2 A = CD2 ρ (V + ωR)2 Acup 2 2 So

(4) (

ωR 2 ) (CD1 − V

CD2 ) − 2

ωR (CD1 + V

CD2 ) + (CD1 −

CD2 ) = 0

which then results in 

ωR V

 =

 1+

CD2 CD1



 1−

 D2 −2 C CD1 

(5)

CD2 CD1

  =1 For CD2 = 0, so no drag on the “returning cup”, this equation (5) results in ωR V as is should be because then the right side cup will run with the same speed  wind. For realistic values CD1 = 1.17 and CD2 = 0.40, Equation (5) yields asωRthe = 0.26, so the rotational speed at the cup is around 25% of the wind speed. V A drag-driven windmill or drag-driven wind turbine operates according to the same principle. The power that can be extracted can be calculated from the equations (3a) and (3b) by multiplying the forces with the arm (lever) R and the rotational speed ω:

1 1 2 2 P = (D1 − D2 )ωR = CD1 ρ (V − ωR) A − CD2 ρ (V + ωR) A ωR (6) 2 2 With the definition of the power coefficient CP =

P 1 3 2 ρV A

, where A is the total

projected frontal area of the device (in the case of the cup anemometer, A = 2Acup ), the expression for the power coefficient yields      ωR 2 ωR 2 ωR 1 CP = − CD2 1 + CD1 1 − 2 V V V

(7)

With the definition λ = ωR V where the coefficient λ is named the tip speed ratio (sometimes also written as TSR though a little bit strange here in the case of a cup anemometer since there is no “tip”), Equation (7) boils down to CP =

1 3 1 λ (CD1 −CD2 ) − λ2 (CD1 +CD2 ) + λ (CD1 −CD2 ) 2 2

(8)

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By differentiating Equation (8) with respect to the variable λ, the value λ = λopt for which the power coefficient is maximal is obtained: 3 dCP 1 = λ2 (CD1 −CD2 ) − 2λ (CD1 +CD2 ) + (CD1 −CD2 ) = 0 dλ 2 2

λopt =

2 (CD1 +CD2 ) −

(9)

4 (CD1 +CD2 )2 − 3 (CD1 −CD2 )2 3 (CD1 − CD2 )

(10)

When the drag of the returning cup is set to 0, CD2 = 0, the optimum value becomes λopt = 13 , and then the maximum power coefficient is CP max =

1 1 1 2 CD1 (CD1 ) − (CD1 ) + (CD1 ) = 54 9 6 27

(11)

And with a value CD1 = 1.17, this leads to the conclusion that the maximum power coefficient for a cup anemometer used as a wind turbine is CP max ∼ 0.09. When a semi-two-dimensional concave surface is used instead of a circular cup, the drag coefficient can be as high as CD1 = 2.30, and then the following max power coefficient can be achieved: CP max ∼ 0.17. 4 Sometimes the expression CP max = 27 CD is seen, but this expression is only valid for the case that the area of the retreating vane becomes zero. Then the reference area is equal to the area of the vane moving with the wind (A = Avane ), and in such cases, with a maximum value for CD = 2, 3, the maximum power coefficient will be around CP max ∼ 0.34. In practice, though, these values are never realized, since the returning vane will have a certain drag, which will have a negative effect on the performance. Even when the returning vane is shielded from the wind by an encasement, it will experience a certain amount of resisting drag when moving inside the encasement. The same holds by drag-driven systems in which the returning vane is minimized in frontal area, such as with flexible, folded, or hinged sails. And on top of that, the folding or hinging system will add parasitic drag during rotation which has a detrimental effect on the performance.

Momentum Theory Momentum theories are based on the three basic conservation laws in mechanics, the conservation of mass, momentum, and energy. Though the interest and the first descriptions of flows and fluids were already seen in early Greek times as well in China, the fundamental understanding of fluid mechanics started with the insights and modelling efforts of Isaac Newton, Daniel Bernoulli, Jean d’Alembert, and Leonhard Euler in the last part of the seventeenth and the early eighteenth century. Conservation of mass was the first fluid mechanics principle and already known in its principle of continuity by the Greek philosophers when analyzing the flow through flows and fountains.

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Gerard J. W. van Bussel

With water as a fluid, the continuity equation is probably rather easy to comprehend, but it even took quite some time before Isaac Newton realized, when he was experimenting with a perforated water barrel, that the amount of water discharging from the holes was about 50% of the expected amount of water based upon the total area of the holes. Once he found out the “vena contracta” effect first described by Evangelista Torricelli was responsible for the reduced volume of water escaping through the holes, the continuity of the water, as well his understanding of pressures in fluids, was confirmed. Understanding the continuity equation, together with the conservation of momentum in subsonic aerodynamics, emerged with the research of Daniel Bernoulli and Leonhard Euler. Bernoulli’s famous book Hydrodynamica was published in 1738, (see Fig. 5) and among others, it carefully describes the mass conservation law as well as the famous Bernoulli equation for the relation between pressure and velocity in an incompressible fluid: 1 p0 + ρV 2 = constant 2

(12)

Equation (12) is basically an energy conservation law. It states that the sum of the static pressure and the dynamic pressure does not change when following a particle of air along its way. But before the time of Bernoulli, the concept of energy was not completely clear. Gottfried Wilhelm Leibniz (1646–1716) defined a kinetic energy conservation law in classical mechanics by stating that in a mechanical system, the  sum of the masses, multiplied with the square of their respective velocities mi Vi2 , remains constant. The conservation of energy law in classical mechanical sense with discrete masses was first described by the female French philosopher and mathematician Émilie du Châtelet in 1740, where at the same time this principle was adopted and applied for fluid mechanics problem by Bernoulli together with the potential energy concept. Jean ‘d Alembert, a French physicist and mathematician, can be held responsible for deriving the momentum equation. In his Traité de Dynamique written in 1743, he comes up with what is also known as d’Alembert’s principle: “the sum of the forces acting on a system and the time derivatives of the momenta of the system is zero” or written in equation form 

(Fi − mi v˙i − m˙ i v).δri = 0

(13)

i

where the δri denotes nowadays the well-known local derivative of the displacement vector, a property defined by d’Alembert as “virtual displacement” where the word “virtual” is a synonym for “infinitesimal.”

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Fig. 5 The book Hydrodynamica by Daniel Bernoulli about the derivation of basic fluid mechanics equations, published in 1738

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Gerard J. W. van Bussel

Next Leonhard Euler showed, for the first time, the power of combining the first two conservation laws, in his article "Principes généraux du mouvement des fluides” (General principles of moving fluids), published in Mémoires de l’Académie des Sciences de Berlin in 1757. Euler was educated and strongly influenced by Johann Bernoulli, a famous mathematician and the father of Daniel Bernoulli, and was well aware of the work of his co-fellow working in fluid dynamics. The mathematical treatment of infinitesimal changes in moving fluids led to the following expressions, nowadays known as the (differential form of the) Euler equations 

dq dt



 +

d.qu dx



 +

d.qv dy



 +

d.qw dz

 =0

(14)

the continuity equation

P−

1 q



dp dx



 =

du dt



 +u

du dx



 +v

du dy



 +w

du dz



          dv dv dv dv 1 dp = +u +v +w Q− q dy dt dx dy dz           dw dw dw dw 1 dp R− = +u +v +w q dz dt dx dy dz

(15)

and the momentum equation. In these equations, q stands for the density of the fluid, P Q and R are the forces in the xy and z direction, and pis the pressure (Euler used the word elasticity instead, an intrinsic property of the fluid particle which should be counterbalanced by the external fluid pressure). In general, Euler and Bernoulli focused on confined flows, such as flows in pipes and channels. The conception of stream tubes in unconfined flows, such as the case in wind energy, emerged later with the work of Lagrange (two-dimensional streamlines in 1781) and Stokes (three-dimensional streamlines in 1845). Apart from the “stream tube concept,” also another important conceptual step had to be made to be able to formulate the momentum theory for wind turbines. This is the “actuator disk” representing the energy extracting action of a wind turbine rotor in operation. This concept only evolved by the end of the nineteenth century and is generally ascribed to William Rankine and R.E. Froude, though also Alfred Greenhill is mentioned in this respect. Some more detail about actuator disk modelling and the use of an actuator disk for determination of forces and performance of wind turbines can be found in subsection “The Navier-Stokes Equation” of this contribution. A more detailed description on this concept and its use is found in the contribution “The actuator disk concept” by Gijs van Kuik [1] that can be found in Section 1 of the handbook.

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The Navier-Stokes Equation The work of Leonhard Euler and Daniel Bernoulli was a great breakthrough in fluid mechanics. A mathematical description of what is sometimes called a perfect fluid, which is a fluid without any friction, was presented. And with the corresponding equations, a lot of flow phenomena could be very well described and hence clarified. Leonhard Euler though realized that “his” equations could not yet grasp friction, and he started with efforts to try to relate fluid friction to the local pressure. These efforts were published in 1761 as research notes in “Novi Commentarii Academiae Scientiarum Petropolitanae” (New notes of the Imperial Academy of Sciences in St. Petersburg) but did not provide a proper base for correct friction modelling in fluid mechanics. The real breakthrough in the modelling of real fluids came in the beginning of the nineteenth century, when Claude-Louis Navier (1785–1836) picked up the challenge again to try to model fluid friction. Claude-Louis Navier was a French scientist and a mechanical engineer working at the famous École Nationale des Ponts et Chaussées in Paris (now a branch of ParisTech), and he was initially interested in elastic solids. But then he realized that his equations for elastic solids could well be extended for use in viscous fluids. From the existence of the laws of equilibrium and motion of fluids written in terms of differential equations with partial derivatives as developed by d’Alembert and Euler, he follows Laplace in the investigation of fluid particle actions in the fluid motion described by what we now call the Euler equations (see previous subsection). And after an extensive study, he concludes that no viscous forces are present when fluid particles are in common motion but that these only arise when there is a relative motion between the particles. And as a follow-up on the earlier attempt from Euler, he further concludes that there is an increase of pressure related to particles that have a relative movement toward each other, and a decrease in pressure when particles move away from each other, but that this has no net effect on the magnitude of the fluid pressure as such. The end result of the research done by Navier can be presented according to the following formulae:      2       du dp d u d 2u d 2u du du du =X +  −u −v −w + 2+ 2 − 2 dx dt dx dy dz dx dy dz       2       2 2 dv d v d v d v dv dv dv 1 dp =Y + −u −v −w + 2+ 2 − 2 ρ dy dt dx dy dz dx dy dz      2        2 2 dw d w d w d w dw dw dw 1 dp − =Z+ −u −v −w + + ρ dz dt dx dy dz dx 2 dy 2 dz2 (16) 1 ρ



The  in these equations is what Navier called the “adherence” of the fluid (later identified as viscosity).

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Gerard J. W. van Bussel

Remember there was no vector/operator notation in those days nor the habit to denote partial derivatives using the δ symbol; hence, this rather extensive x, y, z component notation in Equation (16). The way George Stokes came into the picture in developing what was later branded the Navier-Stokes equation is not completely clear. Stokes was an Irish mathematician who became the Lucasian Professor of Mathematics at Cambridge University at the age of 30 in 1849. He worked at Cambridge University until his death in 1903. In mathematics, he is most known from the “Stokes theorem” used to reduce a multiple integral of a differential function over an area to the integral of the function itself along the boundary of the area. He showed great interest in what we now call mathematical physics and had many important contributions to the theories of light and fluorescence and even chemical engineering. With respect to fluid mechanics, he combined existing theoretical knowledge with carefully designed and performed experiments and in more practical engineering problems, such as the internal friction of fluids on the motion of pendulums and the terminal velocity of a falling sphere. He came up with “Stokes’ law” an expression for the forces on a moving sphere in a fluid, and he published an article in the Transaction of the Cambridge Philosophical Society titled “On the theories of the internal friction of fluids in motion and of the equilibrium and motion of elastic solids” in 1845, when he was a fellow of Pembroke College, so even before he became professor in Cambridge. In this article, he basically picks up the concepts of Claude-Louis Navier and elaborates further on them and includes internal fluid friction into the fundamental equations of hydrodynamics. By applying methods similar to Cauchy’s and Poisson’s, he arrived at the N-S equation by saying that this equation and the equation of continuity “[...] are applicable to the determination of the motion of water in pipes and canals, to the calculation of friction on the motions of tides and waves, and such questions” (Bistafa 2018). The incompressible Navier-Stokes equation, often used in wind turbine aerodynamics, then yields in index notation common in the Stokes era 

∂u ∂t



 + uk

∂uj ∂xk



1 = gj − ρ



∂p ∂xj



 +v

∂ 2 uj ∂xi ∂xi

 (17)

where gj is the acceleration due to a body force in the xj direction and ν is the “kinematic viscosity” defined as the ratio between the dynamic viscosity μ and the fluid density ρ: v = μρ .

Actuator Disk Theory An actuator disk (AD) is an imaginary infinitively thin permeable surface that is able to extract (wind turbine) or inject (propeller) energy to the flow. The actuator disk is placed perpendicular to the incoming flow, and the exchange of energy is represented by a (uniform) pressure jump over the actuator disk. In a steady flow, the stream tube around the actuator disk can be obtained by following (“staining”) the particles of air passing just across the edge of the actuator disk. In Fig. 4, Fig. 6

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Fig. 6 Actuator disk and corresponding stream tube

such a stream tube around the actuator disk Ais depicted. Flow particles outside the stream tube will stay outside the stream tube. So they will not cross the actuator disk, but will pass along the side. The incoming flow velocity is V , the area of the actuator disk is A,and F is the force applied by the disk upon the flow. Since the force is acting against the wind direction, the flow will retard, and because of continuity, the stream tube has to expand. The fact that the “end” area A2 is larger than the area of the actuator disk A was already conceived ago in the end of the nineteenth century by Rankine and Froude, but the insight that the expansion of the stream tube commences already upfront of A was only gained in the beginning of the next century when Betz and Lanchester started thinking about optimal performances of wind turbine rotors. If we now apply the continuity equation for the stream tube shown in Fig. 6, the result is ρV A0 = ρV1 A = ρV2 A2

(18)

It is difficult to pinpoint the moment in time at which the idea was conceived that low-speed aerodynamics can be treated with the “incompressibility” argument, as is done nowadays, certainly in wind turbine aerodynamics. Most probably that insight came in the time Laplace lived, from 1749 to 1827, but it might already been implicitly used by his famous predecessors Bernoulli and Euler in the period 1740–1760. Laplace is of course the key in the further development of aerodynamic theory since he came up with a thorough mathematical basis for potential flow aerodynamics. Using incompressibility (ρ = constant), the continuity equations for wind turbine aerodynamics narrows down to V A0 = V1 A = V2 A2

(19)

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Gerard J. W. van Bussel

The most simplest way to implement the conservation of momentum law in wind turbine aerodynamics is to assume that the pressure everywhere around the stream tube depicted in Fig. 6 is equal to atmospheric pressure p0. According to Bernoulli’s law (12), the velocity around the stream tube should then be equal to V everywhere, and that is of course something that needs further investigation. But under this assumption, the following relation holds according to the law of conservation of momentum: ρV A0 V − ρV2 A2 V2 = F

or

F = ρA0 V 2 − ρA2 V22

(20)

When using the continuity equation (14), Eq. (15) results in F = ρA V 1 (V − V2 )

(21)

And this equation tells us that the force F applied by the AD equals the loss in velocity (V − V2 ) of the mass flow (the mass per second) “processed through” the AD: ρAV 1 . This understanding was already quite familiar to both Bernoulli and Euler, but they lacked the concept of the actuator disk and could not comprehend in detail the consequences on the flow velocities. Fair enough, but as yet not sufficient to either draw a conclusion about the magnitude of the force or the velocity reduction in the wake of the actuator disk. Yet there is another way to calculate the force on the actuator disk, and this comes from the fact that the force should be equal to the pressure jump over the AD multiplied by its area: F = (p1 − p2 )A

(22)

And then there is Bernoulli’s law (12) to calculate the pressures upfront and downstream of the AD:

and

1 1 p0 + ρV 2 = p1 + ρV12 2 2 1 1 p2 + ρV12 = p0 + ρV22 2 2

(23)

where p is the ambient pressure, p1 the pressure just in front of the AD, and p2 the pressure directly behind the AD. From (18), it can be directly seen that the pressure difference over the AD is equal to

7 History of Aerodynamic Modelling

223

p 1 − p2 =

1 ρ(V 2 − V22 ) 2

(24)

And hence the force on the AD equals F = (p1 − p2 ) A =

 1 1  2 ρ V − V22 A = ρ(V + V2 ) (V − V2 ) A 2 2

(25)

Combining the two equations for the force on the AD yields V1 =

1 ρ(V + V2 ) 2

(26)

This important result states that the velocity through the AD is the average of the incoming and the end velocity in the stream tube, or equivalent, half of the ultimate velocity reduction is “felt” at the location of the AD. And finally the power extracted by the application of the force F , often branded as thrust, on the incoming flow with velocity V is equal to the total energy loss in the stream tube: P = P1 − P2 =

1 1 ρ(V 2 A0 V − V22 A2 V2 ) = ρ(V 2 − V22 )AV1 = F V1 2 2

(27)

And this is an important, interesting, and also plausible result since it states that the power P extracted by the AD equals the thrust F on the AD multiplied by the local velocity V1 at the AD. The introduction of non-dimensional properties as became common practice in aerodynamics in the end of the eighteenth century and its first use is attributed to Otto Lilienthal (1889 Der Vogelflug als Grundlage der Fliegekunst Figure 7). With the introduction of an induction coefficient a = 1 − VV1 , the coefficient representing the velocity reduction induced by the presence of the actuator disk, the key non-dimensional momentum equations for wind turbine aerodynamics (16) and (22) become CT =

F 1 2 2 ρV A

= 4a(1 − a)

(28)

= 4a(1 − a)2

(29)

and CP =

P 1 3 2 ρV A

for the thrust coefficient and the power coefficient, respectively. It is not exactly clear when these equations were first presented, but most probably it was in the same time as Lilienthal wrote his book, since he also expressed a lot of other aerodynamic equations in a non-dimensional fashion.

Fig. 7 From the book Der Vogelflug als Grundlage der Fliegekunst by Otto Lilienthal (1889)

224 Gerard J. W. van Bussel

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225

From (29), it is fairly easy to come up with a value for the maximum value of CP , simply by differentiation of the expression with respect to the axial induction factor a. The result is an optimal value a = 1/3 and a max power coefficient CP max = 16/27. From the continuity law, it can be concluded that with a=1/3, the amount of air through the actuator disk at optimum operation is 2/3 of the incoming mass flow. And from this mass flow, 8/9 of the kinetic energy is extracted at a thrust coefficient CT = 8/9. It is also easy to conclude that at optimal operation, the “end” velocity in the stream tube of Fig. 6 will be equal to (1-2a)V=1/3V, so the velocity “caused” by the actuator disk found in the wake equals 1/3 of the incoming flow velocity. This optimal value was found more or less simultaneously by Betz and Lanchester. With this result, it is already possible to design close to optimal wind turbine rotors. Using the above axial momentum theory result, the local flow conditions for rotor blades are known. The incoming velocity is the resultant of the local rotational component caused by the rotation of the rotor blades and the axial component which is 2/3 of the incoming wind speed. The local chord and the angle with respect to the local incoming flow at the rotor blade aerfoil should then be such that the axial component of the lift generated, multiplied with the number of blades, should be equal to the optimal local thrust force Eq. (30):  Bl (r) cos

2 3V

r



 2  2 8 1 2 V = • ρ ( r) + 9 2 3

(30)

where l(r) is the local lift generated by the aerfoil at radial station r and B is the number of blades.

Potential Flow Theory/Vortex Modelling Apart from the development of the momentum theory, also the mathematical description of flows became an important asset for wind turbine aerodynamic modelling. Though Pierre-Simon Laplace is known as the founder of potential flow theory, based upon “his” Laplace equation, the history on this is not evident on his side only. Leonhard Euler published his General Principles of the Motion of Fluids in 1757, and in this book, the velocity potential was mentioned, while Laplace used it in his memoirs from 1773, where he showed that the scalar potential function is a solution of the Laplace equation. The term potential flow is attributed to Daniel Bernoulli, in his book Hydrodynamica published in 1738, I(fig 5) and stems from the situation where the difference in potential energy in a flow depends only on the positions, but not upon the path taken to travel from one position to the other. Potential flow modelling is valid in situations where the flow can be considered incompressible and inviscid.

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Gerard J. W. van Bussel

Two of the key aspects of potential flow theory are: (a) Solutions of the Laplace equation are scalar functions. (b) Linear superposition of solutions is possible. In other words, when two solutions are known, any linear combination of solutions is also a solution of the Laplace equation. The boundary conditions in a given flow problem then determine which “basic potential flow solutions” apply and which amplitudes of these solutions need to be taken to fulfil these conditions. The key equations for a two-dimensional potential flow solution are the 2D Laplace equation

ϕ =

∂ 2ϕ ∂ 2ϕ + 2 =0 2 ∂x ∂y

(31)

and the equation providing the relation between the scalar function φ and the − → velocity vector V ∂ϕ − → ∂ϕ − → − → V = ∇ϕ = 1x + 1y ∂x ∂y

(32)

− → The arrows are used here to emphasize the vector character of the velocity, and 1x − → and 1y are the unit vectors in x and y direction, respectively. Moreover, and ∇ are the Laplace operator and the gradient operator, respectively, and their definitions are implicitly presented in the above equations. A very well-known and important solution for these two equations, the sink/source potential reads ϕ = A ln r where r is the radial distance to the origin of the coordinate system r = and A is its amplitude. The resulting velocity distribution, following Eq. (32), reads → − → A− V = 1r r

(33)

x2 + y2

(34)

The velocity is inversely proportional to the radius and proportional to the amplitude A. When A is positive, the velocities are pointing radially outward (a velocity source located in the origin), and a negative A represents a velocity sink in the origin. With a distribution of sinks and sources, and the addition of a parallel flow field potential, it now is possible to generate a velocity field around many different shapes. The potential function of a parallel flow along the x axis is very obvious: ϕ = Ax

(35)

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227

and any linear combination of (32) and (35) will provide the (inviscid incompressible) potential flow solution ϕ = A1 x + A2 ln r

(36)

of the flow around a (two-dimensional) bluff body. When a sequence of sources and sinks are located along the x axis, it is possible to generate potential flows around symmetric aerfoil shaped bodies. A problem arises though when calculating the (pressure) force around such closed bodies. The pressure differences are directly related to velocity differences by the Bernoulli equation (12), and integrating these pressures along the body (the separating streamline depicted in red in Fig. 8), the end result is no resulting force. This might be expected, since in a potential flow, a symmetric body will not generate lift and drag, but the fact is that it is still true for streamlined bodies under an angle of attack. The integral of the pressure forces along the body will result in a zero force as well. The underlying velocity potential function is a little bit complicated but can be obtained through a solid body rotation of the coordinate system. This phenomenon is known as d’Alembert’s paradox to honor Jean le Rond d’Alembert, following an article he wrote in 1749 when he was working on a prize problem of the Berlin Academy on fluid drag (Fig. 8). It is evident of course that something needs to be added in order to represent real 2D flows, for example, those present in wind tunnel measurements on aerfoils (see Section 3.2 of the handbook Measurement Techniques for Airfoil Measurements by Nando Timmer)[2]. The answer is the introduction of a two-dimensional potential flow solution representing a circulatory velocity around the aerfoil. Equation (37) provides the velocity potential function of a 2D discrete vortex: ϕ = A arctan

y x

(37)

Taking the divergence of equation (see Equation (32)), this results in u=

dϕ 1 −y y y =A = −A 2 = −A 2 2 2 2 y dx x +y r 1+( ) x

and

x

v=

1 x 1 x dϕ =A =A 2 =A 2 2 y 2x dy x + y r 1+( )

(38)

x

The velocity induced by this vortex potential function (37) is hence inversely proportional to the distance from the origin. Along the positive x axis, the u component is 0, and the v component is pointing upward (for positive A). Along the vertical axis, the u component is negative and v component is 0. Along circles around the origin, the velocity is tangential and constant in magnitude, and the magnitude decreases proportional to 1/r:

Fig. 8 Left: streamlines (velocity contours) around a bluff body created by the combination of a parallel flow and a source term in the potential flow model. Right: flow around a “Rankine oval” where a sink term of equal strength has been added to the solution of the left figure. (Courtesy Cambridge-MIT MDP Resource Library: http://www-mdp.eng.cam.ac.uk/web/library/)

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7 History of Aerodynamic Modelling

A − → − → − → V = 2 (−y 1x + x 1y ) r

229

(39)

This circular flow of which its magnitude decreases inversely proportional to the distance of the vortex is often referred to as circulation, and the symbol used is often A

where = 2π . Analogous to the distribution of sources and sinks, also vortices can be distributed along the center line of a symmetric aerfoil. These vortices will generate a velocity difference parallel to the aerfoil surfaces, and hence a pressure difference is created, which, when integrated, leads to the lift on the aerfoil. The amount of lift can still be chosen arbitrarily through the amount of vorticity present along the center line such that a tangential flow is established along the aerfoil surface. The question still remains what the strength should be of the vorticities along the center line. This was solved in 1902 by the German mathematician and aerodynamicist Martin Wilhelm Kutta, and his “solution” is known as the Kutta condition. It states that the flow must be parallel to the center line of the aerfoil at the location of the trailing edge. This can be interpreted as the implementation of a viscous phenomenon into inviscid potential flow theory. In viscous incompressible flows, it is impossible to establish attached flows around sharp edges, unless, of course, the direction of the flow is parallel to the surfaces close to the surface. And with sharp trailing edges, normally found on aerfoils, this tangency condition is reflected through application of the Kutta condition. This Kutta condition then determines the magnitude of , where is usually the integrated value of the vorticity along the center line. Its value can be easily obtained by integrating the velocity along a closed contour about the aerfoil; hence, is often branded as “the circulation.” Circulation and vorticity are often erroneously mixed up, where circulation is the consequence of the existence of vorticity. For increasing angles of attack, the value of will increase proportionally. Since flow separation is a viscous phenomenon, not represented in potential flow theory, the lift of an aerfoil keeps increasing for increasing angles of attack. For symmetric aerfoils, the following relation holds: 1 L = 2π α ρV 2 = ρV Γ 2

(40)

where α is the angle of attack (in radians), ρ is the density of air (~1.225 kg/m3 at sea level and at 15 degrees Celsius), and V is the velocity of the incoming flow. Non-symmetric aerfoils, hence aerfoils with a curved center line (or “mean line”), can also be represented in potential flow theory. In such cases, sources/sinks and vorticity are distributed along a curved center line. Nikolay Yegorovich Zhukovsky (1847–1921) (often spelled as Joukowski) is called the father of Russian aviation. He is renown from the Joukowski transformation, in which a conformal mapping procedure is used. In this transformation a complex plane (z=x+iy plane) including the flow potential around a circle (a twodimensional cylinder) is mapped into another plane, resulting in the potential flow around an aerfoil.

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Fig. 9 Joukowski transformation, the conformal mapping of an eccentric circle into an aerfoil

The classical Joukowski aerfoils always have a cusped trailing edge, and in the conformal mapping, the rear stagnation point of the circle is transferred into the trailing edge of the aerfoil. Hence, the Kutta condition is always satisfied, and tuning the circulation/vorticity to a smooth trailing edge flow on any sharp edged aerfoil is also called the Kutta-Joukowski condition. For those interested in the mathematics of potential flow theory and vortex modelling, the books of Prandtl and Tietjens are classical must-reads/must-haves that have been reprinted many times. Also the book of Anderson Fundamentals of Aerodynamics provides, among many other issues, a good introduction to these topics (Fig. 9). In modern design tools for wind turbine aerfoils, the shape of the aerfoil is usually generated by different means then by using distributed sources and sinks inside the aerfoil body. Nowadays often NURBS (non-rational uniform B-splines) or Bezier curves are used to generate a wide selection of aerfoil shapes. The surfaces of such shapes are then usually equipped with “panels,” which are in fact (very) small straight line sections, and on such panels, either a given amount of distributed vorticity is assumed or a discrete vorticity strength at either edge of the panel A non-crossing (parallel) velocity boundary condition is then applied at the panel to assure tangential flow. On top of this, a boundary layer model is added to assure close resemblance to reality. More information about boundary layer modelling can be found in Section 2.4 of the handbook Interacting Boundary Layer Methods and Applications by Hossein Ozdemir [3]. A very often used tool for aerfoil design was originally developed by Mark Drela at MIT in 1989 and is called XFOIL. XFOIL is an interactive program for the design and analysis of subsonic isolated aerfoils. It consists of a collection of menu-driven routines, and it is open-source software.

BEM Development The combination of momentum theory and aerfoil performance characteristics can be used for the design of rotor blades. The first attempts of developing a theory for the aero-/hydrodynamic design of rotor blades were made by William Froude and Stefan Drzewiecki in the end of the nineteenth century where both were working on the design of ship propellers. By dividing a rotor blade into a number of sections in radial direction, they were able to calculate the forces on each “blade element” and then integrate them to obtain the global forces on the rotor.

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A number of complicating issues prevent the straightforward implementation of such blade element theory (BET) for accurate predictions of the performance. The most important issue is the fact that at the plane where the rotor is located, the incoming velocity has a different value than the velocity far ahead of the rotor. By combining blade element theory with actuator disk theory (see subsection “The Navier-Stokes Equation”), which provides a method to derive the (axial) induced velocity at the rotor disk, given the amount of thrust on the rotor, a design method can be determined. In an iterative fashion, the thrust on the rotor is first “guestimated” using actuator disk theory, which results in an induction velocity at the rotor disk. This (axial) induction velocity is then used in the blade element approach to determine the local forces. Integration over all the blade elements and multiplication with the number of blades then results in an (integrated) thrust. This thrust may now be used to recalculate the induction velocity in the rotor plane, and hence, in an iterative way, the loads on a real rotor can be determined. For the determination of local forces, knowledge is needed about the lift and the drag of the aerfoil used at the respective blade element. This may come from wind tunnel experiments on “twodimensional” wing sections, from potential flow and/or vortex filament modelling or, in more recent times, from aerfoil design codes such as XFOIL (see previous subsection “Actuator Disk Theory”). A second complication of using the combination of blade element theory and momentum theory is the fact that the result from the local load calculation at a given blade element is an axial force, contributing to the thrust on the rotor, but also a tangential (in-plane) force contribution to the torque. This tangential force will evoke a reaction on the flow, resulting in a tangential induction velocity. This tangential induction velocity should then be taken into account for the next iteration, since it has an effect on the lift and drag of the aerfoil. This is usually taken into account by extending the actuator disk theory with a tangential induction velocity as well. Then the results from the blade element load calculations are transferred to the extended actuator disk theory, which then provides “new axial and radial induction factors” for the next blade element calculation. The simple actuator disk theory in this iterative BEM calculation procedure is replaced by an actuator annulus approach where axial and tangential induction velocities are calculated in concentric annuli coinciding with their respective blade element radial sections. With the occurrence of the computer in the 1970s, it became possible to develop numerical codes for implementation of the iterative procedure for designing wind turbine rotors, as well as for determining the performance of existing rotor design over a large part of the operational envelope. The most famous software tool developed in these days was the PROP code by Robert Wilson and Peter Lissaman, presented in their “Applied Aerodynamics of Wind Power Machines” in 1974. A European version of this FORTRAN code PROPSI was developed soon after and used in the countries operating with the SI system of units, in contrast to the original version which used feet, pounds, and miles per hour.

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More details about further BEM code development and use can be found in Section 2 of the handbook in the chapter “Pragmatic Models: BEM with Engineering Add-Ons” by Gerard Schepers [4].

The Emergence of Lift-Driven VAWTs Apart from the “standard” horizontal axis wind turbine most seen in present days, a different design principle has been developed over the past hundred years. The idea of a vertical axis wind turbine (VAWT) with rotor blade operation according to the lift principle (in contrast to drag-driven machines) came from a French aeronautical engineer Georges Jean Marie Darrieus, who filed the patent of his Darrieus rotor on October 1, 1926. A Darrieus rotor usually has two or three rotor blades with a (close to) parabolic shape. The official name of the ideal shape is troposkein (Greek “shape of a swinging rope”), and this shape is chosen to minimize the bending moments along the rotor blades caused by centrifugal loads during rotation (Fig. 10). In practice, this blade shape is often approximated by two straight end parts and a circular midsection; see Figure 11b. The Darrieus rotor operates according to the lift principle, which means that the torque generated by the rotor blades is originating from lift acting upon the (streamlined) cross section of the blades. Figure 12 shows a schematic of the lift acting on Darrieus rotor blades at its cross section at the mid-span position. It can be seen that the vector sum of the inflow velocity and the rotational velocity results in a non-zero angle of attack and hence in a lift force acting upon the blades, at least on the windward and the leeward side of the rotor. When the rotor blade is moving away from the wind, or toward the wind (the three o’clock and the nine o’clock positions), the relative flow is such that no lift acts on the blade. Hence, the

Fig. 10 Drawing from the original Patent of G.J.M. Darrieus. Left: the troposkein shape version, the one that made Darrieus famous. Right: his straight bladed variant

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Fig. 11 Left: Darrieus wind turbine components. Right: two-bladed Darrieus wind turbine installed in Martigny, Switzerland. (Source Creative Commons, picture by Lysippos)

aerodynamic loads on the rotor blades experience a periodic variation during each revolution (Fig. 12). A second thing to notice is that the suction side and the pressure side of the rotor blade change roles during one rotation. At the six o’clock position, the inner side of the aerfoil acts as the suction side, whereas at the 12 o’ clock position, the outer side of the aerfoil “feels” the lowest pressures. This also leads to the fact that (virtually) all lift-driven VAWTs are equipped with symmetric aerfoils, in order to deal with both negative and positive angles of attack during a revolution. Sometimes asymmetric aerfoils are seen in VAWT designs, for example, to be able to better deal with lift generation along the “undisturbed” front passage across the incoming wind, where the downward passage of the rotor blade experiences all kinds of flow disturbances, caused by the (dynamic) wake of the windward blade and the central column. The major advantage of vertical axis machines is their insensitivity to wind direction changes and hence the absence of a yaw system to orient the rotor in the direction of the wind. A very successful company in the 1980s was FloWind. The company installed more than 500 Darrieus wind turbines in California in the 1980s with a total capacity of around 100 MW. Dynamic loads on VAWTs are intrinsically present as has been explained above. Hence sticking to the troposkein shape of the rotor, as in the original Darrieus concept, does not provide the ideal solution in real designs. Moreover the fact that the diameter of the rotor varies with height also implies that the amplitude of the angle of attack varies along the span of the blades, which makes them less effective. Larger Darrieus rotors also suffered from the fact that gravity loads on the lower blade connection point introduce fairly large bending moments. Hence, a quest for

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Fig. 12 Working principle of a Darrieus rotor. Right-hand side cross section at mid-span position; the central axle is omitted to allow better clarification of the lift generation process

more optimal designs was initiated as a follow-up from the classical design. Though already conceived by Darrieus in his patent (Fig. 10), straight bladed designs started popping up in the 1980s in order to increase the swept area and allow for a more optimal energy extraction. The disadvantage though is that straight bladed designs (see Figures 10b and 13) always need struts to fix the rotor blades at their radial position, and the drag on such struts may have a significant detrimental effect on the performance of the rotor. Not only the drag of the rotating strut(s) themselves(s) but also the interference drag between the blade and the strut at the connection point contributes to the performance loss. An advantage is that it opens the possibility to pitch the rotor blades during a revolution, which leads to a further improvement of the aerodynamic performance. And it makes better power control possible. A classical Darrieus rotor and its straight bladed counterpart do not experience a starting torque in uniform inflow, and at high wind velocities, only RPM control is available for power limitation. But VAWTs with pitchable blades also have their drawback since it takes the design away from its elegant simplicity and makes it again wind direction sensitive. Over the years, a number of innovative designs popped up from which the VAWT450 is probably the most prominent (Fig. 13).

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Fig. 13 The VAWT-450 a two-bladed variable geometry vertical axis wind turbine located at Carmarthen Bay in the UK Left: the complete machine installed with reefed rotor and drive train located on the bottom of the concrete tower Right: the blade system in power control (reefing) mode and in full power (H-Darrieus) mode, respectively

This variable geometry vertical axis wind turbine, a concept conceived by Peter Musgrove from Reading University which was realized in 1986, had a rotor diameter of 25 m, two blades with a length of 18 m, and a swept area of 450 m2 . Its rated power was 130 kW and the rated wind speed was 11 m/s. The reefing system was designed to be able to maintain a constant power output and to shut down the system at wind speeds above the cut-out wind speed of 30 m/s (Fig. 13). After a few years of testing the machine at its location in Carmarthen Bay, South Wales, UK, the conclusion was drawn that the reefing system complicated the design a lot and that there were no real advantages, as power control could well be established through RPM control, thanks to the (periodic) stalling of the rotor blades at low tip speed ratios. After a few decades of declining interest in vertical axis wind turbine, new interest in VAWTs was emerging in the beginning of this century. A number of companies worked on the development of urban VAWTs, intended to be used in the built environment. As such, more clarification about the aerodynamics of liftdriven VAWTs can be found in Section 6 of the handbook Vertical Axis Turbines by Delphine de Tavernier and Carlos Simao Ferreira [5].

The Number of Blades Issue As long as the development of modern wind turbine rotors has been ongoing, there always was the issue of the number of blades to be used. Since most of the modern wind turbines designed in the last century were for electricity generation, and electrical generators usually run at high speed, it makes sense to design the wind turbine rotor such that its rotational speed is high as well. This is to avoid a too large gear ratio between rotor speed and generator speed. When looking further back in history, this high RPM limited number of blades approach was not seen yet. The starting point for those electricity-generating wind

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turbine designs was their more classical use, such as for water pumping, irrigation, and grinding wheat. The American wind pumps had a large number of blades, usually between 24 and 36. This choice was made to be able to generate sufficient (starting) torque to get the piston pump running. In the Netherlands, the elevation height was usually quite small (around 1 m), and here, traditionally four-bladed windmills were applied together with a water wheel and later also an Archimedes screw to pump the water out of the polders into the canals that further transported the water to the rivers and the seas. In the development of “modern” electricity-generating wind turbines, a clear trend can be seen. In the national research programs that were initiated and implemented in the 1970s and 1980s, a preference was seen for two-bladed designs, at least in the USA, the UK, Germany, and the Netherlands (Fig. 14). Denmark followed a different track, by always sticking to three-bladed designs since the 1970s. Countries with a history in helicopter design have also opted for one-bladed wind turbine rotors, for example designs by the Italian company Riva Calzone and by MBB, a (former) German helicopter manufacturer. Throughout time, when the industry matured, the three-bladed (upwind) design came out as the winner. The main reasons for this design choice are their relative low noise generation (certainly when compared to high TSR one-bladed rotors) and a more aesthetic appearance. The latter may sound somewhat surprising, but once confronted with

Fig. 14 Two famous two-bladed wind turbine designs with a teetered hub connection. Left: the Smith-Putnam wind turbine, Grandpa’s Knob, USA, a 53 -m-diameter down-wind design (1941) Right: NASA-MOD2, a 91-m-diameter 2500 KW upwind design (1980) with pitchable outer blades

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two-bladed wind turbine rotors in the field, this issue is immediately clear. Seen from a perspective different from a frontal view (or a view from the back), a twobladed rotor seems to “pulsate” when rotating, while three (or more) rotating blades are always visually associated with their circular rotor plane, no matter from which direction they are seen. And this is perceived as a much more quite, steady view when compared to a two-bladed rotor. A third advantage is that the rotor is also aerodynamically more balanced, although the unbalance of a two-bladed design can be mitigated by adding a flexible blade-hub connection, such as a teeter. The advantage of such a teeter, a hinged connection between rotor blades and main shaft, is that the large bending moments due to rotor thrust are not carried over to the main shaft. This also means that any unbalance in blade root moment between the two is counteracted by a teeter movement. This results in a tip path plane that is slightly skewed with respect to the plane perpendicular to the main shaft (Fig. 14b). More information about such flexible blade construction and its consequences for rotor aerodynamic loads can be found in Chapter 1 “Alternative Concepts” of Section 6 of the handbook [6]. The main disadvantages of a three-bladed design are more material usage and a larger, heavier gearbox. The latter is needed in order to meet the high nominal rotational speed of the generator. In terms of aerodynamic efficiency, there is, in first instance, no real difference between a large number of slowly rotating blades and a few fast running blades. High tip speed ratio rotor blades need a more careful design choice regarding their aerfoils, since there is a much larger drag penalty. The torque generated by a rotor blade is the resultant of the sine component of lift and the cosine component of drag. And for well-designed high tip speed ratio rotors, the incidence angle (the angle between the rotor plane and the local blade aerfoil) is small. As such also the sine component of the lift is small, and hence its contribution to the torque is then easily outweighed by a somewhat too large drag. This will then result in a bad to non-performing rotor. This does not hold, at least to a much lesser extent, for very low tip speed ratio (multiple blade) designs, since the incidence angles of such rotors are a lot larger. For a more detailed discussion about the relation between the number of blades, the tip speed ratio, and the corresponding design requirements as well as the aerodynamic efficiencies of the different designs, the reader is referred to Section 1 of the handbook, Chapter 4 “Rotor Blade Design, Number of Blades, and Performance Characteristics” by Georgios Sieros [7].

Dedicated Aerfoil Designs for Wind Turbine Applications The search for dedicated aerfoils for wind turbine rotor blades started in the 1990s of the last century. Almost all commercial wind turbines in those days were using stall control and were running at a constant (or at dual) rotational speed. The quest at those days was to increase the size of the wind turbines; every other year, a wind turbine was put on the market that surpassed the size and capacity of the machines with a large step.

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That led to the fact that the general aviation aerfoils used so far (such as the NACA 4412, NACA 23012, and NACA 64-618 aerfoils) could not anymore meet the requirements for wind turbine rotor blade designs, mainly because of their limited thickness. Stall-controlled (fixed-pitch) wind turbine rotor blades experience a large flapwise bending moment (a bending moment in the direction of the incoming wind), and with increase in size, the 12% thickness of the classical aerfoils was not sufficient anymore. And apart from the demand for a larger thickness (that could have been provided following the classical NACA series, since most or the designs also provide an 18% and sometimes even a 24% thickness version), also two other design requirements had to be complied with. Design requirements that could no longer be met using general aerfoil series. These requirements are a benign stall behavior and the (relative) insensitivity for roughness (read accumulation of dirt and insects). The benign stall behavior, a gradual reduction of lift with increasing angle of attack after clmax is reached, was key in those days needed to realize a (fairly) constant power output in the post stall (high wind speed) region, and the roughness insensitivity design requirement obviously was coming from the fact that wind turbines operate in various environmental conditions and cannot be cleaned every other day. Hence the search for thicker aerfoils meeting both the smooth stall behavior as well as the roughness insensitivity requirement. That resulted in the DU-W series from TU Delft and the aerfoil series from the FFA in Stockholm, Sweden, where the DU91W250 became the working horse for many European wind turbines. The dedicated wind turbine aerfoil was designed and tested in the LTT wind tunnel in 1991 and has a thickness of 25%, where the following coding is used: W for wind turbine, 91 for the design year 1991, and 250 for the 25.0% percent thickness. Later on, more aerfoil designs popped up for even larger thicknesses up to the DU97-W300 and beyond (Fig. 15). The FFA aerfoils follow a similar type coding, though the design year is not explicitly in the name. Early Swedish aerfoils are the FFA-W1252 and the FFA-W2-252. Both aerfoils have a thickness of 25.2%, and the code W stands for wind turbine blade used. The W1 have a low cl/cd ratio at rather high

Fig. 15 Series of dedicated HAWT rotor aerfoils designed by TU Delft, NL, in the period 1991–2000

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values of cl, whereas for the W2 aerfoils, this so-called drag bucket is occurring at lower cl values (Fig. 15). Interesting is that in these days the XFOIL code became available, and since it was open source, it was picked up by many research groups active in wind turbine aerfoil design. In the 1990s of the last century, and in fact over the whole period until somewhere around 2010, it was standard practice to complement aerfoil design with validation through testing in dedicated wind tunnels. In about the same time, James Tangler and Dan Somers worked at NREL on dedicated aerfoils for wind turbines that resulted in the S801-S815 series Fig. 16. NREL designed this series of dedicated aerfoils for different blade sections, as well as for different power control options (passive and active stall control and pitch control). Their commonality was insensitivity to roughness. In a later stage, the series was extended up to S823, but no designs were made with a thickness above 26% since this was seen to result in “unacceptable performance characteristics.”

Fig. 16 Example of a dedicated aerfoil family designed for a wind turbine blade. Tangler et al

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Sometimes an extension A is seen (e.g., S806A) which indicates a modified (adapted) S806 aerfoil (see Fig. 16). Usually such adaptation was the result of validation experiments in a wind tunnel. Also Glasgow University was active in designing thick aerfoils, such as the 25% thick GU25-5(11)8, though their interest was more in application in ultralight aircraft and unsteady aerodynamics. Nowadays wind tunnel testing has become less standard. Two reasons for this are the fact that aerfoil design codes are getting more advanced and provide reliable results and the shear sizes of wind turbine rotor blades. The blade chords and hence the corresponding Re numbers are getting so high (Re ~8–12*10^6) that proper wind tunnel testing facilities in which such high Re numbers can be reached as well are hardly available. At present, aerfoil design codes for wind turbines are no longer strictly twodimensional. This means that inevitable 3D effects when applied on rotating aerfoils are already taken into account in the design phase. And this evidently also makes it impossible to do full validation tests in wind tunnels. An example of such a design code is RFOIL, a further extension of the XFOIL code of Drela, where typical threedimensional effects experienced in real rotating conditions such as a smoother stall behavior as well as stall delay are taken into account in the aerfoil design code. More information about the design of aerfoils for wind turbine application can be found in the chapters “The Designing of Airfoils” by Christian Bak [8] in Section 1 of the handbook and “2D Airfoil Aerodynamics”[9] in Section 2 of the handbook. At present also aerfoils at root sections are applied with thicknesses that go far beyond the traditional use in aircraft. But such very thick aerfoils (40–45%) need to be designed taking full consideration of the three-dimensional flow along the inboard part of wind turbine blades. Also a proper behavior, in the sense that very early flow separation and hence large drag are avoided, can only be achieved by boundary layer manipulators such as vortex generators. The chapters “Thick Sections” by Francesco Grasso [10] and “Modeling Add-Ons” by Kristian Godsk [11] in Section 2 of the handbook provide more information about the design of very thick aerfoils and add-ons on wind turbine blades to manipulate the separation of the boundary layer.

Yawed Flow Aerodynamics, Shear Flow Modelling, and Wind Farm Aerodynamics Apart from the aerodynamics of the wind turbine rotor as such also the environment of the rotor that needs to be modelled to be able to make a proper assessment of the aerodynamic performance of a wind turbine rotor. And this is all about the inflow of the rotor. In the BEM method introduced above, the implicit assumption is the existence of a uniform inflow perpendicular to the rotor disk. As a first start in the theoretical modelling, such assumption is quite defendable, but in practice, such an ideal situation will never occur. Of course wind tunnel setups can come real close to this situation, and wind tunnel experiments are still key for further development of wind power technology; read, for example, the chapter “Wind

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Tunnel Measurements” by Carlo Botasso [12] in Section 5 of the handbook. But in order to come closer to the aerodynamic modelling of wind turbine rotor blades operation in real atmospheric conditions as present in the atmospheric boundary layer along the earth surface, more advanced approaches are needed. The first category of non-uniform flow conditions to address are situations with a yawed or sheared inflow. Yawed inflow conditions are conditions in which the incoming flow is still uniform (or can be considered close to uniform), but this flow then interacts with a rotor whose rotational plane is not perpendicular to the wind. In practice, this yawed or sheared flow case occurs quite often. Many wind turbines are designed with a rotor shaft slightly tilted upward from the horizontal position. This is done on purpose to create sufficient tower clearance for the blades when passing the wind turbine tower. A second common operating condition where a non-uniform inflow exists is when a wind turbine in operation is not perfectly aligned with the incoming wind direction. In such tilted or yawed rotor cases, the inflow of the wind, seen by the rotating blade, is a function of its azimuthal position over the rotor plane. The modelling of such yawed or tilted flows is not that evident and easy. Let’s first simplify the case further and assume there is no wind shear nor wind veer present. In other words, the incoming flow is indeed uniform in time and space, but the rotor is not aligned with the wind direction. As a start, two effects can then be identified. There is a cosine component acting perpendicular to the rotor plane – this component does not change with the azimuth angle of a rotor blade – and a sine component which is dependent on the position of the blade. This type of yawed inflow is also found on aircraft propellers, for example, when the trajectory of the aircraft is not aligned to the wind. This of course happens quite often, though the deviation from perpendicular inflow is limited, since the traveling speed of the aircraft is often far larger than the wind speed. Hermann Glauert, a famous British aerodynamics engineer and researcher born in 1892 in Sheffield (his father was a naturalized British citizen of German birth), extensively looked, among other interesting phenomena, at the flow about rotors, both experimentally and theoretically. The interest not only came from yawed inflow of propellers but also from the flow about helicopter and autogiro rotors, where the rotors are subjected to large skewed inflow angles. In a chapter “Aircraft Propellers” of an extensive book edited by W.F. Durand and first published by Springer in 1935, Glauert comes up with an expression for the thrust on rotors with a misalignment angle ψ:

CT =

F 1 2 2 ρV AR

 = 4a sin2 ψ + (cosψ − a)2

(41)

where CT is the thrust coefficient perpendicular to the rotor disk and a is the normal component of the (average) induction coefficient in the rotor plane. The corresponding expression for the power coefficient of such a yawed/tilted rotor then becomes

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Fig. 17 Inflow and induced velocities for a wind turbine rotor with a yaw/tilt angle ψ

CP =

P 1 3 2 ρV AR

  = 4a sin2 ψ + (cosψ − a)2

(42)

In order to better understand the basic assumptions of Equation (41), it makes sense to look at the underlying vector equation and to Figure 17. −  − → → → − → → − n . Vi ) n → n . Vi  (− T = 2ρAR  V + −

(43)

→ n is the unit normal where T is the rotor thrust, AR is the physical rotor area, − − → vector perpendicular to the disk pointing in downwind direction, and Vi is the induced velocity vector at the rotor disk. The absolute value operator ||stems from the fact that Glauert’s equation was also derived for situations with a very small to zero incoming wind speeds, such as propeller thrust at standstill (before takeoff), and for hovering/ascending helicopters. And in such cases, the magnitude of the induced velocity exceeds the incoming velocity magnitude and can have an opposite direction. What is interesting from Equations (41) and (42) is that the thrust T and the corresponding power P are based on the physical rotor swept area AR and not on the projected rotor surface perpendicular to the incoming wind speed V (Fig. 17). That insight stems from the flow around autogiros and helicopters, where the amount of fluid accelerated by the action of the rotor is usually correlated to a sphere with a diameter equal to the rotor diameter. In wind turbine aerodynamics, there is a lot of discussion about the best way to take into account the cosine effect, i.e., the fact that the projected area of the rotor surface perpendicular to the incoming wind direction equals AR cosψ. In order to incorporate such “loss of area” into the performance, several “cosine corrections” have been proposed using the above projected area correction, but also

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cos2 ψ and even a cos3 ψ factor. Following Glauert, as can be seen from Equation (42), a first-order correction (for small ψ) for the power should be approximately cos2 ψ. Figure 17 obviously shows a rather simplistic view on the effects of yaw and tilt on the aerodynamics. Image a blade rotating along the rotor plane. One can imagine that the tilted rotor disk results in a normal inflow component that varies with azimuth angle, but also the tangential (in-plane) component varies with azimuth. One of them shows a cosine variation and the other a sine variation. Sometimes the normal component variation is referred to as the component due to the “advancing blade” or complementary “retreating blade,” where the blade “advances” to the wind in the lower half of the rotor blade plane of Fig. 17 and “retreats” when moving along the upper half of the rotor disk. But on top of that, there is also the fact that the induced velocity Vi is not constant over the rotor plane but varies with both blade azimuth angle and blade span-wise position. This makes the calculation of loads and performances of real wind turbine rotors under such non-rotational symmetric conditions quite a challenging task. The first BEM approaches in which such yaw/tilt correction was implemented are presented in the well-known book of Robert Wilson and Peter Lissaman Applied Aerodynamics of Wind Power Machines, published in May 1974 by Oregon State University, USA, under a grant from the National Science Foundation. This book also provided, in the Appendix, a Fortran code named PROP for the (numerical) determination of performance and loads on a horizontal axis wind turbine rotor. This Appendix provided one of the first numerical codes based upon BEM theory, with a number of engineering modifications to take into account the limited number of rotor blades, blade root losses, corrections for very high loading, etc. And since this was an open-source code, PROP and its follow-ups, such as PROPSI (its SI units equivalent), became very popular at many research institutes. Over time, a number of assumptions and engineering approaches were added the different users, and they became participants in a number of international benchmark exercises, organized since 1977 under auspices of the IEA (through several “Wind Tasks” of the International Energy Agency). In the chapter “Pragmatic Models: BEM with Engineering Add-Ons” by Gerard Schepers [4], more information can be found on more modern ways to take into account yawed flow in BEM calculations. This section also provides ample information about other correction methods nowadays applied in wind turbine rotor loads and performance calculations based upon BEM. With respect to aerodynamic wind farm research, the history goes back to the 1970s of the last decade. Particularly in the UK and in the Netherlands, research programs were initiated with the aim to look for alternative energy suitable for the next millennium, and that led in both countries to research projects on wind turbine wake and farm aerodynamics. In the UK, the Central Electricity Generating Board had a suitable wind tunnel at their CERL research laboratory in Leatherhead, and at the same time, TNO in the Netherlands used their environmental (atmospheric boundary layer) wind tunnel for wind turbine and wind farm wake measurements.

Fig. 18 Far wake structure results, up to 25D, from wind turbine far wake measurements in the CERL boundary layer wind tunnel

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The major issue was to determine wake losses, and indeed these experiments provided insight into the power loss of wind turbine operation in the wake of another wind turbine. Key authors in those days were David Milborrow and John Ainslie from CEGB/CERL and Paul Vermeulen and Peter Builtjes from TNO. The result from the measurements started to appear in scientific publications in the well-known BWEA (British Wind Energy Association) conferences, which were organized since 1977 at major UK university premises. These were the major scientific meetings on wind energy in Europe in these days. One of the nice results from the early wind turbine wake and wind farm wake research was that wakes could extend up to ten times the diameter, before a wind velocity recovery of 90% was achieved, and that at such differences, also the turbulence intensities were up to 50% higher than the ambient turbulence level. The measurements also led to wind farm wake loss models, but it took another 20 years before these rather primitive and wind tunnel result-based models could be validated in practice. In the EU Fifth Framework Project ENDOW (2000–2003), the existing models for wind farm wake effects were validated with measurements from two early near-shore wind farms in Denmark (Vindeby and Bockstigen). And it was the first comprehensive validation effort of offshore wake models. The general conclusion was that the various numerical models provided a rather wide range of results, which was mainly caused by the way in which turbulent entrainment was modelled. The project led to improvements of the parametrizations related to turbulent entrainment in wind turbine wakes. ENDOW was succeeded by the EU Sixth Framework Project UPWind in which, among others, further validation and wind turbine wake model improvements were performed with extensive experimental results from the first large (80 wind turbines) Danish offshore wind farm Horns Rev. And the results were quite shocking since none of the existing wind farm wake models was able to accurately predict the power losses of wind turbines operating in the wake of previous wind turbine(s) (see Fig. 19). After extensive research and analysis, one of the main causes of the discrepancy was found to be originating from what is nowadays called wake meandering. The wake behind a wind turbine is, under most conditions, not moving straight downward, but was meanders the wind farm, causing less losses for wind turbine farther downstream in the wind farm (Fig. 19). Since the beginning of this millennium, wind farm aerodynamics became a completely new and fast expanding research area. The interaction of a wind farm with the atmospheric boundary layer and the aerodynamic interference effects inside wind farms are key elements in this new field of wind energy aerodynamics. But also the cumulative effects of multiple wind farms in vicinity of each other and modern and novel wind farm control strategies aiming at mitigation of wake losses and/or more intensive mixing with the external flow are important issues of research in this field.

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Fig. 19 Calculated and measured power wake loss along a line of wind turbines inside a wind farm. [Gaumond et al 2012]

Wind Turbine Augmentation Principles and Devices Ever since the emergence of “new” wind energy extraction devices, there have been thoughts and principles to try to augment the performance of wind turbines. One of the first wind turbines in which it was tried to concentrate the wind speed in order to extract its energy with a smaller rotor was designed and built by the British company Enfield based upon a patent of the French engineer Jean Edouard Andreau. It was erected at St. Albans in 1954 and had two rotor blades and a diameter of 24.5 meter (Fig. 20). The rotor blades of this machine are hollow and have an air vent at the tip. A dedicated hub/nacelle design connects the hollow blades to the tower, and as a result, the air, which is drawn through the rotor blades to the tip vents as a result from centrifugal effects, is sucked up in the tower. The air intakes are situated at the lower part of the tower, and just above them, a fast running rotor is positioned driving a fast running 100 kW generator. In the end, the total efficiency of the system and the complexity of the nacelle/hub/blade connection turned out to be one of the factors why the concept was not further developed, but one of the advantages was that a high-speed generator

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Fig. 20 The Andreau-Enfield wind turbine erected in 1954 in St. Albans, UK

Fig. 21 Wind speed augmentation principles based on large vortex generation. Left: wing tip vortices system by John Loth, West Virginia University (1975) Right: delta wind side edge vortical system by Pasquale Sforza, Polytechnic Institute, New York (1977)

running at a constant speed providing 50 Hz AC (alternating current) could be used, which allowed grid coupling without the use of a gearbox and complicated wind turbine control systems. Other ideas about augmenting the flow first before extracting its energy content came from the principle of concentrated vortices. Vortices in the flow are generated when relatively large pressure differences exist over an edge. For example, along the tip of an aerfoil-shaped wing or along the sides of a delta wing when subjected to an inclined inflow. At such locations, large vortices are emanated, and from such vortices, energy can be extracted with relatively small rotors. Figure 21 shows two

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examples of concepts conceived by John Loth from West Virginia University and from Pasquale Sforza from the Polytechnic Institute in New York, and both of them got their concept patented in 1977. Different views exist about the amount of energy that can be extracted from such vortical systems. From measurements, one can extract that the maximum speed in the direction of the incoming wind speed (so more or less aligned with the core of the vortex) is about twice its value. However, that does not mean that the local energy content that can be extracted is proportional eight times the energy from a rotor in free, unaugmented flow. The main reason for this is that inside a vortex, a significant lower pressure exists, and trying to extract the additional (potential) energy present through the pressure reduction may lead to a complete distortion of the vortex system. A different way of viewing this is that potential energy is converted to kinetic energy inside the vortex, but in the end, after the energy extraction through the rotors has taken place, the flow needs to be “dumped” at ambient pressure, so the additional energy present locally though conversion from potential energy to kinetic cannot be extracted, at least not to its full extent. The amount of additional energy that can be extracted depends on the energy entrainment from the “outer flow” (outside the vortices) into the wake behind the rotor(s). Another big disadvantage of these systems is that these demand the realization of a large amount of auxiliary structure that also needs to be rotated into the direction of the wind. These types of structures never made it to a proof-ofconcept or a proof-of-principle design of a scale beyond that of a wind tunnel model (Fig. 21). A third type of augmentation that has been researched to quite an extent is the addition of a structure around the rotor with the aim to increase the wind speed in the rotor plane. Several types of structures have been considered in the past, such as circular plates, venture tubes, funnels, and diffusers. The rotor in a hole in a plate device is the easiest to understand from an aerodynamic perspective, since it directly makes use of the pressure difference over the plate when the plate is oriented perpendicular to the wind. Also buildings can be used as such a plate, where a tunnel is realized through the building which contains a rotor. The amount of energy that can be extracted is maximized by the pressure difference and the cross-sectional area of the hole. With a maximum pressure difference equal to the dynamic pressure of the incoming wind, the aerodynamic efficiency is hence equal to 1. This is 33% better than the Betz limit, but of course, the whole building contributes to generating the pressure difference between the windward and the leeward side. A disadvantage is obviously that the system works only optimal for one (or two opposite) wind direction. An example of such system is the Strata Tower in London, where three wind turbine rotors are mounted inside three tunnels (Fig. 22a). During the rebuilding discussion of the twin towers in New York, the addition of such building-integrated wind turbines was also seriously considered. But in the end the idea was abandoned because it was realized that renting out the necessary area as office space was much more profitable than generating wind power, even at that heights.

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Fig. 22 Wind turbine augmentation principle based upon the “hole in the plate principle.” Left: Strata Tower, South London, UK. Right: brimmed diffuser wind turbine, Kyushu University, Fukuoka, Japan

The concept of adding a circular plate around a rotor is researched extensively by Yuji Oya from Kyushu University, Japan. (Fig. 22). He calls his structure a “brimmed diffuser” or “wind lens,” and its principle is equal to the hole in the plate concept described above. An aerodynamically more complicated concept is that of a ducted wind turbine, where the duct surrounding the rotor has an aerodynamic cross section. The cross section can be a single- or a multi-element aerfoil with the capability of generating a large amount of lift. The principle follows the use of ducted propellers that became popular for ship propulsion and later also for aircraft propellers. The idea was developed by Luigi Stipa (1931), an Italian aeronautical engineer. The first research and application for wind turbine rotors came from Lilley and Rainbird from the College of Aeronautics from Cranfield University in the UK in 1956 followed by Ozer Igra from Ben-Gurion University in Israel in 1975 and Foreman Gilbert and Oman from Grumman Aerospace in 1977 (Fig. 23a). By adding an aerfoil-shaped duct, such “diffuser-augmented wind turbine” (DAWT) is able to generate more power. The added power comes from an increased wind speed at the location of the rotor, generated by the surrounding duct. This additional wind velocity is the result from the under-pressure generated by the aerfoil-shaped duct, which is designed such that the suction surface is pointing inward. The design of such DAWTs is not straightforward because of the strong interaction between the aerodynamics of the duct and the rotor. A well-known pitfall of the system is the occurrence of flow separation along the inner side of the duct, which can eliminate the augmentation effect. Augmentation ratios between 2 and 4 have been reported, though the latter number should be treated with care, since many quotes come from wind tunnel experiments, where the confined conditions inside a wind tunnel test section can easily lead to overestimation of the performance. Prototypes of such DAWTs were built; the most well known was the Vortec 7 built near Auckland in New Zealand in 1997 (Fig. 23b).

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Fig. 23 DAWTs – diffuser-augmented wind turbines. Left: conceptual setup using a multielement aerfoil duct design Right: the Vortec 7 prototype built near Auckland, New Zealand

These types of augmented wind turbines never became really commercial, certainly not on the scale of the Vortec 7 (7 meter rotor diameter) and beyond, mainly because the auxiliary structure needed to achieve power augmentation needs to be able to resist extreme loads. Also yawing the complete system into the wind is complicated, and in the end, it did not pay off economically (Fig. 23). More details about the aerodynamic design and performance of DAWTs can be found in two references (R. Bontempo & M.Manna, 2020, and M. Werle 2020). A final concept that is worth mentioning with regard to wind turbine augmentation principles is the “tipvane” concept conceived by Theo van Holten, at Delft University of Technology, NL, in 1975. The aerodynamic principle is similar to a DAWT, be it that the circular duct around the wind turbine rotor is replaced by its aerodynamic equivalent, namely, a free-floating vortex ring. Or maybe slightly better described, a zigzag-shaped circular “free” vortex around the rotor, generated by the presence of two rather small wings (the tipvanes) mounted on the tip of the rotor. The principle of the tipvane system is depicted in Fig. 24. A common misunderstanding is that these tipvanes are equivalent to winglets in their mode of operation. Winglets are meant to displace the location of the tip vortex of a wing toward a more upward and sideward position and hence have a lesser influence on the induced velocities at the wing. Tipvanes need a dedicated span and a dedicated skew angle when mounted on the tip of a rotor blade to assure that the tip vortex emanated from the windward side of the tipvane is exactly cancelled by the downwind side of the tipvane on the next rotor blade (see Fig. 24b). The principle was demonstrated theoretically, and a proof of principle was given through several

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Fig. 24 The tipvane rotor system. Left: geometry of a tipvane rotor Center: working principle of the tipvanes and tip vortices forming a closed circular vortex around the rotor Right: open air test facility of the tipvane rotor system, rotor diameter 8.5 m, at the TU Delft test field

wind tunnel experiments. Based on these positive results, the decision was made to build a proof-of-principle wind turbine rotor with tipvanes under real atmospheric conditions. An outdoor test facility was built which was equipped with a 10 -mdiameter rotor (Fig. 24c). Extensive experiments at TU Delft showed that under real atmospheric conditions, the stability of the zigzag vertical system could not be maintained. This led to an extensive amount of additional drag, and hence power augmentation factors found in the wind tunnel (up to a factor of 2) could not be achieved.

Cross-References  Interacting Boundary Layer Methods and Applications  Pragmatic Models: BEM with Engineering Add-Ons  Rotor Blade Design, Number of Blades, Performance Characteristics

References Anderson JD (1991) Fundamentals of aerodynamics. McGraw-Hill series in aeronautical and aerospace engineering, 1991, LCCCN 90-42291 Bistafa SR (2018) On the development of the Navier–Stokes equation by Navier. History of physics and related sciences. https://doi.org/10.1590/1806-9126-RBEF-2017-0239 Bontempo R, Manna M (2020) Diffuser augmented wind turbines: Review and assessment of theoretical models. J Appl Energy 280:2020 van Bussel GJW, van Holten T, van Kuik GAM (1982) Aerodynamic research on tipvane windturbines, report LR-355 1982, Delft University of Technology

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Drela M (1989) XFOIL: An analysis and design system for low Reynolds number airfoils, Conference on Low Reynolds Number Aerodynamics, University of Notre Dame, 1989 Drzewiecki S (1893) Méthode pour la détermination des éléments mécaniques des propulseurs hélicoïdaux. Bulletin de l’Association Technique Maritime No. 3 - Session de 1892 (1893), pp. 11–31 (in French) Froude W (1878) On the mechanical principles of the action of propellers. Trans. Inst. Nav. Archit. 19, 47 Gaumond M et al (2012) Benchmarking of wind turbine wake models in large offshore wind farms, DTU Wind Energy, Proc. Torque 2012 Conference, Oldenburg GE Hau E (2000) Wind turbines, fundamentals application technology economics. Springer, 2000, LCCCN 2011943801 Hoerner SF (1965a) Fluid dynamic drag. Hoerner Fluid Dynamics, 1965, LCCCN 64-19666 Hoerner SF (1965b) Fluid dynamic lift. Hoerner Fluid Dynamics, 1965, LCCCN 64-19666 Notebaart JC (1972) Windmühlen, der Stand der Forschung uber das Vorkommen und den Ursprung. Mouton Publishers Den Haag-Paris, LCCCN 70-152977 Phillips DG, Flay R, Nash TA (1998) Aerodynamic analysis and monitoring of the Vortec 7 Diffuser-Augmented wind turbine 1998, University of Auckland Prandtl L, Tietjens OG (1957a) Fundamentals of hydro and aeromechanics. Dover Publications, 1957, LCCCN 57-4859 Prandtl L, Tietjens OG (1957b) Applied Hydro and Aeromechanics. Dover Publications, 1957, LCCCN 57-4858 van Rooij R (1996) Modification of the boundary layer calculation in RFOIL for improved airfoil stall prediction. Technical Report, Report IW-96087R, 1996 Delft University of Technology NL Tangler JL, Somers SM (1995) NREL Airfoil Families for HAWTs, NREL TP-442-7109, Jan 1995 Werle MJ (2020) An enhanced analytical model for airfoil-based shrouded wind turbines. Wind Energy Wilson RE, Lissaman PBS (1974) Applied aerodynamics of wind power machines. Oregon State University May 1974

8

Interacting Boundary Layer Methods and Applications Hüseyin Özdemir

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Layer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solutions of IBL Equations with Prescribed Boundary Conditions . . . . . . . . . . . . Similarity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow Over a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Over Airfoil Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscous-Inviscid Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solutions for Interacting Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Over Airfoil Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations of Interacting Boundary Layer Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter the derivation of the viscous integral boundary layer equations is presented in an unsteady, two-dimensional form. Closure sets for both laminar and turbulent flow conditions together with a laminar to turbulent transition method are given. The solution methods for the inviscid region and the viscous-inviscid interaction coupling scheme are briefly discussed. The numerical solution of the integral boundary layer equations are first presented

H. Özdemir () TNO Energy Transition, Petten, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_11

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assuming a prescribed solution for the inviscid flow region and then for the coupled viscous-inviscid interacting boundary layer method. Keywords

Boundary layer · Integral boundary layer method · Panel method · Viscous flow · Inviscid flow · Viscous-inviscid interaction scheme · Discontinuous galerkin method · Rotor aerodynamics · Wind energy

Introduction The flow around wind turbines are usually unsteady due to natural time-dependent changes in the (wind) flow field, i.e., flow separation from the blades or body, high turbulence levels in the flow field, or changes in the position or orientation of the blades, caused by control actuators or by interactions between the fluid and the elastic structure. Important unsteady aspects involve not only the kinematic changes in boundary conditions caused by the motion of a body but the influence of an unsteady wake, especially in wind farm configuration, when many turbines have to operate relatively close to each other. Unsteady effects are expected to become important near separation since the timescale of the boundary layer increases and the characteristics form envelopes around separated regions and information propagates more slowly. High pitch rates, aeroelastic behavior, and wake interference further decrease the timescale at which unsteady effects play a role. Another reason to search for unsteady solutions is that steady solutions may simply not exist for separated flows (e.g., the Von Karman vortex street behind airfoils with a thicker trailing edges). In many practical cases, we are interested in flows around solid bodies of fluids with high Reynolds numbers. These problems can be attacked by the limiting solution Re = ∞, but in this case the no-slip condition is not satisfied, i.e., the velocities at the wall are not zero but finite. In order to satisfy the no-slip condition, viscosity must be taken into account which realizes the transition from the finite value of the velocity close to the wall to the value of zero at the wall. At large Reynolds numbers, this transition occurs in a thin layer close to the wall which is called boundary layer by Prandtl (1904) who has introduced this concept. According to the boundary layer concept, flows at high Reynolds numbers can be divided into two regions (see Fig. 1). In the outer region, viscosity can be neglected and an inviscid approximation is valid, while the inner region consists of a very thin boundary layer at the wall where the viscosity must be taken into account. This division of the flow field leads to considerable simplifications. Using order of magnitude analysis on the terms of Navier-Stokes equations, it can be determined that the pressure is nearly constant across the boundary layer and only diffusion in the wall normal direction is significant. The inviscid flow in the outer region feels the presence of the boundary layer through a displacement effect, and the viscous flow in the inner region senses the presence of the outer inviscid flow through the pressure field at the boundary layer edge.

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Fig. 1 Viscous and inviscid regions of the flow field

In practical cases mentioned above, we are not interested in the details of the velocity field inside the boundary layer but rather certain integral quantities of the boundary layer. These integral values depend on the lengths x and y in three dimensions (depends only on x in two dimensions) and are called global values with respect to any dependence on the direction normal to the no-slip surface, z. For the global description of the boundary layer, these integral values are obtained by integrating the boundary layer equations in the normal to the solid surface direction over the boundary layer thickness. Integral boundary layer (IBL) equations have been used widely for the global description of the flow (von Kármán 1946; Rosenhead 1966; White 1991) especially in engineering applications for aircraft aerodynamics. The methods based on these equations have the advantage of lowering the space dimension by one, and as a consequence there is no need for a volume grid. This leads to a reduction in the computational costs and input efforts. At the start of the CFD era, these methods therefore were used as airfoil analysis tools, and even today some successful applications are based on this approach (Drela 1989). Integral boundary layer equations have been commonly discretized with finite-difference schemes (Drela 1985; Swafford 1983b). More recently also finite element and finite volume discretizations are employed (Mugal 1998; Özdemir and van Boogaard 2011; Özdemir et al. 2017; Zhang et al. 2019). The integral boundary layer equations have been analyzed in detail and a lot of research has been carried out on laminar, and turbulent closure relations and laminar-turbulent transition models. Within this chapter we would like to present the derivation of the integral boundary layer (IBL) equations in unsteady two-dimensional form for the incompressible flow conditions starting from the equations of motion. We will also briefly discuss about the inviscid flow field and the viscous-inviscid interaction schemes which play a role as a coupling mechanism between these two flow regions. The numerical solution of the IBL equations will be briefly discussed, and some examples will be given first without the presence of interaction scheme where inviscid solution would act as a boundary condition for the IBL equations. Since the solution of the complete IBL equation system (including the interaction scheme and the inviscid flow region) is well known from widely used industrial standard tools XFOIL (Drela 1989) and RFOIL (van Rooij 1996), we will mainly focus on the limitations of these solutions and possible improvements. Although there are several attempts (including the author of this chapter) to solve the complete system of IBL equations in unsteady form (Özdemir and van Boogaard 2011;

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Garcia 2011; Özdemir et al. 2017; Zhang et al. 2019), these solutions are more in the preliminary phase, and therefore these results are not included in this chapter.

Governing Equations In this section we would like to show the derivation of integral boundary layer equations for the incompressible viscous boundary layer flow. The aim here is to present a simpler form of this derivation and supplement additional remarks for the special cases when necessary. The reasoning for this choice is that in most of the cases we are interested in analyzing aerodynamic properties of two-dimensional airfoil sections where well-defined simplifications can be made in the governing equations. Furthermore, considering the specific wind energy applications of interest, it would be safe to assume that the wind turbines are operating in the incompressible flow regime where flow Mach number is usually well below 0.3 around a wind turbine blade. When the current large wind turbine blades are considered, the Mach number might reach around 0.3 (or above) locally at the tip region of the blade where incompressible flow assumption still holds. The derivation of the complete set of integral boundary layer (IBL) equations can be achieved starting from two different reasoning. One can assume that considering the flow around a rigid body, the flow behaves inviscid on the large part of the flow field. Thus, with necessary assumptions, the equations of motion can be reduced to potential flow equations, and the viscous effects can be included as means of IBL equations. We would like to follow the other path in this chapter that when zoomed in to the flow around solid surfaces, the viscosity effects are quite dominant, and starting from the same equations of motion with the appropriate assumptions, the IBL equations can be derived. It will be clear from the analysis of these equations that a boundary condition will be necessary to prescribe the behavior of the inviscid solution (e.g., velocity distribution) at the edge of the boundary layer region. This boundary condition can be prescribed analytically for some cases like laminar flow over flat plate, stagnation flow, etc., but in general cases there is no analytical definition exists. So, most logical conclusion, to include the effect of this inviscid flow, is through the solution of this inviscid flow field which leads to the same coupled viscous-inviscid solution. In the literature this coupled system is named in various ways, but we would like to call this whole coupled system as interacting boundary layer method since (especially the unsteady solution is considered) these two regions are continuously interacting with each other. We will start the derivation of the IBL equation from the equations of motion and then introduce the assumptions that would lead to reduced set of these equations called boundary layer equations. Consider steady, two-dimensional continuity equation can be written for incompressible flow in unsteady form in two dimensions (see Fig. 2):

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Fig. 2 A sketch of a viscous boundary layer developed over a flat plate

∂u ∂v + = 0, ∂x ∂y

(1)

where u and v are the components of the velocity vector in x- and y-directions respectively and x being parallel to the free-stream and y normal to it. Momentum equation can be written as follows for a steady two-dimensional flow in laminar form: ∂u ∂u ∂u 1 ∂p ∂τij +u +v =− + , ∂t ∂x ∂y ρ ∂x ∂x ∂v ∂v ∂v 1 ∂p ∂τij +u +v =− + , ∂t ∂x ∂y ρ ∂y ∂y

(2)

with p is pressure and where the body forces are neglected. Assuming a constant viscosity, the stress tensor can be written as follows:  τij = ν

∂uj ∂ui + ∂xj ∂xi

 .

(3)

Please note that for compressible equations, the viscous stress tensor contains and additional term called second or bulk viscosity (λ + (2/3)μ). This term vanishes for the truly incompressible flows but has some effect for nearly incompressible flows. Please see, e.g., White (1991) for a detailed discussion on this topic. Although it is common practice to start from these laminar governing equations (Eqs. 1 and 2) to derive the integral form of the boundary layer equations and cover the turbulent flow conditions with the dedicated closure set (as weill see in section “Turbulent Closure Set”), the connection between these equations and the closure for turbulence is lost since there are no terms representing the turbulence effects in these equations. This connection can be constructed if the governing equations are written for the turbulent flow conditions. In order to do this, one would typically decompose the flow variables into time-averaged mean component (¯) and fluctuating component ( ): u = u¯ + u , v = v¯ + v  , p = p¯ + p ,

(4)

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and substitute these functions into the governing equations (Eqs. 1 and 2) and apply the time averaging (using the averaging rules) to obtain the turbulent form of the momentum equation (please see, e.g., (White 1991) or any text book on turbulence for detailed derivation and further reading). This turbulent momentum equation contains an extra term, namely, the spatial derivative of the turbulent inertia tensor, ui uj . The components of this tensor can be determined by the fluid physical properties and the local flow conditions like velocity, geometry (curvature, surface roughness, etc.), and upstream history of the flow (inflow conditions, flow separation, etc.). In a two-dimensional turbulent boundary layer, as in the current case, u v  will be the only significant term which will still require a (empirical) modeling because of the lack of the knowledge of the detailed turbulent structure. The additional turbulent inertia term is treated typically as a stress term and combined with the viscous stress tensor to give:  τij = v

∂uj ∂ui + ∂xj ∂xi



− ui uj ,

(5)

where the first term on the right-hand side represents the laminar viscous stresses and the second one is the turbulent stress tensor and is called turbulent shear in the two-dimensional turbulent boundary layer (−u v  ).

Boundary Layer Approximation When the flow considered is at a very high Reynolds number or with a very small viscosity, we have the flexibility of making simplifications to the Navier-Stokes equations. Prandtl showed in 1904 how viscosity affects the Reynolds number, and for this limiting case, he derived the boundary layer equations. In this section we will briefly discuss the assumptions to be made and the concepts that lead to the boundary layer equations and show the resulting set of equations. Consider a flow of a fluid with very low viscosity around a body with a solid surface (see Fig. 2). The velocities in the flow field are of the order of magnitude of the free-stream velocity except in the immediate vicinity of the solid surface. The fluid on the surface does not slip but attaches to the surface. The velocity is zero at the surface, and it reaches to its maximum value at a certain distance from the wall with a viscosity-related transition mechanism. The mentioned transition takes place in a very thin layer called boundary layer. Within this thin layer, the velocity gradient normal to the wall is very large, and a very small viscosity can play an important role since the shear stress (τ = ν∂u/∂y) can reach large values. Outside this thin layer, the velocity gradient is not large and viscosity does not play an important role. In this outer region, the flow has no friction and can be described by the potential flow conditions. We would like to determine the simplifications obtained by the asymptotic solution of Navier-Stokes equations at high Reynolds numbers by solving the

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simplified differential equations instead of first solving the complete Navier-Stokes equations and then taking the limit of Re → ∞. The thickness of the boundary layer, δ, is defined to be the distance from the no-slip surface where the velocity of the viscous flow is about 99% of the original flow (e.g. Schlichting and Gersten 2000; White 1991). It is assumed that this 1% difference of the velocities is an acceptable value to be neglected. In the simplifications of Navier-Stokes equations, the thickness of the boundary layer is assumed to be very small compared to a characteristic length of the body, l, which is to be specified: δ  l, which leads to an asymptotic behavior of the solutions of the boundary layer equations. The components of velocity vector, uref , of the inviscid flow are assumed to be in the order of 1 (O(1)). The u-component of the velocity vector, u, of the viscous flow can also be assumed of the order O(1) for y → δ, because of the smooth transition from viscous to inviscid regions. Here, one should note that x-coordinate is defined as parallel to the body surface and y-coordinate is defined as normal to it. Also the spatial and time derivatives of the u-component of the velocity vector, which is parallel to the no-slip surface, do not vary very much within the boundary layer and is assumed to be O(1) as well. Considering the continuity equation, we can draw the same conclusion; ∂v/∂y = O(1) for y → δ or for x → 1, where it leads to v = O(δ) with no-slip condition. Using again the no-slip condition and  the mean value theorem, we can assume ∂u/∂y = O (1/δ), and ∂ 2 u/∂y 2 = O 1/δ 2 . Also the same mean value theorem tells us that ∂v/∂x = O(δ) because x can grow up to the order 1. Similarly we can conclude that ∂v/∂t = O(δ) and ∂ 2 v/∂x 2 = O(δ) hold as well. Using the free-stream velocity uref and a characteristic dimension of the body l as reference values, it can also be shown that  Re = O

1 δ2



  1 , ,δ=O √ Re

(6)

with Re =

uref l . ν

(7)

Above relationship tells us that the boundary layer thickness tends to zero as the Reynolds number (Re) increases. Before the governing equations are reduced with the arguments of the order of magnitudes discussed above, we would like to introduce the following nondimensional variables to obtain the final nondimensional version of the boundary layer equations: x∗ =

x u ρ U , u∗ = , ρ ∗ = = 1, t ∗ = t , L U ρ0 L

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H. Özdemir

y∗ =

y v ν p , v∗ = , ν ∗ = = 1, p∗ = . δ V ν0 ρ0 U 2

Inserting these parameters into the governing equations given above and working out the order of magnitude analysis (neglecting all terms of which the magnitude is much smaller than order 1) would lead to the final form of the boundary layer equations. We would like to note that v ∼ u Lδ and v  u as a final remark and leave the derivation steps to the reader and give the two-dimensional boundary layer equation for unsteady laminar flow conditions as follows: ∂u ∂v + = 0, ∂x ∂y

(8)

∂u ∂u ∂u 1 ∂p ∂ 2u +u +v =− +ν 2, ∂t ∂x ∂y ρ ∂x ∂y 0=−

1 ∂p . ρ ∂y

(9) (10)

The last equation above (Eq. 10) states that the pressure does not change in the direction normal to the surface, within the boundary layer. This implies p = pe in the boundary layer, and this relation can be used instead of the y-momentum equation. The external layer is inviscid, so the flow is assumed to be parallel to the surface, which makes v zero at the boundary layer edge. With these two results, the x-momentum equation at the boundary layer edge becomes: 1 ∂pe ∂ue ∂ue + ue =− . ∂t ∂x ρ ∂x

(11)

Since there is no change in pressure in the y-direction within the boundary layer, the pressure gradient in x-direction from (9) must be the same as this pressure gradient at the boundary layer edge. Therefore (9) and (11) can be combined into: ∂u ∂u ∂u ∂ue ∂ 2u ∂ue +u +v = + ue +ν 2. ∂t ∂x ∂y ∂t ∂x ∂y

(12)

The combination of Eqs. (8) and (12) is the field form of the boundary layer equations and can be solved for the variables u and v if the edge velocity ue and the kinematic viscosity ν are known. This problem requires a discretization in two dimensions. Please note that the statement made in Eq. (10) is a very strong statement. Noting the assumptions made to derive this equation, this is only valid for thin boundary layers without large curvature effects. Considering the flow problems of interest (especially for wind energy applications), we are mostly interested in solving flows around two-dimensional airfoil sections with thicker profiles where at the tip of the wind turbine blade, minimum thickness commonly starts from 18% and reaches to

8 Interacting Boundary Layer Methods and Applications

261

around 60% at the blade root. Also massive separation is quite common especially at the inboard region of the blade. Considering these examples, the statement of Eq. (10) that the pressure is constant in the boundary layer normal to the surface is hardly valid in these flow conditions. The same holds in the vicinity of the trailing edge of the airfoil sections where the streamline curvature (or impingement of a shock wave for compressible flows) can cause a significant normal pressure gradient in the boundary layer where whole boundary layer approximation breaks down. Additionally, if we start the derivation of the boundary layer equations from the turbulent version of the momentum equations, then Eq. (10) would include an additional turbulent term (−v  2 ) on the right-hand side to represent a slight variation of pressure due to velocity fluctuations. These violations usually reflects back in the numerical solutions where the convergence of the solution for these thick(er) airfoils is hard to obtain or simply not possible. These convergence issues are usually solved by some practical numerical treatments at the implementation level. Recently some studies (Ramanujam et al. 2016; Ramanujam and Özdemir 2017; Vaithiyanathasamy et al. 2018) focused on these problems and suggested some improvements on prediction of the flow solution which are also summarized in section “Numerical Solutions for Interacting Boundary Layers.”

Some Remarks on Three-Dimensional Effects In two-dimensional flows, the streamlines are aligned along the surface tangential direction, called stream-wise direction. In three-dimensional flows, the streamlines of the external flow can be curved in the transverse direction when observed in a plane parallel to the surface. The curved nature of an external streamline generates a centrifugal force, and this can be balanced by a pressure gradient in the transverse direction. Therefore at any point on the external streamline, the pressure gradient has two components, one in stream-wise direction and another in the transverse direction. The transverse pressure gradient gives rise to a velocity component in that direction which is called cross-flow or secondary flow in the boundary layer. The cross-flow is directed toward the center of the external streamline curvature. On the wall surface the cross-flow component becomes zero due to the no-slip condition. The external streamlines have an inflection point when the crosswise pressure gradient changes sign. The change of sign of the crosswise pressure gradient first occurs near the wall where inertia is smaller. These types of boundary layers are called pressure-driven three-dimensional boundary layers. In the boundary layer equations, the sub-characteristics or streamlines play a role in determining the main flow structure. In the boundary layer, diffusion occurs only in the direction normal to the body surface. There is a definite region upstream of a point where the solution at that point is influenced by any disturbances (or conditions) within that region. This region is called domain of dependence. A disturbance at a point is first felt at the line that is passing through that point and normal to the flow and then is convected downstream by all streamlines crossing this line. This definite region downstream of the point is called range of influence. In obtaining numerical solutions of boundary layer equations, the concept behind

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these domains of dependence and influence is essential, since within the flow field, these zones vary from point to point and cannot be predicted in advance. In order to select a suitable discretization technique, this concept has to be kept in mind. The boundary layer equation, (12), is essentially parabolic in character but have somewhat different physical properties. This equation physically represents the diffusion due to viscosity along the boundary layer in the direction normal to the body surface and the convective propagation along the streamlines through each point where the coordinate normal to the body surface is constant with a finite speed. The diffusion phenomenon is controlled by the term in the right-hand side of equation with the second derivative in normal direction of the velocity. The terms in the left-hand side where there is no second derivative suggest advection or propagation of properties.

Integral Boundary Layer Equations For large varieties of flow conditions, the Navier-Stokes equations together with turbulence models deliver accurate solutions for both steady and unsteady cases. The main issue arises when large-scale differences exist in the flow field and the time dependency cannot be avoided where solving the Navier-Stokes equations become computationally too expensive. Especially when the design phase of wind turbine blades are considered, these computational times become critical and make the use of full flow-field solutions impractical. As stated in previous sections, the integral boundary layer equations (IBL) can be derived starting from the unsteady, two-dimensional boundary layer equations with a fundamental assumption that the only effect of the boundary layer and the wake is to displace the inviscid flow away from the physical body to create an effective displacement thickness. Lighthill (1958) showed that this assumption is accurate when the ratio of the boundary layer thickness to streamline radius of curvature is small. As discussed above this assumption is arguable for some cases of the flow conditions around the wind turbine blades. To obtain these useful integral values for the global description of the boundary layer, the boundary layer equations should be integrated with respect to the transverse direction (normal to the no-slip surface) over the boundary layer thickness. The integral boundary layer equations can be derived either starting from a control volume and setting up the mass and momentum balance or from the boundary layer equations and integrating over the boundary layer thickness. In the literature the derivation of the integral boundary layer equations is given in various ways where one is slightly differing from another (i.e., see references Tetervin 1947; Matsushita et al. 1984; Swafford 1983b). It is interesting to note that in a recent study, Seubers (2014) showed the derivation of the integral boundary layer equations with a streamline-based control volume approach including terms for nonclassical (separating) boundary layers. Integrating the following equation for n = 0, 1, . . . with respect to the transverse direction, y, (normal to the no-slip surface) over the boundary layer thickness, we

8 Interacting Boundary Layer Methods and Applications

263

can derive the n-th moment of boundary layer equation (Matsushita et al. 1984): momentum equation × (n + 1)un − continuity equation × (un+1 − un+1 ), e in which the free-stream velocity ue = ue (x, t) is presumably known from a potential flow analysis. In the current study, the 0-th (momentum integral) and 1-st (kinetic energy integral) moments of momentum equations are used to obtain the global quantities such as displacement, momentum, and energy thicknesses of the boundary layer as given below: Cf ∂δ ∗ ∂ue ∂ δ ∗ ∂ue + (ue θ ) = ue − (δ ∗ + θ ) − , ∂t ∂x 2 ∂x ue ∂t

(13)

∂ θ ∂ue ∂(δ ∗ + θ ) ∂ue + (ue δ k ) = CD ue 2δ k −2 , ∂t ∂x ∂x ue ∂t

(14)

where δ ∗ is the displacement thickness, θ is the momentum thickness, δ k is the kinetic energy thickness, Cf is the friction coefficient, CD is the viscous diffusion coefficient, and ue is the velocity at the edge of the boundary layer. The displacement thickness, δ ∗ , is the distance by which the surface would have to be moved parallel to itself toward the reference plane in an ideal fluid stream of velocity, ue , to give the same volumetric flow as occurs between the surface and the reference plane in the real fluid (Fig. 3). In other words to define displacement thickness, we should simply state conservation of mass in steady flow. The definition of the displacement thickness given as: δ∗ =



1−

0

Fig. 3 Displacement thickness

∞

u ue

z

δ*

 (15)

dy,

ue

u

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H. Özdemir

with ∞ taken outside of boundary layer thickness δ. The above equation gives the modified shape of the body as it is experienced by the external flow and holds true for any incompressible laminar and turbulent flow, with constant or variable pressure and with constant or variable temperature. Since variations in the y-direction are integrated away, δ ∗ is a function only of x. Its exact value depends upon the distribution u(y). Similar to the displacement thickness, we can introduce transverse distance (momentum thickness, θ ) over which the solid wall has to be displaced such that an inviscid flow produces the same momentum transport: 

∞

θ= 0

u ue

  u 1− dy. ue

(16)

Kinetic energy thickness is defined as follows: 

∞

δ = k

0

u ue

  u2 1 − 2 dy. ue

(17)

The skin friction coefficient can be defined as follows: cf =

τw , 1 2 2 ρue

(18)

where the shear stress along a surface, τw , is τw = μ

∂u |wall . ∂z

(19)

Finally the dissipation coefficient CD is given as: CD =

D 1 3 2 ρue

,

(20)

∞ where the viscous dissipation, D = 0 τ (∂u/∂y)dy. The system of Eqs. (13) and (14) is accurate for laminar flows and for turbulent flows when the turbulence production and dissipation are in near equilibrium. In such a scenario, the dissipation coefficient, CD , depends only on local boundary layer parameters. However, experiments (e.g., Goldberg 1966) show that there are significant upstream history effects on Reynolds stresses for flows with adverse pressure gradients. These effects are included by introducing Reynolds stress an additional unknown in a stress-transport equation as described by Green’s lagentrainment method (Green et al. 1977). Özdemir (2010) derived the unsteady form of this so called shear-lag equation, and in a recent study, it is further simplified by Ye (2015) to take the following form:

8 Interacting Boundary Layer Methods and Applications

∂

265

 ue

 1 Cτ ) ∂ ue Cτ ) Cτ ue  12 ∂ue ue 2Cτ ∂ue + = Kc CτEQ − Cτ2 ) − − Cτ , ∂t ∂x δ u ue ∂t ∂x

u

(21)

with Cτ the shear stress coefficient and CτEQ shear stress coefficient for the equilibrium flow conditions and Kc = 2a1

ue δ , u L

(22)

and   1.72 δ = θ 3.15 + + δ∗. H −1

(23)

The commonly used values are uue = 1.5 and Lδ = 12.5, but Thomas (1984) takes in to account the dependency on the shape factor and gives uue = H3H +2 , which is more accurate for separated flow profiles. Vector form of Eqs. (13), (14), and (21) can be formulated as ∂u ∂ + [f(u)] = s(u) , ∂t ∂x

(24)

where the new conservation variable u, flux f(u), and source s(u) have been introduced: ⎤ ⎡ ⎤ δ∗ ue θ u = ⎣ δ ∗ + θ ⎦ , f(u) = ⎣ ue δ k ⎦ , ue ue Cτ u Cτ ⎡ ∗ Cf e e ue δ ∗ + θ ∂u − uδ e ∂u 2 ∂x ∂t ⎢ θ ∂ue e CD ue − 2δ k ∂u s(u) = ⎢ ∂x − 2 ue ∂t ⎣ 1 1  ∂ue ue 2Cτ ∂ue Cτ ue 2 2 δ Kc CτEQ − Cτ ) − u ue ∂t − Cτ ∂x ⎡

⎤ ⎥ ⎥. ⎦

(25)

The above equation set forms a conservation form when the interaction scheme is not considered and the system is shown to be hyperbolic (van den Boogard 2010; Passalacqua 2015). This system is derived for the turbulent flow conditions, and in the laminar system, the last line of Eq. (25) doesn’t exist. As it will be shown in the sections below, in the laminar case, an equation can be added for the transition condition and can be replaced with turbulent lag entrainment equation when the flow transitions from laminar to turbulent.

Eigenvalues of the Laminar System Splitting the spatial flux vector f(u), in Eq. (25), in the known edge velocity ue ˜ depending on the unknown variables present in the temporal times a vector f(u),

266

H. Özdemir

flux vector F and using the definition of the shape factor for the kinetic energy thickness (H ∗ = δ k /θ ) gives: f(u) = ue

      θ θ (δ ∗ + θ ) − δ ∗ = u = u = ue ˜f (F ) . e e δk θH∗ ((δ ∗ + θ ) − δ ∗ ) H ∗ (δ ∗ , δ ∗ + θ ) (26)

Substitution into Eq. (24) gives the convective formulation: ∂ue ∂ ˜f ∂F ∂F + ue = S + ˜f , ∂t ∂F ∂x ∂x

(27)

where ue

  ∂ ˜f −1 1 = A. = ue ∗ ∗ ∂H ∗ −H ∗ + (H + 1) ∂H ∂F ∂H H − H ∂H

(28)

The characteristic equation is given by det(A − λI ) = 0, where λ are eigenvalues and I is the identity matrix. Resulting eigenvalues are given in Eq. (29) and are graphically presented in Fig. 4. These eigenvalues represent the directions or the characteristics of the boundary layer flow.

Fig. 4 Eigenvalues for the Laminar IBL equations as a function of the shape factor H . A negative eigenvalue arises if the flow passes H = 4.19808 (at the circle), which is the separation point

8 Interacting Boundary Layer Methods and Applications

267 ∗

Concluding from this analysis, there is a point at ∂H ∂H = 0, H = 4.19808 where one of the characteristics is zero, see Fig. 4. This point is the separation point, and inverse flow downstream of this point is the cause of the negative direction of one of the characteristics:

λ± = ue

   ∗ ∗ 2 ∗ H ∗ − 1 − H ∂H ± 1 − H ∗ + H ∂H + 4 ∂H ∂H ∂H ∂H 2

(29)

At the point of separation, matrix A is singular. This singularity was already noted by Goldstein (1948) and prevents a straightforward solution procedure using the IBL equations for separating flows. A steady solution method will require the inversion of matrix A to find the unknowns F and will diverge when A is singular. It could therefore be advantages to take F to be the unknowns, to remove the need for a matrix inversion. However, in this case, the two equations considered are not coupled with H ∗ (H ), but with H (H ∗ ) which is a double-valued function as can be seen in Eq. (43) and Fig. 5, with one value of H before separation and one after separation, for the same value of H ∗ . A straightforward solution cannot be found. For unsteady calculations, the matrix A will not be inverted, so it should not lead to a diverging solution. However the unsteady method will not be able to reach the correct steady solution for separating flows (see, e.g., van Dommelen and

Fig. 5 A plot of the closure relation for the H − H ∗ . Note that for certain H ∗ values (around 1.53–1.55), there are two corresponding H values one before separation and one after separation

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H. Özdemir

Shen (1980)), as there will still be a singularity at finite time. A viscous-inviscid interaction scheme coupling the viscous boundary layer solution with the inviscid potential flow solution will introduce an additional equation and should prevent the matrix A from being singular at the point of separation. In this case the correct steady solution can be found also in an unsteady manner.

Laminar to Turbulent Transition The most commonly used transition model for integral boundary layer methods, eN (originally e9 ) method, was introduced by van Ingen (1965) and Smith and Gamberoni (1956). Based on the linear stability theory (Ye 2015), this semiempirical method provides a good practical prediction of natural transition onset for incompressible two-dimensional boundary layers. The prediction of the natural transition is based on the evolution of small disturbances introduced in the laminar flow. An exponential decay of disturbances with time implies the mean flow remains laminar and is stable. However, if the disturbances increase with time, the flow is considered unstable and transitions to turbulent flow (Schlichting and Gersten 2000). The N in the eN method stands for the maximum amplification rate of the disturbances at a given location. The flow is said to transition to a turbulent regime, at that location, when this amplification rate (N) exceeds a critical value (Ncrit ) determined from experiments. The value of this critical amplification rate is (Drela 1995): Ncrit = −8.43 − 2.4ln

Tu . 100

(30)

T u is the free-stream turbulence intensity nominally taken as 0.07% for airfoils in wind tunnel conditions. The amplification rate N at any location is determined by: N=

dN (Reθ − Reθcrit ), dReθ

(31)

where Reθ is the Reynolds number based on momentum thickness, θ , Reθ =

θ ue . ν

(32)

The slope is given by (Drela 1985): 1 2 dN = 0.028(H − 1) − 0.0345e−(3.87 H −1 −2.52) . dReθ

(33)

Ye (2015) slightly modified the Reθcrit that is based on Arnal’s experimental work (van Ingen 2008) to find the onset as follows:

8 Interacting Boundary Layer Methods and Applications

 log(Reθcrit ) =

269

   12.7886 0.267659 + 0.394429 tanh − 8.57463 H −1 H −1 +

3.04212 + 0.6660931. H −1

(34)

For similar flows, H is constant and Reθ is uniquely related to the stream-wise coordinate x. Equation (30) immediately gives the amplification factor N as a unique function of x. There is less known about the unsteady flow effects on the transition, and there are no models including these effects. Obremski and Fejer (1967) performed experiments for the oscillating boundary layers with a flat plate where the velocity of the flow is varied with a sinusoidal function around a mean. They have defined an unsteady Re number based on the amplitude and the frequency of the varying velocity. Later on Obremski and Morkovin (1969) introduced a quasi-steady theory about laminar to turbulent transition. Although there are numerous studies on unsteady transition to the knowledge of the author, there is no unsteady model exists to complement or replace the steady eN theory. In order to complement to the unsteady form of the IBL equations, a naive approach could be just to add a time derivative to the amplification rate: ∂N ∂N t + x, ∂t ∂x

(35)

  ∂Reθ ∂N ∂Reθ t + x . ∂Re ∂t ∂x

(36)

N = to give N =

For the solution of the IBL equations, the last line of Eq. (25) is replaced by the Eq. (36) for the laminar region of the flow field to check the transition to the turbulent flow. The complete system can be written in vector form for the laminar system as follows: ⎡

δ∗

⎤

u = ⎣ δ∗ + θ ⎦ , 0

⎡

⎤

ue θ f(u) = ⎣ ue δ k ⎦ , N

⎡

⎤ δ ∗ ∂ue e ue δ ∗ + θ ∂u − ∂x ue ∂t ⎥ ⎢ θ ∂ue ⎥ k ∂ue ⎢ s(u) = ⎢ CD ue − 2δ ∂x − 2 ue ∂t ⎥ . ⎣ ∂N ∂Reθ ⎦ ∂Reθ ∂Re ∂t t + ∂x x Cf 2

(37) When N value is equal to the defined Ncrit value, the onset of transition is triggered and the equations will switch to the turbulent ones (Eq. 25). Alternatively the equation for N can be applied as a check after each time step.

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H. Özdemir

Closure Set The integration over the direction normal to the surface applied in order to derive the IBL equations removes some physical information. In Eq. (37) the displacement thickness δ ∗ , kinetic energy thickness δ k , momentum thickness θ , friction coefficient Cf , viscous diffusion coefficient CD , and the edge velocity ue are unknowns. Assuming that the edge velocity is provided by the potential flow solver, there are still three more unknowns than the number of equations. In order to close this system, the so-called closure relations are introduced. These closure relations can be derived by modeling unknowns in terms of other (un)known terms using experimental data or analytical solutions of representative test cases under certain assumptions. In the literature there are many researchers contributed to the development of these closure relations for various flow conditions and under various assumptions. A selection of these major works can be found in the following literature (Drela 1985; Swafford 1983a; Nishida 1997; Smith 1972; Whitfield 1979). The selection of the closure equations depends strongly on the problem under consideration. To give a simple example, consider shape factor, H , defined as the ratio of displacement thickness, δ  , and the momentum thickness, θ . When a flow over a flat plate is considered, this ratio will be different then for turbulent flow case (dependent on Re number) and will also be different then laminar flow over an airfoil section (dependence on geometry) leading to different shape factor values for each of these flow conditions. With this reasoning, one should be careful in selecting and using these equations. Another good example to demonstrate the importance of the closure relations is adaptation of the well-known aerodynamic design tool XFOIL (Drela 1989) based on the interacting boundary layer method for the wind energy applications to develop RFOIL (van Rooij 1996). In the next two sections, the widely used forms of the closure set for laminar and turbulent flow conditions are given. Laminar Closure Set The laminar closure relations presented in this section are mainly combined from Nishida (1997) and Drela (1985). The shape factor, H , for the displacement thickness and the shape factor, H ∗ , for the kinetic energy thickness are defined as follows: H =

δ∗ , θ

H∗ =

δk . θ

(38)

Although the closure set for laminar system can be found from the literature for the sake of completeness, they are given here as well: ⎧ (H − 4.35)2 (H − 4.35)3 ⎪ ⎪ 0.0111 − 0.0278 ⎪ ⎪ ⎨ H + 1.0 H + 1.0 ∗ H = +1.528 − 0.0002[(H − 4.35)]2 ⎪ ⎪ 2 ⎪ ⎪ ⎩0.015 (H − 4.35) + 1.528 H + 1.0

H < 4.35 , H ≥ 4.35 , (39)

8 Interacting Boundary Layer Methods and Applications

⎧ (5.5 − H )3 ⎪ ⎪ − 0.07 H < 5.5 , ⎨0.0727  H + 1.0 2 Reθ Cf = 1 ⎪ ⎪ ⎩0.015 1.0 − − 0.07 H ≥ 5.5 , H − 4.5 ⎧ ⎪ ⎨0.207 + 0.00205(4.0 − H )5.5 Reθ CD = (H − 4.0)2 ⎪ H∗ (4.0 − H )5.5 ⎩0.207 + 0.0016 1.0 + 0.02(H − 4.0)2

271

(40)

H < 4.0 , H ≥ 4.0 . (41)

The accuracy of the IBL system is bounded by the accuracy of the closure relations. The analytical models used for the closure relations do not include unsteady effects, and experimental data are correlated with trial and error curve fits, which might bring non-negligible fit-errors. Due to the complex form of the closure relations, the high-order numerical method applied will also have limited order of accuracy, as it will be discussed later. The laminar system is analyzed in details previously (see van den Boogard 2010; Passalacqua 2015), and it is shown that both eigenvalues are positive for low values of H until a point of separation, where ∂H ∗ /∂H = 0 and H = 4.19808 (see the plot of Eq. (39) in Fig. 5). Downstream of this point the flow is separated and one of the characteristics travels in the upwind direction.

Turbulent Closure Set Turbulent boundary layers have a two-layer structure (and an overlap region), the thickness of which scales differently with the local Reynolds number Reθ . A one-parameter velocity profile is not adequate to describe all turbulent boundary layers, and such a dependency on Reθ must be considered. Many closure sets can be found in literature for the turbulent closure set as well. The closure sets presented in this section are mainly from Drela and Giles (1987) and Nishida (1997), with small corrections suggested by Ye (2015) and neglecting compressibility effects. The skin friction coefficient Cf is given by:

Cf = 0.3e−1.33H



log Reθ 2.3026

−1.74−0.31H

  + 0.00011 tanh 4.0 −

  H −1.0 . 0.875 (42)

For the energy thickness shape factor H ∗ , the following relation is used:

272

H ∗=

H. Özdemir

⎧    4 H0 − H 2 1.5 4 ⎪ ⎪ ⎪ 1.505+ , + 0.5 − ⎪ ⎪ Reθ Reθ ⎡ H0 − 1.0 H + 0.5 ⎪ ⎨

⎤

⎢ log Reθ 0.015 ⎥ 4 ⎪ ⎪ ⎥, ⎪ 0.007 + (H − H0 )2 ⎢ + 1.505 + ⎪ ⎣ ⎪ 4 Reθ H ⎦ ⎪ ⎩ H − H0 + Reθ

H 0 represents favorable pressure gradients where β < 0 the adverse pressure gradients. In Fig. 6 the analytic solutions for some positive values of m (left) are compared with the numerical solution obtain by solving the steady √IBL equations (right) for Reynolds number independent displacement thickness δ ∗ Re∞ .

8 Interacting Boundary Layer Methods and Applications

277

Laminar and Turbulent Flow Over a Flat Plate The edge velocity prescribed for a flat plate is ue = V∞ , where V∞ is the free-stream velocity. The unsteady simulation is hence performed until a steady solution is obtained for a Reynolds number of Re = 105 for the laminar flow based on the plate length, L. The numerical results presented in this section and in section “Viscous-Inviscid Interaction” are obtained by employing discontinuous Galerkin method to discretize the IBL equations together with a nonconservative flux scheme and a multistage low-storage Runge-Kutta scheme for the time integration (Özdemir et al. 2017). The discontinuous Galerkin method is a highly compact formulation where the computational domain is divided into nonoverlapping (hexahedral) elements and the solution in each element is approximated via a local basis function set. Thus the governing equations are solved in a finite element space of discontinuous functions. The degree of the approximating polynomials (local basis function set) determines the order of accuracy of the method, and, if desired, the degree of the polynomials used can be easily changed from element to element. The effectiveness of the applied numerical method can be perceived by a simulation using very few elements for the spatial discretization. On the left of Fig. 7, the solution converged to a steady state obtained at the end of the simulation with N = 10 elements for the basis functions of degree p = 0 and p = 1 is shown (order of accuracy is p + 1), together with the analytical solution. As can be seen from the left of Fig. 7, already a very coarse mesh gives results very close to the exact solution. The first element is of course the most problematic and where the numerical solution is less accurate due to the infinite slope which cannot be well represented by low-order polynomials. The problem is reduced by p-refinement or h-refinement. For p = 0 the discontinuous Galerkin method

1.8

δ∗

1.4

Re∞

1.2 1 0.8 0.6 0.4

1.2 1 0.8 0.6

θ

0.4

0.2 0

DG p = 1, N = 400 Exact

1.6

δ ∗ Re∞ , θ

Re∞

1.4

δ∗

1.8

p=0 p=1 Exact

1.6

0.2 0

0.2

0.4

0.6

X

0.8

1

0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

X

Fig. 7 Comparison of the numerical solution (discontinuous Galerkin method) with the analytical solution for the flow over a flat plate for the displacement thickness, δ ∗ , and for the momentum thickness, θ on a coarse mesh (N = 10, left) and a fine mesh (N = 400, right)

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actually reduces to a finite volume method, but an increase on the degree of the basis function increases the accuracy of the solution faster than a mesh refinement. On the right of Fig. 7, finally the displacement thickness δ ∗ and momentum thickness θ for p = 1 and the finest grid with N = 400 elements are shown. The results agree very well with the analytical solution. For turbulent flow over a flat plate, the experimental data used to validate the numerical solution are from Wieghardt and Tillmann (1953) as identified with the flow number 1400. Free-stream velocity is ue = 33 m/s with no pressure gradient along the 5-meter-long flat plate. The corresponding Reynolds number is Re = 1.0927 × 107 again based on the plate length L. For the unsteady simulations, the simulation time is chosen to be long enough such that the boundary layer may be considered to converge to a steady state. In Fig. 8 comparisons of the numerical solution for the displacement thickness δ ∗ and friction coefficients Cf to the experimental data are shown.

Laminar Flow Over a Cylinder Flow over a cylinder is a test case where the need for a (strong) interaction scheme can be demonstrated due to the large separation. Van Dommelen and Shen (1980) obtained the solution for the field method in Lagrangian coordinates showing the displacement thickness, δ ∗ for various time instances. The simulation is performed for the flow over a cylinder with unit radius, for which the upper half has a length L = π . Kinematic viscosity is taken as unity, ν = 1. The edge velocity (obtained by the potential theory) around a cylinder is given by: ue (x) = V∞ 2 sin(x).

(53)

A comparison of the numerical solution obtained by employing DG method with the results obtained by van Dommelen and Shen (1980) is shown in Fig. 9. There is a discrepancy between both numerical solutions especially for later time instances where the separation becomes larger. The IBL simulation will not be able to reach 5.5

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8 Interacting Boundary Layer Methods and Applications 16

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to a steady-state condition due to the singularity at the separation point. This large discrepancy and the singularity problem might be overcome by coupling the IBL system with inviscid solution through an interaction scheme.

Flow Over Airfoil Sections In this section the unsteady solution of the IBL equations will be demonstrated by developing boundary layer flow over airfoil profiles. As in the previous examples, the focus here is the solution of the viscous integral boundary layer equations, and therefore the velocity distribution at the edge of the boundary layer, ue , is prescribed. This distribution of the edge velocity is obtained separately by the potential flow solution over the same airfoil (e.g., running either XFOIL (Drela 1989) or RFOIL (van Rooij 1996) in inviscid mode to obtain the inviscid velocity along the airfoil surface). This is a good way to verify the numerical solution of the IBL equations. As a first example flow, over a NACA0009 airfoil profile is presented. The airfoil is considered to set into motion instantaneously where the boundary layer is developed in time. A steady solution is obtained with the aerodynamic design software XFOIL in order to compare the results with the ones obtained by the present method. NACA0009 is a symmetric airfoil with maximum thickness of 9% of the chord. Reynolds number for the numerical simulation is kept relatively low (Re = 104 ) in order to ensure the flow to be fully laminar. A simulation with zero angle of attack is performed. Since the airfoil geometry is symmetric, only one side of the solution is presented. In Fig. 10 the evaluation of the solution for δ ∗ in time is presented. Since the initial and boundary conditions are not time dependent, it is expected that the solution converges to a steady-state solution as can be seen from the comparison with the solution obtained by XFOIL. This match of the solutions is expected since the prescribed velocity distribution in the IBL solution is obtained by running XFOIL in inviscid mode and verifies the IBL method.

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Fig. 10 Unsteady simulation for NACA0009 at Re = 104 and α = 0◦ . Time evaluation of the displacement thickness δ ∗ is compared to the results obtained by XFOIL. (a) Initial condition. (b) t = 0.2 s. (c) t = 2.0 s. (d) Steady solution

In order to have some asymmetries on suction and pressure side, a cambered airfoil NACA2205 is then simulated for 0 deg angle of attack and for the Reynolds number of 104 . Distributions of displacement and momentum thickness along the cord for the solution converged to a steady state are shown in Fig. 11 together with XFOIL results which show a good agreement. Similar to the above example, the prescribed edge velocity is obtained from XFOIL, and this matching results verifies the IBL method solution.

Viscous-Inviscid Interaction Since the introduction of the boundary layer theory by Prandtl in 1904, the approach of splitting of the flow domain into viscous and inviscid regions has been used in many variants, since both regions allow a range of models to be selected, which can be combined using a range of interaction methods (see, e.g., Cebeci and Cousteix 2005). Historically, the two regions were solved separately and glued together (weak or no interaction). This approach works for attached flows but leads to singularities

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Fig. 11 A comparison of the results converged to steady state for NACA 2205 airfoil profile for Re = 104 and α = 0◦ obtained by the unsteady simulation with the results obtained by XFOIL. (a) Displacement Thickness δ ∗ . (b) Momentum Thickness θ

associated with flow separation, both for steady (Goldstein 1948) and unsteady flows (van Dommelen and Shen 1980). The latter result was interpreted by Matsushita and Akamatsu (1985) as the result of discontinuities in the boundary layer (analogous to shocks in compressible flow). They used a dissipative finite-difference method to capture separating flow including the singularity. Catherall and Mangler (1966) solved the singularity problem by inverting the solution approach. Prescribing the displacement thickness (instead of velocity) as a boundary condition, they were able to integrate the boundary layer equations through the separation point without encountering any singularity evidence. The convergence of this inverse method has been proven to be very slow. Due to the hierarchy assumed in all these methods, only a weak interaction is provided. An improvement to these methods introduced by Le Balleur (1978) where the outer flow is calculated directly and the boundary layer in an inverse way (semiinverse method). A similar approach was used by Carter (1981) for transonic turbulent separated flow. Later approaches take into account the two-way coupling of the viscous and inviscid regions during computation (strong interaction), thereby successfully extending the validity to (mildly) separated flow. The interaction of the two systems is essentially stabilizing: the breakdown due to singularity does not occur until more strongly separated flow regimes are entered, as can be seen in the analysis of Coenen (2001). These approaches saw major development in the 1980s, with the quasi-simultaneous (Veldman 1981) and fully simultaneous (Lees and Reeves 1964) approach. The fully simultaneous method is used by Drela (1985); Drela and Giles (1987) to couple and Euler method with the integral boundary layer equations for calculation of two-dimensional transonic flows. Although there are many studies on steady two-dimensional flow problems for unsteady flows, a very limited number of studies concerning coupling schemes are documented. Cebeci et al. (1993) studied initiation of dynamic stall by solving boundary layer equations with an inverse finite-difference method employing as

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interaction law the Hilbert integral and the algebraic eddy-viscosity formulation of Cebeci and Smith (1974). Other authors instead mostly focused on the simultaneous solution of the complete viscous and inviscid models, e.g., Drela (2013a). The equations of motion in each of the subdomain provide a relation between the velocity along the edge of the boundary layer ue and the displacement thickness δ ∗ . A symbolic notation is here used to denote these relations:   external flow: ue = E δ ∗ ,   boundary-layer flow: ue = B δ ∗ ,

(54) (55)

where E denotes “external” and B “boundary layer.” The inverse form of the boundary layer formulation has been indicated with B since it is assumed that this formulation does possess a (unique) solution after discretization even at separation points, while the direct one does not (Veldman 2008). If a steady solution is sought, the two edge velocities for given displacement thickness need to match. A justification of this requirement is also given by the functional approach theory for steady flows, extensively treated by Brune et al. (1974) and Williams and Smith (1990). In order to decrease the computational effort required by the simultaneous method but preserving the asset to allow a strong interaction between the layers, based on the triple-deck theory, the quasi-simultaneous method was developed by Veldman (1981). The idea behind this scheme is to combine the advantages of the direct and simultaneous methods and avoid the singularity problems at separation. Instead of solving viscous and inviscid regions simultaneously, the boundary layer calculations are performed together with an approximation for the external inviscid flow. The boundary layer is therefore informed instantaneously about how the external flow reacts on changes inside the boundary layer itself through a simple but sufficiently accurate approximation of the external flow, called interaction law, and denoted here with I . The following iterative process is therefore created:       ⎧ ⎨ue (n+1) − I δ ∗ (n+1 ) = E δ ∗ (n) − I δ ∗ (n) ,   ⎩u (n+1) − B δ ∗ (n+1) = 0 . e

(56)

Since the goal of the interaction law is to make the numerical calculations survive the separation point, a good description of the local interactive physics is essential. When this requirement is fulfilled, a fast convergence can be pursued. It is remarked by Veldman that the choice of the interaction law does not influence the converged solution. This is in fact only determined by the choices made for the inviscid and viscous flow operators E and B in Eqs. (54) and (55). The interaction law only controls the rate with which the viscous-inviscid solution is obtained (Veldman 2009).

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In the quasi-simultaneous method, the boundary layer is solved together with an approximation of the inviscid flow. Every time-step, iterations are performed in order to reach a converged solution between the edge velocity provided by the boundary layer modeling together with the interaction law and the complete inviscid modeling described by Eq. (56). The stopping criterion can be either based on the difference between the two edge velocities or on a fixed number of iterations. Cebeci et al. (1993) assumed that 20 viscous-inviscid iterations even for flow involving stall are sufficient, while less iterations are required when stall does not occur. Unless an unsteady panel method based on Helmholtz theorem of conservation of vorticity is developed (Basu and Hancock 1978), a fixed number of iterations are performed. The time advance is kept fixed during the iteration process by fixing the initial conditions to the solution at the previous time-step. Only when the iteration procedure is terminated, the solution computed is used as the new initial condition for the next iterations. Veldman (2009) suggests that a simple thin airfoil approximation of the external inviscid flow is sufficiently useful for the development of the method. Other choices are possible. The interaction law implemented has the form: 1 ue (x) = ue0 (x) + π

 χ

d  ∗  dζ ue δ , dζ x−ζ

(57)

where ue0 is the edge velocity distribution without displacement effects and χ is the region where strong interaction is required, as indicated by triple-deck theory. When the edge velocity distribution ue is considered as an additional unknown and the interaction law Eq. (57) is added to the set of Eq. (25) (or to the laminar version of the mentioned equation), the nature of the problem changes. This can be shown just looking at the integral boundary layer equations, independent of the interaction law chosen. The terms where the derivatives of ue appear cannot be considered sources any more, and by indicating the unknown vector as u = (u1 , u2 , u3 )T = (δ ∗ , δ ∗ + θ, ue )T , the integral boundary layer equations are written in the form: Cf ∂u1 u1 ∂u3 ∂ ∂u3 + + u2 + u3 , [u3 (u2 − u1 )] = ∂t u3 ∂t ∂x ∂x 2 (58)  2(u2 − u1 ) ∂u3 ∂  ∂u2 ∂u3 + + 2H ∗ (u2 − u1 ) + u3 H ∗ (u2 − u1 ) = CD u3 . ∂t u3 ∂t ∂x ∂x (59) Already the above equations present the problem that they contain nonconservative products that cannot be transformed into divergence form, i.e., the equations in the form ∂t u + ∂x f (u) + g(u)∂x u = s(u) cannot be written as ∂t u + ∂x h(u) = s(u). This causes problems once the solution becomes discontinuous, because the weak solution in the classical sense of distributions then does not exist.

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Therefore in order to couple the system described by the integral boundary layer equations with the interaction law, a numerical method for nonconservative systems is required.

Numerical Solutions for Interacting Boundary Layers One of the earliest methods that combined the inner solution (integral boundary layer (IBL) equations) with outer solution (inviscid flow) is developed by von Kármán and Milikkan (1934), and later on a similar method is introduced by Stratford (1954). But the most famous solution of the complete interacting boundary layers is by Drela (1985) which became an industry standard aerodynamic design tool (XFOIL). This tool which have been made open source (Drela 2013b) together with the pioneering work by Drela and co-workers (Drela 1985, 1989; Nishida 1997; Mugal 1998) is still helping the community to develop these ideas further. It is also worth to mention the recent work of Drela (2013a) on an attempt to extend the solution of interacting boundary layers to three dimensions. The XFOIL is extended (to RFOIL) by van Rooij (1996) to be able to analyze wind turbine blade sections improving the prediction for the maximum lift coefficient and including a method for predicting the effects of rotation on airfoil characteristics (Snel et al. 1993b, a). A finite element discretization of the IBL equations in three dimensions was employed by Nishida (1997), Milewski (1997), and Mugal (1998) where (lower order) approximation of the inviscid flow is coupled in a fully simultaneous way. A more recent method (Garcia 2011) includes an unsteady formulation which seems very promising, but no validation results have emerged yet. Özdemir developed a discontinuous Galerkin (DG) method (Özdemir and van Boogaard 2011; Özdemir et al. 2017) to solve the unsteady two-dimensional IBL equations strongly coupled with an inviscid panel method, with linear strength vortex distribution, through a quasi-simultaneous viscous-inviscid interaction scheme (Veldman 1981). It should be noted that the nonconservative effects, arising due to (quasi-)simultaneous integral formulation, affect the unsteady behavior and the location of equilibria. In effect this causes the resulting solutions to depend not only upon the physics of the problem but on the numerical strategy as well (Seubers 2014). To prevent this undesirable condition, specialized nonconservative numerical scheme was applied (Seubers 2014; Passalacqua 2015; Özdemir et al. 2017). In a recent study, Zhang et al. (2019) also employed a DG solution where a strong coupling is achieved by fully simultaneous interaction method for the flows with free transition. Few truly unsteady applications are also noted: a dynamic stall simulation by Cebeci et al. (1993) and a flutter computation by Zhang et al. (2004), both of which show significant effects due to the viscous part. In the next section, steady solutions obtained by XFOIL and RFOIL will be presented for the sake of completeness since the solutions and discussions on these

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tools are widely available in the literature. In the remainder of the section, the main focus will be the shortcomings of these methods and possible improvements.

Flow Over Airfoil Sections In Fig. 12, typical solutions obtained from XFOIL (left) and RFOIL (right) are presented where the numerical solutions are compared with experimental data for the CL − α. From XFOIL (similar for RFOIL), it is possible to obtain the boundary layer and wake parameters (like displacement thickness, momentum thickness, etc.) as well as the solution for lift, drag, momentum, etc. coefficients dependent on flow parameters like angle of attack, Re number, etc. It is also possible to simulate airfoil sections with natural transition or set a transition position manually to force the turbulent solution beyond that position. These tools allow a complete analysis and design of two-dimensional airfoil sections for large variety of flow conditions. For the numerical solution of the IBL equations, several practical choices can be made. One such choice is whether the wake would consist of a single layer (e.g., XFOIL (Drela 1989), RFOIL (van Rooij 1996)) or allow a double-layered wake. In the single-layered wake case, the solution on both sides of the wake line is assumed to be symmetric independent of the airfoil geometry and angle of attack (Drela 1989). By introducing two layers of wake line, the solution is allowed to be different on both sides of the wake line and obtained separately (Özdemir et al. 2017). Another advantage of introducing two layers of wake is that it would be possible to change the separation location of each individual wake line to simulate flows with high separation and increase the accuracy of the lift prediction. An example of such a double wake implementation on the steady solution is given in section “Double Wake Implementation.” In Fig. 13a a comparison of the unsteady numerical solution for the displacement thickness, δ ∗ is shown to the numerical solution obtained by XFOIL where the flow naturally transitions from laminar to turbulent conditions for NACA63418 airfoil for Re = 3.0 × 106 and α = 1◦ . The unsteady DG method is simulated for a long enough time to achieve a steady-state solution. The wake simulation in the unsteady method is performed with introducing two layers of panels (i.e., can be interpreted as pressure and the suction sides of the wake). The effect of this implementation can be seen in Fig. 13a clearly. The upper (suction side) end location of the wake is located to x = −1 and continues until x = 0 where the trailing edge of the suction side starts. The leading edge of the airfoil is around x = 1, and the pressure side of the airfoil geometry continues until around x = 2 reaching the trailing edge of the body on the pressure side. Finally the lower side of the wake (pressure side) extends until x = 3. Within this setting when the upper and lower sides of the wake is examined, a difference in the solution (displacement thickness) can be noticed. This result is expected for nonsymmetric airfoils. Furthermore from the comparison, it can be seen that the transition onset and the behavior of the transition are different for both solutions. Although the transition model applied in DG method involves unsteady terms (as presented in section “Laminar to Turbulent Transition”), this model needs

Fig. 12 Comparison of the experimental data of CL − α polar with the numerical solution obtained by XFOIL for NACA2415 airfoil for Re = 302,500

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a further attention. In order to avoid the use of transition model, the same test case is considered for a fully turbulent flow condition where the solution is compared to the XFOIL result as presented in Fig. 13b. Considering the differences between the present method and XFOIL, one naturally does not expect to see an exact match in the solutions. It can be concluded that the solution obtained with the unsteady DG method is in good agreement with XFOIL.

Limitations of Interacting Boundary Layer Methods As we have discussed the details above, XFOIL is developed to analyze and design two-dimensional transonic airfoil sections. Naturally the necessary choices have been made specifically for these cases (i.e., Mach number correction, etc.) and tuned to obtain best results for these specific types of problems under consideration. Later on, in the beginning of the 1990s when a similar need emerged to analyze and design two-dimensional airfoil sections for the wind turbine blades, a natural choice was to start from then available XFOIL. Soon after the necessary modifications have been introduced to XFOIL (where details explained above) to cover the specific needs for the type of flows under consideration (e.g., low Mach number, thicker airfoil sections, Re number range, rotational effects, etc.). Similarly, in the current practices, some other aspects of these problems become dominant considering the recent developments in the wind energy world. Most common of these new problems where we are seeking solutions are thicker airfoil sections especially with thick (flat-back) trailing edges, dominant unsteady effects, and need for a design of blade add-ons. Thicker airfoil sections (starting from around 18% at the blade tip to about 60% at the blade root) with or without thick trailing edges are more frequently used for larger wind turbine blades to satisfy the structural requirements while maintaining optimal aerodynamic performance. These large-sized blades are more flexible, leading to stronger fluid-structure interaction effects, that in turn (together with the sections with thick trailing edges) generate flow conditions where unsteady effects become dominant. To maintain the aerodynamic performance of these large blades, so-called blade add-ons (vortex generators, blade root spoilers, etc.) are adapted, e.g., to overcome or delay separation, increase lift performance, etc. One of the advantages of these add-ons is that they can be applied to existing wind turbine blades. In any ways the analysis and design of new wind turbine blade (sections) need to include the effects of these add-ons in the design phase already. With all these mentioned new requirements in mind, some selections of the proposed improvements are summarized in the sections below. First, improvements to drag (Ramanujam et al. 2016) and lift (Ramanujam and Özdemir 2017) prediction methods are presented. The improvements proposed in both of these methods can easily be applied any interacting boundary layer method (XFOIL/RFOIL like methods). Final section is dedicated to double wake (Vaithiyanathasamy et al. 2018) implementation (in RFOIL).

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Some other recent developments on modeling the effects of vortex generators for interacting boundary layer methods are also worth to mention here. De Tavernier et al. (2018) showed a modification where the turbulent boundary layer formulation is modified using a semiempirical source term. In a more recent work, Ravishankara et al. (2019) presented some preliminary results of a theoretical approach where the effects of the vortex generators are modeled with the help of theory of plane mixing flows.

Drag and Lift Correction Ramanujam et al. (2016) derived the correction for the drag prediction formulas as follows:      Ue Ue 2 θe 1 + 1 − . (60) cd = (gH1 − 1) − 1 c U∞ U∞ x=∞ with the mass flow shape factor correction, g, defined as: g=

A2 B . 42.084(1 + xCτEQ )

(61)

where the value of factor A2 B is equal to the same value used to develop the IBL method under consideration with and A and B are the G − β equilibrium locus coefficients. The proposed drag correction method has been implemented in XFOIL and RFOIL and has been tested for a range of airfoils in order to verify the accuracy of drag prediction. There was a significant improvement observed for all the test cases considered. The analysis presented in this section is computed for incompressible flow conditions (M∞ = 0) for conditions of natural transition with a free-stream turbulence intensity of 0.07% (Ncrit = 9). Even for thinner airfoils, there is a noticeable improvement in drag prediction (Ramanujam et al. 2016). A similar trend is also observed for the NACA 633 418 airfoil as seen on the left-hand side of Fig. 14. For this airfoil, the drag underprediction is approximately 12% at the minimum drag point when using the original XFOIL and RFOIL results. For thicker airfoils, the underprediction in drag becomes more severe. The levels of under-prediction in drag can become as high as 30% for very thick airfoils such as DU 99-W-405LM which has a maximum thickness of 40.5% as shown on the right-hand side of Fig. 14. The drag correction yields excellent results in the linear lift regime although the drag in the post stall regime is not very well predicted. This is to be expected since the viscous-inviscid interaction method of XFOIL and RFOIL is not designed to handle massively separated flows. Similar to drag underprediction, current interacting boundary layer methods (i.e., Rfoil, Xfoil) overpredict the lift around 5–10% around (l/d)max for a wide range of angle of attacks for the thick airfoils. Ramanujam and Özdemir (2017)

Fig. 14 Comparison of Lift-Drag polars for thick wind turbine airfoils with and without the proposed drag correction to the experimental data for NACA 633 -418 (left) and DU 99-W-405LM (right)

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argued that the wake geometry in RFOIL and XFOIL is determined from the inviscid calculations which seems to give rise to this problem, and they proposed a scheme for IBL methods and implemented in RFOIL, to include the effects of viscous flow on the wake geometry which lead to improved lift prediction for thick airfoils. Although this approach is a practical improvement for the implementation procedure rather than a theoretical solution, it demonstrated slightly better lift predictions for the thick airfoils. The left-hand side of Fig. 15 presents a case where lift discrepancy is observed in RFOIL predictions and the updated wake results in an improved prediction. The agreement with experimental results is significantly improved. Both the magnitude of lift coefficient and the lift slope are in better agreement when compared with the experimental data (Timmer 1998). It is worth to repeat the discussion in Ramanujam and Özdemir (2017) since it gives an insight to the improvement of the lift prediction and reveals the related drag prediction problem. With the updated wake geometry, the drag prediction becomes inaccurate. The drag prediction is reasonable at lower angles of attack as long as the drag does not rise significantly with increasing angle of attack. The drag discrepancy occurs as soon as the lift curve becomes nonlinear. As the flow approaches to a separation, the shape factor is artificially restricted in order to avoid numerical difficulty. This is achieved by prescribing a shape factor value as soon as flow separation is approached. This prescribed shape factor in turn modifies the pressure gradient that the equations see, resulting in a smooth convergence. This approach is supported by the argument that in case of flow separation, the accuracy of the solution tends to decline anyways and a small modification would not significantly affect the result. However, when the wake geometry is updated with the modified vorticity distribution, this results in limiting the resulting shape factor below the value corresponding to flow separation. This artificial restriction makes the method behave as if the flow never separates. Although the lift follows the same behavior as before, the drag never rises post stall. Since this issue arises because of workaround to overcome the separation singularity problem, the drag is not updated when using the updated wake geometry. The right-hand side of Fig. 15 presents the drag curve with the corrected drag and improved lift formulation. It is also worth noting that since the solution procedure has been modified, the drag correction argued above needs to be recalibrated. Using the drag value from the first wake iteration also avoids this exercise.

Double Wake Implementation As discussed earlier in the traditional panel methods, a single wake is released from trailing edge. The integral boundary layer equations are then solved using source terms on the airfoil surface and the wake to estimate the boundary layer parameters. The single wake concept leads to inefficiencies in predictions for separated and stalled flows. The prediction at angles of attack for which separation occurs can be improved by releasing a secondary wake from the separation point. The idea of double wake implementation can be extended to both conventional sharp trailing edge and blunt trailing edge airfoils.

Fig. 15 DU 97-W-300: Lift and drag prediction with and without wake geometry modification compared with experimental data (Timmer 1998)

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Several authors in literature provided different approaches for the double wake modeling. Vezza and Galbraith (1985) modeled an unsteady, incompressible separated flow where N+1 linear vortices are considered on N airfoil panels; for the near wake, a panel at separation point and one at the trailing edge are included; for the far wake, discrete vortices model the asymptotically steady separated flow. For moving airfoils, Voutsinas and Riziotis (1999) and Riziotis and Voutsinas (2008) modeled unsteady double wake with N sources and two distributed vortices representing attached and separated regions with strong interaction of unsteady boundary layer. The model predicts accurately the separation location up to moderate stall. Zanon (2011) adopted Riziotis model for vertical axis wind turbines with modifications in the orientation of near wake panel. Ramos García (2011) modeled unsteady 2D flow with linear vortices and a source. This yields to a good agreement for the predicted aerodynamic lift compared to experimental data, while drag is under predicted. Steady double wake for separated flows are modeled by Maskew et al. (1984) and Marion et al. (2014) with the length of the wake sheets determined by a wake factor and a wake height. Maskew determines the initial wake shape with a parabolic curve, whereas Marion considers an experimental wake factor. These models show good agreement with experimental data with some discrepancies in deep stall region. In this section we will present the results obtained by Vaithiyanathasamy et al. (2018) where he first demonstrated the converged wake shapes for the inviscid flow. The initial wake shape used for constructing the constant strength singularity vortex element and final converged wake shape (when Cp over airfoil surface is interpreted), for NACA63415 airfoil at the chosen AoA, is shown in the left-hand side of Fig. 16. From the figures in Fig. 16, it can be seen that the final wake shape representing the converged constant wake sheet from the double wake inviscid model and the streamlines enclosing the separated flow from CFD simulation are nearly similar in shape. Figure 17 shows the comparison of pressure coefficients between the inviscid double wake method with the numerical models CFD, XFOIL, and experimental data (Sagmo et al. 2016; Aksnes 2015) for S826 airfoil at Re = 1.0 · 105 and angle of attack of 14.5◦ . The separation point for the inviscid double wake method is located at x/c = 0.43 and is obtained from experimental data. It can be seen that the inviscid double wake method can replicate result closer to CFD, XFOIL viscous solutions, and experimental data in the separation region. However, the peak suction pressure is highly overpredicted as it is an inviscid solution. Further, the release of wake from separation point gives a small oscillation at the separation point. Figure 18 shows the results including the viscous solution of the test cases with blunt trailing edge airfoils for lift coefficients (left) and lift-drag polar (right) for AH 93-W-300 airfoil at Re = 1.5 · 106 . It can be seen that the lift coefficient with RFOIL double wake model is predicted better than the RFOIL single wake version in the separated, stalled, and deep stall regions. Also, the complete polar is predicted more accurately than XFOIL and closer to experimental data. This leads to good prediction of lift to drag coefficient (polar plot) with the RFOIL double wake model compared to other numerical methods. The drag coefficients are not shown in the

Fig. 16 Initial and final wake shape of NACA63415 airfoil (left) and vorticity plot from SU2-CFD simulation with streamlines enclosing the separated region at Re = 1.6 × 105 and at AoA of 16◦

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Fig. 17 Comparison of pressure coefficient of inviscid double wake method for S826 airfoil with experimental data, CFD and XFOIL viscous solutions at Re = 1.0 · 105 and at angle of attack of 14.5◦

result for these airfoils as they remain the same as that of the RFOIL with single wake model. Figure 19 shows the plot of lift coefficients and lift drag polar for NACA 633 − 418 airfoil at Re = 3·106 . It can be seen that the lift prediction for NACA 633 −418 is improved with double wake model and closer to experimental data compared to single wake method of RFOIL and XFOIL. Further, the RFOIL double wake method predicts accurate lift coefficients in the deep stall region where the XFOIL fails to converge and the RFOIL single wake method underpredicts the lift coefficient.

Future Work Current trend in the wind energy world is to increase the size of the wind turbine rotor diameters aiming to harvest more energy for a given square kilometers of wind park area. This increase in the sizes of these structures brings their own challenges as also mentioned in previous sections. From the aerodynamic design and analysis point of view, current dedicated tools are being extended to their limits and to obtain accurate analysis results becoming more and more difficult. One can think of addressing higher fidelity tools to overcome some of these challenges, but those tools are typically not suitable for a daily design tool chain. Additionally those high fidelity tools also suffer from many challenges which the details are beyond the scope of this chapter. Referring to the discussion in section “Limitations of Interacting Boundary Layer Methods,” some challenges we are facing today are thick airfoil sections, blade add-ons, surface roughness (erosion, corrosion), etc. These challenging applications extend our current design tools to their limits.

Fig. 18 Comparison of lift coefficients and lift drag polar for AH 93-W-300 at Re = 1.5 · 106

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Fig. 19 Comparison of lift coefficients and lift drag polar for NACA 633 − 418 at Re = 3 · 106

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Flow over thick airfoil sections especially with a thicker (flat-back) trailing edges is highly unsteady where a von Kármán vortex street is formed behind the thick trailing edge. From the interacting boundary layer method point of view, the challenge here is not only the violation of the boundary layer theory due to the high curvatures in the geometry and high gradients within the boundary layer and the wake but also the highly unsteady behavior of this flow field. As mentioned above one way to increase the energy production is to increase the sizes of the wind turbine blades, but another more proactive way is to increase the efficiency of the blades already in the design phase. Blade add-ons are quite promising to help improving the efficiencies of the blades, but currently as the name suggests, they are introduced as post-modifications to already designed blades. Developing our tools to include the effects of these blade add-ons (already in the design phase) would allow us to optimize the design of the airfoil sections including these blade add-ons. Another, a bit more passive way to increase the efficiency is to understand and predict the performance of the turbines in real operating conditions. These structures are constantly facing climate and environmental effects that degrade the blade surfaces and causing problems like leading edge erosion and overall roughness of the surface, etc. Knowing the effects of these off-design conditions would help us optimize these blades not for peak performance points (maximum lift to drag ratio, etc.) but an optimum performance band. Increasing sizes of the blades lead to more flexible structures leading to more pronounced effects of the interaction between the structure and the fluid (wind) around it. This interaction also leads to stronger unsteady flow field as well as structural challenges and problems with vibration and (aero)acoustics. From the interacting boundary layer method point of view, this field might be rather new and challenging but definitely worth putting more effort in it.

References Aksnes NY (2015) Performance characteristics of the NREL S826 airfoil. MSc Thesis – Norwegian University of Science and Technology Atkins HL, Shu CW (1996) Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations. No. 96–1683 in AIAA-paper Basu BC, Hancock GJ (1978) The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow. J Fluid Mech 87(1):159–178 Brune G, Rubbert P, Nark TC Junior (1974) A new approach to inviscid flow/boundary layer matching. In: 7th Fluid and Plasma Dynamics Conference, AIAA, Palo Alto Carter JE (1981) Viscous-inviscid interaction analysis of transonic turbulent separated flow. In: 14th Fluid and Plasma Dynamics Conference, AIAA, Palo Alto Catherall D, Mangler KW (1966) The integration of the two-dimensional laminar boundary-layer equations past the point of vanishing skin friction. J Fluid Mech 26:163–182 Cebeci T, Cousteix J (2005) Modeling and computation of boundary-layer flows, 2nd edn. Springer, Berlin/Heidelberg Cebeci T, Smith AMO (1974) Analysis of turbulent boundary layers, Academic Press

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Cebeci T, Platzer MF, Jang HM, Chen HH (1993) Inviscid-viscous interaction approach to the calculation of dynamic stall initiation on airfoils. J Turbomach 115(4):714–723 Cockburn B, Shu CW (2001) Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. J Sci Comput 3(16):173–261 Coenen EGM (2001) Viscous-inviscid interaction with the quasi-simultaneous method for 2D and 3D aerodynamic flow. Ph.D. thesis, Rijksuniversiteit Groningen. ISBN: 90-367-1472-9 Curle N (1967) A two-parameter method for calculating the two-dimensional incompressible laminar boundary layer. J R Aeronaut Soc 71:117–123 De Tavernier D, Baldacchino D, Ferreira CS (2018) An integral boundary layer engineering model for vortex generators implemented in XFOIL. Wind Energy 21(10):906–921 Drela M (1985) Two-dimensional transonic aerodynamic design and analysis using the Euler equations. Ph.D. thesis, Massachusetts Institute of Technology Drela M (1989) XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils. In: Mueller TJ (ed) Low Reynolds Number Aerodynamics. Lecture Notes in Engineering, vol 54. Springer, Berlin, Heidelberg Drela M (1995) Mises implementation of modified Abu-Ghanam/Shaw transition criterion. Report, MIT Aero-Astro Drela M, Giles B (1987) Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA J 25(10):1347–1355 Drela M (2013a) Three-dimensional integral boundary layer formulation for general configurations. In: Fluid Dynamics and Co-located Conferences. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2013-2437 Drela M (2013b) XFOIL: subsonic airfoil development system. https://web.mit.edu/drela/Public/ web/xfoil/ Falkner VM, Skan SW (1931) Solutions of the boundary layer equations. Philos Mag J Sci 12(80):865–896 Garcia NR (2011) Unsteady viscous-inviscid interaction technique for wind turbine airfoil. Ph.D. thesis, DTU Goldberg P (1966) Upstream history and apparent stress in turbulent boundary layers. Tech. rep., DTIC Document Goldstein S (1948) On laminar boundary-layer flow near a position of separation. Q J Mech Appl Math 1(1):43–69 Green J, Weeks D, Brooman J (1977) Prediction of turbulent boundary layer and wakes in compressible flow by a lag-entrainment method. ARC R&M Report 3791, Aeronautical research council Katz J, Plotkin A (2001) Low speed aerodynamics, 2nd edn. Cambridge University Press, Cambridge Lees L, Reeves BL (1964) Supersonic separated and reattaching laminar flow. AIAA J 2(11):1907– 1920 Lighthill MJ (1958) On displacement thickness. J Fluid Mech 4(04):383–392 Le Balleur JC (1978) Couplage visqueux-non viscqueux: Méthode numérique er applications aux Écoulements bidimensionnels transoniques et supersoniques. La Recherche Aérospatiale 183:65–76 Marion L, Ramos-García N, Sørensen JN (2014) Inviscid double wake model for stalled airfoils. J Phys Conf Ser 524(1):012132 Maskew B, Vaidyanathan TS, Nathman J, Dvorak FA (1984) Prediction of aerodynamic characteristics of fighter wings at high angles of attack. Tech. rep., Office of Naval Research, N00014-82-C-0354, Arlington Matsushita M, Akamatsu T (1985) Numerical computation of unsteady laminar boundary layers with separation using one-parameter integral method. JSME 28(240):1044–1048 Matsushita M, Murata S, Akamatsu T (1984) Studies on boundary-layer separation in unsteady flows using an integral method. J Fluid Mech 149:477–501 Milewski W (1997) Three-dimensional viscous flow computations using the integral boundary layer equations simultaneously coupled with a low order panel method. Ph.D. thesis, MIT

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Moran J (2013) An introduction to theoretical and computational aerodynamics. Dover Publications, New York Morino L, Kuo CC (1981) Subsonic potential aerodynamics for complex configurations: a general theory. AIAA J 12:19 Mugal BH (1998) Integral methods for three-dimensional boundary-layers. Ph.D. thesis, MIT Nishida BA (1997) Fully simultaneous coupling of the full potential equation and the integral boundary layer equations in three dimensions. Ph.D. thesis, MIT Obremski HJ, Fejer AA (1967) Transition in oscillating boundary layers. J Fluid Mech 29(part 1):93–111 Obremski HJ, Morkovin MV (1969) Application of a quasi-steady stability model to periodic boundary-layer flows. AIAA J 7(7):1298–1301 Özdemir H (2006) High-order discontiuous Galerkin method on hexahedral elements for aeroacoustics. Ph.D. thesis, University of Twente Özdemir H (2010) Development of a discontinuous Galerkin method for the unsteady integral boundary layer equations. Tech. Rep. ECN-X–10-107, Energy Research Centre of the Netherlands, Petten Özdemir H, van Boogaard E (2011) Solving the integral boundary layer equations with a discontinuous Galerkin method. EWEA-2011, Brussels Özdemir H, Hagmeijer R, Hoeijmakers HWM (2005) Verification of the higher order discontinuous Galerkin method on hexahedral elements. Comptes Rendus Mecanique 333:719–725 Özdemir H, Garrel Av, Ravishankara AK, Passalacqua F, Seubers H (2017) Unsteady interacting boundary layer method. Texas, USA, AIAA SciTech Forum 35th Wind Energy Symposium Passalacqua F (2015) Implementation of unsteady two-dimensional interacting boundary layer method. Master’s thesis, Politecnico di Milano Prandtl L (1904) Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg. In: Krazer A (ed) Teubner, Leipzig, Germany, p. 484 Ramanujam G, Özdemir H (2017) Improving airfoil lift prediction. Texas, USA, AIAA SciTech Forum 35th Wind Energy Symposium Ramanujam G, Özdemir H, Hoeijmakers HWM (2016) Improving airfoil drag prediction. J Aircraft. https://doi.org/10.2514/1.C033788 Ramos García N (2011) Unsteady Viscous-Inviscid Interaction Technique for Wind Turbine Airfoils. Ph.D. thesis, Technical University of Denmark Ravishankara AK, Özdemir H, Franco A (2019) Towards a vortex generator model for integral boundary layer methods. California, USA, AIAA SciTech Forum 2019 Wind Energy Symposium Riziotis VA, Voutsinas SG (2008) Dynamic stall modelling on airfoils based on strong viscous – inviscid interaction coupling. Int J Numer Methods Fluids 56:185–208 Rosenhead L (1966) Laminar boundary layers. Dover, New York Sagmo KF, Bartl J, Sætran L (2016) Numerical simulations of the NREL S826 airfoil. J Phys Conf Ser 753:082036 Schlichting H, Gersten K (2000) Boundary layer theory, 8th edn. Springer, Berlin, Heidelberg Seubers JH (2014) Path-consisten schemes for interacting boundary layers. Master’s thesis, Delft University of Technology, Delft Smith PD (1972) An integral prediction method for three-dimensional compressible turbulent boundary layers. Tech. Rep. RAE R&M No. 3739 Smith A, Gamberoni N (1956) Transition, pressure gradient and stability theory. Tech. rep., Douglas Aircraft Company, El Segundo Division Snel H, Houwink R, Bosschers J (1993a) Sectional prediction of lift coefficients on rotating wind turbine blades in stall. Tech. Rep. ECN-C–93-052, ECN Snel H, Houwink R, Bosschers J, Piers W, van Bussel GJW, Bruining A (1993b) Sectional prediction of 3-D effects for stalled flow on rotating blades and comparison with measurements. Tech. Rep. ECN-RX–93-028, ECN Stratford BS (1954) Flow in the laminar boundary layer near separation. ARC R&M Report 3002, HMSO, London

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Swafford T (1983a) Three-dimensional, time-dependent, compressible, turbulent, integral boundary layers equations in general curvilinear coordinates and their numerical solution. PhD thesis, Mississippi State University Swafford TW (1983b) Analytical approximation of two-dimensional separated turbulent boundary layer velocity profiles. AIAA J 26:923–926 Tani I (1954) On the approximate solution of the laminar boundary-layer equations. J Aeronaut Sci 21:487–495 Tetervin N (1947) Boundary-layer momentum equations for three-dimensional flow. Tech. rep. Thomas J (1984) Integral boundary-layer models for turbulent separated flows. In: AIAA 14th Fluid and Plasma Dynamics Conference, Snwomass Thwaites B (1949) Approximate calculation of the laminar boundary layer. Aeronaut Q (1): 245–280 Timmer WA (1998) Ontwerp en windtunneltest van profiel DU 97-W-300. Tech. rep., TU Delft, in Dutch Vaithiyanathasamy R, Özdemir H, Bedon G, van Garrel A (2018) A double wake model for interacting boundary layer methods. Florida, USA, AIAA SciTech Forum 2018 Wind Energy Symposium van Dommelen L, Shen S (1980) The spontaneous generation of the singularity in a separating laminar boundary layer. J Comput Phys 38(2):125–140 van Garrel A (2016) Multilevel panel method for wind turbine rotor flow simulations. Ph.D., University of Twente van Ingen JL (1965) Theoretical and experimental investigations of incompressible laminar boundary layers with and without suction. Tech. rep. van Ingen JL (2008) A new en method for transition prediction. historical review of work at TU delft. AIAA J 3830:2008 van Rooij RPJOM (1996) Modification of the boundary layer in XFOIL for improved stall prediction. Report IW-96087R, Delft University of Technology, Delft, The Netherlands van den Boogard E (2010) High-order discontinuos Galerkin method for unstready integral boundary layer equation. Master’s thesis, Delft University of Technology Veldman AEP (1981) New, quasi-simultaneous method to calculate interacting boundary layers. AIAA J 19(1):79–85. https://doi.org/10.2514/3.7748 Veldman AEP (2008) Boundary layers in fluids. Lecture notes in applied mathematics, University of Groningen Veldman AEP (2009) A simple interaction law for viscous–inviscid interaction. J Eng Math 65(4):367–383. https://doi.org/10.1007/s10665-009-9320-0 Vezza M, Galbraith RAM (1985) An inviscid model of unsteady aerofoil flow with fixed upper surface separation. Int J Numer Methods Fluids 5(6):577–592 von Kármán Th (1946) On laminar and turbulent friction. Tech. Rep. TM-1092, NACA von Kármán Th, Milikkan BC (1934) On the theory of laminar boundary layers involving separation. Tech. Rep. 504, National Advisory Committee Aeronautics, Washington Voutsinas SG, Riziotis VA (1999) A viscous–inviscid interaction model for dynamic stall simulations on airfoils. In: 37th Aerospace Sciences Meeting and Exhibit, pp 154–163 White FM (1991) Viscous fluid flow, 2nd edn. McGraw-Hill, USA Whitfield DL (1979) Analytical description of the complete two-dimensional turbulent boundary layer velocity profile. AIAA J 17:1145–1147 Wieghardt K, Tillmann W (1953) On the Turbulent Friction Layer for Rising Pressure. Technical memorandum 1314, National Advisory Committee for Aeronautics, translation of ZWB Untersuchungen und Mtteilungen, Nr. 6617, 20 Nov 1944 Williams BR, Smith PD (1990) Coupling procedures for viscous-inviscid interaction for attached and separated flows on swept and tapered wings. In: Cebeci T (ed) Numerical and Physical Aspects of Aerodynamic Flows IV. Springer Berlin/Heidelberg, pp 53–70 Ye B (2015) The modeling of laminar-to-turbulent transition for unsteady integral boundary layer equations with high-order discontinous Galerkin method. Master’s thesis, Delft University of Technology

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Zanon A (2011) A vortex panel method for VAWT in dynamic stall. Ph.D. thesis, Università degli Studi di Udine Zhang Z, Liu F, Schuster D (2004) Calculations of unsteady flow and flutter by an Euler and integral boundary-layer method on cartesian grids. In: Proceedings of the 22nd Applied Aerodynamics Conference, AIAA. https://doi.org/10.2514/6.2004-5203 Zhang M, Drela S, Galbraith MC, Allmaras SR, Darmofal DL (2019) A strongly-coupled nonparametric integral boundary layer method for aerodynamic analysis with free transition. California, AIAA SciTech Forum 2019 Wind Energy Symposium

9

CFD Simulations for Airfoil Polars ˘ Uzol, and Ezgi Orbay-Akcengiz Nilay Sezer-Uzol, Oguz

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Near and Post-stall Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reynolds Number Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter on CFD simulations for airfoil polars, we focus on studies that are relevant to wind turbine airfoils, which have been investigated at reasonably high Reynolds numbers (i.e., Re>~1×106 ). We specifically focus on topics such as solution approaches, grid characteristics, effects of turbulence

N. Sezer-Uzol Department of Aerospace Engineering, Computational Aerodynamics Lab, METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey e-mail: [email protected] O. Uzol () Department of Aerospace Engineering, METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey e-mail: [email protected] E. Orbay-Akcengiz Department of Aerospace Engineering, Middle East Technical University (METU), METU Center for Wind Energy Research (RÜZGEM), Ankara, Turkey © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_12

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models, near/post-stall behavior predictions, transition modeling, and Reynolds number effects. We include a sample group of selected studies covering relevant airfoils in wind energy research. More research papers can be found using the references given in this paper as a starting point. The main objective is to provide some guidance to the reader regarding how to set up a good CFD simulation to obtain airfoil polars by giving relevant examples from the literature. Keywords

Wind Turbine Blade Aerodynamics · Airfoils · Airfoil Aerodynamics · Aerodynamic Polars · Computational Fluid Dynamics · CFD

Introduction In wind energy industry, a successfully optimized wind turbine blade design almost exclusively depends on accurate determination of sectional (airfoil) aerodynamic characteristics of the wind turbine blade. The main reason is that the aerodynamic design process of the turbine blade heavily relies on blade element momentum (BEM) theory-based fast design codes, which of course require accurate lift and drag polars as an input in a wide range of Reynolds numbers that correspond to the sectional operational conditions of a wind turbine blade. Though fully threedimensional computational fluid dynamics (CFD)-based integrated airfoil+blade design approaches are being investigated in recent years (e.g., Zhu and Shen 2013; Dhert et al. 2017), BEM-based aerodynamic/aeroelastic codes are still the workhorse of the industry. A typical nonlinearly twisted and tapered three-dimensional wind turbine blade shape is generally constructed by properly distributing a family of carefully selected airfoils along the blade span. Finding the optimum twist and chord distributions as well as the thickness-to-chord (t/c) ratio and design lift coefficient distributions is not a straightforward task in order to generate an optimized blade geometry. Therefore, determining the optimized airfoil distribution, i.e., the distributions of aerodynamic coefficients for a variety of different airfoil shapes and thicknesses, along the radius is a critical issue for blade design and performance. This requirement of course brings in the necessity of accurately obtaining airfoil lift and drag coefficients variations with angle of attack (i.e. airfoil aerodynamic polars) for those airfoils intended to be used in the blade design. Airfoil aerodynamic characteristics are generally obtained through a combination of numerical simulations and wind tunnel tests during both airfoil design and blade design processes. Numerical results are generally obtained either using fast tools based on integral boundary layer methods such as XFOIL (Drela 1989), RFOIL (Van Rooij 1996), or MSES (Drela and Giles 1987) or using two-dimensional CFD simulations based on Navier-Stokes equations. All these simulations are validated in general against wind tunnel data though in limited Reynolds number ranges.

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Because of the limited availability of wind tunnel data, obtaining accurate numerical results, therefore, is of vital importance. In addition to relatively faster panel method-based aerodynamic analysis tools like XFOIL and RFOIL, 2-D CFD simulations solving steady or unsteady Reynoldsaveraged Navier-Stokes (RANS) equations have also become sufficiently fast enough to be embedded into airfoil and blade aerodynamic design environment. Although their prediction capability is better in general compared to panel-based methods, there are still a number of challenges that need to be addressed to increase the accuracy of these CFD simulations in predicting airfoil aerodynamic polars. Table 1 presents a summary of airfoils used in numerical or experimental investigations relevant to wind turbine blades. These include various NACA airfoils used in early wind turbine blade designs to more recent modern airfoils used in various research and commercial turbines. These airfoils usually have relative thickness between 18% and 30% and an operational Reynolds number range of 1–15 M. Table 2 presents a list of various research reference wind turbines as well as the airfoils used for these turbines. As one might expect, the topic of airfoil CFD simulations is a very wide research area, and there are hundreds of publications that can be found in the literature. Here, we tried to focus mostly on recent studies that are relevant to wind turbine airfoils, which have been investigated at reasonably high Reynolds numbers (i.e., Re>~1x106 ). Also, we tried to include a sample group of selected studies covering airfoils that are listed in Tables 1 and 2. Table 3 presents a summary of these research studies. More research papers can be found using the literature given in Table 3 as a starting point. The objective of this chapter is to provide some guidance to the reader regarding how to set up a good CFD simulation to obtain airfoil polars by giving relevant examples from the literature. We specifically focus on topics such as solution approaches, grid characteristics, effects of turbulence models, near/poststall behavior predictions, transition modeling, and Reynolds number effects in the following sections.

Solution Approaches 2-D and 3-D CFD simulations for airfoils have been performed for steady-state or unsteady, compressible or incompressible flows by using finite volume approach with different turbulence modeling methods as can be seen in Table 3. RANS and URANS simulations are mostly done for 2-D, whereas DES, DDES, LES, and DNS simulations are done for 3-D, thus making them computationally expensive. Most of the investigations in literature are based on solutions of steady-state RANS equations. This approach can in general predict variations of aerodynamic coefficients in a satisfactory manner in a wide range of angles of attack (except around stall and post-stall), especially for relatively thin airfoils (i.e., ~15%> 1, which means the AOA over a revolution may be approximated with: α=

(1 − a) cos ψ λ

(4)

where ψ represents the azimuth angle. It is thus clear the experienced AOA may become very large, i.e. larger than the static stall angle, depending on the operational tip-speed ratio. This is in stark contrast with modern variable-pitch variable speed HAWT, which seldom experience AOA beyond the static stall angle during normal operation. As such, AOA fluctuations induced by atmospheric turbulence may produce DS events during VAWT normal operation. Perhaps even more important than atmospheric turbulence, the impingement of shed vorticity on VAWT blades (Fujisawa and Shibuya 2001) may induce DS. Due to the geometric configuration of these machines, the azimuthally dependent AOA results in a continuously varying bound circulation and thus permanent

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vortex shedding. The vortices released on the upstream half of the revolution (when 90 deg < ψ < 270 deg –  Chap. 41, “Vertical-Axis Wind Turbine Aerodynamics”) are convected with the wind speed and may thus ‘hit’ the VAWT blades on the downstream half of the revolution, resulting in ‘quick’ fluctuations in AOA at the blade sections (McLaren 2011). As shown in Eq. 4 the operational AOA is largest at 270 deg < ψ < 90 deg, i.e. on the downstream half of the revolution, and thus this vortex impingement often results in DS at the VAWT blade sections. Unlike HAWT, wind shear and yaw misalignment are not expected to induce DS on VAWT blade sections; this is because for VAWT wind shear results in AOA variations along the span, not over each revolution, and obviously since VAWT are never misaligned with respect to the wind direction. Ultimately, and similarly to HAWT, accurate prediction of DS events and associated loads on VAWT requires dedicated aero-servo-structural computations, though the aerodynamics of VAWT is generally harder to simulate (Ferreira 2009).

Experimental Data on Dynamic Stall The insight gained by the scientific community on the complex DS phenomena and its occurrence on WE machines has been achieved through decades of wind tunnel testing and data analysis. Experimental studies on dynamic stall date back to (at least) 1932 (Kramer 1932), with characterization of different physical quantities including balance measurements, surface pressure taps (Ramsay et al. 1995), smoke flow visualization (Galbraith et al. 1996) and flow field determination with particle image velocimetry (Ferreira et al. 2009). Numerous experimental studies have been conducted to analyse the dynamic stall phenomenon in a two-dimensional configuration. This is both due to the relatively simple experimental apparatus required and because 2D performance may be (cautiously) used to (Guntur 2013) extrapolate towards actual 3D flow behaviour. Initially considered in the airplane wing context (Kramer 1932), dynamic stall in helicopter operation has received a lot of attention (Carr and Chandrasekhara 1996; Leishman and Beddoes 1986, 1989; Tran and Petot 1981) as it may occur during forward helicopter flight, i.e. in normal operation. From the early 1970s onwards, 2D experiments included a range of different tests on airfoils in pitching, heaving and ramp-up conditions, and a combination of surface pressure taps and flow visualization provided insight on the topology of the DS phenomenon, namely, on the formation, detachment and convection of LE vorticity. This knowledge was used to interpret load balance measurements and ultimately inspires many of the DS models proposed (Leishman and Beddoes 1989; Tran and Petot 1981; Gangwani 1984). Within the context of WE, dedicated dynamic stall experimental research started somewhat later, around the early 1990s. A few experimental studies tested the performance of specific WE airfoils under dynamic conditions, (Ramsay et al. 1995; Janiszewska et al. 1996; Fuglsang et al. 1998; Galbraith et al. 1992), which contributed to existing unsteady airfoil performance databases

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(National Renewable Energy Laboratories; Gobbi 2010). The motivation for testing WE airfoils was that their geometry is somewhat different (thicker sections) from those used in helicopter blades, and as such the exact DS behaviour may differ too. Again, the remark is made here that the particular experimental conditions may reflect the specific quantities sought by a modelling philosophy and research group. Several experimental efforts studying DS phenomena on rotating blades have been conducted (Boorsma et al. 2011; Hibbs 1986; Schreck et al. 2001). Such studies are significantly more challenging than two-dimensional configurations but provide much more insight on real HAWT performance, particularly on the actual occurrence and extent of DS when several effects are combined. Usually in HAWT wind tunnel tests DS is enforced by imposing a rather large yaw misalignment angle. This leads to large amplitude AOA variations at high reduced frequencies (Table 1) for the blade inboard sections, but obviously the DS events taking place under these conditions are also heavily influenced by the rotational augmentation (Guntur 2013). In order to gain insight on the overall flow topology, HAWT experiments aiming at studying DS may include pressure taps at different spanwise stations of the blade and also flow visualization. A few studies have addressed DS on VAWT machines specifically (Fujisawa and Shibuya 2001; Paraschivoiu et al. 1988; Ferreira et al. 2009) and have, namely, investigated the influence of blade tip geometry, aspect ratio and tip-speed ratio on the DS characteristics of vertical axis machines (Fig. 3).

Dynamic Stall Modelling From the previous sections, it is clear DS encompasses a complex set of interlinked phenomena, all the more in rotating, wind energy machines where aerodynamic effects, such as rotational augmentation (Guntur 2013) and tip vortex interactions (Paraschivoiu et al. 1988), come together with blade structural vibrations. In short, modelling DS effects in WE machines is not straightforward at all. As hinted throughout the previous sections, aerodynamicists, particularly people involved in helicopter and later WE machine’s design and analysis, have been working on the topic(s) for a few decades (Gangwani 1984; Leishman and Beddoes 1989; Sheng et al. 2008). Different types of DS modelling have been used, ranging from simple, purely empirical approaches, i.e. curve fitting of experimental data, to complex computational fluid dynamics (CFD) simulations. Though generally speaking more complex methods will capture more of the DS phenomena physics and thus yield more accurate predictions, it is noted proper CFD simulations can only be obtained using the Navier-Stokes equations with a suitable turbulence model, given the nature of the interacting phenomena, and even then it is not clear whether URANS simulations (Guilmineau and Queutey 1999) always outperform simpler methods compared to experimental data.

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Fig. 3 Normal force coefficient variation with azimuth angle for HAWTs in yawed flow conditions – simulated from different dynamic stall models (Snel 1997; Tran and Petot 1981; Leishman and Beddoes 1989) and compared with New MEXICO measurements, for different spanwise locations and yaw misalignment angles. (Adapted from Khan 2018)

Beddoes-Leishman Type of Dynamic Stall Modelling In practice, the compromise between computational effort and required accuracy of DS modelling usually leads to the employment of the so-called semi-empirical methodologies. Such approaches do model the main physical phenomena associated with dynamic stall but rely on experimental data to tune the specific values of the modelling parameters. Additionally, the ease of implementation of semiempirical methods in aeroelastic WE codes (Jonkman et al. 2021; Rezaeiha et al. 2017b) simulating WE machines using BEM theory has greatly contributed to the dissemination of such DS models. Several semi-empirical models developed specifically with wind energy applications in mind have been proposed (Snel 1997; Øye 1990; Sheng et al. 2008), though models originally developed for helicopter rotors (Tran and Petot 1981; Leishman and Beddoes 1989) are also commonly used for estimation of loads on

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WE machines. Due to its very extensive employment (Gonzalez and Munduate 2007; Jonkman et al. 2021), dedicated subsequent model refinement studies (Mert 1999), good agreement with experimental data and relative simplicity, we will now focus on the Beddoes-Leishman (Leishman and Beddoes 1989) dynamic stall model. For further information and a more detailed overview of existing DS models, the reader is referred to Pereira (2010) and Khan (2018). The Beddoes-Leishman method is capable of assessing the unsteady lift, pitching moment and drag, and it is characterized by having a rather complete physical representation of the unsteady aerodynamic problem. It is composed of three subsystems, briefly explained here for the normal force coefficient (CN ): 1 – An attached flow module for the unsteady linear aerodynamic loads, which C ) (Theodorsen 1935) and impulsive (Fung 1993) considers both the circulatory(CN I (CN ) components. At instant n the circulatory component is obtained by computing an equivalent AOA (αEq,n ) and using indicial (or deficiency) functions (Xn ,Yn ). The impulsive component is typically significantly smaller for WE machine operation, and it is thus sometimes neglected (Barlas 2011) in wind turbine aerodynamic P ) modelling. The total, instantaneous attached flow normal force coefficient (CN,n is simply the sum of both circulatory and impulsive components. C C N,n = CNα αEq,n = CNα (αn − Xn − Yn ) Xn = Xn−1 e−b1 ΔS + A1 Δαn e−b1 C P N,n = C C N,n + C I N,n

ΔS 2

(5)

(6)

(7)

where b1 and A1 are constants fit to the linear range (attached flow) experimental airfoil performance and Yn is defined analogously to Xn . ΔS represents the distance (in semi-chords) travelled by the airfoil over two consecutive instants (n, n − 1). 2 – A separated flow module for the nonlinear aerodynamic loads, which uses Kirchhoff/Helmholtz (reviewed in Thwaites (1961)) static trailing edge separation chordwise position f (α) and the angle of attack history to calculate the instantaneous flow separation point (f ") and the associated (viscous) aerodynamic force f coefficient CN . C N = CNα

 1 + √f 2 2

(8)

where CNα is the slope of the (steady) normal coefficient experimental data. Since there is a lag in the leading edge pressure response with respect to the attached flow module, an intermediate normal force coefficient C  N,n is calculated with: C  N,n = CN,n − DP ,n

(9)

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from this intermediate normal force coefficient, a new effective AOA is calculated with: αf,n =

C  N,n − CN0 CNα

(10)

where CN0 is the (static) normal force coefficient at zero incidence. From this effective AOA αf,n , the instantaneous chordwise separation point is finally calculated by applying another deficiency function to account for the lag in the response of the boundary layer: f  n = f  n − Df,n

(11)

where deficiency function Df,n depends on the empirically derived (viscous) time constant Tf . Finally the effective chordwise separation point is used in Eq. 8 to compute the instantaneous (viscous) normal force coefficient C f N at instant n according to:

C

f

N,n

= CNα

 1 + f  2 2

αEq,n + C I N,n

(12)

3 – A dynamic stall module for the aerodynamic loads induced by the leading edge vortex, which simulates the LE concentrated vorticity build-up, detachment and convection across the airfoil’s upper surface and into the wake. The LE vortex detachment event is determined by comparison with a critical condition (typically based on equivalent LE pressure (Leishman and Beddoes 1989) or maximum normal V to force coefficient Pereira et al. 2012), whereas the actual vortex contribution CN V the instantaneous airfoil loads is estimated using the vortex travel time τ . As long as the vortex is situated over the airfoil, the normal force coefficient will increase, and this increment is estimated with: CV ,n = C

C

N,n (1 −

1+

 f  ) 4

(13)

The total vortex contribution at instant n is calculated with: C V N,n = C V N,n−1 e

−ΔS TV

+ (CV ,n − CV ,n−1 )

−ΔS 2TV

(14)

where TV is the empirical constant that regulates the vortex contribution’s decay. The total, instantaneous normal force coefficient CN,n is then finally computed with: CN,n = C f N,n + C V N,n

(15)

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Fig. 4 left – Flowchart of Beddoes-Leishman type of dynamic stall models and right – decomposition of Beddoes-Leishman model showing the contribution of each module component. (Adapted from Pereira et al. 2012)

From the expressions shown above, it is clear that the so-called time constants (TP ,Tf ,TV ) have a major role in the instantaneous aerodynamic forces. For a discussion on typical values for these constants, the reader is referred to Mert (1999) and Pereira et al. (2012). The equations displayed above are used to estimate the normal force coefficient. The instantaneous tangential force coefficient CC,n , i.e. along the chordwise direction, is estimated with:  CC,n = ηCNα α 2 Eq,n f  n

(16)

where η represents the so-called recovery factor, which is introduced because the airfoil does not realize 100% of the chordwise force attained in potential flow. This factor is also obtained empirically and has a typical value η ≈ 0.95. Figure 4 illustrates the different modules that comprise the Beddoes-Leishman type of modelling approach and main variables used in the computation of instantaneous loads, together with a graphical representation of the contribution of each module to the total loads experienced. Like any physical model, the Beddoes-Leishman method does have limitations; for example, typically in the implementation of this model, only one LE-originated vortex structure is considered simultaneously, and the contribution of the LE vortex is neglected when it is convected ‘far enough’ from the airfoil. However, when very high reduced frequencies are considered, multiple LE vortex structures may be present, which can lead to abrupt load variations, particularly ‘spikes’ in the lift coefficient as observed in the experimental results from Fig. 2 around the

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maximum AOA. To accurately capture such secondary vortex structures, specific delayed detached eddy simulation approaches may be used (Guntur et al. 2015). Within semi-empirical, engineering DS models, and as hinted before, more complex Beddoes-Leishman type of approaches have been proposed, which can yield better agreement with experimental data. Such methods (Sheng et al. 2008; Tran and Petot 1981) however typically require a very significant number of experimental parameter values (around 20 whereas the Beddoes-Leishman uses only 4) derived under 2D conditions, which are specific to each airfoil section geometry and unsteady forcing condition. This may become slightly impractical and, in this author’s opinion, somewhat defeats the purpose of employing engineering models. Additionally, and as mentioned before, DS in WE machines is never strictly a 2D aerodynamic event, and the coupling of different phenomena may have a larger influence (Guntur et al. 2015) than specific improvements of accuracy in 2D configurations bring.

Modelling Dynamic Stall in Wind Energy Machines Since DS models are typically derived and/or tuned to match 2D experimental (airfoil section) data, one must be careful when attempting to model actual WE machines’ observed behaviour. As hinted before, one of the main ‘culprits’ for DS occurrence on HAWT is yaw misalignment, particularly for inboard sections. At blade stations close to the root, however, there will also be ‘static’ stall delay. Several models for the so-called rotational augmentation have been proposed (Snel et al. 1993; Chaviaropoulos and Hansen 2000), which typically correct the airfoil section 2D polar based on the local chord-to-blade-radius ratio, in practice prolonging the linear portion of the lift curve to larger AOA. Such a correction however means the maximum (static) lift Cl and normal Cn force coefficients will be increased. Since typically the sectional rotationally augmented static polar is used as input to semi-empirical DS models (Jonkman et al. 2021; Guntur 2013), the instantaneous load estimation will be directly affected. The maximum sectional (static) Cn is associated with the magnitude of LE suction peak and extent of the separated flow region, and is often used (in some form) as the criterion (Pereira et al. 2012; Khan 2018) for LE vortex detachment in dynamic conditions, and thus essentially controls the contribution of LE shed vorticity to the total instantaneous load experienced. In addition to the maximum (static) lift or normal force coefficient, the fact that blades are rotating may affect the dynamics of trailing edge flow separation as well. Due to the radial pressure gradient, blade spanwise flow component and associated Coriolis force, the effective time necessary for the flow and specifically TE separation location (f ) to adapt to, e.g. variations in AOA is changed with respect to a 2D, non-rotating scenario. In terms of DS modelling, this might require an adaptation of the time constants (Mert 1999) controlling the TE separation

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point evolution. This aspect is particularly relevant for smart rotor ( Chap. 6 “Blade Design with Passive Flow Control Technologies”) applications which typically rely on TE flaps to control the flow and thus regulate (Barlas 2011; Bergami 2014) the blade-integrated aerodynamic loads. It must be noted that although rotational augmentation and dynamic stall are quite different physical phenomena which obviously have a complex interaction, typically simple model superposition is used. Nevertheless, reasonable agreement (Guntur et al. 2015; Pereira et al. 2012; Khan 2018) is usually obtained, at least in terms of the sectional load amplitude over a rotor revolution, as illustrated in Fig. 3. As for DS modelling on VAWTs, and since the topology of such machines does not easily lend itself to accurate BEM variant formulations, semi-empirical methods are perhaps employed more seldom than for horizontal machines. Instead, often CFD approaches (Rezaeiha et al. 2017a; Almohammadi et al. 2015) are used to estimate the instantaneous aerodynamic loading, which precludes the need for engineering DS methods. Nevertheless, specific studies have recently been published in which the Beddoes-Leishman model is adapted to the VAWT environment and coupled to a BEM formulation (Pirrung and Gaunaa 2018). Earlier studies also tested different semi-empirical DS models (Cardona 1984) in freevortex VAWT aerodynamics’ formulation, in which the relevance of including flow curvature (Migliore et al. 1980) to accurately capture the instantaneous loads in vertical axis machines was highlighted. As mentioned before, DS models usually rely on the Cl , Cd and Cm polars, obtained under static conditions. However, even under static conditions, there will be stall hysteresis (Timmer 2008), meaning that around the stall AOA the value of the (steady) lift coefficient will depend on whether the AOA is increasing or decreasing. For practical purposes, particularly when simulating the loads on WT blades, it is recommended to use the upper branch of the lift and drag polars as input to DS models, since otherwise the instantaneous aerodynamic loads will be underestimated. A final remark is made on the influence of the surface roughness and its impact on DS modelling. As mentioned before, real operating conditions are often emulated by imposing transition on the LE, and this ‘rough’ configuration will lead to a smaller maximum static Cl than ‘clean’ conditions. As DS models often use the maximum static lift coefficient to define the onset of LE vortex shedding, it is clear that having ‘rough’ instead of ‘clean’ conditions will have a significant impact on the load amplitude (Bousman 2000) occurring during DS phenomena. It is again noted however that the load dynamics, which is the delay in the unsteady loading with respect to the instantaneous AOA, seems to be relatively insensitive to LE transition.

Cross-References  Pragmatic Models: BEM with Engineering Add-Ons  Vertical-Axis Wind Turbine Aerodynamics

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References Almohammadi KM, Ingham DB, Ma L, Pourkashanian M (2015) Modeling dynamic stall of a straight blade vertical axis wind turbine. J Fluids Struct 57:144–158 Barlas T (2011) Active aerodynamic load control on wind turbines: aeroservoelastic modeling and wind tunnel experiments. Ph.D. Dissertation, TUDelft Bergami L (2014) Smart rotor modelling – aero-servo-elastic modeling of a smartRotor with adapptive trailing edge flaps. Springer Bertagnolio F, Sorensen N, Johansen J, Fuglsang P (2001) Wind turbine airfoil catalogue – R-1280. Risø, Technical Report Boorsma K, Schepers JG et al (2011) Final report of IEA task 29, Mexnext (phase 1), analysis of Mexico wind tunnel measurements. ECN-E-12-004, Energy Research Center of the Netherlands Bousman G (2000) Airfoil dynamic stall and rotorcraft maneuverability. NASA/TM-2000-209601 Cardona JL (1984) Flow curvature and dynamic stall simulated with an aerodynamic free-vortex model for vawt. Wind Eng 8(3):135–143 Carr LW, Chandrasekhara MS (1996) Compressibility effects on dynamic stall. Prog Aerosp Sci 32(6):523–573 Cermak JE, Horn JD (1968) Tower shadow effect. J Geophys Res 73:1869–1876 Chaviaropoulos P, Hansen M (2000) Investigating three-dimensional and rotational effects on wind turbine blades by means of a quasi-3D Navier-Stokes solver. J Fluids Eng 122:330–336 Connel JR (1982) The spectrum of wind speed fluctuations encountered by a rotating blade of a wind energy conversion system. J Solar Energy 29(5):363–375 Coton FN, McD Galbraith RA, Green RB (2001) The effect of wing planform shape on dynamic stall. Aeronaut J 105(1045):151–159 Daley DC, Jumper EJ (1984) Experimental investigation of dynamic stall for a pitching airfoil. J Aircr 21(10):831–832 Ferreira CS (2009) The near wake of the VAWT: 2D and 3D views of the VAWT aerodynamics. TUDelft Ph.D. Thesis Ferreira C, van Kuik G, van Bussel G, Scarano F (2009) Visualization by PIV of dynamic stall on a vertical axis wind turbine. Exp Fluids 46(1):97–108 Fuglsang P, Antoniou P, Bak C, Madsen H (1998) Wind tunnel test of the RISOE-1 airfoil. RisoeR999(EN) Fujisawa N, Shibuya S (2001) Observations of dynamic stall on darrieus wind turbine blades. J Wind Eng Indus Aerodyn 89:201–214 Fung YC (1993) An introduction to the theory of aeroelasticity. Dover Phoenix Editions, pp407 Galbraith RAM, Caton FN, Jiang D, Gilmour R (1992) Summary of pressure data for thirteen airfoils on the university of glasgow airfoil database. Glasgow University Report 9221 Galbraith RAM, Coton FN, Jiang D, Gilmour R (1996) The dynamic stalling characteristics of a rectangular wing with swept tips. In: Proceedings Conference 22nd European Rotorcraft Forum Gangwani S (1984) Synthesized airfoil data method for prediction of dynamic stall and unsteady airloads. Vertica 8:93–118 Gobbi G (2010) Analysis and reconstruction of dynamic-stall data from nominally twodimensional aerofoil tests in two different wind tunnels. Ph.D. Thesis – University of Glasgow Gonzalez A, Munduate X (2007) Unsteady modelling of the oscillating S809 aerofoil and NREL phase VI parked blade using the Beddoes-Leishman dynamic stall model. J Phys Conf Ser 75(1):012020 Greenblatt D, Ben-Harav A, Mueller-Vahl H (2014) Dynamic stall control on a vertical-axis wind turbine using plasma actuators. AIAA J 52(2):456–462 Guilmineau E, Queutey P (1999) Numerical study of dynamic stall on several airfoils sections. AIAA J 37:128–130 Guntur S (2013) A detailed study of the rotational augmentation and dynamic stall phenomena for wind turbines. DTU Ph.D. Dissertation Guntur S, Schreck S, Sørensen N, Bergami L (2015) Modeling dynamic stall on wind turbine blades under rotationally augmented flow fields. NREL Technical Report – TP-5000-63925

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Thick Sections

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Francesco Grasso

Contents Introduction and General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Tunnel Testing Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354 354 355 363 368 371 372

Abstract

A key aspect in blade design is the selection of the airfoils to be adopted. This means generally selecting or designing section families where the percentage thickness can vary between 12% of the chord at the tip and 100% of the chord at the very root, where the blade is connected to the hub. The present chapter provides an overview about the main aspects concerning thick sections and the major challenges around their development. It appears in fact on simulation side that the tools have some limitations in predicting thick airfoil performance, which introduces larger uncertainty on controlling their behavior. At the same time, a rich and reliable experimental database of wind tunnel tests is not available for the inner region airfoils. The wind tunnel testing itself of thick sections offers several challenges and makes more difficult to model this class of shapes.

F. Grasso () Aerodynamics and Acoustics, Vestas Blades Technology UK LTD, Newport, UK e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_16

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Finally, an overview of most popular features to improve the aerodynamics of these shapes is included in the chapter. Keywords

Root-end · Thick airfoils · Thick sections · Flat back · CFD · Modelling · Wind tunnel testing · Splitter plates · Base drag · Swallow tail · Wake rake · Cove · Shed vorticity · Inner region · Bluff body

Introduction and General Aspects The airfoils play a very important role in the blade design to achieve a sound rotor performance. The ones installed at the tip and middle part of the blade have major influence on Annual Energy Production (AEP) and noise, while the ones in the inboard region of the blade affect the structural response/integrity of the blade more than the aerodynamics. Because of this, they have the maximum percentage thickness (t/c) ranging from 30% to 100% (i.e., the cylindrical section at the connection with the hub). Several airfoil sets can be found in literature, which include thick sections (Tangler 1995; Althaus 1996; Björk 1990; Timmer et al. 2003) and recent European projects such as INNWIND (www.innwind.eu) that investigated this subject with a dedicated task (2.1.2, http://www.innwind.eu/ publications/deliverable-reports). For small and older turbines, it is easy to divide the blade in portions and conclude that thick sections should satisfy mainly structural requirements. However, as the modern blades are becoming larger and larger, the blade weight is one of the most relevant parameters to be controlled, as it has direct impact on the cost of the rotor. The selection of materials is probably the most obvious ingredient to obtain lightweight blade, but also, the design of the airfoils contributes to the final mass distribution of the blade. In fact, thicker sections have intrinsically higher sectional inertia moment and so less material is needed. Ideally, it would be desirable to extend the thick sections regions toward the middle portion of the blade, but this implies that the thick sections become an example of multidisciplinary design problem as they must be designed for aerodynamics as well. In the next section, different aspects of thick sections development are discussed so that the reader can have an overview of the subject complexity.

Development Challenges As mentioned in the introduction, the development of thick airfoils is challenging for different reasons. In this section, some of them are analyzed. The order in which they are presented is purely driven by the need to schematize things, as in reality all the aspects are interconnected and affecting each other.

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Design Challenges The thick airfoils play traditionally strong role on structural side and a secondary role on the aerodynamic side (Fig. 1). This introduces a first challenge for the designer since multidisciplinary scheme should be ensured over more traditional approach, where each discipline is separately considered. Although more complex, a multidisciplinary framework would allow to capture the best compromise between conflicting requirements. As it can be easily understood in fact, the need for structural efficiency will push the design toward very thick sections with wide internal area to increase the moment of inertia of the section. This would provide strength with no need of using large quantities of expensive material such as carbon fibers. Conversely, the aerodynamics would drive the process toward thinner shapes. In terms of aerodynamic requirements, the aerodynamic efficiency is the driving parameter for the sections installed at the outer and middle region of the blade. For thick sections, the lift itself is more relevant parameter to secure good rotor performance (e.g., Annual Energy Production – AEP). This is because maintaining optimal level of axial induction factor at the root is more effective way to obtain sound blade design than privileging high aerodynamic efficiency at all costs and axial induction is directly related to lift. From different perspective, the root airfoils work at high lift coefficient conditions (e.g., sometimes quite close to stall) because the local velocity is very low and so high lift coefficient is needed to compensate and obtain good amount of lift force; in those conditions, the drag is naturally quite high, so some compromise on this must be understood and accepted. As the drag plays secondary role, it might seem that the design of the airfoil is somehow easier as only lift is relevant to be optimized and traded against structural properties. This might seem even more true as other parameters such as the moment coefficient are not relevant. Drag cannot be neglected of course, but larger drag can be tolerated at the root region with no or little impact on rotor performance, while it becomes a critical parameter at the tip of the blade. What makes challenging the design process is the airfoil performance robustness against roughness and/or rough conditions. During its operative life, the blade accumulates dirtiness (e.g., sand, insects) or it is affected by rain erosion, which deteriorates the surface quality of the leading edge area (see Fig. 2). The airfoil lift reduces, stall phenomena are anticipated, and the drag increases. As effect, the rotor AEP gets significantly penalized. In case of thick airfoils, early stall is the major issue and the stall angle reduction can be quite large. Looking at the airfoil geometrical parameters, the camber line is the primary ingredient which drives the lift performance and stall behavior of the airfoils. Larger camber and/or more aft-centered camber lines shapes would produce more lift, but they are also more sensitive to early stall and abrupt stall response. This is simply because the suction side of the airfoil will be more pronounced and the slope of the contour in the rear part of the airfoil will be steeper to connect the maximum thickness point with traditionally sharp trailing edge and this favors flow separation and sharp stall. More friendly suction side in regard to the separation would mean shallower slopes, which mean the airfoil shape being more pronounced on the

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Fig. 1 Disciplines influence change along the blade. The number of dots indicates the level of importance of a specific discipline/performance for certain blade region

Fig. 2 Example of leading edge erosion (on the left side) and insect accumulation (on the right side)

Fig. 3 Example of thick airfoil with positive camber line (on the left side) and thick airfoil with partially negative camber line (on the right side)

pressure side and so the camber line to have negative camber around the airfoil maximum thickness location. This is illustrated in Figs. 3 and 4. A traditional way to compensate the lift reduction is to install vortex generators (VGs) on the suction side and/or on pressure side (see Fig. 5). Their effect is to generate vortices (from which the name derives), which energize the flow and delay undesired separations which might compromise the rotor performance (see Figs. 6 and 7). The adoption of VGs makes possible focusing the airfoil shape design on structural requirements, while the aerodynamics can be partially guaranteed by the VGs. Additionally, spoilers and gurney flaps (GFs) can be deployed to increase the lift produced by the airfoil and complement the

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Fig. 4 Comparison between thick airfoils with different slope on rear part of the suction side. Note: the arrows reflect the slope on the suction side of the contours

Fig. 5 Example of root vortex generators (RVGs) installed in the root region of wind turbine blade

Fig. 6 Effects of vortex generators on the airfoil lift performance. Note: the sketch is representative of VGs installed on suction side. For VGs installed on the pressure side, negative stall delay should be expected

VGs in “boosting” the airfoil behavior. These devices (see Figs. 8 and 9) generate additional circulation and so additional lift. Similarly, to VGs, the extensive use of GFs during the design phase could allow more efficient airfoil shapes for structural purposes with contained aerodynamic compromise. The drawback of VGs and GFs or spoilers is the fact that they are additional parts installed on the blade, which increase the blade cost and could break, which will compromise the durability of the performance and require periodical (additional) maintenance (with consequent

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Fig. 7 Effects of vortex generators on the airfoil drag performance. Note: the sketch is representative of VGs installed on suction side. For VGs installed on the pressure side, negative stall delay should be expected

Fig. 8 Basic operating principle for a gurney flap Fig. 9 Effect of the gurney flap on the airfoil lift performance. Note: this is just sketched to highlight the primary trend as impact on stall characteristics is not illustrated

additional costs, the so-called OPEX costs) to restore the nominal characteristics. In offshore environment, these costs could be quite relevant. Large maximum thickness coupled with relatively sharp trailing edge may introduce steep geometrical gradients which favorite flow separation and abrupt stall. For the same reason, the airfoil performance could degrade significantly in rough conditions. The 34.3% thick FX77-W-343 airfoil (Althaus 1996) is a good example of this scenario (Figs. 10 and 11). The so-called flat back airfoils offer a solution to improve the aerodynamic performance of the section, while gaining structural integrity. They can be obtained

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Fig. 10 FX77-W-343 airfoil contour. (Courtesy of Institute of Aerodynamics and Gas Dynamics, Stuttgart University, Germany)

Fig. 11 Lift characteristics of the FX77-W-343 as measured at Stuttgart LWK wind tunnel. 3 million Reynolds number, free transition. Note: the arrows indicate the change in response depending on angle of attack change direction. (Data from Althaus 1996. Courtesy of Institute of Aerodynamics and Gas Dynamics, Stuttgart University, Germany)

by cutting part of a conventional airfoil in the rear area to obtain a blunt trailing edge face or by gradually opening the two sides of the airfoil at the trailing edge for the same scope. The first system is sometimes preferred in manufacturing line as it is just about cutting part of the chord. For instance, the FX77-W-343, FX77W-400, and FX77-W-500 airfoils have been obtained in this way starting from the FX77-W-270 shape. The second system requires a partial modification of the design before the manufacturing phase, but it has the advantage to preserve the camber line and leading edge characteristics (i.e., the leading edge radius and the general proportions). Figures 12 and 13 illustrate some of the flat back shapes designed at UC Davis University by gradually opening the trailing edge equally on both sides of the section and their lift performance (from van Dam et al. 2008a; Baker et al. 2008). As it can be observed, these shapes make use of generous blunt trailing edge, which helps to achieve larger sectional area and so higher sectional moment of inertia. Less material for the skin of the blade would be needed because of this, which should lead to substantial weight and cost savings. At the same time, the large panel at the trailing edge introduces more gentle shape variations in the rear portion of

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Fig. 12 Example of flat back airfoils. The FB3500-XXXX series developed by Prof. C.P. van Dam (van Dam et al. 2008a). Note: “XXXX” defines the trailing edge thickness. (Courtesy of Prof. C.P. van Dam)

the component, and this would help the flow to follow the airfoil’s contour with no separation; even at the high angles of attack where typically the root area of the blade operates, an overall improvement can be observed in stall angle of attack and maximum lift. The reduced separation then makes these airfoils more robust also in rough conditions. The rough performance in terms of maximum lift coefficient would be still penalized compared to the free transition condition but significantly improved when compared to the conventional shapes (see Fig. 13). The flat back designs improve significantly the overall characteristics of the section, but they introduce at the same time a new challenge, which is the shed vorticity generated behind the flat trailing edge. As the flow reaches the trailing edge in fact, it tries to follow the contour of the corner, but the abrupt change in curvature leads to a sudden separation. Because of the flow trying to follow the shape, vortices are generated, which alternate between suction side and pressure side and have unsteady behavior with characteristic frequencies. These vortices are of course undesired as they increase drag and noise. Furthermore, they can generate vibrations (i.e., vortex-induced vibrations (VIV)) which can influence the operative life of the blade under fatigue-related loads.

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Fig. 13 Comparison between lift curves in free and fixed transition for different values of trailing edge thickness (van Dam et al. 2008a). (Courtesy of Prof. C.P. van Dam)

Several literature studies (Bearman 1965; Metzinger et al. 2012; van Dam et al. 2008a, e.g.,) and patents can be found, which focus on ways to reduce the intensity of the shed vortices. The most common is the splitter plate (see Fig. 14a), which basically consists of a panel forming 90 degrees with the flat panel of the trailing edge. Its function is basically to separate the flow coming from the suction side from the one on the pressure side. The plate introduces a guide for the flow, which limits the growth of the vortices (i.e., their intensity) and their interaction. Following the same principle, the double splitter plates (see Fig. 14b) are installed near the edges to further reduce the intensity and size of the vortices. In this way, the vortices are more localized and self-confined in small areas of the airfoil. Comparing these two solutions, one of the advantages of the double splitter plates idea is the reduced length of the plates compared to that one of the single plates in order to be effective in dividing the flows. It appears in fact that being the double plates expected to be closer to the edges of the trailing edge need smaller size to “capture” and contain the vortices before their growth. From literature (Winnemoller and van Dam 2007), the size of the splitter plate is roughly the same as the size of the flat back panel. Considering modern large wind turbines, the size can be around one meter of standing thin plate

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Fig. 14 Examples of solutions to reduce flat back shed vorticity. (a) single splitter plate, (b) double splitter plates, (c) cove

Fig. 15 “Swallow tail” concept. (From Grasso and Ceyhan 2015)

for several meters in span direction. Reinforcements will be needed to prevent the part to fail or break. Refinement in the plate geometry and their orientation lead to the creation of the cove configuration in the back of the airfoil (see Fig. 14c), which has more stable and thinner wake. This is because the flow will separate at the edges of the plates and naturally take most effective trajectory, trying to converge to the flow on the other side, for each actual angle of attack. Further evolution of the cove concept, by meaning of an asymmetric cove, forms the basis of the “swallow tail” concept (Grasso 2013; Grasso and Ceyhan 2015). In this case, the sharp corners at the trailing edge generate wakes which automatically adjust their trajectory based on the actual local angle of attack. For positive values of the angle of attack (i.e., where these sections are expected to operate), given the larger extension of the suction side corner, the vortex generated on the pressure side will tend to merge with the one on the suction side. This should reduce the overall wake size and the amount of shed vorticity. The data obtained during recent wind tunnel tests at TU Delft test facility (Ceyhan and Timmer 2018) confirmed the results of the numerical simulations (Grasso and Ceyhan 2015). These show (see Fig. 16) that for the same angle of attack, not only the overall size of the wake is reduced but also the core intensity and size of the vortices generated. All the concepts presented, target primarily the aerodynamics as they aim to reduce the base drag by limiting the shed vorticity generated. However, as secondary effect, they have impact also on the noise produced since the reduction in vortices intensity and drag means noise alleviation for the blades equipped with

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Fig. 16 CFD simulation for conventional flat back airfoil and swallow tail solution (Grasso and Ceyhan 2015) Fig. 17 Example of trailing edge serration for noise reduction

flat back sections. From literature and investigations (e.g., the European project SIROCCO, https://cordis.europa.eu/project/id/ENK5-CT-2002-00702), it is known that the outer region of the blade is the most influent contributor to the aerodynamic rotor noise since the noise is proportional to the fifth power of the local speed and that is larger at the tip. Based on this, there is generally no strong concern about the aerodynamic noise contribution coming from the root area of the blade. The evolution of wind turbines toward larger and larger sizes, however, pushes the designers toward lightweight solutions to get cost-effective solutions, which are competitive on the market. This could lead to designs where thick sections are moved outboard. This was already anticipated during the INNWIND project where the reference machine was equipped with 24% thick airfoils up to the very tip. As consequence, thick airfoils (either flat back or conventional ones) could be deployed in noise-sensitive regions of the blade. Suddenly, noise reduction ideas such as serrated trailing edges would be needed for thick airfoil too or even more advanced solutions like flow control to reduce the drag penalty Xu et al. (2018).

Modeling Challenges Normally, panel codes like the very popular Prof. Drela’s XFOIL Drela (1989) are widely used in wind energy to simulate the aerodynamic performance of the airfoils during the design stage. Their ease to use and high computational speed even with normal laptops make this class of tools very attractive, especially in

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combination with optimization algorithms, where many iterations are necessary before completing a design. The basic idea behind these tools is that the airfoil contour is discretized in (generally flat) panels (hence the name for the class of tools) connecting the coordinates (corner points) of the airfoil and the inviscid flow around the contour is simulated for each angle of attack by assuming a distribution of the so-called aerodynamic discontinuities (see Katz and Plotkin 2012). This solution is then refined by introducing the viscosity via a boundary layer scheme, which is coupled with the inviscid solver. Clearly, the coupling scheme plays a critical role in providing reliable simulations, but the viscous model in itself is key to predict the airfoil behavior, especially in terms of transition and separation. Inaccuracies in modeling the separation and/or the transition would compromise the overall accuracy, leading often to drag underestimation and/or maximum lift overprediction, together with stall shape which is smoother and gentler than in real life. European projects like AVATAR (https://www.eera-avatar.eu) have been very important to improve the tool accuracy, and some of these, like RFOIL (van Rooij 1996), produce results which are reasonably comparable to the wind tunnel results, both for thin and thick airfoils (up to 40% relative thickness). In this, the availability of reliable wind tunnel measurements helped a lot by offering large variety of test cases for tool benchmarking. Unfortunately, as it will be explained in the next section, the availability of wind tunnel tests for thick sections remains limited, especially at high Reynolds numbers, resulting in large uncertainties in numerical simulations. In general, the modeling quality via panel codes may be compromised for thick sections as these shapes can be more sensitive than the thin ones to early separation and quick change in the transition location, as the angle of attack changes. As said in the previous paragraph, this is driven by the contour’s slopes and large concavities/convexities and can generate local separation (i.e., separation bubbles) which the solver might find hard to detect or to model correctly. These local separations have an unsteady behavior, and little disturbance in the flow could cause the sudden extension of the separation to significant area of the shape. Furthermore, thick sections may be affected by other unsteady phenomena like alternate shed vorticity, as discussed in the previous paragraph in regard to flat back airfoils. Shapes with rounded trailing edge region such as the FX79-W-660 airfoil (Althaus 1996) can also be particularly sensitive to this phenomenon. Unless specifically modeled, XFOIL, RFOIL, and most of the panel codes available in literature are meant to capture steady aerodynamics, and certain assumptions might be in place to numerically stabilize the solution. One of these is assuming that the transition region collapses into a single point rather than small separation bubble (where there is local separation). As anticipated, this would result in overoptimistic predictions, which later will hardly find any support from wind tunnel test measurements, while compromising the real wind turbine blade performance. In case of flat back solutions, panel codes, which are assumed to work for trailing edge thickness less than 2.5% of chord per Drela, predict the correct trend in terms of lift improvement, and the solutions are generally more stable numerically. This is because the sharp corners at the trailing edge help to identify a well-defined and numerically stable separation point. Overall in terms of lift, the accuracy in

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Fig. 18 FX79-w-660 airfoil. (From Althaus 1996. Courtesy of Institute of Aerodynamics and Gas Dynamics, Stuttgart University, Germany)

Fig. 19 Example of computational grids. (a) C-grid, (b) O-grid, (c) H-grid

Opening

40c

Outlet Inlet

80c

Opening

y x

Fig. 20 Example of CH-grid topology with domain size and boundary conditions. (Taken from Grasso and Ceyhan 2015)

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Fig. 21 Example of CH mesh around flat back shape with and without “swallow tail” concept. (Taken from Grasso and Ceyhan 2015)

capturing the stall behavior depends on the specific implementation. In terms of drag instead, the limitations mentioned above still hold and a general underprediction should be expected. In case of flat back shapes, the inaccuracy level could be larger than conventional shapes because of the base drag contribution, which is generally not modeled or not completely modeled. This is in fact a general limitation of the panel codes. The system of equations to be solved, which link all the corner points, is completed by an additional (closure) equation, which implements the Kutta condition imposing the so-called “non-slip” condition (Katz and Plotkin 2012, see). In practice, this equation links the first and the last of the corner points by constraining the components of the local velocities. The general formulation of the Kutta condition is usually valid for sharp trailing edge shapes or with limited amount of bluntness as it starts to lose its realism for thick trailing edge panels, resulting in drag underprediction. In more practical terms, the base drag is not properly modeled, which is in this case relevant because of the flat back shape. The equation from Hoerner (1950) and Hoerner and Borst (1985) or the work from Grasso (2014) could be implemented to account for that. Despite the differences, all these models depend on the baseline drag and on geometrical parameters such as the airfoil thickness and the trailing edge thickness. Still about the trailing edge face, the airfoil corner points are assumed not to cover that face and so any details about that cannot be modeled via panel codes unless some specific modification of the contour is introduced to emulate the effect of those details. Some of them have been illustrated in the previous section, when gurney flap has been shortly presented or when splitter plates have been mentioned in regard to flat back airfoils. A local modification of the coordinates could be deployed as practical trick to introduce the gurney flap or the spoiler in panel codes. In case of splitter plates, such modification is more difficult since the 90 degrees between panels would create numerical issues, while smoother angles would risk modifying too much the actual configuration. Similarly, the cove or the swallow tail concepts would require the capability to assign sharp corners a concave region at the trailing edge. Tools like XFOIL or RFOIL have an embedded routine to spline the contour through the points, which would make impossible to capture the sharp corners. In

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Fig. 22 Example of O mesh around the splitter plate for flat back airfoil. (Taken from Winnemoller and van Dam 2007)

addition, it would be hard to conciliate the abovementioned Kutta condition with the cove-specific topology. As an alternative to panel codes, CFD-based tools should be adopted to obtain more accurate modeling of thick sections. In fact, CFD aims to resolve the full set of Navier-Stokes equations without any simplification or specific engineering model (like instead the panel codes). Because of this, CFD should be able to investigate and address any complex shape with full details. As it can be imagined, the potential improvement does not come for free; the model is more complex and results in an increase in computational time. It normally requires computational clusters rather than normal PCs. Also, CFD requires to preprocess the geometry to be analyzed by creating the so-called mesh around the shape and set up the characteristics of the domain to be investigated. In case of thick sections, the specific characteristics of this class of airfoils pose additional challenges. The abovementioned vorticity being an unsteady and time-dependent phenomenon suggests the usage of more advanced CFD approach such as URANS (Unsteady Reynolds Averaged Navier-Stokes); abrupt stall capture is often addressed by LES (large eddy simulation) approach. By all means, this increases furthermore the complexity of the problem and the computational time. In case of advanced features such as splitter plates to reduce the shed vorticity, the flow behavior can only be realistically modeled by using CFD. The traditional CFD methods with turbulence models also overpredict maximum lift with delayed stall. Menter introduced and implemented in ANSYS CFX the high lift model to address such overprediction with some success. Quite large literature can be found on CFD development and deployment to investigate thick airfoils. Besides Ansys CFX, other tools have been used such as Fluent, OVERFLOW (Nichols and Buning 2008), ARC2D (Pulliam 2005), or, more recently, OpenFOAM. As the mesh plays an important role in CFD analysis,

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many investigations have been focused on the grid generation for thick airfoils (Cooperman et al. 2010). The general conclusion suggests that O-grids are often better suited than C-grids for this class of problem although the specific software has some influence on the results and the coupling between mesh topology and solver should be considered. The addition of split plates or other solutions make the trailing edge region meshing even more complex as discussed in Grasso and Ceyhan (2015) or Metzinger et al. (2018). The figures below show examples of mesh refinement taken from literature.

Wind Tunnel Testing Challenges If modeling of thick airfoils by meaning of numerical simulations appears to be challenging, performing wind tunnel tests for this class of sections is neither an easy task. A chapter is dedicated to measurements and wind tunnel tests in this book; however, some words are spent here to highlight limitations and issues this activity could encounter when the airfoil thickness increases. As anticipated in the previous paragraph, despite progresses in computational aerodynamics, wind tunnel testing is still to be considered the standard way to verify numerical predictions and thus validating new design performance. In order to obtain meaningful validations, the measures should be executed in operating conditions which are representative of the real scenario. Ideally, they should perfectly match; however, given the size of modern wind turbines, this is hard to achieve for thick airfoils. The Reynolds number measures the relative importance of inertia forces compared to viscous forces, while the Mach number measures the disturbance propagation speed compared to the sound speed, which is the compressibility effect of the flow. Provided that these two parameters are well representative of the real scenario, it is possible to scale the experiment. This is of course attractive for large wind turbines, as it states the possibility to scale down the chord of the wind model, provided that the wind tunnel speed compensates to maintain the real Re number (or close enough to be realistic representation). However, the wind tunnel speed has clear limits for each wind tunnel facility and so there are practical limits to what can be achieved in the attempt. In fact, the scale-down of the model chord has clear (lower) boundaries if the target Re number must be achieved. For very large offshore wind turbines, this can reach 10 million or more, even for the thicker sections deployed toward the blade root, where the chord can be above five meters. The upper boundary to the wind tunnel model chord size is imposed by the solid blockage of the model in the tunnel. In case of two-dimensional tests (such as the ones for isolated airfoils), the socalled wall-to-wall models illustrated in Fig. 23 are used. There is an interaction between the model and the wind tunnel, which needs to be minimized in order to obtain reliable data representative of the section in free flow. The solid blockage is the ratio between the airfoil frontal area and the test section area (Barlow et al. 1999); when the blockage is greater than certain threshold level, significant effect on

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Fig. 23 Wall-to-wall wind tunnel model for 2D tests. (Courtesy of Institute of Aerodynamics and Gas Dynamics, Stuttgart University, Germany)

measurements could be expected and corrections should be applied to compensate. Thick airfoils are naturally more affected by solid blockage parameter; when the limit is hit, the most immediate solution is to reduce the model chord, which will reflect immediately in Reynolds number reduction from the abovementioned ranges. In alternative, one could think of modifying the flow characteristics in order to obtain desirable Reynolds number. The cryogenic wind tunnels (e.g., https://www. dnw.aero/wind-tunnels/kkk/) or the pressurized ones (e.g., https://www.dnw.aero/ wind-tunnels/hdg/) allow to obtain high Reynolds number in spite of the airfoil thickness as they “unlock” the flow properties (i.e., density and viscosity) which play a role in Reynolds number definition. The pressurized wind tunnels need indeed quite small wind tunnel model as the idea is to keep the flow running in the tunnel at high pressure levels. In general, cost saving to produce the model could be expected but instead is more expensive as the quality level (i.e., the surface roughness) needs to be finer to avoid that very small defects result in large penalty during the tests. The small scale makes more difficult to manufacture details on the models. Thick airfoils on wind turbines are often equipped with vortex generators (VGs) to ensure robust performance against roughness sensitivity and early stall phenomena. Normally such parts are attached on the wind tunnel model and tests are performed by changing size and/or location of these add-ons. In case of pressurized wind tunnels, the parts are so small that even the thickness of the part could introduce to large drag penalty. Embedded or integrated parts should be instead deployed, which on the other hand, compromise the flexibility of the tests. The example of VGs can be extended to any small parts to be installed on the model. Different is the case of cryogenic wind tunnels as the scale of the model is comparable with standard wind tunnels. In this case, the challenge is introduced by the intermittency of the tests as the tanks with cryogenic fluid should be refilled periodically during the tests.

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Fig. 24 Example of wake rake. (Courtesy of Institute of Aerodynamics and Gas Dynamics, Stuttgart University, Germany)

Besides the solid blockage, thick airfoils are also more affected by the wake blockage. Any airfoil produces a wake behind the trailing edge as consequence of the flow separating from the airfoil contour. Larger separated areas generate larger wake. As generally thick airfoils are more prone to early separation or sharp stall, large wake effect could be expected, which could compromise the measurement itself as explained later in the paragraph. Testing in rough condition for thick airfoils is particularly challenging as the severity of the separation can be even larger and so the impact on wake. Regarding the interaction between the wind tunnel and the airfoil model, the interference coming from the walls should be considered, especially in case of wallto-wall models. In those areas, there is interaction between wind tunnel and model boundary layers. As the angle of attack increases during the tests, the interaction will become stronger, and three-dimensional effects can generate out of the walls, which should be eliminated from the measurement data. Large airfoil thickness just increases the risk of having these effects contaminating the measurements so blowing system or jets are normally in place to reduce the wall effects. In practice, the flow rate of such devices is limited by the installed power, so it is good practice to discuss the test plan and test setup with the wind tunnel facility test engineers before the test campaign. In terms of measurement systems, force balances and/or pressure scans (either on the model or the tunnel walls) are normally used. The wake rake (see Fig. 24) is particularly suited for drag measurement in the linear region of the lift curve. As the separation becomes relevant (i.e., approaching stall points), the wake becomes larger and thus more difficult to be captured by the rake. Body pressure or force balance can be more appropriate for separated flows. In case of thick airfoils, the wake is naturally larger and so the wake rake can lose its function quickly even if larger wakes are deployed. Tests in rough conditions are even more challenging as the wake is larger; for those tests, the wake rake could be completely unusable and need to be removed to avoid that strong vibration due to separated flow might damage it. Rounded shapes toward the trailing edge would be more prone to such

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vibration as the separation point might not be steady located but instead moving back and forth. In addition to this, larger distance between wake rake and the model could be necessary for thick airfoils in the attempt to reduce the large flow fluctuations and obtain more reliable measurements. This can be limiting factor in adopting the device at all, and in fact, overall, body or wall pressure measurements are recommended to have good testing on thick airfoils. If at the least a couple of data points could be secured for conventional thick airfoils by the wake rake before the wake becomes too large for the wake rake, in case of flat back shapes, surface pressure measurements for the drag could be the only way to acquire drag as the blunt trailing edge and the base drag contribution could make the wake rake totally meaningless. In designing the wind tunnel model, enough pressure ports should be placed along the trailing edge face, based on the actual thickness. Overall, all the points presented above have an impact on the repeatability of the measurements on thick airfoils. In fact, for the same wind tunnel model, the same wind tunnel and full-operative instrumentation, large fluctuations could be found for the same tests. The drag would be particularly sensitive to such issue as it is affected by any disturbance in the flow or on the surface. In case of manifest unsteady phenomena (e.g., the moving separation point mentioned above for rounded shapes), this could be even more pronounced, resulting in large measurement deviations. Uncertainty bars should be perhaps included in postprocessing the data to capture this.

Conclusions The root region of the blades is a key element in providing structural efficiency to the whole rotor. Historically, not much has been attempted in using these airfoils to contribute to the rotor aerodynamic performance. However, in modern wind turbines every portion of the rotor needs to be optimized to contribute in achieving optimal performance. This poses challenges during the design of the thick airfoils, so that multidisciplinary optimization (MDO) is recommended to capture all the different aspects connected to this class of problems. Already the simple analysis and wind tunnel testing pose, however, challenges, which require specific tools and/or solutions. The present chapter offers an overview about these challenges and suggestions about most common solutions to overcome such challenges and obtain reliable data. Some of these can be introduced at design stage, such as flat back idea and the addon devices to reduce the base drag of these sections. Some others are relevant in running reliable simulations, while some others are specific for wind tunnel testing activities.

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References Althaus D (1996) Stuttgarter Profilkatalog II: Niedriggeschwindigkeitsprofile, Vieweg&Sohn Braunschweig, Stuttgart Baker JP, van Dam CP, Gilbert BL (2008) Flatback airfoil wind tunnel experiment, Sandia National Laboratories, SAND2008-2008 Barlow JB, Rae WH, Pope A (1999) Low-speed wind tunnel testing. Wiley-Interscience. ISBN-10: 0471557749, ISBN-13: 978-0471557746 Bearman PW (1965) Investigation of flow behind two-dimensional model with blunt trailing edge and fitted with splitter plates. J Fluid Mech 21(2):241–256 Björk A (1990) Coordinates and calculations for the FFA-W1-xxx, FFA-W2-xxx and FFA-W3-xxx series of airfoils for horizontal axis wind turbines. FFA TN 1990-15, Stockholm Ceyhan O, Timmer WA (2018) Experimental evaluation of a non-conventional flat back thick airfoil concept for large offshore wind turbines. In: 2018 Applied Aerodynamics Conference, AIAA AVIATION Forum, Atlanta. AIAA 2018-3827. https://doi.org/10.2514/6.2018-3827 Cooperman AM, McLennan AW, Chow R, Baker JP, van Dam CP Aerodynamic performance of thick blunt trailing edge airfoils. In: 28th AIAA Applied Aerodynamics Conference, 28 June–1 July 2010, Chicago. AIAA 2010-4228. https://doi.org/10.2514/6.2010-4228 Drela M (1989) XFOIL: an analysis and design system for low reynolds number airfoils. In: Conference on Low Reynolds Number Airfoil Aerodynamics, University of Notre Dame. https://www.eera-avatar.eu/ Grasso F (2013) Swallow tail airfoil, patent application. WO2014025252Al, CN104704233, US20180238298A1 Grasso F (2014) Modelling and effects of base drag on thick airfoils design. In: AIAA SciTech Forum, 32nd ASME Wind Energy Symposium, 13–17 Jan 2014, National Harbor Grasso F, Ceyhan O (2015) Non-conventional flat back thick airfoils for very large offshore wind turbines. In: 33rd Wind Energy Symposium, AIAA SciTech Forum, Kissimmee. AIAA 20150494. https://doi.org/10.2514/6.2015-0494, http://web.mit.edu/drela/Public/web/xfoil/ Hoerner SF (1950) Base drag and thick trailing edge. J Aeronaut Sci 17(10):622–628 Hoerner SF, Borst HV (1985) Fluid-dynamic lift, hoerner fluid dynamics. Bricktown, pp 2–10, 2–11 Katz J, Plotkin A (2012) Low speed aerodynamics. Cambridge University Press. ISBN:9780511810329. https://doi.org/10.1017/CBO9780511810329 Metzinger C, Baker J, Grobbel J, Van Dam CP (2012) Effects of splitter plate length on aerodynamic performance & vortex shedding on flatback airfoils. In: 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville. https://doi.org/10.2514/6.2012-235 Metzinger CN, Chow R, Baker JP, Cooperman AM, van Dam CP (2018) Experimental and computational investigation of blunt trailing-edge airfoils with splitter plates. AIAA J 56(8). https://doi.org/10.2514/1.J056098 Nichols RH, Buning PG (2008) User’s manual for OVERFLOW 2.1, version 2.1t. NASA Langley Research Center, Hampton Pulliam TH (2005) Solution methods in computational fluid dynamics. NASA Ames Research Center Stone C, Barone M, Lynch CE, Smith MJ (2010) A computational study of the aerodynamics and aeroacoustics of a flatback airfoil using hybrid RANS-LES, ASM 2010 Tangler JL, Somers DM (1995) NREL airfoil families for HAWT’s. In: Proceedings of WINDPOWER’95. Washington D.C., pp 117–123 Timmer WA, van Rooij RPJOM (2003) Summary of the Delft university wind turbine dedicated airfoils. AIAA-2003-0352 van Dam CP, Mayda EA, Chao DD, Berg DE (2008a) Computational design and analysis of flatback airfoil wind tunnel experiment, Sandia National Laboratories, SAND2008-1782

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van Dam CP, Kahn DL, Berg DE (2008b) Trailing edge modifications for flatback airfoils. Sandia National Laboratories, SAND2008-1781 van Rooij RPJOM (1996) Modification of the boundary layer calculation in RFOIL for improved airfoil stall prediction, Report IW-96087R TU-Delft Winnemoller T, van Dam CP (2007) Design and numerical optimization of thick airfoils including blunt trailing edge. AIAA J Aircr 44(1). https://doi.org/10.25114/1.23057 Xu HY, Dong QL, Qiao CL, Ye ZY (2018) Flow control over the blunt trailing edge of wind turbine airfoils using circulation control. MDPI, Energies. https://ocw.mit.edu/courses/ mechanical-engineering/2-29-numerical-fluid-mechanics-spring-2015/lecture-notes-andreferences/MIT2_29S15_Lecture22.pdf

The Effect of Add-Ons on Wind Turbine Blades

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Kristian Godsk

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Add-Ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gurney Flaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Reduction from Serrated Trailing Edge (STE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attachment of the STE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Reduction Effect of the STE on the Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Add-ons improve power production and reduce noise on wind turbine blades. Appling vortex generators and Gurney flaps to inner part of blade increases lift force and increases power production with 0.3–3% more AEP. Design of vortex generators depends mainly on fin height, fin angle, chordwise location, shape of fin, and spacing between fin pairs. Design of Gurney flaps depends mainly on height. A winglet at the tip of the blade reduces tip-loss and increases power production with 0.5–1% more AEP. Design of winglet depends mainly on height of winglet and if it is an upwind or downwind pointing winglet. Serrated trailing edge applied at the outer part of the blade reduces trailing edge turbulence boundary layer noise with 2–3 dB. Design of winglet depends mainly on relative length of STE of airfoil chord, aspect ratio of tooth length, and the STE flap inclination to airfoil chord line. K. Godsk () Vestas Wind System, Ry, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_17

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Keywords

Add-ons · Gurney flaps · Noise reduction · Production increase · Serrated trailing edge · Vortex generators · Winglet · Wind turbine

Introduction The use of add-ons like vortex generators (VG), Gurney flaps (GF), and less common variants like T-spoilers, chord extension (CE), winglet, stall fences, etc., has a long history in the wind turbine industry. In the 1980s and 1990s, the use of VGs was typically to improve poor performance of the blade due to either stall induced noise on the outer part of the blade (Boeing MOD-5B) or extensive stall in the root region. Today, the use of add-ons is more a design feature to improve the blade performance. Many sub-suppliers and wind turbine manufactures offer add-ons as an upgrade for power boost of the turbine. There is a large variation in the claims of the effect of the add-ons, but typically the AEP increase is in the range of 0.3–3% more AEP that has a huge impact on the business case for the turbine owner. Another add-on is serrated trailing edge (STE) for noise reduction, and the principle was described in the beginning of the 1990s and is now widely used on wind turbine blades. The figure below (LM Wind Power presentation Mathew et al. 2016) illustrates the benefits of the different add-on works:

Selection of Add-Ons The main add-ons used in the industry today are VGs, GFs, winglets, and STE.

Vortex Generators The thick root sections used in a wind turbine blade that has an early stall and the lack of induction because of max chord constrains can be improved by adding VGs in long rows along the root span. The effect of VGs is to reenergize the boundary layer and delay the flow separation so the maximum lift increases and drag from stall separation is reduced. The design of VGs is typically a triangular fin with an angle to the flow direction: Each fin pair creates a trail of counterrotating vortices downstream of the fins that reenergizes the boundary layer: The flow visualization done by LM Wind Power clearly shows the effect of VGs where the left side of the airfoil is without VGs and shows separated flow pattern

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and the right side is with VGs where the vortex trails and attached flow pattern are clearly shown: The design of VGs has mainly been experimental, but the use of computational fluid dynamic (CFD) is increasing as computer power has increased. One simple numerical approach is to simulate the VG as sources that pump energy into the boundary layer. Simulating the VGs using actual geometries on the blade increases the necessary cell counts substantially and complicates the meshing and time spent on the CFD simulation. Fundamentally, the standard SST turbulence model and all the derivatives in CFD simulation still overpredict the stall point for airfoils both without and with VGs so the actual benefits of VGs based on numerical results are doubtful. Another tool is the panel-based Xfoil with modifications to make VG simulation available. Although results are promising, it is not clear if the results are valid in general or only in specific cases. From experiment, the effect on the lift and drag for a DU97-W-300 is shown in the following (Timmer and Rooij 2003): The VGs also make the airfoil much more robust to surface roughness as seen in the below figure: The shape of VGs comes in many variants as illustrated in the figure below: The most important design parameters are the chordwise location along the airfoil and relative height of the VGs. The VGs should be placed just in front of the boundary layer separation point on the airfoil, and the effectiveness of the lift increase will be a factor of how much the VG is perturbating the boundary layer. This will also increase drag, and the height of the VG should be minimized. Typically, VG height is 0.5–1% of the local chord for the outboard section on a wind turbine blade, while the inboard VG is around 1 to 2% of the local chord dimension.

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The spacing of the VGs is another factor that influences the effectiveness of the VGs, and a ratio of z/h of 5 to 7 yields glide ratio (Cl/Cd) on par with a clean airfoil but with much larger max lift. Below figures (LM Wind Power presentation Madsen 2017) show the performance of VGs as function of chordwise position, VG height, and VG spacing. Another example of VG design parameter is vane angle, length, spacing, and shape that is from Baldacchino (2019). For a comprehensive explanation of the test configuration, please refer to the paper: The performance increase of using VGs is hard to generalize since it depends on actual blade design (the relative blade thickness) and the surface condition of the blade (a rough surface will improve more from VGs than a clean surface).

Gurney Flaps The effect of GFs is to increase the effective camber of the airfoil by adding a flat plate or wedges perpendicular to the pressure surface along the trailing edge of an airfoil. The effect of the Gurney flap is an increase in the maximum lift coefficient, a decrease in the angle of attack for zero lift while the slope of the lift curve remains relatively constant, and an increase in the nose-down pitching moment. The flap has a typical size of 1–5% of the local chord, which increases the lift and moderately increases the drag when its height is less than 2% of the chord. An increase in height beyond 2% of the chord yields an increase in lift that is obtained at the cost of substantially increased drag.

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Investigations from Liebeck and Jeffery show that the wake of the Gurney flap consists of a von Karman vortex sheet of alternately shed vortices. The vortex shedding increases the suction of the trailing edge on the suction side of the airfoil. On the pressure side of the airfoil, the Gurney flap decelerates the flow and thus increases the pressure (Fig. 1). From experiment the effect on the lift and drag for a DU93-W-210 (Timmer and Rooij 2003):

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Fig. 1 Illustration of flow near a Gurney flap as suggested by Liebeck Jang et al. (1998)

The pitching moment also increases with increased Gurney flap height. An example from wind tunnel test of a NACA 4412 where AoA is 0 degree is shown in the below figure (Jang et al. 1998): The performance increase of using GFs will typically be in the range of 0.2–0.5% AEP. Another add-on like the T-spoiler has the same effect as GFs.

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Winglet The winglet is a 90-degree (or close to) bend of the tip of the blade and will effectively reduce the tip vortex. A winglet decreases the induced drag from the blade by changing the downwash distribution but also adds profile drag from the winglet itself, so an optimal winglet design has a positive total effect on the power production. The prediction of winglet performance would require CFD analysis of the whole rotor. Some investigations have been carried out by researchers from Risoe/DTU. The parametric study is done by designing a normal blade and then bending the tip section by 90 degrees either to upwind or downwind direction and varying the length of the winglet in terms of length/rotor radius. The results are illustrated below (Fig. 2):

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The design of the winglet is most effective when bend to the downwind side (toward the tower) but will also reduce tower tip clearance and that is why many manufactures choose the upwind side.

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Fig. 2 Winglet study from Risoe/DTU (Gaunaa 2007)

The effect of a winglet is proportionally the same as an extension of the blade length for small winglet heights below 1% of the blade length, while larger winglet will gradually be less and less effective compared to a blade extension. Extracting the lift force on the tip section with and without a winglet, the lift curve will change as function of AoA. A typical example of how the winglet influences the tip section aerodynamic lift is illustrated below. The lift is reduced in the tip region from the 3D flow, and the winglet reduces the effect of the lift reduction and effectively increases lift force and power production. The manufacturing and transport of blades with winglets is more complicated and costly than blades without winglet, and the overall benefit of winglets is in many cases less attractive than a longer blade. The most important design parameter is the height of the winglet and to what side the bend of the winglet is placed. Other design parameters are illustrated in the below figure (Fig. 3).

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Fig. 3 STE design parameters

Noise Reduction from Serrated Trailing Edge (STE) The total noise from a wind turbine consists of various sources where the airfoil self-noise is generated by the blade movement through the airstream. Depending on

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the flow conditions, several self-noise mechanisms may occur as illustrated in the below figures. The most dominant self-noise under normal wind turbine operation is the turbulent boundary layer-trailing edge noise (TBL-TE). The most effective reduction of TBL-TE noise seems to be serrated trailing edge (STE). Airfoil trailing edge noise can be reduced by modifying the trailing

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edge geometry so that the efficiency by which vorticity is scattered into sound is reduced. Noise reduction occurs over a certain frequency range. However, beyond this frequency, a noise level increase is also observed. This noise increase thus undermines the overall noise performance of the trailing edge serration. Wind tunnel measurements have shown up to 10 dB reduction. But on full-scale wind turbine measurements, with the total sum of all noise sources presented, the reported noise reduction is in the order of 3 dB at some wind speeds while lower at other wind speeds.

Design Parameters The most important design parameters of the STE are the absolute length of the STE in terms of chord percentage of the airfoil and the aspect ratio of the width and

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length of the STE. Another important design parameter is the STE flap angle to the airfoil chord line (Fig. 4). Normally, the length of the STE is given as 2 h where h is half the length of the STE and λ is the distance between each sawtooth, and the aspect ratio of the STE is 2h/ λ. The STE flap angle ϕ is the angle between the airfoil chord line and the flap inclination to this line. According to the published studies, longer STE improved the reduction of noise, and the aspect ratio seems less significant as long it is larger than 1, and a good compromise would be aspect ratio around 2 for easy

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Fig. 4 Terminology of the STE design

Fig. 5 Streamlines around an airfoil that the flap angle needs to be aligned with

manufacturing and less fragile STE teeth. Since the STE will increase the forces on the blade and especially on the thin trailing edge and the strength of the baseplate of the STE has a limitation, then STE total length can probably not exceed 10–20% of the airfoil chord. Under operation, the STE flap angle must be aligned with the streamlines at the trailing edge to avoid any crossflow. The below figure shows a CFD calculation of the flow passing an airfoil, and the streamlines are shown, and at the trailing edge of the airfoil, the STE flap angle must follow the same angle (Fig. 5). This flow direction will change as operation condition in terms of AoA changes on the turbine so the alignment of the flap angle is normally chosen where the noise reduction is most critical but a compromise must be done if the flow direction

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Fig. 6 Example of lift and drag on an airfoil as function of angle of attack (AoA) with and without STE

changes a lot and the STE noise reduction disappears at other critical operation points. The flap angle must also be optimized to reduce the loads on the turbine blade and other components. The additional chord area from the STE and the flap angle can increase the lift force on the blade significantly and increase the drag as well. This can be reduced or eliminated by setting the flap angle accordingly (Fig. 6).

Attachment of the STE The STE is usually attached to the pressure side of the blade. Serration placement along the span, shown in Fig. 7, targets the regions where noise sources are dominant, which is generally the last 30% outboard region. Increasing the serrated span beyond 30% is not advisable since it increases the additional loads on the turbine while providing insignificant noise reduction. Each color indicates a serration panel of different lengths and flap angles, with longer serrations occupying inboard regions and shorter serrations at the outboard regions to match the optimal percentagewise STE length of the airfoil as close as possible.

Noise Reduction Effect of the STE on the Blade Attachment of the STE on, e.g., outer 30% of the blade reduces the noise level locally on that part of the blade. If it is measured in the wind tunnel that on the

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Fig. 7 Schematic of sample serration placement on the blade (LM Wind Power illustration Madsen 2017)

Fig. 8 Noise reduction of STE as function of blade radial position assuming 5 dB noise reduction from STE

given operation angle of attack (AoA) the STE reduces the noise level with 5 dB on an airfoil level, then the noise reduction as function of span is in ideal condition as illustrated in Fig. 8. The values shown are only for illustration purposes. The total noise reduction in this example is 4.6 dB. The inflow AoA will vary under turbine operation and for different wind speeds. Typically, the turbine operates with the highest AoA around rated power where the highest noise level is usually reached. The STE should be most effective at this condition, but the AoA does change for other wind speeds especially for high wind speeds where it goes to negative AoA on the outer part of the blade. If the noise level is restricted at high wind speeds, care must be taken to ensure that STEs also work sufficiently under those conditions. Further research in the design of STE is ongoing,

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Fig. 9 STE with additional comb-like features in between (Siemens Wind power illustration)

and both the shape of the sawtooth profile and additional serrations between the sawtooth profiles have been presented in the wind industry (Fig. 9). It is still not clear how to design an optimal STE and how much noise reduction that can be expected, and many investigations are inconclusive and often show the opposite trend as expected from previous tests.

References Baldacchino D (2019) Vortex generators for flow separation control wind turbine applications Gaunaa M (2007) Determination of maximum aerodynamic efficiency of wind turbine rotors with winglets Jang CS, Ross JC (1998) Cummings RM Numerical investigation of an airfoil with Gurney flap Madsen J (2017) Advances in aerodynamics of wind turbine blaDES. LM Wind Power Mathew J, Singh A, Madsen J, León CA (2016) Serration design methodology for wind turbine noise reduction. LM Wind Power Timmer WA, van Rooij RPJOM (2003) Summery of the delft university wind turbine dedicated airfoils

Pragmatic Models: BEM with Engineering Add-Ons

13

Gerard Schepers

Contents Definition and the Need for Engineering Models in Rotor Aerodynamics . . . . . . . . . . . . . . . Description of Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tangential Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainties and Assumptions in BEM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2D (Basic) Airfoil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of Annular Independency, Axi-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of Actuator Disc Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of Stationary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption of 2D Airfoil Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yawed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cone Angle, Tilt Angle, and Unconventional Blade Shapes . . . . . . . . . . . . . . . . . . . . . . . . Tower Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assessment of Engineering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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G. Schepers () TNO Energy Transition, Netherlands Organisation for Applied Sci, Petten, The Netherlands Hanze University of Applied Science, Zernikelaan, Groningen, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_19

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Abstract

This chapter discusses several aspects related to engineering methods in wind turbine design codes. Current engineering models for rotor aerodynamics topic are built around the Blade Element Momentum (BEM) theory. The Blade Element Momentum theory in itself is very basic, e.g., it is derived for two-dimensional, stationary, homogenous, and non-yawed conditions. For this reason, several engineering models have been developed which overcome these simplifications and which act as add-ons to the basic BEM theory. This chapter describes the BEM theory, the most important engineering add-ons, and an assessment of BEM with engineering add-ons with results from higher fidelity models and measurements. Keywords

Wind turbine aerodynamics · Blade element momentum method · Engineering methods · Wind turbine aerodynamic measurements

Definition and the Need for Engineering Models in Rotor Aerodynamics This chapter discusses several aspects which are related to engineering models in wind turbine rotor aerodynamics. It is largely copied from Schepers (2012) which in turn is largely based on analysis and validation with: 1. Detailed aerodynamic field measurements which have been taken in the 1990s. These measurements are analyzed in IEA Tasks 14 and 18 (Schepers et al. 1997, 2002a) 2. Wind tunnel measurements which have been taken in the year 2000 within NREL’s Phase VI experiment on a turbine with a diameter of 10 m placed in the large NASA-Ames wind tunnel. These measurements are analyzed in IEA Task 20 (Schreck 2008) 3. Wind tunnel measurements which are taken in the Mexico experiment on a turbine with a diameter of 4.5 m placed in the German-Dutch Wind Tunnel DNW. These measurements are analyzed in the first phase of IEA Task 29 (Schepers et al. 2011). In the present chapter, new results from among others the New Mexico experiment, i.e., “updated” Mexico experiments taken in 2014 within the German-Dutch Wind Tunnel, are included. These are analyzed in the third phase of IEA Task 29 (Boorsma et al. 2018). Also results from the EU project AVATAR (Schepers 2018) are included. Note that more information on the wind tunnel measurements can be found in Boorsma (2020). The term engineering method often indicates a simplified and generalized representation of a complex physical phenomenon (like wind turbine rotor aerodynamics)

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which is difficult to comprehend with a more complex aerodynamic model. At the same time, it indicates that the calculational time remains within acceptable limits for design calculations. In this respect, it is very important to realize the role of calculational time which, for wind energy calculations, is much more crucial than it is for most other areas of technology. This is in particular true for the calculation of a design load spectrum: A large number of 10 min time series have to be calculated and combined into an overall load spectrum in order to reflect the statistics of the wind over the entire 20 years’ lifetime of a wind turbine. This can make the number of time steps for such calculations some 7.2 million based on 120 time series (i.e., 6 realizations at 20 wind speeds) of 10 min and a time step of 0.01 s. Bearing in mind that every time step requires an aerodynamic calculation of the turbine and the flow field upstream and downstream of it, this puts severe constraints on the computational efficiency of the aerodynamic model. As such the performance of wind energy aerodynamic models is inextricably connected to their computational effort. This has made the so-called blade element momentum (BEM) theory the most popular model for rotor aerodynamics. Although this theory is a computational efficient model indeed, it is also a very simplified model which in principle is valid for stationary, 2D, and non-yawed conditions only. These simplifications are (partly) overcome by “engineering add-ons” which cover these deficiencies. Such engineering add-ons are often destillated from more advanced aerodynamic models and/or measurements with several tuning factors. They are still of a simplified character, and when added to the BEM theory, they do not significantly increase the calculational effort. It is these engineering add-ons which are the subject of the present chapter. The chapter starts with a description of the basic BEM model after which several engineering models are described, where possible validations, limitations, and the practical importance of the engineering add-ons are touched upon. The present chapter limits itself to engineering models for 3D rotor aerodynamics. The need for engineering models in wind farm aerodynamics is equally obvious, but these models will be discussed in Schmidt (2020). Moreover, we limit ourselves to conventional rotors without flow devices for which 2D steady airfoil polars at the appropriate conditions should be available either from measurements or from calculations; see Kodsk (2020).

Description of Blade Element Momentum Theory As explained in section “Definition and the Need for Engineering Models in Rotor Aerodynamics”, most of the present wind turbine design codes are based on the socalled blade element momentum (BEM) theory. The BEM theory can be considered as a combination of the blade element theory, which models the blade aerodynamics, and the momentum theory which models the induction aerodynamics. It was first described by Glauert (1935). Since then, it has been reported in many textbooks on wind turbine technology which is the reason why this chapter only gives a concise description of the basic BEM theory. In sections “Axial Momentum Theory”

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to “Axial Blade Element Momentum Theory”, the axial BEM equations are discussed where the energy extraction process is based on axial forces and velocities only. In section “Tangential Blade Element Momentum Theory”, it is explained how an equation is added for the forces and velocities in inplane (rotational) direction.

Axial Momentum Theory The axial momentum theory applies the conservation laws on a 1D streamtube (see Fig. 1) in axial direction. The rotor is modelled as an actuator disc, which can be seen as a hypothetical semi-transparent disc which exerts an axial force (Fax ) on the flow. The flow within the disc plane is assumed to be uniform which is the reason why an actuator disc is often described as a rotor with an infinite number of blade, since a finite number of blades would make the flow within the rotor plane nonuniform. Note that the concept of an actuator disc is described in more detail in van Kuik (2020). The positions 1, 2, 3, and 4 refer to the locations far upstream, just upstream, just downstream, and infinitely far downstream of the rotor plane. The velocity at position 1 (U1 ) is the free stream wind speed Vw . The locations 2 and 3 are both at an infinitely small distance upstream and downstream from the disc by which U2 = U3 = Ud with Ud the disc velocity. This disc velocity is written as the free stream wind speed minus the so-called axial induced velocity, where the axial induced velocity is the velocity reduction in the rotor plane due to the energy extraction of the actuator disc, denoted as u. Hence, Ud = Vw – ui . The axial force on the rotor implies a pressure jump (p2 – p3 ) over the disc. The main aim of the axial momentum theory can then be seen as finding a relation between the axial induced velocity and the axial force on the disc. Thereto, the following conservation laws are applied:

Fig. 1 Streamtube; also indicated are the axial velocities as used in the axial momentum theory

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• Conservation of mass (flux): m ˙ = ρU1 A1 = ρUd AR = ρU4 A4

(1)

• Conservation of axial momentum (flux): m ˙ = (U4 − U1 ) = Fax

(2)

Note that the net axial force from the pressure forces on the streamtube can be shown to be zero (see, e.g., Hansen 2008 and Snel and Schepers 1992) by which the total force on the streamtube in axial direction, i.e., the right-hand side of Equation 22, is formed by the axial force which the actuator disc exerts on the flow. (Fax ) • Conservation of energy (flux, i.e., power): 

1 2 1 2 U − U m ˙ 2 1 2 4

 = Pdisc = Fax Ud

(3)

This shows that the kinetic energy which is extracted from the streamtube equals the energy absorbed by the actuator disc (Pdisc ). By combining Equations 1, 2, and 3, it is found that: Ud =

1 (U1 + U4 ) 2

(4)

properties are expressed Now U1 is written as the wind speed Vw , and all relevant  in terms of the unknown axial induction factor a = ui Vw . Hence: Ud = Vw (1 − a)

(5)

and the mass flow in the streamtube (Equation 1) can be written as: m ˙ = ρVw (1 − a)Ar

(6)

U4 = Vw (1 − 2a)

(7)

and Equation 4 yields:

Eventually this results in the following relation between the axial force and axial induced induction factor: Fax = ρV2w 2a(1 − a)Ar which more conveniently is written in the following form:

(8)

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CDax = 4a(1 − a)

(9)

with CDax the axial force coefficient CD.ax =

Fax 1 2A ρV w r 2

(10)

The power which the actuator extracts from the flow in the streamtube is found from Equations 3, 4, and 8 and yields: P = ρV3w 2a(1 − a)2 Ar

(11)

The most convenient way of writing this equation is in the form of: CP = 4a(1 − a)2

(12)

in which CP is the power coefficient which is defined as: CP =

P 1 3 2 ρVw Ar

(13)

Many references explain the momentum theory with Bernoulli’s equation instead of Equation 3 (the conservation of energy). Such approach obviously results in the same outcome since Bernoulli’s equation is derived from the conservation of energy but it provides direct information on the pressures within the streamtube. The equations as described until now model the global flow field around the wind turbine rotor in terms of the induced velocities as a function of the axial force on the rotor (Equation 8 (or 9)). In wind turbine design codes, Equation 8 (or 9) is applied on concentric rings (annuli) with radial extent dr; see Fig. 2. Then the rotor area Ar is replaced by the area of the annular ring dAr with dAr = 2π rdr. It must be noted that by dividing the actuator disc in annuli, the pressures acting on the boundaries of these annuli should in principle be added to the momentum balance.

Fig. 2 Division of streamtube into annuli

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However, Sørensen and Mikkelsen (2001) finds on the basis of CFD calculations the contribution from these pressure forces to be small.

Blade Element Theory For the axial force from the momentum theory, Equation 8 is applied on annular ring and set equal to the axial force from the blade element theory as the sum of the forces in axial direction on all blade elements located in an annular ring. Thereto, the lift and drag forces on the blade elements are decomposed in axial direction, where the lift and drag forces are calculated from the airfoil coefficients, i.e., the lift (c1 ) and drag (cd ) coefficients. These airfoil coefficients are a characteristic of the airfoil shape, and they are a function of Mach number, Reynolds number, and angle of attack (i.e., the angle between the incoming velocity and the chord line). For wind turbine applications, the Mach number dependency can generally be neglected where the dependency on the Reynolds number is relatively weak by which many codes prescribe the airfoil characteristics in tabular form as a function of the angle of attack only. The tables of airfoil characteristics are usually known from wind tunnel measurements. Note that more information on the uncertainties on airfoil characteristics, including those at high Reynolds numbers and Mach numbers, are described in section “2D (Basic) Airfoil Data.” Then the lift force (per unit length) is given by: 1 L = c1 (α) ρV2eff c 2

(14)

1 D = cd (α) ρV2eff c 2

(15)

and the drag force is given by:

The angle of attack and effective velocity in these equations are found from the so-called velocity triangle (see Fig. 3), using the induced velocity or axial induction factor by which the axial velocity equals Vw (l – a). Note that Fig. 3 shows the inplane velocity to be r but section “Tangential Blade Element Momentum Theory” will explain that a tangential induced velocity should be added.

Fig. 3 Blade element with lift, drag, and velocity diagram

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Hence: V2eff = V2w (1 − a)2 + 2 r2

(16)

α =φ−

(17)

   φ = atan Vw (1 − a) r

(18)

where

with φ the inflow angle:

and  the twist angle (in the case of non-zero pitch angles, the pitch angle should also be subtracted from the inflow angle in Equation 17). Furthermore, Fig. 3 shows the inflow angle φ to be the angle between the axial and lift direction with which the lift and drag forces can be decomposed in axial direction: dFax = [Lcos(φ) + Dcos(φ)] dr

(19)

However, it is very important to note that, although the actual axial force on the annulus obviously contains a (small) component from the drag, this component is generally not included in the axial force balance from Equation 8. The main reasoning behind this approach lies in the idea that the drag force leads to a velocity change in the viscous wake behind the blade element which is not considered to be part of the induced velocity; hence, the axial force on the annulus is written as: dFax = Lcos(φ)dr

(20)

Axial Blade Element Momentum Theory In the previous sections, the axial momentum theory (Equation 8) and the axial blade element theory (a combination of Equations 20, 14, 16, 17, and 18) are described. Both theories form a relation between the axial induction factor and the axial force which are combined into the blade element momentum (BEM) theory: 1 2a(1 − a)ρV2w dAr = Bc ρV2eff c1 cos(φ)dr 2

(21)

4a(1 − a)V2w = σ V2eff c1 (α) cos(φ)

(22)

which reduces to:

 using dAr (the area of the ring) = 2π rdr and σ (the local solidity) = Bc (2π r)

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The effective velocity and inflow angle in the right-hand side of Equation 22 are found from Equations 16 and 18 and expressed in terms of the axial induction factor using the (known) tip speed ratio, which is defined as: R Vw

(23)

r r = λ Vw R

(24)

λ= and the local tip speed ratio, defined as: λr = This yields:

V2eff = V2w (λ2r + (1 − a)2 )

φ = arctan

1−a λr

(25)

(26)

Then the angle of attack is known from Equation 17, and the axial induction factor in Equation 22 can be solved iteratively. With the resulting lift and drag force on the blade element (Equations 14 and 15), the forces can be decomposed in each direction, and the overall forces and moments along the entire rotor blade are known by summing over the relevant elements.

Tangential Blade Element Momentum Theory Sections “Axial Momentum Theory” to “Axial Blade Element Momentum Theory” describe the BEM equations in axial direction. The main result is Equation 22. It is a relation between the induced velocity in axial direction (or axial induction factor) and the axial force on a blade element. In a similar way, a second momentum equation can be derived for the relation between the force on a blade element in inplane direction (dFinp1ane , where dFinplane r represents the contribution of the blade element to the torque dQ) and the induced velocity in this direction. This tangential (inplane) induced velocity is denoted as ut or ωr with ω the induced rotational speed. The system is closed with a second blade element relation for dFinplane using a velocity diagram to which the tangential induced velocity is added. Thereto, the positions 2, d, and 3 are defined. Position 2 is located just upstream of the rotor plane, d represents the disc value, and position 3 is just downstream of the rotor plane, respectively; see Fig. 4. The tangential induced velocity just downstream of the rotor plane (denoted as ut3 = ω3 r with ω3 the wake rotation) is calculated with the conservation of angular moment from the torque dQ: dQ = ρVw (1 − a)ω3 r2 dAr

(27)

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Fig. 4 Streamtube concept for BEM equation in inplane direction Fig. 5 Blade element with axial and inplane force

The torque (or inplane force) is found from the blade element theory by decomposing the lift force in inplane direction with the inflow angle φ (note that the drag forces are again not included in the calculation of the induced velocities). dFinplane =

dQ = L sin(φ) r

(28)

See Fig. 5. The determination of the lift (from the effective velocity and angle of attack) and the inflow angle is then almost similar to the procedure described in section “Axial Blade Element Momentum Theory”; the only difference is the addition of the tangential induced velocity in the velocity triangle. This velocity triangle however does require the tangential velocity induced in the rotor plane (i.e., at position d) where Equation 27 only includes the rotation at position 3, just downstream of the rotor. According to de Vries (1979), the tangential induced velocity in the rotor plane is half the tangential induced velocity at point 3; hence: ωd =

1 ω3 = a  2

(29)

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in which a’ is the tangential induction factor. As such, a’ relates the tangential induced velocity in the rotor plane to the rotational speed of the rotor. Then, by combining Equations 27, 28, 29, and 14, it is found that: 4a (1 − a)Vw r = σ V2eff c1 (α) sin(φ)

(30)

From the axial and tangential BEM equations, the axial and tangential induction factors are solved iteratively.

Uncertainties and Assumptions in BEM Theory As explained in section “Definition and the Need for Engineering Models in Rotor Aerodynamics”, a basic BEM model consists of Equations 22 and the auxiliary expressions 2.25, 2.26, and 2.17. Airfoil coefficients (c1 (α) and cd (α)) should be prescribed as input to the BEM model. In the derivation of the BEM equations, a large number of assumptions and simplifications have been made. Furthermore, it is important to realize that the airfoil coefficients are generally measured (or calculated) under 2D and steady conditions. This 2D, steady environment differs considerably from the situation on a wind turbine blade in the free atmosphere which is highly three-dimensional and unsteady. In order to overcome these deficiencies, several engineering methods have been developed which need to be added to the BEM model or to the airfoil data. In this chapter, the main assumptions and simplifications from the BEM theory are described together with the way how they are covered in nowadays design codes. Before discussing the add-ons in more detail, it should be realized that the uncertainties are treated as if they are independent but there are many interferences between them, e.g., there is an uncertainty on the variation of induced velocity at yaw which interferes with the uncertainty on dynamic stall at yaw. Still these uncertainties are discussed as if they are independent. In the next sections, the various assumptions are discussed.

2D (Basic) Airfoil Data It must be noted that the models which are applied as corrections to the 2D, steady airfoil characteristics obviously rely on the validity of these 2D, steady characteristics. However, these basic characteristics are often only measured for a limited angle of attack range around zero angle of attack and at Reynolds numbers which are (much) lower than the Reynolds number on modern wind turbine blades. Modelling of airfoil data at large angles of attack with CFD was found to be challenging; see Sørensen and Timmer (2017) in particular with RANS. Airfoil characteristics calculated with both steady and unsteady RANS show large discrepancies with measured airfoil data. On the other hand, DDES models showed encouraging results.

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A method to extend the basic airfoil data for the entire angle of attack range from −180 to +180 degrees is offered by the ATG program (Bot 2001), where flat plate data are used at high angles of attack. The use of flat plate data for actual airfoils at large angles of attack seems to have some validity as shown by Khan (2018) where flat plate airfoil data are compared with the airfoil data measured at five different radial positions from the New Mexico experiment at standstill conditions (Boorsma 2020). A very good agreement in lift over drag is found for angles of attack say larger than 25–30 degrees. Another problem with basic 2D airfoil data lies in the fact that they are generally measured in wind tunnels at Reynolds numbers which are generally not above 6M. Large wind turbines can have Reynolds numbers above 20M. It could then be argued that 2D airfoil data calculated with CFD at the actual Reynolds number could serve as input for the basic 2D airfoil data in BEM. In AVATAR (see Pires et al. 2016 and Ceyhan et al. 2017), several 2D airfoil prediction methods were validated with measurements in a pressurized wind tunnel in which Reynolds numbers could reach 15 million. It was found that standard CFD models which very often rely on correlation-based transition methods cannot capture the physics of the aerodynamics at those large Reynolds numbers. Surprisingly enough, the very old semi-empirical eN method (van Ingen 1956) performs well but requires calibration of a so-called N factor which is not a priori known.

Assumption of Incompressible Flow The assumption of incompressibility (i.e., constant density) is made both in the momentum theory and in the blade element theory. In principle, compressibility corrections on airfoil characteristics √ are known, e.g., the Prandtl-Glauert correction, using a compressibility factor 1 − M2 with M the Mach number. Another method is the more complicated correction from KarmanTsien: Cp = √

Cp,0  √  1 − M 2 + [M 2 (1 + 1 − M 2 ]Cp,0 2

(31)

in which CP is the corrected pressure coefficient and CP,0 the uncorrected value. Alternatively, the airfoil characteristics can be calculated with a compressible airfoil design code like RFOIL; see Montgomerie et al. (1997) and van Rooij (1996). With this code, the airfoil characteristics can be calculated for the actual free stream Mach number. However, compressibility corrections to airfoil characteristics are hardly applied because the tip speed (which is the maximum relevant velocity for wind turbine applications) is generally limited to around 80 m/s (mainly due to noise consideration), by which the free stream Mach number remains lower than 0.25. This Mach number makes the compressibility corrections small enough to be neglected.

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Current offshore wind turbines may have tip speed in the order of 100 m/s (in view of the fact that noise does not play a role), but even then, the compressibility corrections remain limited in relation to other uncertainties. Nevertheless, in the EU project AVATAR (Schepers 2018), calculations were done with compressible and incompressible CFD codes on a representative offshore reference wind turbine with a tip speed of 103.4 m/s. The (light) effect of compressibility could be distinguished when comparing the results from the compressible and incompressible codes. This enabled the recommendation of the Karman-Tsien compressibility corrections above the more common Prandtl-Glauert rule. For more information on this subject, see Jost et al. (2018).

Assumption of Inviscid Flow Viscous effects are accounted for in the blade element theory through the drag coefficient. As a matter of fact, drag losses are one of the reasons why the power production of a wind turbine will always be below the Betz limit. Viscous effects will also disturb the simplified streamtube concept from Fig. 1 and the accompanying text in section “Axial Momentum Theory.” This concept goes together with a constant low velocity in the streamtube and a higher velocity outside. The fact that the momentum theory is generally applied on annular ring level does allow some radial variation in induced velocity, but it does not prevent an (unrealistic) discontinuity in velocity (and pressure) at the edge of the streamtube. The resulting high shear at that position will lead to production of turbulence. Nevertheless, wind tunnel measurements from the project Mexico (Boorsma 2020) still confirm that the velocity in the near wake follows this streamtube concept well, at least in the near wake which is the determinant for the induced velocities. They show the velocity in the wake to be constant at a lower value than the free stream velocity, where the tip vortices induce a rather abrupt increase toward the free stream velocity at the edge of the wake more or less in agreement with the streamtube concept. Hence, although the inviscid assumption in the momentum theory is difficult to assess, it is not expected to be a significant source of deviations. As a matter of fact, viscous mixing in the wake is mainly believed to be a parabolic process, i.e., it influences the downstream flow, but its upstream effect on the induction in the rotor will be limited. As such, the impact of viscosity mainly lies on the field of wind farm aerodynamics and less on the field of rotor aerodynamics.

Assumption of Annular Independency, Axi-Symmetry A main assumption in the momentum theory is the division of the streamtube in independent annuli. For helicopter flows, Bramwell (1974) finds on the basis of calculations with a more detailed flow model that the induced velocity mainly depends on the local pressure jump in the annular ring which then confirms the

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annular independency. However, this is not expected to be true when small (and inevitably) yaw errors occur since this will lead to a radial flow component along the blade. Furthermore, in Sørensen and van Kuik (2011), the equations of motions are analyzed showing a large radial pressure gradient in particular at low rotational speeds and resulting in large tangential induced velocities. This radial pressure gradient obviously violates the assumption of annular independency. Moreover, the induced velocities within an annular ring are assumed to be azimuth angle independent, i.e., they are assumed to behave 2D (axi-symmetric), and they depend on streamwise and radial coordinate only. This assumption is violated by the finite number of blades which leads to a non-uniform flow between the blades. This is explained in more detail in section “Assumption of Actuator Disc Concept.” Another violation of the azimuthal independence assumption comes from noncoherent inflow, e.g., through wind shear or turbulence and yaw. The implementation of the induced velocity calculation in BEM for sheared and (non-coherent) turbulence is discussed in section “Shear and Turbulence.” Yaw leads to an azimuthal dependency of the induced velocity which is discussed in section “Yawed Flow.” A related problem which until now got little attention is addressed by Snel et al. (2008). There it is pointed out that, for three-bladed rotors, BEM methods should account for the effect of the inflow at a blade induced by the bound vortex of the other blades. For axi-symmetric flow, the net effect is zero, because the other blades have equal but opposite effects. However, for wind shear and yaw, the bound vortex depends on the azimuth; hence, there is a non-zero effect. In Snel et al. (2008), a clear non-negligible velocity is induced by the other two blades for a yaw angle of 30 degrees, and recommendations are given to include this effect in BEM methods.

Shear and Turbulence It should be realized that for practical situation with shear and non-coherent turbulence, the way how the induction factors are solved in the BEM equations has a huge impact on the results, even at relatively mild sheared and turbulent situations. This is explained through the axial BEM Equation 22 (note that similar observations can be made for the equation in tangential direction). This equation solves the axial induction factor. The axial induction factor in the momentum part (the left-hand side) is straightforwardly given, but it is hidden in the blade element part (the right-hand side), since the angle of attack α, the resultant velocity Veff , and the inflow angle φ depend on the axial induction factor through Equations 17, 25, and 26. Now consider the case of shear which gives different wind speeds throughout the annulus of Fig. 2 and so different wind speeds at the elements of the different blades. Then a question arises how to solve the momentum balance in this annulus. A first possibility would be to apply an annulus approach, i.e., to average the wind speed over the annulus. Then this averaged wind speed is applied in the left-hand side of Equation 22 to solve the averaged induction factor. Then the right-hand side of Equation 22 uses an angle of attack, inflow angle, and Veff based on the averaged

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Fig. 6 Possible sector approach

induction factor with the local wind speed at the blade (which in the case of shear is different from blade to blade; note that a slight complication appears by the tip loss factor which might make the local blade induction factor different from the averaged induction factor and which may even be different from blade to blade). Another solution method would be to apply an element approach, i.e., the axial induction factor is solved for all three-bladed elements separately using the local wind speed at the element. Alternatively this method could use the wind speed averaged over the sector around the blade element (Fig. 6). In this figure, the averaging sector is an azimuth interval 120 degrees, but smaller intervals could be considered too which makes it closer to the local approach. Hence, the element approach uses in the left-hand side of Equation 22 either the local wind speed or the sector-averaged wind speed where for both variants the local axial induction factor at the blade is solved. In the right-hand side of Equation 22, the angle of attack α, the inflow angle φ, and the resultant wind speed Vres use the local axial induction factor as solved for each blade separately with the local wind speed at the blade element. In Boorsma et al. (2016), aero-elastic calculations at turbulent inflow were carried out with a BEM code which used an element-local approach. An overprediction in fatigue loads of approximately 15% was found. The results from an annulus approach lead to an even larger overprediction in fatigue loads in the order of 20%. As stated above, the element approach may be applied in two different ways, and the question remained whether the wind speed Vw in the left-hand side should be the actual wind speed at the blade element or some sort of sector-averaged wind speed. In Fig. 7, a comparison is made between the induced velocities along the blade of the AVATAR reference wind turbine at four azimuthal positions at a heavy shear

Fig. 7 Calculation with ECN’s AeroModule for the AVATAR rotor under heavy shear: induced velocity calculated with element approach and local or sector wind compared with AWSM results for four azimuth angles

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with exponent 0.75 calculated with ECN’s AeroModule. AeroModule is a code with an easy switch between a free vortex wake method (AWSM) (van Garrel 2003) and a BEM method. This allows a straightforward comparison between different aerodynamic models to be made. In the investigated case, the BEM method applies an element approach with the local wind speed and the sector wind speed. Generally speaking, the results using the sector wind speed compare better to the ASWM results. Hence, most results call for an element approach to solve the induced velocities in the case of shear and turbulence. On the other hand, the application of a yaw model like the one from section “Yawed Flow” requires an annulus averaged approach. This implies a dilemma when modelling yawed flow at turbulent inflow, and so a generalized implementation model for the calculation of the induced velocity is needed. The definition of such approach requires further investigation. Hence, a very important observation lies in the fact that THE BEM model does not exist and it is not sufficient to describe a BEM model through its engineering add-ons alone, as is often done. Equally important is to describe the solution procedure for the induced velocities where an element approach is preferred for the prediction of fatigue loads under turbulent sheared inflow and an annulus approach for the prediction of yawed flow.

Assumption of Actuator Disc Concept One of the most important simplifications in the momentum theory is the representation of the rotor by an actuator disc. Such actuator disc is a hypothetical concept which to some extent can be seen as a rotor with an infinite number of blades since the flow in the rotor plane is assumed to be uniform. However, the fact that a real rotor has a finite number of blades makes the actual flow in the rotor plane non-uniform. This non-uniformity is generally covered with the Prandtl tip loss correction F (or modifications to it; see, e.g., Shen et al. 2005). In its basis, the Prandtl tip loss factor gives the ratio between the local axial induction factor at the blade (as applied in the blade element theory) and the azimuthally averaged axial induction (as applied in the momentum theory). The Prandtl tip loss factor takes the following form: F=

2 arccos(exp(F1 )) π

(32)

−B(R − r) 2rsin(φ)

(33)

with F1 =

There exist different implementations of the Prandtl tip loss factor into the BEM equations, but usually it changes Equation 22 into:

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4aF(1 − aF)V2w = σ V2eff c1 (α) cos(φ)

(34)

The left-hand side of Equation 34 (i.e., the momentum theory term) uses the annulus averaged induction factor aF. In the right-hand side (i.e., the blade element term), the angle of attack, the effective velocity, and the inflow angle are calculated with Equations 17, 25, and 26 and based on the local axial induction factor a. Prandtl derived the factor in the pre-computer era (1919). This necessitated the use of a very simplified vortex wake concept by which it was possible to derive Equation 32 analytically. In Fig. 8, the helical vortex wake structure behind a three-bladed rotor is sketched. Note that this figure is a simplification already. It neglects wake expansion and assumes constant circulation along the blades, with vortices which are trailed only from the tip (and root) of the blades. This model is further simplified by Prandtl into a system which consisted of vortex planes which move with a constant transport velocity Vw (1 – a), i.e., the velocity in the rotor plane based on the local axial induction factor. Flow “wipes” in and out from the free stream into the streamtube; see Fig. 9. This brings the actual wake velocity to (Vw (1 – aF)), i.e., a value between the “inner” wake velocity (Vw (l – a)) and the free stream velocity (Vw ). Hence, the annulus averaged induction factor is decreased, or, vice versa, the local axial induction factor at the blade is increased; see also Fig. 10. This then reduces the local inflow angle according to Equation 26 and hence the angle of attack and the resulting aerodynamic loads. As the name tip loss factor already indicates, the correction is stronger at the tip. This is consistent with Equations 32 and 33 which show the Prandtl tip loss factor to approach zero toward the tip. The correction is less strong for a shorter distance between the vortex planes (i.e., for a large number of blades and a high tip speed ratio). This is again consistent with Equations 32 and 33 which show F to approach 1 for an infinite number of blades, in agreement with the actuator disc concept. The same happens for a fast rotating

Fig. 8 Helical vortex wake structure behind a three-bladed rotor; wake expansion is ignored and constant circulation along the blades is assumed with vortices trailed only from the tip of the blades (From Burton et al. 2001)

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Fig. 9 Wake model used to derive the Prandtl tip correction, from Burton et al. (2001) (U denotes the free stream wind speed and d the distance between the vortex planes)

Fig. 10 Local and azimuthally averaged induction factor, from Burton et al. (2001)

turbine (i.e., λ → ∞ which implies φ → 0 according to Equation 26). The Prandtl factor was derived to model tip effects, but a similar effect occurs at the blade root. Thereto, the same loss factor is often used where the tip radius R in Equation 32 is replaced by the root radius. It should then be noted that such root radius is less well defined than the tip radius. The practical importance of the tip loss factor is assessed in Branlard (2011) where a CP calculated without tip loss factor is found to be some 15% higher than a CP calculated with tip loss factor. As explained above, the Prandtl tip loss factor inevitably had to rely on a very simplified wake model due to the fact that it was derived in 1919, i.e., in the pre-

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Fig. 11 The angle w between the helical vortex sheet and the axial direction, from Branlard (2011)

computer era. One hundred years later, within the EU project AVATAR (see Ramdin 2017), the Prandtl tip loss factor has been assessed with higher-fidelity models. It then has to be realized that F1 from Equation 33 is a generalized expression from Prandtl’s derivation in which many details on the precise implementation are hidden. Thereto, Ramdin (2017) writes F1 in a more basic form: F1 =

−π(R − r1 ) d

(35)

In (35), d is the distance between the vortex sheets, and r1 is the radial position where the correction is applied. The distance d is given in Fig. 11 with w which is the angle between the helical vortex sheets and the rotor plane. This gives:  d = h Bcos(w )

(36)

with h the axial distance between the helical sheets: h = 2π r2 sin(w )

(37)

(The variable h is written with a generalized radial position r2 since it is not necessarily considered at the tip location.) By assuming w to be equal to the inflow angle from Equation 18, the factor F can be written in the following form: F=

2 arccos(arg) π

(38)

with arg = exp(

√ −B(R − r1 (V2n + V2t ) ) 2r2 Vn

(39)

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In Equation 39, Vn and Vt are the axial and tangential components of the wake velocities which are assumed to be constant and equal to the values in the rotor plane, i.e., Vw (1 – a) ar(1 + a ), respectively; see Fig. 5. It can easily be seen that Equation 38 with 39 returns to Equation 32 with 33 when r2 and r1 are written in a generalized radial position r. Equation 38 with 39 makes it clear that a large number of variations are possible for the practical implementation of the tip loss factor in design codes. As a matter of fact, Ramdin (2017) counts 72 variants where knowledge is lacking on the bestperforming implementation. The freedom lies in the values of r1 for which usually (but not always!) the local value is taken but in particular for r2 which is sometimes taken as local value but also as tip value. For the axial induction factor in Vn , the local value can be used but also the tip value where moreover a difference lies in applying the azimuthally averaged value or the value at the blade section. In some codes, the induction factor is even ignored, or an average is taken between the wake value and free stream value. Finally different implementations are possible in Vt . In order to find the best choice for these factors, the ASWM code is applied which has a much more physical basis to calculate the induction than the simplified wake representation from Fig. 9. With this code, a tip loss factor is determined as the annulus average induction divided by the local induction factor. Calculations with the free vortex wake code AWSM have been performed for five different rotors, i.e., the small wind tunnel rotors from the Mexico (D = 4.5 m) and the NREL Phase VI (D =10 m) experiment and the 10 MW reference wind turbines as designed in the EU projects AVATAR and INNWIND.EU. Moreover, a rotor has been designed as a variant of the Mexico rotor with a constant loading along the blade which is in agreement with the basic concept applied by Prandtl. Figure 12 shows the effect of different implementations of the Prandtl tip loss factor for the constant loading variant of the Mexico rotor at a low tip speed ratio (λ = 4.9) and a high tip speed ratio (λ = 11.9) showing a large difference caused by the different implementations.

Fig. 12 Different implementations of Prandtl tip loss factor as a function of radial position compared with tip loss factor from AWSM for a constant loading variant of the Mexico variant. Left λ = 4.9 and right λ = 11.9)

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It is then found that for all cases the best results are obtained when the distance r2 and Vn (i.e., the axial induction factor) are evaluated locally, i.e., at the particular radial position of the blade element. Moreover, the axial induction factor should be applied on blade level instead of on annulus averaged level. The dependency on the precise implementation of Vt turns out to be very limited.

Turbulent Wake The conservation laws as discussed in section “Axial Momentum Theory” assume positive flow velocities in the streamtube of Fig. 1, or in other words, the flow direction in this figure should be from left to right. This is however not true anymore for a >0.5 which yields negative values for U4 according to Equation 7. This situation is called the turbulent wake state. The invalidity of the streamtube concept for large values of the axial induction factor can be illustrated by considering the situation at a = 1.0 which implies the velocity in the rotor plane (Ud ) to be 0, i.e., a fully blocked flow in the rotor plane. Nevertheless, this goes together with a zero axial force coefficient according to Equation 9. This leads to a controversy since a zero axial force coefficient implies no blockage at all! For this reason, the momentum relation 2.9 is generally replaced by an empirical turbulent wake relation between the axial force coefficient and the axial induction factor. A large number of these relations have been proposed, e.g., Wilson (1981) applies the following relation for a > 0.38: CD.ax = 0.96aF + 0.58

(40)

Another turbulent wake relation is given in Anderson et al. (1982): CD.ax = 1.93aF + 0.425

(41)

Also the correction from Glauert (1935) is well known: CD.ax = 4aF[(1 − (.025(5 − 3a)a)]

(42)

This equation is applied for a >0.33 The fact that there exists such a large variety of turbulent wake corrections is seen as an indication for a big uncertainty in these models. However, this uncertainty may be of less relevance since the practical importance of the turbulent wake state is limited. Thereto, it should be realized that high axial induction factors go together with high tip speed ratios which are uncommon for variable speed turbines (such turbines generally operate near the optimal tip speed ratio which implies that a ≈1/3). Constant speed turbines may operate at high tip speed ratios, but this then corresponds to very low wind speeds which contribute little to the energy production and load spectrum. As a matter of fact, the main reason for including a turbulent wake model in a BEM code is to guarantee its robustness since a standard BEM

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model without it will face convergence problems at high axial induction factors. Turbulent wake corrections may also be important near the tip of the blade, where the Prandtl tip loss effect, as described in section “Assumption of Actuator Disc Concept,” yields very high local induction factors. As such research on the turbulent wake state generally has not gotten much attention, but in IEA Task 29, PIV measurements in the Mexico experiment are taken at very high axial induction factors. A comparison is made with results from several CFD codes. In the fourth phase of IEA Task 29, DanAero field measurements (Madsen et al. 2010) are analyzed, and some of them are taken at relatively high axial induction factors too.

Assumption of Stationary Conditions Wind turbines operate at a very unsteady environment due to, e.g., turbulence, wind shear, deflections, control actions, etc. Nevertheless, the BEM theory is derived for stationary conditions. The assumption of steady flow is made in the blade element theory through the use of (calculated or measured) steady airfoil data and in the momentum theory. Unsteady phenomena are most conveniently explained in terms of a vorticity representation of the wake (and the blades). The wake vorticity exists of shed vorticity and trailing vorticity, both time dependent. Unsteady effects in the momentum theory depend mainly on the trailing vorticity (i.e., the vorticity related to the spanwise variation of the bound vortex) which is transported with a velocity in the order of the wind speed. The characteristic length scale for the bound vorticity is in the order of the rotor diameter. Unsteady effects on the airfoil aerodynamics depend on the shed vorticity (i.e., the vorticity related to the unsteady variation of the bound vortex). This vorticity is transported with a velocity in the order of Veff , and it has a characteristic length scale of the chord length. This makes the time scale of unsteady airfoil effects in the order of c/Veff (≈ c/r)) where the time scale of unsteady effects in the momentum theory is in the order of D/Vw . Hence, the time scale of unsteady effects in the momentum theory is much slower than the time scale of the unsteady airfoil aerodynamics, by which the phenomena can be considered as independent.

Unsteady Airfoil Aerodynamics Unsteady effects on the airfoil characteristics are especially important at high angles of attack where dynamic stall occurs, but even at attached conditions, there is an unsteady effect. • The unsteady effects at attached conditions can be modelled with the model from Theodorsen (1935). This model can be seen as an unsteady extension of thin airfoil theory which implies that it has been derived for inviscid conditions. It basically describes the effects associated with the acceleration of the flow around the airfoil and the angle of attack variations as induced by the shed vorticity.

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• The classical Theodorsen model is however of a 2D character where the shed vorticity is represented by infinite long vortex lines. For a wind turbine situation, the shed vorticity can more realistically be represented by vortex “spokes”; see Snel (2004). Lack of good validation material hampers the further development of such model as pointed out by Hur (2011). • Dynamic stall is a viscous phenomenon which is initially caused by an accumulation of vorticity near the leading edge of the airfoil as the angle of attack increases and the separation point approaches the leading edge. This causes an overshoot in lift followed by an abrupt decrease in the lift when the vortex is convected downstream of the airfoil. It altogether leads to a hysteresis loop on the c1 (α) curve with high c1 in the upstroke and a low c1 in the downstroke. It goes together with very large moment variations due to center of pressure movements. In the EU Joule project Dynamic Stall and Three-Dimensional Effects (see Björck 1995), an overview is given of engineering dynamic stall models. Dynamic stall effects are often expressed as a (time-dependent) correction to the steady-state cl (α) in the form of an ordinary differential equation in time: τ

dα d2 α dc1 + f(α)c1 = g(c1 , α, , ) dt dt dt2

(43)

Dynamic stall models of this form are generally developed and validated using steady two-dimensional c1 (α) coefficients as a basis. In section “Assumption of 2D Airfoil Aerodynamics”, it will be explained that the time-averaged airfoil coefficients on a wind turbine blade (in particular at stalled conditions) are exposed to strong rotational effects for which several correction methods are invented. Therefore, dynamic stall effects for wind turbine situations are often modelled according to Equation 43 but applied to rotationally corrected steady-state c1 (α) characteristics. Experimental validation of dynamic stall effects in a field environment suffers from the uncertainty in the stochastic inflow which complicates the interpretation of the measurements. This uncertainty (together with the uncertainty on angle of attack) makes field measurements less suitable for a direct validation of dynamic stall models. An example is given in Figs. 13 and 14. They show measurements on the RISØ test facility as used in IEA Task 14/18 (Madsen 1991) of the normal force coefficients at 68% span as a function of angle of attack (around a low and a high angle of attack). A very disorderly pattern can be observed at high angle of attack by which these measurements cannot be used for a direct validation of dynamic stall models. To some extent, the latter problem is overcome by selecting data according to the baseline criterion as applied by TU Delft and NREL (Bruining 1993) and (Shipley et al. 1995). This technique aims to select measurements which are taken at conditions which are as steady as possible. Thereto, data sets are selected in which the variations in wind speed and yaw error are limited over three subsequent rotations. The middle cycle data has been averaged for all the azimuth positions and

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Fig. 13 Normal force as a function of attack (low angle of attack) as measured by RISØ in IEA Task 14/18

Fig. 14 Normal force as a function of attack (high angle of attack) as measured by RISØ in IEA Task 14/18

is the final baseline result. The baseline data generally show less scatter than the data from the full base of measurements. It is obvious that wind tunnel measurements, e.g., the NREL Phase VI and Mexico experiments, do not suffer from these uncertainties in the inflow. In particular, wind tunnel measurements at yaw are more useful since they are

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exposed to a very well-defined excitation. Several examples are given in literature where wind tunnel measurements at yawed conditions are analyzed in terms of dynamic stall. Examples are Santos et al. (2011), Khan (2018), and Adema et al. (2019). A complicating factor when using yawed wind tunnel measurements for the interpretation of dynamic stall effects lies in the interference of dynamic stall with the (partly unknown) effects from yaw; see section “Yawed Flow.” The practical importance of unsteady airfoil aerodynamics lies on the impact it has on load variations (i.e., fatigue loads) and its impact on the aerodynamic damping. Aerodynamic damping is determined by the energy transfer between the blade and the airflow for a vibrating blade. During a flapping motion in stall, the use of steady airfoil characteristics may lead to negative aerodynamic damping, which is not observed in measurements; see, e.g., Björck (1995). These instabilities could then be prevented by dynamic stall which changes the phase of the aerodynamic forces during a flapping cycle. Possibly the most important impact of unsteady airfoil aerodynamics, which until now did not get very much attention, lies on the effect it has for storm loads of large wind turbine blades at standstill. Under these conditions, very large angles of attack may occur with or without a significant cross-flow along the blade. Current stateof-the-art dynamic stall models were not developed for such large angles of attack with cross-flow. As a result, within the AVATAR project, a worrying observation was made: Some dynamic stall models were found to give a stable solution for a large wind turbine blade at standstill, but others predicted very large vibrations where it is not known which set of models is correct! Since then, some progress has been made on modelling dynamic stall at standstill (see Khan 2018 and Adema et al. 2019), in which several dynamic stall models are updated to make them suitable for the prediction of storm loads but several uncertainties have not been solved yet. A different effect from the turbulent wind field on the airfoil aerodynamics may be through the transition point, i.e., the location where the transition from a laminar to a turbulent boundary layer takes place. Some commonly used airfoils have a large part of laminar flow along the airfoil, at least when they are measured in the low turbulence environment of a wind tunnel. The question can be asked whether the turbulent environment in which a wind turbine operates causes “by-pass transition,” i.e., a significant forward shift of the transition point on the suction side. This would yield a much higher drag and consequently a considerably lower power. Within IEA Task 29, Phase IV, a special activity is devoted to boundary layer transition, among others, using results from the DanAero experiment (Madsen et al. 2010) led by DTU and the German Aerodynamic glove experiment led by UAS Kiel (Schaffarczyk et al. 2017).

Dynamic Inflow The steady assumption in the momentum theory is apparent through the fact that the induced velocity follows the load situation (i.e., CD.ax ) instantaneously. This is often called the equilibrium wake assumption. However, when the loading situation changes (due to, e.g., a change in pitch angle, wind speed, or rotor speed), the induced velocity will lag behind, since an appreciable amount of air

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Fig. 15 Wake with “mixed” vorticity as a result of, e.g., a pitch angle step

must be accelerated or decelerated. In such cases, the wake behind the turbine and consequently the induction will achieve steady-state conditions after a certain delay. This phenomenon is commonly called “dynamic inflow” (alternatively, the names “dynamic wake” or “dynamic induction” are sometimes used). In Fig. 15, the dynamic inflow effect is explained by means of a (trailed) vorticity representation. The trailed vorticity is formed at the blade and convected downstream with the local total velocity, partly wake induced. Then a change in CD.ax (e.g., through a change in pitch angle) modifies the bound vorticity and hence the trailed vorticity. Due to the fact that the trailed vorticity is convected with a finite velocity, the resulting wake becomes a mixture of “old” and “new” vorticity. Consequently, the velocity induced by such wake includes a contribution from the “old” and the “new” situation. As soon as the “old” vorticity has travelled a distance of some 2 to 4 diameters behind the rotor, its influence is hardly felt anymore in the rotor plane, and the new equilibrium situation is reached. However, before the vorticity has travelled this distance, a gradual change of the induced velocity takes place from its old equilibrium value to its new equilibrium value. It is this gradual change in induced velocity which is the essential characteristic of the dynamic inflow phenomenon. It was studied extensively in the 1990s within two European “Dynamic Inflow” projects (see Snel and Schepers 1994 and Schepers and Snel 1995), where it was shown that the delay in induction is the cause of a temporary enlargement of the forces, in particular during fast pitching steps. In these EU projects, the effect of dynamic inflow was generally modelled by adding a first-order time derivative on the axial induced velocity to the momentum theory relation 2.9, i.e.:  τ dui dt + 4ui (1 − ui ) = Vw CD.ax

(44)

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G. Schepers

with τ a time constant which decreases toward the tip and increases with rotor diameter. Note that in an equilibrium situation this equation returns to Equation 9. A similar equation has been derived for the inplane component; see Snel and Schepers (1994). The assessment of dynamic inflow effects in the EU projects generally relied on measurements from integrated rotor loads (blade root bending moments, rotor shaft torque). These measurements hide the local variation of dynamic inflow effects along the blade. Later experiments, like the NREL Phase VI and Mexico experiment gave more detailed information since they measured the dynamic transient of loads at different radial positions which then provides information on the radial dependency of dynamic inflow. An example of such analysis is given in Schepers (2007b). They showed some discrepancies with the commonly assumed radial dependency of dynamic inflow in engineering methods in the sense that the assumed rapid decrease time constant toward the tip appears to a much smaller extent in the experiments. On the other hand, results from higher-fidelity vortex wake methods were generally found to be in good agreement with the experiments. It was then shown that two time constants are needed to describe the dynamic inflow process; see, e.g., Pirrung and Madsen (2018) and Wei (2018): The first is a short one which represents the time delay effects on the induced velocities from the near wake. This short time constant is a function of radial position. Next to that, there is a long time constant which reflects the time delay effects on the induced velocity from the far wake and which is assumed to be invariant from radial position. It must be noted that this two-time-constant approach was followed already by one of the participants in the abovementioned EU projects; see Øye (1986). Hence, although the dynamic inflow measurements from the NREL Phase VI and Mexico experiment have added value compared to measurements of integrated rotor loads, they remain indirect in the sense that they rely on the transients of loads where dynamic inflow is determined by velocities. Much more detailed information is provided by velocity measurements in the wake during dynamic inflow events; see, e.g., Wei (2018) and Berger and Kühn (2018). Alternatively information from high-fidelity methods can be used. In order to assess the practical importance of dynamic inflow, it must first be noted that it is driven by a change in axial force coefficient. This happens among others during a pitch angle transient. Therefore, dynamic inflow effects are often related to pitch angle variations only. However, dynamic inflow effects can be expected during a change in wind speed and/or rotor speed as well. Dynamic inflow effects during a rotor speed transient were observed in the New Mexico experiment indeed; see Boorsma et al. (2018). Dynamic inflow effects at wind speed transients are often believed to have little practical impact. This is based on the cases studied in the EU projects where it was found that, although the axial induction factor changes with wind speed, the induced velocity itself is hardly affected. This could be explained with a linearized BEM model; see Snel and Schepers (1994). Also measurements at wind speed changes carried out by the University of Delft in their Open Jet Facility showed negligible dynamic inflow effects. Still a recent study on dynamic inflow effects at wind speed

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variation showed that these effects are not fully negligible for modern wind turbines (van Deijl 2018). Perhaps more important than the impact of dynamic inflow effects on the loads is the impact it has on the aerodynamic damping characteristics with regard to the design of pitch control algorithms; see van Engelen and van der Hooft (2004). This is expected to be even more relevant for larger wind turbine since the time constant from Equation 44 depends on rotor size.

Assumption of 2D Airfoil Aerodynamics The airfoil characteristics as used in the blade element theory are generally based on 2D wind tunnel measurements. Blade rotation and three-dimensional geometrical effects (taper, twist, different airfoils) violate this two-dimensional assumption. The main effect from rotation on the airfoil coefficients is a so-called stall delay as first noted by Himmelskamp (1950) who carried out experiments on propellers. It was found that rotation postpones the separation of the boundary layer in particular at the inner part of the blade. Hence, the lift increases until a larger angle of attack than expected from 2D experiments. The practical importance of stall delay for wind turbine situations mainly lies in the fact that power and loads are higher as predicted with 2D airfoil data for conditions where the inner part of the blade is stalled. This may happen at gusts for pitch-controlled turbines and at high wind speeds for stall-controlled turbines. In Snel et al. (1993), the stall delay effect is explained by considering the rotating boundary layer equations: δu v δv δw + + + =0 δs r δr δz

(45)

u

δu δu 1 δp 1 δτs uv δu +v +w =− + + 2v − δs δr δz ρ δs ρ δz r

(46)

u

δv δv δv 1 δp 1 δτr (u − r2 ) +v +w =− + + δs δr dz ρ δr ρ δz r

(47)

The first equation gives the continuity equation (in which s, r, and z denote the coordinates in chordwise, radial, and “boundary layer direction,” respectively, and u, v, and w denote the corresponding velocities). The second equation denotes the momentum equation in chordwise direction, and the third equation denotes the momentum equation  in radial direction. The latter equation includes the radial pressure gradient (dp dr) and centrifugal force [(r)2 r] which act on the boundary layer giving it an outward radial velocity component v (on the suction side). For an attached boundary layer, an order of magnitude analysis showed the v component to be much smaller (order (c/r)) than the u component. However, in a separated

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boundary layer, the u component is much smaller by which the “residence time” of the air is sufficient to give the boundary layer particles a large radial velocity v, directed outward. Since at a more outboard station, the blade moves faster in chordwise direction (i.e., r is larger), the boundary layer particles that are radially transported from the inboard stations get a relative velocity in chord direction, toward the trailing edge. This relative acceleration is present in the chordwise momentum equation (Equation 46) as the Coriolis force 2 v which works in line with the pressure gradient dp/ds along the chord. As such, the Coriolis force works as a “favorable” pressure gradient, i.e., it reduces the adverse pressure gradient along the chord leading to a thinner boundary layer and an increased lift. An order of magnitude analysis showed the local solidity (c/r) to be the dominant parameter for this effect, i.e., the largest increase in lift is found at the root. A method to generate rotating c1 (α) curves from the 2D characteristics is introduced in Snel et al. (1993) using a factor fcl . This factor is the ratio of the actual increase in c1 (i.e., cl,3D − cl,2D ) and c1 with c1 the difference between the non-viscous cl,inviscid (with “inviscid” slope dcl /dα = 2π ) and the 2D value of cl2D ; see Fig. 16. Hence: fcl =

cl,3D − cl,2D cl,3D − cl,2D = cl,inviscid − cl,2D

cl

(48)

and cl,3D = cl,2D + fcl (cl,inviscid − cl,2D ) = cl,2D + fcl cl

Fig. 16 Correction to rotational effects on lift coefficient

(49)

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In Snel et al. (1993), the following value for fcl is proposed: fcl = 3(c/r)2

(50)

Equation 50 shows the strongest rotational correction at the inboard stations where the chord c is large and the radial position r is small. Since then, many alternative formulations for fcl have been proposed. As an example, Chaviaropoulos and Hansen (2000) found, by matching results of CFD calculations: fcl = 2.2(c/r) cos4 ( + θ )

(51)

The term  + θ can be understood by realizing that this is the angle between the rotor plane and the chord. Hence, the Coriolis term in Equation 46 should in principle be multiplied with cos( + θ ). Experimental evidence for the dependency of stall delay on  + θ is reported in Schepers (2012) based on measurements from ECN and NREL taken in the field and the wind tunnel. As an alternative to Equation 49, or in the absence of any 2D airfoil data, the rotating airfoil data can be calculated with the airfoil design code RFOIL (see Montgomerie et al. 1997, van Rooij 1996, Ramanujam and Ozdemir 2017, and Ramanujam et al. 2018). RFOIL calculates airfoil characteristics from the airfoil geometry and the local solidity c/r as input. It is a modification of the 2D airfoil design code XFOIL from Drela (1989). RFOIL is based on the 3D rotating boundary equations as presented above but written into an integral boundary layer formulation. An order of magnitude analysis is performed on the equations by which rotational effects are again accounted for through c/r. This makes it possible to apply the model in a quasi-2D way. The advantage of using RFOIL above Equation 49 lies in the fact that no measured airfoil characteristics need to be available. However, it also implies that any measured information is ignored. Therefore, instead of using Equation 49, RFOIL is sometimes used to determine the increment on cl , and then this increment is added to the measured 2D lift coefficient. It must be noted that Equation 48 only models the rotational effects on the lift coefficients. A rotational effect on the drag coefficient was proposed in the abovementioned reference from Chaviaropoulos and Hansen (2000). They found the drag to increase. This drag increase was modelled through a factor fcd and the difference between the 2D drag coefficient and the minimum drag coefficient:  cd,3D = cd,2D + fcd cd,2D − cd,2D,min

(52)

In the same reference, fcd was proposed to be similar to fcl ; see Equation 51; hence: fcd = 2.2(c/r) cos4 ( + θ )

(53)

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In Schepers (2012), several studies are summarized in which the expressions for fcl and fcd are validated and refined on the basis of detailed aerodynamic measurements from IEA Task 14/18 and IEA Task 20 (NASA-Ames). Apart from the 3D effects from blade rotation, there are 3D geometrical effects (due to taper and twist). These mainly play a role at the tip of the blade. This is shown in Schepers (2012) where several comparisons are summarized between calculated and measured tip loads from IEA Task 14/18 and IEA Task 20 (NASAAmes). A significant overprediction of the tip loads is found when using the Prandtl tip correction from section “Assumption of Actuator Disc Concept” only. Thereto, it should be emphasized that the Prandtl tip correction addresses the flow nonuniformities from the finite number of blades. As a result hereof, the local axial induction factor is increased (by which the angle of attack and the aerodynamic loads are decreased). However, it does not  affect the airfoil coefficients in itself which remain 2D (i.e., the lift slope dc1 dα ≈ 2π at attached flow conditions according to thin airfoil theory). As such an additional tip correction should be added to the airfoil coefficients. A suggestion for such tip correction is done by Shen et al. (2005).

Yawed Flow The expressions in the blade element theory as well as the expressions in the momentum theory have been derived under the assumption that the rotor plane is perpendicular to the wind direction. This is generally only true on a time-averaged basis. The inevitable wind direction fluctuations around the mean wind direction imply the wind turbine to be in a continuous yawed situation. An inventory of 5 years’ measurements in the ECN Wind Turbine Test Site EWTW (Boorsma 2013) showed the standard deviation of the turbine yaw errors (based on 10-min time series) to vary between 2 and 10 degrees, although much higher values were found in wake operation. Failure of the yaw system might also lead to high yaw errors. The practical importance of yaw partly lies in the fact that the power is expected to decrease with yaw which implies an economical loss. Equally important is the fact that yaw causes an azimuthal variation of the loads on a wind turbine. This affects the power quality but in particular the loads. It is found in, e.g., Schepers et al. (2002b) that yaw can even be design driving in terms of extreme and fatigue blade loads. The importance of yaw aerodynamics has nowadays even become more prominent by the idea that yaw can be applied as a wake reducing concept; see, e.g., Machielse (2011). Thereto, the upstream turbine in the farm is put under yawed conditions. This results in a deflection of the wake behind this upstream turbine. This could most easily be understood by assuming that the lateral velocity component in the skewed wake remains Vw sin φy where the axial component is decreased with the induced velocity; see, e.g., Snel and Schepers (1994). This leads to a so-called wake skew angle (χ ), i.e., the angle between the wake and the rotor axis which differs from the yaw angle:

13 Pragmatic Models: BEM with Engineering Add-Ons

tan χ =

Vw sin φy Vw cos φy − ui

425

(54)

The deflection can then be used as a way to control the wake such that the downstream turbine is exposed to less wake effects by which the overall wind farm production can be increased. The power loss due to yaw is often thought to be proportional to cos3 (φy ) (with φy the yaw angle). Such cubic dependency is expected from the idea that the power behaves as V3axial and the axial velocity component is Vw cos(φy ). However, this would only be true if the induction is unaffected by yaw. Measurements analyzed by Dahlberg and Montgomerie (2005) and Schepers (2001) proved this to be incorrect. This is further explained in Schreck and Schepers (2014) where an analysis of NREL Phase VI measurements shows that yaw can even lead to a power increase! The effect of yaw on the load (variations) was until the beginning of the 1990s only modelled through the advancing and retreating blade effect in combination with stationary lift and the drag coefficients. The advancing and retreating blade effect is explained in Fig. 17 (top). In this figure, the definitions of yaw angle and azimuth angle are also given. For positive yaw, the blade will be retreating in the upper half plane and advancing in the lower half plane with respect to the inplane wind component (Vtan ). This gives a 1P variation of angle of attack and effective inflow velocity, with φmax,α (i.e., the azimuth angle where the angle of attack is maximum) at the 12 o’clock position (φr = 180 degrees) and φmax,Veff (i.e., the azimuth angle where the effective inflow velocity is maximum) at the 6 o’clock position (φr = degrees). It is found (see, e.g., Schepers 2007a) that the effect of the effective velocity on the loading dominates the effect from the angle of attack by which the maximum loading, for positive yaw, occurs at the 6 o’clock position. The advancing and retreating blade effect is symmetric around zero azimuth. Therefore, it will, averaged over a rotor revolution, not lead to a restoring yaw moment as is measured in reality. This problem was covered in the EU JOULE projects “Dynamic Inflow”; see Snel and Schepers (1994) and Schepers and Snel (1995). In these projects, the azimuthal variation of the induced velocity at yawed conditions has been investigated. This variation is a result of the skewed wake geometry on the inflow distribution; see Fig. 17 (bottom): The proximity to the rotor plane of the vortices in the wake strongly influences the inflow. The trailing tip vorticity is on the average closer to the downwind side of the rotor plane, which according to the Biot-Savart law results in a larger value of the axial induction velocity ui . The higher induced velocity means a lower value of the total axial velocity for the downwind half of the rotor plane and (under the assumption of linear aerodynamics) lower blade loads in this part. The resulting load unbalance yields a restoring yaw moment, as illustrated in Fig. 18. Most participants modelled this unbalance in inflow with a sinusoidal variation of the induced velocity as proposed in helicopter theory by Glauert (1926).

426 Fig. 17 Advancing and retreating blade effect (top) and unbalance in inflow induced by the skewed wake (bottom)

Fig. 18 Load unbalance between upwind and downwind side of the rotor plane at yawed condition and sign of yawing moment defined such that a negative moment is restoring

G. Schepers

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   ui = ui,0 1 − Kc f(r R) sin(φr

427

(55)

The formula for Kc will depend on the shape of the wake. Several expressions have been developed in helicopter society; see Section 6 of Snel and Schepers (1994). They depend on the wake skew angle which means that they are all a function of the yaw angle according to Equation 54. For the radial dependency f(r/R), Glauert assumed a simple linear relation. All dynamic inflow engineering models were basically very similar to Equation 55, but there were differences in the modelling of the radial dependency (f(r/R)) and Kc , i.e., the dependency on the yaw angle. The Dynamic Inflow projects showed that the importance of the “skewed wake effect” mainly lies at relatively high tip speed ratios (i.e., low wind speeds), where the advancing and retreating blade effect becomes more important at lower tip speed ratios. Thereto, it should be realized that the skewed wake effect works on the induced velocities which generally speaking are larger at a high tip speed ratio. On the other hand, a high tip speed ratio limits the advancing and retreating blade effect since the inplane component of the wind speed (Vw sin(φy )) is relatively small compared to the rotational component. It is furthermore noted that a high tip speed ratio generally means a small angle of attack by which the dynamic stall effects are limited. At low tip speed ratios (high wind speeds), the opposite is true: The high wind speed leads to a large value of Vtan and consequently to a strong advancing and retreating blade effect, where the low induction factor makes the variation in induced velocity less visible in the load distribution. The large angles of attack lead to strong dynamic stall effects. The research in the Dynamic Inflow projects was followed by a Dutch National project in which the velocities in the rotor plane of a yawed rotor placed in the Open Jet Facility of TUDelft were investigated; see Schepers (1999). From these results and supporting calculations from more advanced models as described by Schepers (2012) and Voutsinas et al. (1993), it became clear that it is not only the tip vorticity which leads to an azimuthal variation of the induced velocity but also the root vorticity. The root vorticity then yields a destabilizing yawing moment at the inner part of the blade. At a later stage, this conclusion was confirmed by Madsen (1999) and Hansen et al. (2010) on the basis of CFD calculations. In Schepers (1999), TUDelft wind tunnel measurements were used to derive a model for the overall variation of the induced velocity around its mean value. It is expressed in the following way: ui = ui,0 [1 − A1 cos(φr − ψ1 ) − A2 cos(2φr − ψ2 )]

(56)

In this equation, ui,0 is the disc (or annulus) averaged induced velocity (see below), and the amplitudes A1 and A2 and the phases ψ1 and ψ2 have been modelled as a function of radial position and yaw angle; see Schepers (2012). Since then, several studies on yaw have been performed. Measurements in the TUDelft OJF have been analyzed in more detail in Sant (2007) and Haans (2011). Furthermore, NREL’s Phase VI (NASA-Ames) measurements are used to study the

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power dependency on yaw and the azimuthal variation of local aerodynamic loads. The same is done using the IEA Task 29(Mexnext) measurements. The IEA Task 29 (Mexnext) PIV measurements are also used to understand the flow field and the tip vortex trajectories. The EU project AVATAR even appointed yaw as one of the most complication aerodynamic model issues. This might partly be caused by the fact that the two tip vortex lines in the bottom Fig. 17 are thought to be parallel and both trailed under the wake skew angle from Equation 54. However, Equation 54 shows that the azimuthal variation in induced velocity goes together with a variable wake deflection. The large induced velocities in the downwind side of the rotor plane lead to a stronger wake deflection. The opposite happens at the upwind side where the induced velocity is small. An experimental confirmation of this variable wake deflection has come from measurements presented in Haans (2011) and also from measurements taken in the Mexico project; see, e.g., Schepers et al. (2010). As such the aerodynamics of a yawed rotor is complicated by the fact that the wake geometry is determined by the azimuthal variation in induced velocities where the wake geometry determines this azimuthal variation. Further complications appear from the azimuthal variations in the angle of attack and effective velocity which are a result of the azimuthal variation of induced velocity and the advancing and retreating blade effects. If these variations are fast enough, the boundary layer around the airfoil cannot follow them instantaneously. As a consequence, the steady relation between the c1 (cd ) and α does not hold anymore and unsteady airfoil effects appear as explained in section “Unsteady Airfoil Aerodynamics.” Still the second-order model with root vortex effects was often found to give significantly better prediction of the loads at yawed conditions than a conventional Glauert-based model. This is illustrated in Fig. 19 which compares the axial induced velocities at 30 degrees yaw for the reference wind turbine designed in the EU project INNWIND.EU at 30% and 95% span. Zero azimuth is defined at the 12 o’clock position, and yaw is defined such that the so-called downwind side of the rotor is between and 180 degrees azimuth. It can be noted that two BEM models are included: a BEM-Glauert model based on a sinusoidal azimuthal variation of induced velocities all along the blade such that the maximum induced velocity appears at the downwind side of the rotor plane. Moreover, results are included from Equation 56 with amplitudes and phases from Schepers (2012). This model is indicated with BEM-rv (root vortex). The results are compared with the more physical Actuator Line (AL) free vortex wake (FVW) model AWSM (van Garrel 2003) (which can be applied as a prescribed vortex wake model too). Although Fig. 19 shows that at 30% span both BEM models differ from the AWSM models, the qualitative agreement between the BEM-rv model and vortex wake models is good, i.e., the maximum velocity is induced at the upwind side and the minimum velocity is induced at the downwind side. The Glauert model fails to predict this trend, i.e., the maximum induced velocity is reached at the downwind side of the rotor plane as induced by a tip vortex wake only. On the other hand, it can be observed that the qualitative behavior of the induced velocity at 95% span from the BEM-rv model compares slightly poorer in particular because

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Fig. 19 Induced velocity as a function of azimuth angle at 95% span (left) and 30% span (right) for INNWIND.EU rotor, 6 m/s and 30 degrees yaw

the dip in induced velocity at an azimuth angle of 90 degrees as a result of the root vortex is still present. This was explained by the fact that the model from Schepers (2012) was tuned on experimental data from a small-sized turbine (two-bladed with the rotor diameter of 1.2 m) with a strong root vortex. The current wind turbine blades are far bigger and have a smoother transition at the root. Therefore, in Schepers et al. (2016), it was suggested to improve the model from Schepers (2012) by altering the radial dependency of the parameters such that the effect of the root vortex is damped out at 95% span. This was established in a cooperation between ForWind and ECN. The new yaw model was based not only on AWSM results but also on results from an Actuator Line (AL)-CFD model developed by NREL and used by ForWind (NWTC information portal). It is noted that the AL-CFD code is not a free vortex wake method but a CFD method in which the rotor is modelled with an Actuator Line approach. The induced velocities from both the AL-CFD and the AL-FVW approach were found to agree extremely well Rahimi et al. 2018) Based on this study, a new model was tuned. It is still based on a second-order Fourier model 3.26, but the coefficients have been tuned on the AL-CFD and ALFVW results for the 10 MW reference wind turbines designed in the EU projects AVATAR and INNWIND.EU and the 5 MW reference wind turbine from NREL. Figure 20 then shows the results for the induced velocity as a function of azimuth

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Fig. 20 INNWIND.EU turbine at 6 m/s and 20 degrees yaw: Axial induction factor as a function of azimuth angle at 30% span (left) and 80% span (right)

angle for the INNWIND.EU turbine. It can be seen that the new model agrees very well with the physical ALM. It is important to note however that the model from Rahimi et al. (2018) is based on a so-called annulus approach for the induced velocity as represented through the ui0 in Equation 56. In section “Shear and Turbulence”, it was shown that for calculations of turbulent and sheared conditions, another approach, a socalled element approach, is preferred. Even if a annulus approach is applied, a further uncertainty in the modelling of yawed conditions lies in the determination of this disc averaged induced velocity (ui,0 ). This value is generally calculated with the model from (Glauert 1926): ˜ w + u˜ i,0 |ui,0 Fax = ρAr |V

(57)

where Vw and ui0 have to be added vectorially and next normed. In Glauert (1926), the model is applied on disc level, but in wind turbine BEM codes, the model is generally applied on annular ring level. The model from Glauert (1926) is based on the fact that Equation 57 is the correct expression for a gyrocopter at fast-forward flight, i.e., a yaw angle of 90 degrees. The rotor disc is then seen as a circular wing on which the resultant force works as a lift; see, e.g., Bramwell (1974). Furthermore, Equation 8, which represents the situation for an actuator disc at aligned condition, is a special form of Equation 57 since it can be written as: Fax = ρAr (Vw − ui,0 )2ui,0

(58)

Hence, Equation 57 is valid for 90 degrees and 0 degrees yaw misalignment, and it is supposed to be true for in-between values. However, the flow situation for an actuator disc at aligned flow and the flow situation around the circular wing are substantially different. At aligned flow, the axial force acts as a drag force where it works as a lift force in the case of a helicopter at fast-forward flight.

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Nevertheless, Haans (2011) finds on the basis of TUDelft OJF measurements that Equation 57 yields a reasonable result for the induced velocity in yaw, also when it is applied on annular ring level.

Cone Angle, Tilt Angle, and Unconventional Blade Shapes The expressions in the blade element momentum theory have been derived for zero cone and tilt angle. The effects from cone angle and tilt angle (and deformations and unconventional blade shapes (e.g., aft swept)) can relatively easily be included in the blade element theory by means of geometrical corrections to the velocity diagram given in Fig. 5. It must be noted that the 3D flow effects which may be expected from geometric “deviations” are not taken into account (e.g., the undisturbed wind vector will have a component along the blade in the case of cone and tilt angle). In principle, there is also an effect from the tilt angle in the momentum theory since a tilt angle leads to an azimuthal variation of the induced velocity, similar to the variation of induced velocity from a yaw angle as described before. Such correction is rarely included in BEM where it should be realized that for common values of the tilt angles ( 5), the transition onset

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Fig. 15 Fig. A.1 from Langtry (2006). Effect of increasing y + for the flat plate T3A test case

location moves upstream with increasing y + (Langtry 2006). As reference guideline for the gridding strategy in case of transition detection simulations, the conclusions of (Langtry 2006) remain to be of quite general validity: to achieve grid-independent results, the maximum y + had to be 1 or less, the wall-normal expansion ratio has to be between 1.1 and 1.2, and at least 100 stream-wise nodes along the length of the boundary layer are needed to properly capture the laminar, transitional, and turbulent boundary layer development (see Figs. 15, 16 and 17). However being possible to relax the requirement for the stream-wise grid spacing, the high sensitivity of the transition onset location to the advection scheme used for the turbulence and transition model equations demands the use of bounded high resolution, i.e., approximately second-order, upwind schemes (Langtry 2006) (see Fig. 18).

DES Grid Requirements As reported in Spalart (2001), in DES, one is not in a position to predict an order of accuracy when walls are involved, since the filtered equation that is being approximated cannot be produced. The full flow field is filtered, with a length scale proportional to Δ, which is the DES measure of grid spacing. Nevertheless, grid refinement is an essential tool to explore the accuracy of DES as for any other numerical approach. Assuming a background gridding practice typical of pure

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Fig. 16 Fig. A.3 from Langtry (2006). Effect of wall-normal expansion ratio for the flat plate T3A test case

RANS calculations, the usage of a too much finer grids would activate LES in these regions, thus requiring the usage of DDES or IDDES. It is required to set a target cell size, say, Δ0 , that corresponds to the size of the vortex structure one supposes to solve within the focus region of the LES portion; see the sketch in Fig. 19. This target size can and should be quite smaller than the regular cell size in the so-called departure region, i.e., in the portion of the domain far from the body where particles are unable to return to the focus region. Since in DES one has Δ = max (Δx, Δy, Δz), the least expensive way to obtain the desired Δ0 is to have cubic grid cells. The numerical motivation relies on the eddy viscosity allowing the steepening to about the same minimum length scale in all directions, statistically (this is true away from walls). As a result, finer spacing in one or even two directions direction is wasted. The physical motivation for cubic cells grounds on the LES premise of filtering out only eddies that are small enough to be products of the energy cascade and therefore to be statistically isotropic. Then, equal resolution capability in all directions is logical. Since there is no unique way to choose Δ0 , the ideal DES study contains results with an initial educated guess for Δ0 and a secondary one with Δ0 /2 avoiding to incur in the computational cost limitation. Following Spalart (2001), the “gray area” between RANS resolved (RR) and LES focus region (FR) leads to concerns in physical terms. The grid design is here not troublesome, because of typical flow becoming LES after separation, e.g., the fluid goes from RR to FR. Thus, the wall-

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Fig. 17 Fig. A.4 from Langtry (2006). Effect of stream-wise grid density for the flat plate T3A test case

parallel spacing (Δx) has no reason to vary wildly from RR to FR. Moreover, a well-resolved FR may be finer than the RR, and usually the separation point is not accurately known in advance. This means that the grid design does not mark the RR-to-FR change significantly. In conclusion, one observes the RR having more points than needed outside the boundary layer and the FR having a few too many very near the wall. In Spalart (2001), it is stated that a sub-grid-scale model is best adjusted to allow the energy cascade to the smallest eddies that can be safely tracked on the grid, such that eddies with a wavelengths of maybe λ = 5 will be identified. To best spend the computing resources, it is necessary to have time-integration errors of the same magnitude of the spatial ones. Thus, a second-order time-integration scheme requires at least five time steps per period. That leads to a local Courant-FriedrichsLewy number equal to 1.

LES Grid Requirements As reported in Davidson (2019), the near-wall grid spacing should be about one wall unit in the wall-normal direction. This is similar to the requirement in RANS using low-Re number models. The resolution requirements in wall-parallel planes for a well-resolved LES in the near-wall region expressed in wall units are approximately 100 (stream-wise) and 30 (spanwise). This enables resolution of the near-wall

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Fig. 18 Fig. A.6 from Langtry (2006). Effect of stream-wise grid resolution (top, coarse grid; bottom, fine grid), for resolving transition due to a leading edge separation bubble for the Zierke compressor

turbulent structures in the viscous sub-layer and the buffer layer consisting of highspeed inrushes and low-speed ejections, often called the streak process. At low to medium Reynolds numbers, the streak process is responsible for the major part of the turbulence production. These structures must be resolved in an LES in order to achieve accurate results. Then the spectra of the resolved turbulence will exhibit the Kolmogorov -5/3 range. In applied LES, this kind of resolution can hardly ever be afforded. In outer scaling (i.e., comparing the resolution to the boundary layer thickness, δ), we can afford δ/Δx1 and δ/Δx3 in the region of 10–20 and 20–40, respectively. In this case, the spectra in the boundary layer will look something like that shown in Fig. 20. Energy spectra are actually not very reliable to judge if an LES simulation is well resolved or not. Different ways to estimate the resolution of an LES were investigated, suggesting that two-point correlation is the best way to estimate if an LES is sufficiently resolved or not. For wall-bounded flows at high Reynolds numbers of engineering interest, the computational resource requirement of accurate

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Fig. 19 Reproduction of Fig. 1 from Spalart (2001). Sketch of flow regions around tilt-rotor airfoil in rotor down-wash during hover

Fig. 20 Reproduction of Fig. 18.13 from Davidson (2019). Energy spectra in fully developed channel flow . δ denotes half channel width. Number of cells expressed as (δ/Δx1 , δ/Δx3 ). – : (10, 20); - - : (20, 20); -·- : (10, 40); ◦ : (5, 20); +: (10, 10)

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LES is prohibitively large. Indeed, the requirement of near-wall grid resolution is the main reason why LES is too expensive for engineering flows, which was one of the lessons learned in the LESFOIL project (Mellen et al. 2003).

Turbulence Models In this section, a brief overview of the most commonly used turbulence models is provided. Pros and cons are described, and the reader is referred to the results of section “Examples of Application” for the quantitative discussion. A further subdivision would cover the condition of: • Fully turbulent flow, via one- and two-equation eddy viscosity models • Laminar to turbulent transition flow, via γ (Menter et al. 2006) and eN -like models (Sørensen et al. 2016). The main aspects of the turbulence modeling via eddy viscosity for one- and two-equation approaches follow.

Spalart–Allmaras As reported on the NASA Turbulence Modeling Resource,3 being a linear eddy viscosity models, the Spalart–Allmaras uses the Boussinesq assumption:

2 1 ∂uk τij = 2μt Sij − δij − ρkδij , 3 ∂xk 3

(15)

where, for one-equation model like this, the last term is ignored due to the lack of direct availability of the turbulent kinetic energy k. In this subsection, one refers to the “standard” reference (Spalart and Allmaras 1994) without the rarely used primary tripping term capable of predicting laminar to turbulence transition. Moreover, here and in section “k −ω SST”, the velocity variable would be indicated simply by u, implying this being the Reynolds time-averaged velocity variable u of section “RANS”. This model is described by the following single equation for the eddy kinematic viscosity:

 ∂ ν˜ ∂ ν˜ cb1  ν˜ 2 + uj = cb1 (1 − ft2 )S˜ ν˜ − cw1 fw − 2 ft2 ∂t ∂xj d κ

 ∂ ∂ ν˜ ∂ ν˜ 1 ∂ ν˜ + cb2 , + (ν + ν˜ ) σ ∂xj ∂xj ∂xi ∂xi

3 https://turbmodels.larc.nasa.gov/spalart.html,

accessed on May 1, 2021.

(16)

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with the turbulent dynamic eddy viscosity computed from μt = ρ νf ˜ v1 , where fv1 = given below:

χ3 3 χ 3 +cv1

and χ =

ν˜ ν.

A definition for the auxiliary field functions is

S˜ = Ω + Ω=

(17)



ν˜

(18)

fv2 , κ 2d 2

2Wij Wij ,

χ , 1 + χfv1  1/6 6 1 + cw3

(19)

fv2 = 1 − fw = g

(20)

,

(21)

g = r + cw2 (r 6 − r),  ν˜ , 10 , r = min ˜ 2d 2 Sκ  ft2 = ct3 exp −ct4 χ 2 ,

∂uj 1 ∂ui . Wij = − 2 ∂xj ∂xi

(22)

6 g 6 + cw3

(23) (24) (25)

The companion constants for the baseline version of this model are as follows: cb1 = 0.1355, σ = 2/3, cb2 = 0.622, κ = 0.41, cw2 = 0.3, cw3 = 2, cv1 = 7.1, ct3 = 1.2, ct4 = 0.5, cw1 =

cb1 1 + cb2 . + σ κ2

k − ω SST The SST (shear stress transport) model of Menter (1994) is a linear eddy viscosity model which includes two main novelties: (1) It is a combination of a k − ω model (in the inner boundary layer) and k − ε model (in the outer region of the boundary layer as well as outside of it). (2) It is a limitation of the shear stress in adverse pressure gradient regions. The k − ε model has two main weaknesses: due to too low dissipation, it overpredicts the shear stress in adverse pressure gradient flows because of too large length scale; it requires near-wall modification in the form of low Reynolds number damping functions or equivalent terms. One example of adverse pressure gradient is the flow along the surface of an airfoil; see Fig. 21.

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Fig. 21 Fig. 9 from Timmer and Rooij (2003). Measured pressure distributions of airfoil DU 96-W-180 at Re = 3e6. Transition free. Three different, increasing angles of attack α. APG region more evident for the highest α

Consider the upper surface (suction side). Starting from the leading edge, the pressure decreases because the velocity increases. At the crest (moving from 35% of the chord toward the leading edge for increasing angle of attack – AoA), the pressure reaches its minimum and increases further downstream as the velocity decreases. This region is called the adverse pressure gradient (APG) region. The k − ω model is better than the k − ε model at predicting adverse pressure gradient flow, and the standard model of Wilcox (1988) does not use any damping functions. However, the disadvantage of the standard k − ω model is that it is dependent on the free stream value of ω. In order to improve both the k−ε and k−ω model, it was suggested from Menter in (1994) to combine the two models. Before doing this, it is convenient to transform the k − ε model into a k − ω model using the relation ω = ε/(β ∗ k), where β ∗ = cμ . This two-equation model written in conservation form looks as follows:  ∂(ρk) ∂(ρuj k) ∂ ∂k , + = P − β ∗ ρωk + (μ + σk μt ) ∂t ∂xj ∂xj ∂xj

(26)

 ∂(ρω) ∂(ρuj ω) γ ∂ ∂ω 2 + = P − βρω + (μ + σω μt ) ∂t ∂xj νt ∂xj ∂xj + 2(1 − F1 )

ρσω2 ∂k ∂ω ω ∂xj ∂xj

∂ui , and τij is given by Eq. 15. where P = τij ∂x j

(27)

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A definition for the auxiliary field functions is given below: ρa1 k , max(a1 ω, ΩF2 )  F1 = tanh arg41 ,   √   k 500ν 4ρσω2 k arg1 = min max , , , β ∗ ωd d 2 ω CDkω d 2

 1 ∂k ∂ω , 10−20 F2 = tanh arg22 , CDkω = max 2ρσω2 ω ∂xj ∂xj  √  k 500ν arg2 = max 2 ∗ , 2 . β ωd d ω μt =

(28) (29) (30) (31)

(32)

Each of the model constants is a blend of an inner (1, or k − ω portion) and outer (2, or k − ε portion) constant, blended via φ = F1 φ1 + (1 − F1 )φ2 , where φ1 represents constant 1 and φ2 represents constant 2. The companion constants for the baseline version of this model are as follows: γ1 =

β1 σω1 , κ 2 β2 σω2 κ 2 − √ ∗ γ2 = ∗ − √ ∗ , ∗ β β β β

σk1 = 0.85, σω1 = 0.5, β1 = 0.075, σk2 = 1.0, σω2 = 0.856, β2 = 0.0828, β ∗ = 0.09, κ = 0.41, a1 = 0.31.

(33)

γ − Reθ t Following the approach of Menter in Menter et al. (2006), one can introduce the intermittency γ , i.e., the probability that the flow is turbulent at a certain location (or time), such that γ = 0 if laminar and γ = 1 if fully turbulent. A transport equation for intermittency can be approximately derived – or heuristically formulated – and used to trigger the transition onset and its length. Although being the intermittency generic, thus able to be combined with any RANS model, it requires a proper viscous sub-layer model. The following definitions of boundary layer (of given height δ) quantities are necessary prior to introducing the model equations: 

δ

Momentum thickness θ = 0

Momentum thickness Reynolds number Reθ =



u(y) u(y) 1− , Uδ Uδ

ρUδ θ , μ

(34) (35)

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ρθ 2 dUδ , μ dx √ 2kδ /3 Turbulence intensity T u = . Uδ

Nondimensional pressure gradient λθ =

(36) (37)

The transition onset is dependent on many factors, such as free stream turbulence intensity, nondimensional pressure gradient, separation, Reynolds number, wall roughness, etc. In Menter et al. (2006), it is assumed the dependency solely on the first two abovementioned factors, i.e., Reθt = f (T u, λθ ), derived from experimental database of Ghannam and Shaw (1980). The transport equation for the intermittency in its conservative form looks like

 μt ∂γ μ+ , σf ∂xj

∂(ργ ) ∂(ρUj γ ) ∂ + = Pγ − Eγ + ∂t ∂xj ∂xj

(38)

where the transition production or source term is Pγ = Flength ca1 ρS(1 − ce1 γ )[γ Fonset ]0.5 and the destruction or relaminarization term is Eγ = ca2 ργ (1 − ce2 γ )ΩFturb . In the previous expressions, Flength indicates the transition length (i.e., the region where the friction coefficient recovers after facing its minimum at the transition offset, till the local maximum for turbulent conditions), and  Fonset Rev is a trigger for the transition offset given by Fonset = max 2.193Reθt , 0 , where 2

Rev = ρyμ | ∂u ∂y | is the vorticity Reynolds number. As reported in Langtry (2006), Fturb is used to disable the destruction/relaminarization source outside of a laminar 4  ρk boundary layer or in the viscous sub-layer and is defined as Fturb = exp − 4μω . The constants for the intermittency equation are as follows: ca1 = 2.0, ce1 = 1.0, ca2 = 0.06, ce2 = 50, σf = 1.0.

(39)

In order to get information from outside the boundary layer into the boundary layer (through diffusion), one needs to introduce a local transport equation for the transition onset trigger as the next one: ∂(ρ Re˜ θt ) ∂(ρUj Re˜ θt ) ∂ + = Pθt + ∂t ∂xj ∂xj



∂ Re˜ θt σθt (μ + μt ) ∂xj

 ,

(40)

 2U 2 Reθt − Re˜ θt (1 − Fθt ), with Fθt =0 where the production term is Pθt = cθt ρ500μ for free stream and 1 in the laminar boundary layer. The larger the value of σθt , the less sensitive the transition model is to history effects. The model constants for the transport equation are as follows: cθt = 0.03, σθt = 2.0.

(41)

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Linear Stability Theory with RANS In this small subsection, a short description of the limited application of the linear stability theory for laminar to turbulent transition (Ingen 2008) applied to wind turbine blades is reported. In this case, the trigger for the intermittency function γ is provided by the linear stability analysis for the boundary layer as for the eN method numerically approximated by Drela and Giles (1987). A brief description of the linear stability theory, i.e., the Orr-Sommerfeld equation solved by means of the small disturbances approach, is provided in the following. For reasonable levels of free stream turbulence, instabilities can develop as Tollmien-Schlichting waves. Starting from observations of a water flow over a table,it can be formulated that turbulence starts such as randomly distributed spots. When those regions expand, they can merge across the whole span. Thus, the transition region starts when the spots appear and ends when they merged as a continuous front. These spots can be highlighted experimentally with hotwire response. The transition onset on the body of interest can be affected by several factors: free stream turbulence, heat transfer, pressure gradient, or the surface parameters such as curvature or roughness.The linear stability theory supposed that the basic laminar flow receives a small perturbation and then evolves into turbulent flow. Due to the gradual amplification of infinitesimal disturbances coming from the surface roughness or the external flow, the laminar boundary layer can become turbulent. Initially, the disturbances are weak and have no impact on the steady flow, but when they increase, they highly affect the characteristics of the flow. The conservation law that governs the stability theory is the Orr-Sommerfeld equation, obtained introducing perturbations for the velocity and the pressure fields in the Navier-Stokes equation, further simplified to 2D, incompressible flows, by means of the dimensionless form of the vorticity equation expressed in terms of the stream function. A valid mathematical solution consists of normal mode ψ = ϕ(z) exp(ik(x − cr t)) exp(kci t), where k is the wave number and c is the complex phase velocity. The normal modes are amplified if kci > 0 and damped if kci < 0. There are two ways to obtain the solution of Orr-Sommerfeld equation and the related boundary conditions: temporal amplification theory, i.e., considering the frequency ω = ωr + iωi as complex and the amplitude varying as exp(−ωi t), and spatial amplification theory, i.e., considering the wave number k = kr + iki as complex and the amplitude varying as exp(−ki x). The spatial amplification theory has to be favored since the amplitude change with distance can be obtained in a steady flow. The couples of eigenvalues (k, Re) or (ω, Re) can be plotted to get the stability diagrams to show in which case the disturbances are amplified, neutral, or damped. The locus where ki vanishes is the neutral stability curve, and the critical Reynolds number Recr is the point on this curve where Re is minimal. When Re is below Recr , all disturbances are stable. The neutral stability curve is the same, no matter with which amplification theory it has been computed. Concerning the application field, it is observed that the approach proposed in Madsen (2002) is based on the initial study of Stock and Haase (1999) and leads to experimental validation on the second round of the MEXICO research project

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(Sørensen et al. 2016) and to a more transversal comparison for large-scale wind turbine rotor blades in Schaffarczyk et al. (2018).

Observations Here it is worth to note the distribution of the turbulence models (TM) depending on the simulation type and reference experimental or numerical wind turbine model. As reported in Fig. 22, one can infer that the large majority of the selected references adopted the two-equation eddy viscosity fully turbulent model k − ω-SST. Only a limited number of researchers focused on the transition from laminar to turbulent flow and mostly only in recent years.

Examples of Application In this section, major examples of application are briefly described and discussed. The most published application is the validation against well-known controlled experiments. For an overview about rotor measurements, the reader is referred to  Chap. 19, “Wind Tunnel Rotor Measurements”.

Mexico and MexNext Experiment The selected results reported here refer to the research work available among others in Schepers et al. (2014).In particular, the comparison of experiment and two different CFD solutions for the pressure distributions along the blade sections located at 30, 47, 63, 80, 95% of the span is shown in Figs. 23 and 24. These two cases are representative of the following operating conditions: U∞ = 5.02m/s, Ω = 90.2rpm, β = 3deg for the X05 and U∞ = 10.04m/s, Ω = 90.9rpm, β = 3deg for the X10. As illustrated in Schepers et al. (2014), “the differences that can be observed at the TE between the two computations are due to a point trailing edge geometry employed at CENER instead the thick trailing edge geometry that was at experiments and also was modeled by DTU.” By means of the integration of the pressure forces on each sections, and using an estimation for the angle of attack, the three-dimensional version of the baseline twodimensional aerodynamic polar input table for load calculations can be derived. An example of such an application is reported in Figs. 25 and 26, where the normal and tangential force coefficients along the spanwise direction are shown, respectively. In this case, the X10 configuration is considered, where large separation regions exist on the blade surface. As reported in Schepers et al. (2014), “the added value of CFD is clearly visible. Here it is noted that the experimental blade aspect ratio is small in comparison to large commercial blades, which will influence the mani-

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Fig. 22 Distribution of the turbulence models (TM, horizontal axis) for different types of simulation (left vertical axis) and different wind turbine experimental/numerical models (right vertical axis) based on the literature reference. Legend for the TM on the horizontal axis: SA stands for one-equation Spalart-Allmaras (1994); k − ω for two-equation k − ω family model, like the shear stress transport formulation of Menter (1994); γ for the transitional model based on the intermittent γ formulation and the two-equation k − ω as in Menter et al. (2006); eN for the transitional model based on the linear stability theory formulated in Ingen (2008); and Smagorinsky for the sub-grid-scale model of Smagorinsky (Smagorinsky 1963). Legend for the references indicated by the diagram entries: RANS portion includes  (Bechmann et al. 2011),  (Sørensen et al. 2014), ◦ (Carrión et al. 2015), • (Rethoré et al. 2011), (Lutz 2011),  (Sørensen et al. 2002),  (Gomez-IradiXabier and Munduate 2014),  (Le Pape and Lecanu 2004),  (Chao and Dam 2007),  (Lanzafame et al. 2013),  (Wu and Nguyen 2017), × (Troldborg et al. 2016), + (Sørensen et al. 2016); URANS portion includes  (Réthoré et al. 2011),  (Lutz 2011), ◦ (Sørensen et al. 2014), • (Sugoi and Xavier 2011), (Sørensen et al. 2002),  (Le Pape and Lecanu 2004),  (Chao and Dam 2007),  (Zahle et al. 2009),  (Sørensen et al. 2017),  (Bangga et al. 2017),  (Jost et al. 2018),  (Heinz et al. 2016); DES portion includes  (Rethoré et al. 2011; Réthoré et al. 2011),  (Sørensen and Schreck 2014), ◦ (Rahimi et al. 2016), (Lynch and Smith 2013),  (Ghasemian and Nejat 2015),  (Troldborg et al. 2015); LES portion includes  (Shen et al. 2012),  (Zhou et al. 2016), ◦ (Rahimi et al. 2018), • (Rahimi et al. 2018)

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Fig. 23 Partial reproduction of Fig. 4 from Schepers et al. (2014). Comparison of pressure distributions for the X05 case: 30% spanwise location on the left; 80% spanwise location on the right

Fig. 24 Partial reproduction of Fig. 5 from Schepers et al. (2014). Comparison of pressure distributions for the X10 case: 30% spanwise location on the left; 80% spanwise location on the right

festation of the tip effect.” In fact, the use of three-dimensional aerodynamic polar curves explains the improved agreement of the BEM codes with the measurements, shown on the right side of both Figs. 25 and 26.

NREL Phase VI Experiment The selected results reported here refer to the research work available among others in Rahimi et al. (2016); Länger-Möller et al. (2017); Zhou et al. (2016).The focus is on the differences introduced by the selection of different levels of fidelity

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Fig. 25 Partial reproduction of Fig. 7 from Schepers et al. (2014). Comparison of normal force for the case X10. Left simulations featuring 2D airfoil data and CFD, right 3D airfoil data

Fig. 26 Partial reproduction of Fig. 9 from Schepers et al. (2014). Comparison of tangential force for the case X10. Left simulations featuring 2D airfoil data and CFD, right 3D airfoil data

in the numerical simulations and the choice of the proper turbulence modeling. Firstly, it is interesting to note the very good agreement with measurement data when an LES approach is employed (Zhou et al. 2016). In Fig. 27, the pressure coefficient distributions along five different spanwise sections of the three-bladed 10-m-diameter wind turbine are shown. Unfortunately, the considered case does not present big challenges even for a lower-fidelity RANS approach, as demonstrated in Fig. 28, where a similar operating condition is reproduced via RANS and fully turbulent assumptions. The net difference due to the use of RANS, URANS, or DDES approach is reported in Rahimi et al. (2016), where the more challenging 25 m/s full separated

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Fig. 27 Fig. 4 from Zhou et al. (2016). Comparison of mean pressure coefficients predicted by LES and experimental results at different span-wise sections at 7 m/s uniform inflow, and five different spanwise locations

Fig. 28 Fig. 10 from Länger-Möller et al. (2017). Pressure distribution at blade sections; BL07 case; error bars indicate standard deviation

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Fig. 29 Fig. 14.3 from Troldborg et al. (2013): Comparison of measurements and the computations by DTU Wind Energy of normal (left) and tangential (right) aerodynamic forces at U∞ = 6.1 m/s

operating conditions are analyzed in detail. In Fig. 9 of Rahimi et al. (2016), it is shown the clear advantage for the necessity of something other than RANS or URANS modeling when large separation occurs.

Validation Against Field Experiments DAN-AERO MW Experiment The selected results reported here refers to the research work available among others in Troldborg et al. (2013) and Madsen (2010). The experience of comparing against field measurement data exploits the difficulties in the assessment of a sharp judgment concerning the ability of one or another simulation approach to reproduce with good approximation the reality. As is shown in Fig. 29, with the spread in the used measurements, it was hard to conclude whether transitional or turbulent computations produce the best results even for an axial operating condition. Is it worth to mention here the nontrivial association of measurement data with simulation results for the full rotor case. As clearly shown in Fig. 30, where two different spanwise sections of the DAN-AERO experiment are considered, both inboard and outboard, the intrinsic variability of the inflow conditions requires the access to multiple simulation data in a consistent range, in order to capture the main feature of the pressure distribution. This is particularly true in the inboard portion of the blade.

Validation for Virtual Models NREL 5MW The selected results reported here refer to the research work available among others in Troldborg et al. (2015).The NREL 5 MW (Jonkman et al. 2009) model has been extensively studied since its adoption as reference virtual model for a

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Fig. 30 Fig. 29 from Madsen (2010): Measured pressure distributions, on the rotor and in the LM wind tunnel, on two sections of the LM38.8 blade, the most outboard section of the blade named sec. 4 and the most inboard section named sec. 1, in comparison with pressure distributions computed with the CFD code EllipSys3D

large community of researchers. In this subsection, the focus lies on a comparison between a fully resolved CFD wind turbine simulation and a lower-fidelity approach represented by both actuator disk and actuator line alternatives. The inflow condition is turbulent, as for the Mann model (Mann 1994). As reported by Troldborg in Troldborg et al. (2015), “The normal forces predicted by the three methods are seen to be in quite good agreement. The tangential loads predicted by the AD and AL methods show good resemblance with each other but are significantly higher than what is obtained in the FR simulation. Other researchers found similar overprediction of power when comparing AD and AL simulations with a traditional blade element momentum method. The reason for the difference is mainly an under-prediction of the induced velocities in the rotor plane when using the AL and AD methods, which in effect increases the angle of attack, but is also to some extent due to disregarding transition modeling in the FR simulation, which reduces the blade loads predicted by this method.” This circumstance testifies the added value of full rotor simulations for the tuning and further development of hybrid methods as actuator disk and actuator line used in conjunction with CFD simulations.

DTU 10MW The selected results reported here refer to the research work available among others in Zahle et al. (2014).In a similar way as in section “DAN-AERO MW Experiment”, the comparison of normal and tangential aerodynamic load spanwise distributions is considered. In this case, the full rotor CFD simulations are employed to adjust the aerodynamic polar curves constituting the input for the BEM calculations. As shown Fig. 15 from Zahle et al. (2014), the adoption of corrected aerodynamic input data allows a lower-fidelity BEM calculation to capture the correct trend, thus approaching the results of higher fidelity simulations and contributing to a reduction of the approximations.

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AVATAR AVATAR is a project initiated by the European Energy Research Alliance (EERA), carried out under the FP7 program of the European Union. Its main goal is the development and validation of advanced aerodynamic models, to be used in integral design codes for the next generation of large-scale wind turbines (up to 20MW).4 The virtual wind turbine rotor model for the AVATAR has been deeply investigated by a large group of researchers. In this subsection, the focus lies on the comparison of different CFD codes against vortex-based methods. In Fig. 31, both thrust and driving force per unit length for the AVATAR rotor are shown. The simulations refer to three operating conditions sharing the same inflow speed equal to U∞ = 10.5 m/s and rotational speed about 9 rpm: axial, 30-degree yaw, and shear with 0.2 exponent. The comparison includes the codes: EllypSys3D (DTU-E), Miras (DTU-M), GenUVP (NTUA), and FLOWer (USTUTT), where only the second one is a vortex based method (Fig. 31). The results demonstrate the very good agreement for the different simulations in the outer 60% of the blade span, while in the inner portion of the blade the larger discrepancies emerge. Thus, it is in this region where the choice of turbulence modeling or simulation approach induces the larger effects, as it would be further exploited in section “Derivation of Rotational Augmentation Models”.Detailed information about further selected results related to this virtual model are available among others in Sørensen et al. (2017).

Derivation of Engineering Models In this section, a brief overview of the possible derivation of so-called engineering models is provided. The readers are referred to  Chap. 14, “Pragmatic Models: BEM with Engineering Add-Ons” for more information about the pragmatic models, where BEM with engineering add-ons is thoroughly discussed.

Estimation of the Angle of Attack The selected results reported here refer to the research work available in Rahimi et al. (2018). As discussed in sections “Mexico and MexNext Experiment” and “DTU 10MW”, one major added value of full rotor simulations relies on the possibility of improving the baseline two-dimensional aerodynamic polar curves serving as input for the BEM calculations. However, this important result is based on a non-negligible approximation derived from the estimation of the angle of attack of the local section; see also the Introduction.

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In this subsection, the results of the application of several methodologies to estimate the angle of attack along the spanwise direction of the large-size rotor AVATAR virtual model are reported. A brief description of the presented approaches follows: • As discussed in Rahimi et al. (2018), “Average Azimuthal method (AAT): This technique is based on the annular average values of the axial velocity by using

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the data at several upstream and downstream locations.The shortcoming of this method lies in the fact that this model is only valid for axial conditions (or for the mean value of axial induction factor over one rotation) and it cannot capture the dynamics of the induced velocity of yawed flows. Furthermore, since the method is based on averaged data, many input points have to be sampled, and therefore the computational cost might be high as compared to other methods. It is also noted that the results might depend on the positions of the monitor point and the interpolation algorithm. Finally, AAT provides an annular averaged induced velocity which is known to differ from the local induced velocity near the tip.” As reported in Rahimi et al. (2018), “3-Point method: This method uses three points along the chord length on each side of a particular section. This great simplicity is the main advantage of this method, which makes the calculation of the AoA very straightforward. By choosing three points at each section, the influence of bound circulation as well as the upwash and downwash effect is eliminated. In addition, unlike the AAT method, this method is able to reproduce the dynamic behavior of the induction and AoA (local induced velocity and AoA) for each azimuthal position and also near the tip and root of the blade which is very important for the yawed flow.” As summarized in Rahimi et al. (2018), Shen 1 and 2 methods represent an iterative determination technique consisting of the following steps: “estimating the lift force by projecting the force along the incoming and normal to the incoming directions; calculating the bound vortex using the Kutta-Joukowski law; calculating the induced velocity by the bound vortex using Biot-Savart’s law; computing the relative velocity at the monitor points by subtracting the induced velocity from the bound vortex; computing the AoA from the relative velocity. This procedure continues until the convergence is reached.” In its second formulation, “an alternative technique was presented [. . . ], where a distributed bound circulation along the airfoil/blade surface is used instead of the concentrated bound vortex at the force center. [. . . ] Another advantage is that this method takes the chordwise variation of aerodynamic forces into account which is neglected when the vorticity is concentrated in a bound vortex. Additionally, this method is not iterative. However, the difficulty of using this method is to find the separation point where the local circulation changes sign.” As explained in Rahimi et al. (2018), “the Line average method determines the AoA by averaging the flow velocities along a symmetric, closed line (as a circle) around the rotor blade.[. . . ] The circle center is placed at the quarter chord position, where like in a lifting line representation the bound vortex is located. The idea is that the induced velocity at opposed points on the circle extinguish each other. By averaging the flow velocities along the circle, the influence of bound circulation is eliminated and the local inflow velocity and AoA can be determined.” As illustrated in Rahimi et al. (2018), the Herráez or Zero-Gamma method “obtains the undisturbed flow rotor velocity by extracting them directly from a position in the rotor plane where the influence of the blade bound circulation from each blade is canceled out by the other blades. In the case of axi-symmetric, homogeneous inflow, this position corresponds to the bisectrix of the angle

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between two arbitrary blades. [. . . ] The difference between the free stream velocity and the axial velocity component obtained in this way for each radial position corresponds to the local axial wake induction. The local tangential velocity extracted from the radial traverse corresponds to the local tangential wake induction after changing its sign. It is demonstrated in Herráez et al. (2018) that this method can predict the AoA satisfactorily from the root until at least 92% of the blade length. For larger spanwise positions, other methods, like e.g. Shen are preferred. The main advantage of this method is its simplicity, which makes the calculation of the AoA very straightforward. Furthermore, in opposition to other methods [. . . ], it is not sensitive to input parameters like engineering correction models, monitoring point location, etc. The main limitation of the method is that its use for non-axisymmetric or inhomogeneous inflow becomes much more complicated because of the dependence of the blade bound circulation on the azimuthal blade position.” As indicated by the authors in Rahimi et al. (2018), “in terms of axial induction factor [. . . ] a very good agreement is observed between all the methods at the midspan (0.30 < r/R < 0.85). However at the root (r/R < 0.30) and at the tip of the blade (r/R > 0.85) discrepancies can be observed.” (see Fig. 32).

Fig. 32 Fig. 14 from Rahimi et al. (2018): Comparison of all the considered methods in terms of a, AoA, Cl , and Cd for the AVATAR turbine at 9 m/s. For a better visibility, the inner part of the subfigures representing the AoA, Cl , and Cd at the outer span location is zoomed for a smaller y-range

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There are a number of reasons that can explain the discrepancies between the results of the several methodologies considered. Following the authors of Rahimi et al. (2018), these causes are as follows: • The methods employed consider various 2D sections along the blade, using only the spanwise oriented bound circulation. In regions with 3D predominant effects, such as root and tip, the chordwise component of the bound circulation is not negligible. • In the root area, due to the large separation, the positions of monitoring points play a major role on the calculated average value of induced velocity. This circumstance disappears moving outward. In this sense, the methods using monitoring points far away from the blade surface are probably more accurate. • In the tip area, if the monitoring point falls close to the tip vortex, as well as if it moves far away in the up- or downstream position, larger discrepancies arise: in the first case due to the local high induction and in the latter because of the lack of the corresponding streamline. • Moreover, at the tip, the azimuthal-average-technique method gives an azimuthally averaged induction, while the methods using monitoring points far away from the blade surface give a value for the induction in the rotor plane, and all other methods represent the local induction. Thus, reducing the discrepancies between all possible AoA estimation approaches remains a decisive factor to further improve the input data as well as the correction models for the BEM calculations.

Derivation of Root and Tip Models The derivation of correction model for the root and tip effects able to improve the classical theoretical approaches, e.g., Prandtl, faces the challenge of discriminate for the prescribed operating condition of analysis between the net contribution due to the rotation of the turbine and that of the finiteness of its blade geometry. For completeness, in the following, both initial Prandtl and the approximated Glauert expression for tip losses are reported:

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approaches to tune the correction factor between two-dimensional aerodynamic loads and the three-dimensional correspondent value, F , originally introduced by Prandtl, have been proposed for the NREL Phase VI experimental model, e.g., a fuzzy-logic approach can be employed discriminating on the flow condition (i.e., attached flow at low inflow speeds and separated flow at higher inflow speeds) and the blade spanwise position. Alternatively, an analytical modification of the original Prandtl proposal has been reported, where the exponent term is modified by an extra term g, depending on the tip speed ratio and validated against NREL Phase VI and Swedish WG 500 experimental sets. This approach was later further modified to introduce a dependency on the blade tip tapering and validated against CFD results for the NordTank 500 kW and the NREL 5 MW turbines (Shen et al. 2014).

Derivation of Rotational Augmentation Models The selected results reported here refer to the research work available in Bangga et al. (2017). The prediction of stall for airfoil shape is far from being a solved issued for CFD modeling. The reader is referred to  Chap. 9, “CFD Simulations for Airfoil Polars” for more information about this aspect. For this reason, the goal of capturing the correct behavior of the inner part of large rotors, where the large separation mostly suffers the presence of strong radial flow, seems to be a hard challenge. However, recent validation campaign against experiments, even in free field environment, testifies the good agreement of full rotor simulations with pressure sectional measurements, as reported in section “DAN-AERO MW Experiment”. In this subsection, the focus lies on the necessity to bridge the gap between the standard two-dimensional aerodynamic polar input curve for BEM calculation and the added value derived from high-fidelity CFD rotor simulations. The use of stateof-the-art rotational augmentation models is based on the assumption that the role of the rotational forces is played almost exclusively within the stall portion of the aerodynamic polar. This leads to a more or less complicated formulation able, for instance, to scale the two-dimensional lift curve to reproduce the three-dimensional case. Nevertheless, the introduction of thicker airfoils at the root section of large wind turbines may introduce further challenges. In fact, as reported in Bangga et al. (2017) for the 10 MW AVATAR virtual model, the differences between the two- and three-dimensional case are far from following a mere scale procedure. As shown in Fig. 33, a large portion up to 25% of the blade span is affected by large rotational augmentation effect. Even if the resolution in terms of angle of attack does not allow a 1:1 comparison of the two- and three-dimensional polar curve, one may argue that the complex stall at very low angle of attack for such thick airfoil shapes is completely disregarded by the full rotor simulation results, whenever a laminar flow portion is present or not.Thus, the majority of the available rotational augmentation models would simply fail in delivering an input for the BEM calculations able to reduce the approximation of the load in the inner part of the blade.

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Fig. 33 Fig. 6 from Bangga et al. (2017): 3D and 2D Cl polars. Variable t/c represents the relative thickness. 3D Cl is remarkably higher than that in the 2D conditions, except near the tip region: (top left) t/c = 0.75, 2D Re=7.1e6; (top right) t/c = 0.57, 2D Re=9.2e6; (mid-left) t/c = 0.46, 2D Re=11.3e6; (mid-right) t/c = 0.36, 2D Re=14.2e6; (bottom left) t/c = 0.25, 2D Re=17.6e6; (bottom right) t/c = 0.24, 2D Re=14.2e6

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Cross-References  Pragmatic Models: BEM with Engineering Add-ons

References Bangga G, Kim Y, Lutz T, Weihing P, Krämer E (2016) Investigations of the inflow turbulence effect on rotational augmentation by means of CFD. J Phys Conf Ser 753:022026 Bangga G, Lutz T, Jost E, Krämer E (2017) CFD studies on rotational augmentation at the inboard sections of a 10 mw wind turbine rotor. J Renew Sustain Energy 9(2):023304 Bangga G, Weihing P, Lutz T, Krämer E (2017) Effect of computational grid on accurate prediction of a wind turbine rotor using delayed detached-eddy simulations. J Mech Sci Technol 31(5):2359–2364 Bechmann A, Sørensen NN, Zahle F (2011) CFD simulations of the MEXICO rotor. Wind Energy 14(5):677–689 Carrión M, Steijl R, Woodgate M, Barakos G, Munduate X, Gomez-Iradi S (2015) Computational fluid dynamics analysis of the wake behind the mexico rotor in axial flow conditions. Wind Energy 18:1023–1045 Celik IB, Ghia U, Roache PJ, Freitas CJ, Coleman H, Raad PE (2008) Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J Fluids Eng 130:078001– 078001–4. https://doi.org/10.1115/1.2960953 Chao DD, van Dam CP (2007) Computational aerodynamic analysis of a blunt trailing-edge airfoil modification to the nrel phase vi rotor. Wind Energy 10(6):529–550 Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approac (1997) Coton FN, Wang T, Galbraith RAMD (2002) An examination of key aerodynamic modelling issues raised by the NREL blind comparison. Wind Energy 5:199–212 Davidson L (2019) Fluid mechanics turbulent flow and turbulence modeling (E-Book). Division of Fluid Dynamics, Department of Mechanics and Maritime Sciences, Chalmers University of Technology Davidson L, Peng SH (2003) Hybrid les-rans modelling. Int J Numer Methods Fluids 43:1003– 1018 Drela M, Giles MB (1987) Viscous-inviscid analysis transonic reynolds number airfoils. AIAA J 25:1347–1355 Drofelnik J, Da Ronch A, Campobasso MS (2018) Harmonic balance navier-stokes aerodynamic analysis of horizontal axis wind turbines in yawed wind. Wind Energy 21(7): 515–530 Fedorov V, Berggreen C (2014) Bend-twist coupling potential of wind turbine blades. J Phys Conf Ser 524:012035 Fröhlich J, Rodi W (2002) Introduction to large eddy simulation of turbulent flows. In: Launder BE, Sandham ND (eds) Closure strategies for turbulent and transitional flows, pp 267–299. Cambridge University Press Abu-Ghannam BJ, Shaw R (1980) Natural transition of boundary layers—the effects of turbulence, pressure gradient, and flow history. J Mech Eng Sci 22:213–228 Ghasemian M, Nejat A (2015) Aerodynamic noise prediction of a horizontal axis wind turbine using improved delayed detached eddy simulation and acoustic analogy. Energy Convers Manag 99:210–220 Gomez-Iradi S, Munduate X (2014) Zig-zag tape influence in NREL phase vi wind turbine. J Phys Conf Ser 524:012096 Haase W, Braza M, Revell A (2009) DESider, volume 103 of Notes on numerical fluid mechanics and multidisciplinary design, 1612-2909. Springer

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M. Sayed, P. Bucher, G. Guma, T. Lutz, and R. Wüchner

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-of-the Art of CFD-Based Simulation of Wind Turbine Aeroelasticity . . . . . . . . . . . . . . State of the Art of Wind Turbine Aeroelasticity Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Turbine Aerodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Turbine Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Solver: Modeling and Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh Deformation Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RBF Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spring Analogy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic Smoothing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Analogy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Navier-Stokes-Based CFD with Large Deformations and Surface Motion . . . . . . . . . . . . . Immersed Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A part of this work has been conducted during the PhD period of the first author at the Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany. M. Sayed MesH Engineering GmbH, Stuttgart, Germany e-mail: [email protected] G. Guma · T. Lutz () Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected] P. Bucher Structural Analysis, Technical University of Munich (TUM), Munich, Germany e-mail: [email protected] R. Wüchner Institut für Statik und Dynamik, Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_22

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Embedded Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arbitrary Lagrangian-Eulerian (ALE) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ALE-Variational Multiscale Methods (VMS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Solver: Modeling and Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Fidelity FE Structural Models for Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of High-Fidelity CSD FE Models to Wind Turbines . . . . . . . . . . . . . . . . . . . . Obtaining Beam Sectional Data from Higher Dimensionality Models . . . . . . . . . . . . . . . . Aspects of Computational Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmatching Grid Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface Coupling Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Procedures for High-Fidelity Wind-Structure Interaction Simulations . . . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Solver FLOWer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure Solver Carat ++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Co-Simulation Environment EMPIRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Turbine Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CSD Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of the Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter focuses on the challenges rising when modeling the aeroelasticity of modern wind turbines utilizing high-fidelity methods. A comprehensive review of the state of the art is presented at the beginning, including engineering models. Since the aeroelastic models consist of a flow and a structural solver, a detailed description of the modeling and simulation techniques is provided, including the basic requirement for coupling a computational fluid dynamics (CFD)based solver with a computational structural dynamics (CSD)-based solver. The challenges related to the simulation of large rotating bodies, as well as moving grids, are described. In the numerical analysis of the aeroelasticity, the blades could be structurally modeled by mainly three different elements. These are beam, shell, and solid elements, by which the accuracy level of the results could be improved. Therefore, different fidelity levels of structural discretization of the wind turbine are discussed in terms of using these elements. To model the blade using beam elements, a cross-sectional analysis tool is needed to extract the beam structure properties out of the full three-dimensional (3D) geometry of the blade. Coupling CFD to CSD needs great attention at the coupling interface between both solvers. Since they have different grid resolution, a mapping grid technique is needed to translate the data at the interface between the nonmatching grids. Moreover, the coupling scheme should be carefully chosen based on the required accuracy level.

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The chapter ends by presenting high-fidelity results of a state-of-the-art wind turbine model. The effect of the geometrical nonlinearity of the wind turbine blades is discussed. Comparisons between the different structural elements are described based on these results. The effect of the aerodynamic model fidelity is introduced. Keywords

Aeroelasticity · Wind turbine · Engineering models · High fidelity Abbreviations

1D 2D 3D ALE BECAS BEM CAD CFD CSD DoF DS DTU EAS ECN EMPIRE FAST FDT FE FEM FSI GDW GH HAWC HAWC2 HAWCStab HAWT IBRA IGA IWES LUH MARINTEK

One Dimensional Two Dimensional Three Dimensional Arbitrary Lagrangian-Eulerian BEam Cross Section Analysis Software Blade Element Momentum Computer-Aided Design Computational Fluid Dynamics Computational Structural Dynamics Degree of Freedom Dynamic Stall Denmark Technical University Enhanced Assumed Strain Energy Research Center of the Netherlands Enhanced Multi-Physics Interface Research Engine Fatigue, Aerodynamic, Structures, and Turbulence NREL’s Aeroelastic Simulation Tool Filtered Dynamic Thrust Finite Element Finite Element Method Fluid Structure Interaction Generalized Dynamic Wake Garrad Hassan Horizontal Axis Wind Turbine Simulation Code Horizontal Axis Wind Turbine Simulation Code 2nd Generation The Aeroelastic Stability Tool for Wind Turbines Horizontal Axis Wind Turbine Isogeometric Boundary Representation Analysis Isogeometric Analysis Fraunhofer Institute for Wind Energy and Energy System Technology Leibniz Universität Hannover Norwegian Marine Technology Research Institute

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MBD MBS MRF MSC NREL NTNU NURBS POD RANS RBF SMM TUM URANS VLM VMS

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Multi-Body Dynamics Multi-Body System Multiple Reference Frame MacNeal-Schwendler Corporation Software National Renewable Energy Laboratory Norwegian University of Science and Technology Nonuniform Rational Basis Spline Proper Orthogonal Decomposition Reynolds-Averaged Navier-Stokes Radial Basis Functions Sliding Mesh Model Technical University of Munich Unsteady Reynolds-Averaged Navier-Stokes Vortex Lattice Methods Variational Multiscale Methods

Introduction Over the past three decades, the capacity and size of commercial wind turbines has increased exponentially, from a rated power of 75 kW and a rotor diameter of 17 m for earlier designs to a rated power of 10 MW and a rotor diameter of more than 170 m for modern machines. The increasing size of wind turbines lowers the cost of generating wind power in terms of levelized energy costs ($/kWh); however, it introduces significant aeroelastic effects caused by the interaction of aerodynamic loads, elastic deflections, and inertial dynamics. The dynamics of inertia play an important role in the correlation between aerodynamic loads, elastic deflections, and the resulting accelerations. Because of the aeroelastic effects, the blade can experience oscillations and may become unstable under harmonic conditions and/or if the damping is negative. Current design trends favor larger turbines in order to achieve the market forecast, making the energy extracted from wind more competitive in the future. However, simply upscaling the wind turbine is not an easy and straightforward task, as it could lead to inappropriate design and heavy structural components placing unreasonable high loads on the rotor bearings and the entire support structure. Without the need to redesign the turbine from scratch, several ways are used to enhance the use of direct upscaling. Contemporary and future low-speed wind turbines are designed with relatively light and flexible slender blades. In addition, with an increased power output of up to 20 MW at present, the diameter of the rotor increases dramatically, resulting in a larger amplitude of the blade deformations due to the wind vertical shear effect. In addition, some of the current and future generations of wind turbines are subject to more diverse environmental conditions, such as complex terrain, resulting in wind instability that can cause disturbed inflow conditions (inclination, higher turbulence, etc.). Wind turbines with large and flexible blades, therefore, require an accurate prediction of

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the aeroelastic response to identify unwanted vibrations or destructive flutter. The state of the art of aeroelastic simulation tools for wind turbines uses engineering models to find the wind turbine’s aeroelastic response. To determine surface deformations, these tools use simplified methods such as blade element momentum (BEM) to find unsteady aerodynamic loads and one-dimensional (1D) structural models. These models are computationally cheap, but they are based on various corrections and simplifications to account for the instability, friction, and 3D effects of a complex flow. They capture the aeroelastic response sufficiently accurate for small turbines and under certain operating conditions. However, for large turbines, if the tip deformations exceed 10% of the blade length, it is of great interest to determine whether these models still capture accurately enough the aeroelastic response or whether high-fidelity models are needed to analyze large slender wind turbines. Such deformations may result in blade profile shape deformations (e.g., decambering), which in turn impacts the aerodynamic characteristics of the blade profile yielding a change in loads, aeroelastic response, and the wind turbine’s overall performance. This chapter is structured as follows: section “State-of-the Art of CFD-Based Simulation of Wind Turbine Aeroelasticity” introduces the state of the art for the aeroelastic simulations of wind turbines. Following are sections “Flow Solver: Modeling and Simulation Techniques” and “Structural Solver: Modeling and Simulation Techniques” which are dedicated to modeling and solution techniques for flow solvers and structural solvers, respectively. The challenges and techniques for coupling a flow solver and a structure solver to perform FSI simulations are discussed in section “Aspects of Computational Fluid-Structure Interaction”. FSI simulation results of a state-of-the-art wind turbine are presented then in section “Simulation Results”. Finally, the discussion and conclusion of the main aspects presented in this chapter are summarized in section “Conclusions and Recommendations”.

State-of-the Art of CFD-Based Simulation of Wind Turbine Aeroelasticity State of the Art of Wind Turbine Aeroelasticity Modeling Aeroelasticity is a multiphysics term denoting the study of the interaction among the main three disciplines of aerodynamics, elasticity, and dynamics as shown in Fig. 1 that was originally introduced by A. R. Collar in the 1940s. In terms of forces, it is the interaction between the aerodynamic, structural, inertial, and gravitational forces (Hodges and Pierce 2011 and Bisplinghoff et al. 2013). Due to this interaction, the structure will either approach a new equilibrium state or diverge and lead to structural failure. In the case of an interaction that can potentially lead to divergence, the turbine exhibits the so-called aeroelastic instabilities that strongly affect the operational life of the turbines (Hodges and Pierce 2011).

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Fig. 1 Aeroelasticity fields

As a response to external forces, the elastic structures will vibrate with certain mode shapes that are defined by their frequency and shape. If the structure reaches an equilibrium state, then the external forces damp out all these vibrations. If these vibrations have the potential to grow continuously, then the system will perform an oscillation with increasing amplitude tending to structural failure. Designers conduct aeroelastic simulations in the wind turbine design process to ensure that these vibrations are damped in such a way that the structure is safe. Wind turbine aeroelastic designs are therefore concerned with increasing structural damping and distinguishing the excitation frequencies from the natural frequencies. Blade and tower natural frequencies, as well as the multiples of the rotational frequencies, are the most important design considerations to be kept apart from each other. In other words, aeroelasticity of wind turbines includes not only the static deformations of the blade but also their dynamic deformations, rotor/tower coupling instabilities, and flutter. For modern multimegawatt wind turbines, the aeroelastic instability is expected to occur and can be divided into two main categories: classical flutter for attached flow conditions and stall-induced vibrations during separated flow. Stall-regulated wind turbines do operate in stall conditions for definition and are therefore more subjected to stall-induced vibrations during regular operating conditions than pitch-regulated wind turbines (Zhang and Nielsen 2014). On the other hand, there are some load cases where on both turbine typologies those vibrations can occur, for example, during installation and commissioning or when they are parked/idling.

Blade Instabilities The blade has in general six degrees of freedom, two of which are the most critical, which are the in-plane (edgewise) and out-of-plane (flapwise) deflections, as shown in Fig. 2. Flapwise deflections are profoundly affected by the blade aerodynamic

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(a) Flapwise

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(b) Edgewise

Fig. 2 In-plane (edgewise) and out-of-plane (flapwise) blade deflections. Dashed lines are the deformed blades (Sayed 2018). (a) Flapwise (b) Edgewise

loads, while the edgewise deflections are influenced more by the blade mass or, in other words, the gravitational loads in addition to the aerodynamic loads. The terminologies flapwise and edgewise are used to define the deformations with respect to the local chord including the local twist angle which varies from the root to the tip. However, the terminologies flapwise and edgewise will be used here to define the rotor out-of-plane and in-plane deflections, respectively. These two deflections result from the two blade force components, which are the normal and the tangential forces. The normal force is an out-of-plane load, and its integration over the blade radius results in the wind turbine thrust. The edgewise deformation is the result of the tangential force (in-plane) or, in other terms, the force which creates the torque to turn the rotor. Lundsager et al. (1981) investigated the flapwise vibration (defined as a critical instability) on the Nibe A wind turbine in the stall region. However, Thirstrup et al. (1998) found that the flapwise vibrations are damped out at most of the wind speeds except the high ones (higher than 15 m/s). As most of the modern large wind turbines are pitch regulated, the flapwise instability is of less concern. Jeong et al. (2014) showed that

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the rotor wakes may have a remarkable influence on the dynamic stability of the edgewise modes but not on the flapwise modes. For stall-regulated wind turbines, flapwise and edgewise vibrations are of great concern, especially for large wind turbines as their sizes increase dramatically. The edgewise instability was explained by Thirstrup et al. (1998) and concluded that this instability is always present. Additionally, due to the local twist distribution of the blade, these vibrations are never completely in the rotor plane but have some flapwise components which damp the vibrations in edgewise direction. Therefore, the blade should be designed to vibrate more in flapwise direction to damp out and control the edgewise vibrations. Chaviaropoulos (1999) investigated the combination of both flapwise and edgewise instability. Higher stability was reported using a simplified blade model with thicker airfoils and smaller damping of the structure.

Rotor/Tower Coupling Instabilities Including the dynamics of a single blade or the rotor in coupled simulations may be not sufficient to conduct exhaustive aeroelastic studies since the vibration of the other turbine components, e.g., tower and nacelle, may couple with the blade vibrations through the connections between them. Van Holten et al. (1999) found several instances in the literature of coupling between the tower translation mode and the edgewise blade vibration. Coupling the advancing lead/lag (edgewise) mode with either the second tower bending mode or the first tower torsion mode is another way for reaching this instability. As the angular velocity of the rotor corresponds to half the frequency of the first tower torsion mode, the first torsion mode of the tower is coupled with the lead/lag mode. Another instability occurred when the rotor angular velocity was corresponding to half the frequency of the second tower bending mode (Pavel and Schoones 1999 and Pavel and van Holten 2000). Flutter Flutter is a two-dimensional instability, much stronger than stall-induced vibrations. Here the combination of the torsional mode along the blade axis with the flapwise modes becomes critical. In fact, the torsion changes the angles of attack which results in a change of the lift in a disadvantageous phase with the out-of-plane deformations. Hansen (2007) suggested that the most important parameters that influence flutter are the flow attachment around the blade, the blade stiffness, tip speeds, and the relative position between center of mass and aerodynamic center for the cross sections. Hansen (2002) performed stability analyses of a large variable speed and pitch turbine showing that flutter is strongly influenced by the dynamic behavior of the entire turbine, and therefore only blade analyses are not enough. Bir and Jonkman (2007) showed how particular idling conditions of the turbine lead to instabilities that are sensitive to rotor azimuth and nacelle yaw positions. For small wind turbines, the flutter speed was determined to be about five times the rated wind speed, while it was found to be two times the rated wind speed for large wind turbines (as shown, e.g., in Lobitz 2004). Therefore, performing aeroelastic simulations is essential for large wind turbines.

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Fig. 3 Aeroelastic models with various aerodynamic and structure models

Aeroelastic Models A wind turbine aeroelastic model must contain an aerodynamic model to determine the aerodynamic loads and a structural model to calculate the structural dynamic response. As described in the previous sections, the components of the aeroelastic modeling of wind turbines are presented in Fig. 3. Most of the existing aeroelastic codes are summarized in Table 1 (Vorpahl et al. 2014; Vorpahl 2015 and Robertson et al. 2013, 2016). As shown, almost all the codes are time domain solution based, requiring post-processing (such as fast Fourier transform) technique to find out the instabilities of the wind turbine. Few codes are frequency domain solution based such as the aeroelastic stability tool for wind turbines (HAWCStab) (Hansen 2004) where the identification of the system instabilities is predicted directly from the results. As summarized in Table 1, to date, the most widely adopted aeroelastic simulation approach for wind turbines is based on BEM theory and MBS, allowing efficient and acceptable predictions of both rotor aerodynamics and nonlinear structural response (Riziotis et al. 2008; Jeong et al. 2011). The BEM-based method is computationally efficient and provides reasonable estimations of the aeroelastic behavior of flexible blades, but some aerodynamic phenomena are not captured accurately. BEM theory is mainly based on the 1D momentum equilibrium, and it neglects any 3D effects. Therefore, several corrections are needed to account for complex unsteady and 3D aerodynamic effects (Madsen et al. 2012). As the rotor diameter increases, including more complex geometries such as pre-cone, rotor tilt, and pre-bended blades, symmetric inflow conditions cannot be assumed anymore. BEM-based models can be extended with dynamic inflow models in order to take into account the time delays due to sudden changes in the inflow velocities, like in Henriksen et al. (2013) and Pitt and Peters (1980). Anyhow strong oscillatory load behavior cannot be predicted as well due to the model’s limitations (Yu and Kwon 2014).

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Table 1 Overview of existing aeroelastic codes Code ADAMS-Aerodyn ADAMSWaveLoads alaska/Wind ADCoS-Offshore ANSYSWaveLoads ASHES BHawC Bladed Bladed Multibody CAsT DeepLinesWT DHAT FAST FAST-Ansys FAST v8 FAST+CHARM3D FAST+OPASS FAST+UOU FEDEM WindPower FLEX5 FLEX5-AQWA

Developer Aerodynamic model MSC+NREL+LUH+IWES BEM or GDW+DS MSC+NREL+LUH BEM or GDW+DS

Structure model MBS MBS

IFM ADC+IWES ANSYS+LUH

BEM or GDW+DS BEM+DS BEM or GDW+DS

FEM+Modal/MBS FEM FEM

NTNU Risø DTU+Siemens GH GH University of Tokyo PRINCIPIA-IFPEN Germanischer Lloyd NREL NREL+Ansys+ABS NREL TAMU+NREL CENER+NREL UOU+NREL Fedem

BEM+DS BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS BEM BEM+DS BEM BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS BEM or GDW+DS

FEM MBS/FEM FEM+Modal/MBS MBS FEM FEM Modal FEM+Modal/MBS FEM+Modal/MBS FEM+Modal/MBS FEM+Modal/MBS FEM+Modal/MBS FEM+Modal/MBS FEM+Modal/MBS

BEM or GDW+DS BEM or GDW+DS

FEM+Modal/MBS Modal/MBS

BEM or GDW+DS

FEM+Modal/MBS

BEM or GDW+DS BEM BEM FWV BEM or GDW+DS BEM or GDW+DS BEM BEM or GDW+DS BEM BEM

FEM+Modal/MBS MBS+FEM FEM FEM FEM MBS/FEM FEM FEM+Modal/MBS MBS+FEM FEM

BEM or GDW+DS BEM, GDW orFDT BEM

FEM/MBS FEM FEM

Risø DTU Risø DTU+Ansys+Senvion FLEX5-ASAS(NL) Risø DTU+Ansys+Senvion FLEX5-Poseidon Risø DTU+SWE+LUH FloaWDyn UPC FOCUS6 Offshore WMC GAST NTUA HAWC Risø DTU HAWC2 Risø DTU HAWCStab Risø DTU MicroSAS-OWT McDermott+NREL+AUTh MoWiT IWES NK-UTWindClassNK Aerodyn14 OneWind IWES OrcaFlex Orcina PHATAS ECN

(continued)

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Table 1 (continued) Code PHATASWMCFem Riflex-Coupled Samcef Wind Turbines SESAM/DeepC Simo Simo+Riflex+ AeroDyn SIMPACK SiWEC SWT USFOS-vpOne 3Dfloat

Developer ECN+WMC

Aerodynamic model BEM+DS

Structure model FEM

MARINTEK PLM

BEM+FDT BEM+DS or UD

FEM FEM/MBS

DNV MARINTEK MARINTEK+NREL

None BEM BEM or GDW+DS

MBS MBS FEM

SIMPACK BEM or GDW+DS Windrad BEM SAMTECH BEM or GDW SINTEF + NTNU+ Virtual BEM+DS Proto-typing IFE-UMB BEM or GDW

MBS MBS+FEM FEM+Modal/MBS FEM FEM

BEM was used in combination with the Raj 3D stall delay model to predict the aerodynamic forces coupled to a blade modeled structurally by 15 degree of freedom (DoF) elastic beam elements (Liangyou et al. 2009). The coupled model was applied to predict the aeroelastic response of the wind turbine under the effect of the aerodynamic and gravitational loads. BEM coupled to a multi-body formulation was also utilized to find the aeroelastic response of the wind turbine (e.g., in Fanzhong et al. 2008). The structural model was based on MBS, and a proper orthogonal decom- position (POD) method was applied to extract the dominant eigenvalues of the system. Classical flutter analyses were conducted, and it has been found that the reference wind turbine used in the simulations does not experience flutter under the design operating conditions. Xiong et al. (2010) employed BEM to find the dynamic response of a horizontal axis wind turbine (HAWT) blade using FEM. The blade dynamic response was obtained under aerodynamic, centrifugal, and gravity forces by modeling the blade as a cantilever two-node beam. Dynamic inflow and dynamic stall were taken into account when calculating the aerodynamic loads. It was concluded that the blade suffers from large deflections and significant vibration that were damped out by the aerodynamic and centrifugal stiffening. Moreover, Hamdi et al. (2014) utilized analytical and numerical dynamic analyses of a single blade under aerodynamic, centrifugal, gravity, and gyroscopic loads. BEM and FEM were also employed to model blade aerodynamics and structure. A new aeroelastic simulation method was presented in Lee et al. (2012). The method used the modified strip theory for predicting the unsteady aerodynamic forces which was coupled with a structural model based on flexible multi-body dynamics. The rotor rotation was included in this study, and the results were compared to other BEM results and showed very similar results. Comparative aeroelastic analysis of the NREL phase VI blade was performed by Sargin and Kayran (2014). Quasi-steady and transient simulations were conducted based on

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BEM aerodynamic model, and the blade was modeled employing FEM composite shell model by using MSC Nastran. The results showed good agreement of the predicted deflections by the two methods. It was concluded that the quasisteady aeroelastic simulation is a good predictor for the preliminary design stage. Aeroelastic instabilities using BEM-FEM model of stall-regulated wind turbines operating around and beyond rated speed were studied by Riziotis et al. (2004). It was concluded that the unsteady aerodynamics and the dynamic stall model play a dominant role in the stability limit prediction and should be included. It was advised that CFD could enhance the predictions of the stability limits of the wind turbines. Currently, FAST (Jonkman and Buhl 2005) developed by the NREL, HAWC2 (Larsen and Hansen 2007) developed by Risø, and FLEX5 developed by Øye (1996) at the Technical University in Denmark are some of the most popular aeroelastic models used in industry. Both have some common features as the same aerodynamic model which is the BEM model with some corrections for 3D effects, dynamic wake, dynamic stall, and skewed inflow. Recently, offshore modeling capabilities were added to both solvers by including a model to determine the interaction between the ocean and seafloor with the turbine support structure. But they have different structural models. A combined mode shape formulation and MBS is employed in FAST. The blades and tower are modeled by mode shapes, and the other components are modeled by MBS formulation. In HAWC2, all the turbine components are modeled by MBS with Timoshenko beam elements for the blades and tower. Hansen (2007) and Hansen et al. (2006a, b) have used HAWC2 and HAWCStab to find out the wind turbine instability characteristics and to draw conclusions about each single aeroelastic instability. The influence of large deformations on the wind turbine load simulations was investigated by modifying the horizontal axis wind turbine simulation code (HAWC) (Larsen et al. 2004). Three different approaches were studied such as including small deflections around an initially large deflected blade, multi-body formulation which includes nonlinear effects, and the third method that is the corotational formulation where a separate coordinate system is defined for each element. Power reduction was found due to the inclusion of the large deformation as a result of the reduction of the effective rotor diameter. The aeroelastic damping as well as the natural frequencies of the Vestas 3 MW wind turbine were determined by HAWC2 (Shirzadeh et al. 2012). Two test cases were considered, and the numerical results showed a good agreement with the experimental measurements. HAWC2 has been also used to perform unsteady aeroelastic simulations in the project AVATAR (Croce et al. 2017), where the InnWind baseline rotor designed by DTU (wind turbine description in Bak et al. (2013a)) has been compared to an aerodynamically and structurally optimized version of it. This one showed a positive performance compared to the baseline, with increased annual energy production delivery while operating within a lower load envelope. From the previous literature, it can be concluded that the simulation results based on engineering models agree well for the structural part but limit the overall fidelity

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of the aeroelastic models to the accuracy of the aerodynamic predictions using BEM. Hence, CFD is a more advanced aerodynamic model to be applied to predict the unsteady aerodynamic loads accurately and then predict the complex interaction of the loads with the elastic deformations of the blades. In principle, all occurring viscous, 3D, and unsteady aerodynamic effects can be captured by CFD. Nevertheless, high-fidelity CFD is much more expensive compared to the traditional BEM method but is capable of resolving the unsteady and complex flow field around the wind turbine including the aerodynamic interaction between all components of the wind turbine and with atmospheric/turbulent inflow. Up to now, in the wind energy literature, high-fidelity CFD-CSD aeroelastic investigations have found little attention; only recently they have come into focus due to increasing rotor radius. CFD-based FSI methods are mainly used in research and development. Many of the recent 3D CFD-CSD investigations have been performed on the generic NREL 5 MW rotor (e.g., Yu and Kwon 2014; Bazilevs et al. 2011b; Hsu and Bazilevs 2012). Yu and Kwon (2014) investigated the aeroelastic response using a loosely coupled CFD-CSD method solving the incompressible Navier-Stokes equations on unstructured grids for the blade forces and nonlinear Euler-Bernoulli beam for the structure. The interaction between the fluid and the structure was carried out after each rotor revolution on a periodic basis. Bazilevs et al. (2011b) and Hsu and Bazilevs (2012) utilized a Fluid Structure Interaction (FSI) between a low-order arbitrary Lagrangian-Eulerian Variational Multi-Scale (ALE-VMS) flow solver and a Non-Uniform Rational Basis Spline (NURBS)-based structural solver. CFD-MBS coupled result of the NREL phase VI rotor was investigated by means of a new developed coupling algorithm between a free general-purpose MBS solver called MBDyn and an Unsteady Reynolds-Averaged Navier-Stokes (URANS) solver called TURNS (Masarati and Sitaraman 2011). Tojo and Marta (2012) implemented a FSI loose coupling approach and applied it to perform coupled simulations between OpenFOAM and a structural solver. The authors’ conclusion was that the blade structure should be modeled using a shell model including the composite material properties to achieve accurate FSI results. In Stettner et al. (2016), BEM with different corrections and dynamic inflow was coupled to different beam implementations in order to analyze the stall-induced vibrations of the AVATAR rotor. It was concluded that the dynamic stall model has the greatest influence on the rotor blade oscillations. Most publications on CFD-based FSI are limited to the coupling of the rotor blades, i.e., the interaction of the elastic blade deformation with the aerodynamic loads. A wide variety of solvers are applied, with almost all methods using a partitioned approach. Two basically independent solvers are coupled together with this approach, one for CFD (see also  Chap. 14, “CFD for Wind Turbine Simulations”) and one for the structural solver. Dose et al. (2018) recently presented such a method, which couples the flow solver OpenFOAM with the FEM-based beam solver BeamFOAM. They investigated different configurations of the NREL 5 MW turbine’s rotor and found that the higher the yaw angle, the higher the deviation of the aerodynamic response between only CFD and coupled simulations.

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Sayed et al. (2016) coupled the flow solver FLOWer and the structure solver Carat++ and investigated the rotor blade deformation of a generic 10 MW turbine. Compared to the pure CFD solution, they showed a deviation in the span-wise aerodynamic loads and a decrease in the mean aerodynamic loads on the rotor. While Dose et al. (2018) and Sayed et al. (2016) chose a weak coupling with exchange of the loads and deformations in each time step, Yu and Kwon (2014a) presented an approach in which the loads and deformations are transferred only once per revolution and are regarded as periodic. They also examined the NREL 5 MW turbine with yawed flow and sheared inflow. They used a Reynolds-averaged Navier-Stokes (RANS) solver and a structure solver with nonlinear Euler-Bernoulli beam theory for the rotor blade deformation. In Heinz et al. (2016), a coupling between the flow solver Ellypsys3D and the aeroelastic solver HAWC2 is applied to NREL 5 MW rotor wind turbine, and the results are compared to the BEM-based results of HAWC2 alone. Sharing the same structural model, it was possible to directly analyze the conditions at which CFD is necessary. They found that for uniform and yawed inflow conditions, both models show good agreement, whereas in case of emergency shutdown and a standstill vibration simulation, some discrepancies can be still recognized. While Heinz et al. (2016) considered only stationary inflow, Li et al. (2017) also considered turbulent inflow that is synthetically generated by means of the Mann model (Mann 1994) described in Li et al. (2015). They coupled a hybrid RANS/large eddy simulation flow solver with a MBS structure solver whereby the structural deformations of blades and tower as well as changes of speed, pitch, and yaw angle can be transferred to the CFD network. In contrast to Heinz et al. (2016), they considered tower and nacelle in the CFD simulation. Using coupled simulations of varying complexity, they examined the drive train dynamics and the wake of the NREL 5 MW turbine, among other things. Streiner (2011), Meister (2015a), and Klein et al. (2018) sequentially developed a coupling of the CFD code FLOWer to the MBS solver SIMPACK to take into account the structural deformation of the rotor blades of wind turbines, including drive train torsion, foundation, and controller input to examine the origin of lowfrequency noise sources and seismic excitations. Besides the coupling of separate solvers, another recent trend is to develop/use tools that are inherently capable of solving problems involving several physics such as Kratos Multiphysics (Dadvand et al. 2010). Thus, many problems arising from the coupling of different solvers can be avoided, such as different discretization techniques or the transfer of data between the solvers. In the following sections, further detail is given regarding the mostly used aerodynamic and structural models.

Wind Turbine Aerodynamic Models BEM is the cheapest and simplest aerodynamic model as applied, for example, in Leishman (2002), Gerhard et al. (2013), and Liu and Janajreh (2012),

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but it relies on airfoil data and empirical correlations for the simulations. See also  Chap. 13, “Pragmatic Models: BEM with Engineering Add-Ons” for a more comprehensive discussion on BEM models. To obtain more physical details and still retain high computational efficiency, 3D inviscid aerodynamic models were introduced, including lifting line methods (e.g., in Whale et al. 1999), panel methods (e.g., in Hess 1975), and vortex lattice methods (VLM) as discussed, for example, in Whale et al. (1999) and Gebhardt et al. (2010). Employing BEM as well as the other simplified methods in the simulations, some discrepancies in the prediction of the performance at some conditions arise because potential flow methods cannot handle viscous effects and separation. In the past decade, motivated by the increase of the computational power, literature showed a growing number of CFD-based aerodynamic analyses of wind turbines. 3D RANS calculations are employed, although they are still mostly utilized in case of fully turbulent flows and fully rely on the applied turbulence model. When modeling a blade (or in general a system) by means of CFD, the surface and the complete volume of the flow field need to be discretized into millions of cells. The CFD solver provides then as output for each surface cell the pressure and friction force (or coefficient). CFD models of wind turbines have been widely used, as in Sørensen et al. (2002), Hansen et al. (2006b), Zahle and Sørensen (2007), and Gerhard et al. (2013), to capture the flow phenomena that cannot be resolved using simplified methods. In addition, the prediction of the wind turbines’ load fluctuations needs accurate, robust, and reliable aerodynamic predictions as shown in Gerhard et al. (2013), Li et al. (2012), Keerthana et al. (2012), Bergmann et al. (2012), and Anjuri (2012). Many studies of large wind turbines by means of CFD were published. 3D-RANS simulations on the NREL 5 MW wind turbine were carried out, for example, by Bazilevs et al. (2011a, b), Yu and Kwon (2014), and Li et al. (2012). Many benchmark studies have proven the ability of the 3DRANS analyses on the NREL phase VI rotor to reproduce the experimental results of the wind turbine as in Duque et al. (2003), Tongchitpakdee et al. (2005), SezerUzol and Long (2006), Sezer et al. (2009), Schmitz and Chattot (2006), and Anjuri (2012). Moreover, a series of computations of the NREL phase VI rotor were conducted by Sørensen et al. (2002). The authors investigated the aerodynamic 3D effects on an isolated rotor by means of RANS simulations using the multiblock, FVM, incompressible RANS flow solver EllipSys3D. In addition to RANS, and for massive separated flows, more costly but most advanced CFD methods are employed such as LES or DES in which the LES is used in the separated regions combined to RANS inside boundary layers. In Johansen et al. (2002), for example, DES simulations of the NREL phase VI have been performed using EllipSys3D and compared to experimental results. DES simulation results also for a 5 MW wind turbine in an unsteady atmospheric boundary layer were presented in Meister et al. (2014). The simulations were performed with 60 s periodic time series of LES data that were fed into the DES simulations. By this approach, the impact of the inflow turbulence on the unsteady rotor loads was examined. Many CFD investigations have been carried out using the CFD, FVM compressible flow solver FLOWer (Raddatz 2009). For example, the influence of complex terrain

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(Schulz et al. 2016), atmospheric turbulent inflow (Schulz et al. 2018), trailing edge flaps (Jost et al. 2017), and the usage of hybrid RANS/LES numerical models (P. Weihing et al. 2018) have been analyzed. In Réthoré et al. (2011), the MEXICO rotor in an open wind tunnel has been modeled by means of CFD in both as full rotor and actuator disk using Ellipsys. For a more comprehensive and detailed discussion of wind turbines’ CFD, see  Chap. 14, “CFD for Wind Turbine Simulations.”

Wind Turbine Structural Models For conducting aeroelastic simulations of wind turbines, a suitable structural model and discretization technique is needed to determine the structural dynamic response. The models can be roughly categorized according to the dominant load-carrying behavior and thus the prevailing kind of DoFs such as 1D equivalent beam models or 3D finite element (FE) models. These approaches vary in their complexity and range of applicability and, hence, resulting numerical effort and accuracy. Sectional beam properties are needed for dimensionally reduced 1D methods, which is one reason for the reduced accuracy and level of detail in comparison to a 3D method. In contrary, 3D elements can be applied to model the wind turbine with either isotropic or anisotropic material properties. The structural analysis might be performed under the effect of various load cases such as the gravitational loads only or the combined aerodynamics and gravitational loads. In industry, the structural model is chosen based on the proposed design level. Due to their low setting up and computational efforts, 1D beam models are the mostly used in wind turbine computations, both in the industry and in the literature. A 1D beam model can be created based on mainly three different approaches that are shortly described in the following: modal shape function, Multi-Body Dynamics (MBD), and FEM.

Modal Shape Function The modal shape function is widely used to model the wind turbine’s structure at low frequencies because of its simplicity and low computational cost. It reduces the total number of the system’s DoFs in the dynamic system to a relatively small amount. The model is essentially based on a linear combination of physically realistic basis functions that describe the deflection shape corresponding to the lowest eigenfrequencies. This is the structural model of the widely used commercial aeroelastic simulation tool FLEX (Øye 1996). The results of this model show good agreement with the measurements, but one has to determine the relevant DoFs needed to sufficiently describe the realistic deformation of the wind turbine components. More details about the method can be found in Hansen et al. (2006b). Multi-Body Dynamics The dynamic interaction between the wind turbine components should be considered for more detailed and accurate structural simulations. This includes the motions of the different turbine parts relative to each other through connections where the loads and deformations are transferred from one component to another (e.g., shown

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in Bauchau (1998)). For this purpose, a numerical simulation approach in which systems are composed of various rigid/elastic bodies has been developed and called MBD. In this approach, force elements (such as spring dampers) and/or kinematic constraints (such as joints) are utilized to model the links between the various bodies. Its computational cost is moderate, and it provides reasonable results. MBD has been widely used in vehicle dynamics and automotive suspension design (e.g., shown in Blundell and Harty 2004). Moreover, it is employed in the biomechanical analysis as the human motion analysis is mostly relying on the use of multi-body formulations applied as kinematic or dynamic tools (e.g., Ambrósio and Kecskeméthy 2007). It is also adopted to be used for comprehensive wind turbine simulations (e.g., Pfeiffer and Glocker (1996), Bauchau (1998), and Bauchau and Hodges (1999)) in many different simulation tools such as FAST (Jonkman and Buhl 2005), ADAMS (McConville and McGrath 1998), BLADED (Bossanyi 2003), and HAWC2 (Larsen and Hansen 2007). In different recent wind turbine studies, the MBD method has been applied and improved. A new structural dynamic analysis method for HAWTs was implemented in Patil et al (2001) and Lee et al. (2002). A multi-flexible body system with mixed rigid/flexible bodies was utilized to represent the wind turbine components. The rigid bodies were modeled using Kane’s method, where geometrically exact nonlinear beam elements adapted from FEM were applied to model the flexible bodies. The developed method predicted successfully the natural frequency of the system. Larsen et al. (2005) applied the MBD-based code HAWC2 on a 2 MW offshore turbine to investigate the effects of the tower dynamics. Masarati and Sitaraman (2011) proposed a free general-purpose multi-body solver MBDyn coupled to an URANS solver and developed for wind turbine applications. Beyer et al. (2013) employed the MBD method to find floating offshore wind turbine dynamics by coupling SIMPACK to a CFD solver. Coupling SIMPACK to FLOWer utilizing explicit and implicit FSI approaches was implemented by Streiner (2011). Meister (2015b) improved and used the coupling between FLOWer and SIMPACK to investigate the interaction between the turbulent atmospheric inflow and a 5 MW offshore wind turbine.

Finite Element Method Generally, wind turbines can be modeled by finite elements with different complexities ranging, e.g., from a beam element model with linear isotropic material properties up to a solid element model including anisotropic material properties. The higher the accuracy, the higher the computational cost, and the 3D FEM can be considered as the most computational expensive and advanced approach applied in wind turbine simulations, by which detailed stress distributions within the components of the turbine can be determined. Aeroelastic simulations based on a 3D FEM shell model coupled to a simplified aerodynamic model such as BEM were utilized (e.g., in Verelst 2009). It is an efficient approach, but it has not been widely used in industry because of its higher computational cost. Recently, a 3D FEM shell model was utilized in the aeroelastic simulations of a wind turbine coupled to CFD (e.g., Yu and Kwon 2014b, Bazilevs et al. 2012, 2011b).

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Flow Solver: Modeling and Simulation Techniques There are some basic requirements needed to couple a CFD solver to a structural dynamics solver. These requirements are introduced in the following sections.

Load Integration The output of RANS simulations usually involves the surface pressure coefficient cp and the friction coefficient cf of the field solution (see also  Chap. 14, “CFD for Wind Turbine Simulations”). These coefficients are presented at the surface faces/cells and cannot be directly utilized for structural dynamic simulations. Therefore, a load integration algorithm is needed to collect the force contributions of all surface grid cells. An example of a load integration algorithm is presented by Kranzinger et al. (2015). In this algorithm, the complete surface (e.g., wind turbine blade) is collected, and then all the cells are disassembled into triangles (grid triangulation) which are sorted afterward according to their center of area into an octree. If surface overlapping is present, it should be taken into account using overset meshing techniques. Therefore, an algorithm was introduced by Kranzinger et al. (2015) based on an octree data structure to find the overset. Assuming that the surface grid is triangulated, the forces and moments can be calculated by the integration over every triangle as follows: fi = (cfi − ((cfi · ni ) + cpi ) ni )Ai

(1)

m  i = fi × C

(2)

where i is the cell number, n is the surface normal vector, f and m are the force and moment vectors, respectively, A is the triangle area multiplied by an overlapping factor, and C is the vector of the element centroid or the reference point about which the moments are calculated.

Mesh Deformation Approaches The flow solver receives, in general, the deformations from the structural solver in the form of surface deformations at the defined coupling interface. This coupling interface could be, for example, predefined node positions along the blade in case of modeling the structure by beam elements or the whole structure surface in case of high-fidelity structural models such as a shell model. The deformations should then be applied to the whole CFD volume mesh to deform it through an appropriate mesh deformation techniques. The mostly used mesh deformation methods are summarized in Fig. 4. They are classified into three groups which are point-by-point methods, mesh connectivity-based methods, and the combined one of the hybrid methods.

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Fig. 4 Existing mesh deformation schemes

Point-by-Point Schemes In these schemes, the mesh is deformed by interpolating the nodal deformations of the interface nodes to the remaining volume nodes. There is no mesh connectivity required to use these schemes which allow to handle the mesh with hanging nodes and can easily be implemented for parallel computing (Luke et al. 2012). Different approaches are classified as point-by-point schemes (Fig. 4). One of these is the Radial Basis Function (RBF) approach (Buhmann 2000; De Boer et al. 2007) which is very popular in wind energy applications. This approach will be discussed in more detail since it is the one used later in Sect. “Simulation Results” presented in this chapter.

RBF Approach In this deformation approach, the CFD volume meshes are treated as discrete data in three-dimensional space where the RBF can be utilized for the interpolation of these data (Buhmann 2000). Based on the surface deformation of a certain moving grid, the RBFs are used for inter- polation of moving 3D multigrid volume meshes. This mesh-deforming technique is independent of grid connectivity which follows any structured, unstructured, and overlapping mesh setup to be easily handled (Kranzinger et al. 2015). Therefore, the volume mesh deformation can be interpreted as an interpolation of the known surface mesh at some discrete points based on the RBF method. The RBFs are real-valued functions where their values are only distance dependent and can be defined as φ ( x , c) = φ (|| ( x − c) ||)

(3)

where φ is the RBF function, x is the vector of the evaluation point, and c is the vector to the reference point. Such a function is defined for each data point, and the total sum of them can be used for the interpolation. In CFD, the volume mesh deformation is defined by a space offset for the translation and a linear function of

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the translation for the rotation in the space. Therefore, the displacement of a certain point p in the volume mesh can be determined as follows: u (p)  =

k max

     wk φ || p − psurf acek || + q (p) 

(4)

k=1

where k represents the current point data in the volume mesh, wk is the weighting factor for every RBF, and q(p)  is an arbitrary linear polynomial. The performance of a RBF-based deformation algorithm depends mainly on the matrix inversion and the application of the RBF coefficients on the whole volume nodes. The matrix inversion could be accelerated by different mathematical matrix formulations such as LU decomposition or Gaussian elimination. Another important factor in choosing a deformation algorithm is its ability to handle overlapping grids in case of the flow solver that uses an overlapping technique to assemble the grids.

Mesh Connectivity-Based Schemes Usually these techniques are based on certain physical analogies, such as elasticity or diffusion. The most popular approaches of this mesh deformation technique are the spring analogy, the elliptic smoothing approach, and the elastic analogy. A short overview of these approaches is presented subsequently.

Spring Analogy Approach The spring analogy is the most popular connectivity-based deformation scheme, where the mesh is represented by a set of springs, in which stiffness is related to certain geometric quantities, such as the edge length (Batina 1990). Because of the so-called edge crossovers, this scheme cannot prevent negative cells. In order to increase the robustness of this scheme, several improvements were introduced such as the torsional (Farhat et al. 1998a), semi-torsional (Zeng and Ethier 2005), ballvertex (Bottasso et al. 2005), and ortho-semi-torsional (Markou et al. 2007) spring analogy. However, the computational cost is increased due to these improvements.

Elliptic Smoothing Approach In this approach, the mesh deformation is modeled as a diffusion problem. This mesh deformation method was developed for wind turbines (Horcas et al. 2014). It is a fast and robust method that uses the so-called elliptic smoothing technique to find the new positions of the deformed grid nodes by solving a system of linear equations:  ∇(  x − xref ) = 0) ∇(ω

(5)

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where x and xref represent the deformed and undeformed position of a mesh point in the volume mesh, respectively, and ω is the local diffusivity factor. The deformation of small 3D volume cells is limited by ω to preserve the quality of the final mesh. Therefore, the elliptic smoothing is not suitable when dealing with large deformations even if the diffusivity is controlled by the local diffusivity factor. The “smoother” used in such approach is mostly the Laplacian equation (Löhner and Yang 1996) which limits the method to handle only small deformations. This limitation was overcome by a study by Helenbrook (2003) utilizing a biharmonic operator to allow a second set of boundary conditions to maintain the near-boundary mesh quality, but it leads to more computational cost.

Elastic Analogy Approach This method was introduced to increase the quality of the deformed mesh results from the elliptic smoothing approach. Based on quaternion algebra, Samareh (2002) introduced this method. Instead of modeling the mesh deformation as a diffusion problem, it is modeled based on the elasticity of the structure (Lynch and O’Neill 1980). In this approach, the CFD mesh is considered as an elastic continuum (Horcas et al. 2014). A heterogeneous distribution of the elastic properties is needed to control the orthogonality of the near-wall mesh and the overall quality of the deformed mesh (Yang and Mavriplis 2005).

Hybrid Schemes Recently, a hybrid (mixed) scheme was developed that combines the advantage of the other two mesh deformation schemes. This combination leads to higher robustness as well as higher efficiency methods. One of these schemes is presented by Lefrançois (2008), in which a robust method is applied to a coarse-level grid and the interpolation is then performed on a finer grid. Therefore, it offers a high robustness at low cost. It has also been shown that hybrid approaches are effective in moving viscous meshes by adapting a different method to the viscous layers than to the rest of the mesh. This enables to handle cells with high-aspect-ratio characteristic (McDaniel and Morton 2009). In general, there is no best mesh deformation method for all applications, and according to the desired robustness and efficiency, the mesh deformation scheme has to be selected. The level of the robustness and the efficiency of some of the mesh deformation schemes are presented in Fig. 5. Changing any parameter in the described schemes might lead to different levels of robustness and accuracy. However, it can be concluded that for small deformations, the mesh connectivity-based schemes converge fast. The point-bypoint schemes converge also fast since they are independent of the deformation compared to the mesh connectivity-based schemes (Uyttersprot 2014). Nevertheless, fast convergence could not be achieved in case of 3D problems, and then these schemes become computationally expensive. The structural analogies could suffer in terms of robustness from poor division of displacements to the far field,

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Fig. 5 Robustness and efficiency of different mesh deformation approaches

especially when the variation in structural properties is not carefully selected (Uyttersprot 2014; Selim et al. 2016). Usually, the RBF delivers the highest quality of the deformed meshes. However, the orthogonality close to the deforming surface may deteriorate in some cases, such as the bending of a beam due to the fact that the displacements are not linked in different directions. For a survey on the recent development of the mesh deformation approaches with their advantages and disadvantages, refer to Selim et al. (2016).

Navier-Stokes-Based CFD with Large Deformations and Surface Motion Treatment of Flows with Changing Boundaries: ALE with Mesh Moving and Embedded Techniques Solving the Navier-Stokes equations by means of CFD methods needs a careful treatment of the flow with changing boundaries. There are different methods introduced to include the effects of the mesh movement in the solution of the flow field. Some of them are shortly presented in this section.

Immersed Boundaries Most mesh moving approaches of Cartesian grids use an Eulerian view of the fluid, while the immersed boundary mesh moving technique is based on a Lagrangian approach. In this method, the obstacles of the fluid flow are not mapped to the grid point but to the grid cells. Therefore, the Navier-Stokes equations are solved on this new clipping domain which is called “inner cells,” and the boundary conditions are applied by modifying these equations (Mittal and Iaccarino 2005). In this method, an additional force term is added to the momentum equation to simulate the drag force of the obstacle. The calculation of this term for a given geometry is often done by some kind of triangulation of the mesh describing the geometry.

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Embedded Boundaries Similar to the immersed boundaries, the embedded approach does not require a boundary conforming mesh. Zorrilla et al. (2020) introduced a method with which the boundaries of an object are reconstructed by cutting it out of a regular background grid. Together with an appropriate refinement toward the boundary, the accuracy can be increased. The method is able to handle surface- and volumebased objects as well as “dirty” geometries/non-waterproof computer-aided design (CAD) models, which are very common in industrial environments, for example, because of tolerances in the models. Moving objects requires updating the cut in the background grid but allows for arbitrary movements. Furthermore, the accuracy of the reconstructed boundary is not as good as with a body-fitted approach, but it can be improved with a higher level of refinement.

Arbitrary Lagrangian-Eulerian (ALE) Method The most freedom is provided by the ALE of grid. Therefore, the grid vertices have no fixed locations determined by a global variable, such as the mesh width in regular Cartesian grids, but they may be arbitrarily placed. This applies not only when the grid is initially created in the beginning but can also be done during the simulation if a suitable mapping is used to transfer the solution between two grids. This allows the grid to follow the boundary movement, thus providing a Lagrangian view of the geometry, while the fluid is still treated in an Eulerian way. This approach is referred to as the ALE approach (Donea and Huerta 2003). This method works well for small grid changes where the vertices are only slightly moved. However, if the grid receives strong changes due to large movements or object rotations, this method is likely to produce a distorted grid leading to an unstable system (Ferziger and Peric 2012). In such cases, the grid can be recreated to avoid this problem, which is a very costly task for unstructured grids and thus has a major impact on run time performance (Mittal and Iaccarino 2005). However, this approach provides highly accurate results for moving boundaries also in cases such as unstructured grids and problems that contain only small movements. On the other hand, it lacks the ability to simulate arbitrary scenarios with arbitrary motion. More details are presented in Donea et al. (2004).

ALE-Variational Multiscale Methods (VMS) This method represents a variational multiscale version of the ALE method (Korobenko et al. 2018). Using this method, high-resolution representation of the boundary layer is obtained by controlling the mesh resolution near walls and interfaces. The mesh is controlled by moving the CFD mesh to trace the fluidstructure interface. ALE-VMS gives highly accurate results in case of turbulent

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flows but requires a relatively fine resolution of the boundary layers. This makes ALE-VMS an expensive approach for wind turbine applications because of the high Reynolds number and the involved turbulent boundary layer (Korobenko et al. 2018). Therefore, an improvement of this issue was overcome by introducing a weakly enforced essential boundary condition formulation in the presence of unresolved boundary layers (Bazilevs and Hughes 2007 and Bazilevs et al. 2007). More details and mathematical formulations of this method are described in Korobenko et al. (2018).

Approaches to Deal with Rotating Components The CFD modeling of a wind turbine involves combinations of rotating components, such as rotor, and nonrotating parts, such as tower and nacelle. The flow solver must be able to model the kinematics of the relative motions properly in order to model the physics correctly. In literature, the following approaches to deal with this issue can be found: 1. Multiple rotating reference frames models (Multiple Reference Frame (MRF) and Mixing Plane Model (MPM)) 2. Sliding Mesh Model (SMM) 3. Overlapping grids using the Chimera technique 4. Embedded approach Multiple rotating reference frames models are used to handle problems that can be approximated with steady-state solutions. In both MRF and MPM, the domain is divided into cell zones separated by an interface. These are then solved with moving reference or stationary frame equations depending on whether the zones are moving or not. The two methods differ on how the interface is treated. In MRF, different translational and/or rotational speeds can be attributed to different cell zones. In this approach, the grids are not moving, and this means that it is assumed to have steady-state flow conditions at the grids’ interface and, in other words, the same velocities at the interface. For this reason, this procedure is called also “frozen rotor approach.” At the interface, a reference frame transformation is applied so that the variable values from one zone can be used for the flux calculation of the next one. This method is suggested in case of steady-flow computations with a weak interaction between the rotating and nonrotating zones, while for unsteady calculations, a more advanced model like the SMM is recommended. When the flow at the interface is not uniform, the MPM should be used in that each fluid zone is treated as a steady-state problem and the data is passed within the zones as boundary conditions that are either averaged or “mixed” at the mixing plane. This procedure of mixing is deleting any kind of unsteadiness at the interface, leading again to a steady-state result. The advantage of MPM is that it can provide reasonable approximations for unsteady time-averaged flows being a good costeffective alternative to the more complicated and time-consuming SMM. The SMM allows a time-accurate solution instead of a time-averaged one. Two different cell zones are bounded by an interface zone, along which they move

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relative to each other in discrete steps. It is recommended for unsteady flows but is also more costly (SHARCNet 2019). The cell zones, and therefore the interfaces, can have any shape. This model is one of the most popular in literature by ANSYS users, for example, Carrion et al. (2014) and Lee et al. (2015), although it can be found also in combination with the MRF, where this one is used to compute a steady-state flow field solution as an initial condition for the sliding mesh calculation (Alaimo et al. 2015). The sliding mesh model is called in OpenFOAM Arbitrary Mesh Interface (OpenFOAM 2019). In Sicklinger et al. (2015), for example, three different sliding meshes are applied, where two were used to allow for the blade pitch, and a bigger one enables azimuthal rotation; see Fig. 6. Those three models are implemented, for example, in the commercial solver ANSYS (SHARCNet 2019). The Chimera technique was firstly introduced by Benek et al. (1983). The main idea is to discretize the computational domain by means of overlapping grids. This means that complex geometries are broken down into single different components, and for each of them an adapted grid is created. By the definition of holes and ensuring a sufficient number of overlapping cells between two grids, it is then possible to define areas in which the computation is either controlled by only one grid or the interpolation of the two (see Fig. 7). This technique has the main advantage that the grids associated with moving structures move with the body, without deforming or warping. The difficulty of this procedure increases of course with the number of structures that need to overlap at the same time. The previously mentioned embedded approach is able to handle arbitrary movements of the embedded object, which includes rotating components. Since it is a recent development, no applications to wind turbines have been done, but based on its capabilities, it is a promising approach.

Temporal and Spatial Resolution for FSI Simulations The accuracy of CFD calculations is strongly dependent on the used numerical discretization in space and time. This discretization at the same time needs to be a compromise between accuracy of the results and computational costs. When generating the CFD model of a system, it is necessary to know the goal of the simulations, in other words which variables are of interest and where, because this is directly correlated with the characteristics of the mesh. For FSI, accurate prediction of the blade loads is decisive, which requires a high resolution of the blade grid and sufficient development of the wake to capture the induction effect. Nevertheless, when analyzing the influence of the wake generated from a turbine on the aeroelastic response of a second turbine, then a background refinement is necessary on the area where the wake is expected to be present. The FSI grid requirements for the blade mesh are usually not different from the URANS-CFD rigid simulation ones. In order to ensure a correct load calculation on the blade, the boundary layer needs to be resolved in the direction perpendicular to the wall. The approach to do this is dependent on the chosen turbulence model. When the turbulence model provides wall functions (empirically derived equations used to satisfy the physics in the wall region), the first cell center has the only

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Fig. 6 Example of use of the sliding mesh model (Sicklinger et al. 2015)

requirement to be positioned in the log-law region. If no wall function is available, the condition y + ≈ 1 for the first boundary layer cell needs to be fulfilled, in combination with a growth rate usually lower than 1.2. The blade mesh is then strongly dependent on the 3D effects that are naturally occurring on the blade. The thick airfoils in the root area cause strong flow separation, while the blade tip is subjected to recirculation and vortices shedding; that is why a refinement of the grid in these critical areas is necessary. When performing FSI simulations, an additional requirement is necessary if the Chimera overlapping technique is applied (see Section “Approaches to Deal with Rotating Components”). In fact, when the blade is subjected to high tip deformations, it is necessary to ensure that at the new blade position, a sufficient overlapping area is guaranteed. In other

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Fig. 7 Sketch of two overset grids (Schwarz et al. 2010)

words, if the background is the overlapping grid to the blade grid, the sizes of the background cells at the new blade tip position need to be comparable to the one at the undeformed blade tip area. The background size has indirect effects on the computed aerodynamic blade loads (see Fig. 8), and in particular two aspects need to be taken into account: 1. The background size in relation to the turbine diameter 2. The cell resolution in order to correctly capture tip vortices and wake The first point is related again to the necessity to ensure that the obtained solution is grid independent. Larger domains result in a higher number of cells, and this increases of course the computational costs. At the same time, the domain size needs to be large enough to minimize the blockage effect (i.e., due to the presence of the turbine itself). Rezaeiha et al. (2017) concluded that using ANSYS to simulate vertical axis wind turbine needs at least 10D distance from the rotor to inlet and outlet and 20D domain width. Klein et al. (2018) using FLOWer for HAWT in CFD-MBD coupled simulations created a 24D long background (the turbine was positioned at 6D from the inlet), with 12D domain width and 8D domain height. Bechmann et al. (2011) using EllypSys3D for the MEXICO rotor placed the outer boundary of the computational domain 7D away, while Länger-Möller et al. (2017) placed the rotor plane for the NREL phase VI 5D from inlet and 10D from outlet. When the grids are ready, it is necessary to perform a grid convergence study to control that the results that are obtained are grid independent. In the literature, it was suggested by Roache (1993) to perform the calculation of a grid convergence index that is depending on a grid refinement error estimator based on the generalized Richardson extrapolation theory. The time step size depends on the desired accuracy and is directly correlated to the calculation stability. Depending on the code, a coarser time step can anyhow lead

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Fig. 8 Example of grid refinement for FSI

to converged results, although not as accurate as from a finer time step calculation. A wind turbine has many sources of unsteadiness like yawed inflow, turbulence, blade flapping, influence of the tower, and separation at the inner region. If unsteady calculations are performed, a well-estimated time step is required in order to capture accurately the unsteadiness. Normally, the definition of the time step is given either in seconds or in terms of azimuthal increment of each time step (0.5°, 1°, 2°, etc.). According to the chosen value, it is defined how often the fluid particle covers the distance of the chord length at a defined radial position. This means that if the time step is so big that the fluid particle covers higher distances than the chord length, then the oscillations due to the unsteadiness will not be fully captured. This is anyhow a minimum requirement, because when dealing with cases that present separation, the flow unsteadiness can only be captured with smaller time steps. For example, Barlas et al. (2012) utilized a time step of approximately 0.0021 seconds to simulate with EllypSys3D, a rotor with trailing edge flaps. The time integration scheme has also influence on the time step, although in literature it was found that most of commonly used CFD code uses the implicit procedure called dual time stepping (FLOWer, THETA, EllypSys3D, WMB). At the beginning of each time step, an estimation of the solution is guessed (according to one or more previous time steps), and the closer it is to the real one, the smaller the necessary number of inner iterations in order to achieve convergence. When a grid is deformed relatively to the local coordinate system, for example, and if the velocity vectors are not accordingly aligned, the convergence behavior of the solution in the time step degenerates. This means that optimizing the initial solution at the beginning of the time step can reduce the number of inner iteration, resulting in smaller simulation time. The size of the time step is also influencing the number of inner iterations that are needed. In general, it can be said that the higher the time step, the higher the number of inner iterations, even if this number is totally code dependent. For

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example, Länger-Möller et al. (2017) performed a time step convergence study for the incompressible URANS code THETA (Dillmann et al. 2014) where a time step equal to 0.5° with at least five inner iterations for momentum, pressure, and turbulence equations was found to be a good compromise between results and costs. Another example is given by Sayed (2018) where a similar analysis has been performed for the URANS code FLOWer and a suitable time step has been found to be 1◦ with 100 inner iterations. In Kim et al. (2016), CENER used, for example, one azimuthal degree with 40 iterations to simulate the AVATAR rotor with the compressible CFD code WMB (Wind Multi-Block), while for the same turbine DTU used for incompressible code EllypSys3D 1200 time steps per revolution, using six sub-iterations in each time step.

Structural Solver: Modeling and Simulation Techniques In order to conduct FSI simulations of wind turbines, besides the aerodynamic model, it is necessary to have a structural model of the turbine to give an appropriate response depending on the desired accuracy of the simulation. Section “Wind Turbine Structural Models” gave an overview of the different available structural models, from linear modal models to full 3D nonlinear FE models. In the following, high-fidelity CSD models based on the FEM and their peculiarities when applied in combination with CFD methods for aeroelastic simulations of wind turbines are discussed. Before this, a brief review over the governing equations, numerical discretization, and suitable element types is given. More information on structural analysis using the FEM can be found in the literature: (Bathe 2006; Belytschko et al. 2013; Hughes 1989; Zienkiewicz et al. 2013b).

High-Fidelity FE Structural Models for Wind Turbines Governing Equations The deformation of a structure can be described by its kinematics. To define the kinematics, the Lagrange description is applied where the physical fields are defined on the structure material points. These points form the structure domain with its initial configuration. When a general structure deforms, a relation between an initial infinitesimal length dx and the new length after deformation dxdef ormed is described by the material deformation gradient F as F =

dx dxdef ormed

(6)

Large deformations and rotations can be described by the Green-Lagrange strain tensor E which is defined as

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E=

 1 T F F −I 2

(7)

where I is the identity matrix. The dynamic equilibrium of a structure is described by the momentum equation given in a Lagrangian frame of reference based on the continuum mechanics assumption. The dynamics describe the force balance on the structure, and for large deformations, where geometrical nonlinearities are not negligible, the following boundary value problem has to be considered (Hojjat et al. 2011): ρs

∂ 2 u = ∇ · (F · S) + ρs bs ∂t 2

(8)

where ρs is the structure’s density, u is the displacement vector of the material point in space, S is the second Piola-Kirchhoff stress tensor, and bs is the specific body force. Equation (8) presents the balance of the inertia, the internal elastic force, and the external body force. A constitutive relation that describes the relationship between stress and strain is required to close Equation (8). For a St. VenantKirchhoff material, the constitutive equation used is S = λs tr (E) I + 2μs E

(9)

where λs and μs ; are the Lamé constants and are directly related to the material properties such as Young’s modulus Es and Poisson’s ratio vs as follows: λs =

Es vs (1+vs )(1−2vs )

and μs =

Es 2(1+vs )

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Numerical Discretization To use FEM for structural simulations, the physical domain must be discretized, and the result is a grid with a predefined number of elements and corresponding nodes. Different elements such as spring, damper, mass, truss, beam, membrane, shell, and solid elements exist including geometrical and material nonlinearity as well as some composite material description for membranes and shells. Different time integration schemes are available such as the implicit Newmark-β and the Generalized-α methods as well as the explicit central schemes; see Belytschko et al. (2013). The following types of structural analysis are used for the simulation of wind turbines: • Static (linear/nonlinear) • Dynamic (linear/nonlinear) — Explicit: central difference — Implicit: Newmark-β, Generalized-α • Eigenfrequency • Linear buckling

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Element Types In principle, three different types of structural elements exist that are suitable for structural models of wind turbines: • Line-element: beam (local-space-dimension: 1D) • Surface-element: shell (local-space-dimension: two dimensional (2D)) • Volume-element: solid (local-space-dimension: 3D) All elements are typically formulated in 3D space (working-space-dimension) to account for the three-dimensionality of the wind turbine. The most common approaches of handling the kinematics of the elements are as follows: • Linear: small strains and small displacements • Small strains with large displacements (translations/rotations): arbitrary rigid body modes can be treated without generating parasitic strains in the element. Typical procedures for treating large displacements are: • • • •

Corotational Total Lagrange Updated Lagrange Fully nonlinear: large strains and large displacements

It shall be further noted that including any type of nonlinearity requires nonlinear solution strategies such as the Newton-Raphson method for solving the resulting nonlinear system of equations which leads to additional iterations within each time step. Depending on the degree of nonlinearity, this can increase the computational cost significantly and should therefore be considered when selecting the modeling approaches to represent the structural model. More on nonlinearity in FEM can be found in Zienkiewicz et al. (2013a) and Belytschko et al. (2013).

Solid Elements Solid elements are directly based on the formulations of 3D continuum mechanics, without any simplifications or dimensional reductions. Hence, they contain every effect that is included in the theory of continuum mechanics. Due to this, they are in principle suitable for every problem. Different ansatz orders are available; most commonly linear or quadratic shape functions are used. The elements can handle different topologies such as triangles and quadrilaterals in 2D or tetrahedral and hexahedral in 3D. They can be formulated for structured as well as unstructured meshes. They can be combined with every constitutive law, since no dimensional reduction is done which can change the way the constitutive laws are used. Even though these elements can represent any structure, they also have their drawbacks such as locking in certain configurations, which requires additional

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element improvements or the higher computational effort compared to the dimensionally reduced elements. Also they need real 3D meshes, which requires a lot of effort from the user, especially for complex models.

Shell Elements Shell elements are typically used to represent structures where one dimension (thickness) is significantly smaller than the other two dimensions, such as thinwalled structures. A set of assumptions and kinematical conditions is used for the derivation of the shell theory. Bischoff et al. (2017) provide a comprehensive overview over shell models. The most important shell models are: • Kirchhoff-Love/three-parameter shell: this shell model neglects the effect of transverse shear strains. • Reissner-Mindlin/five-parameter shell: this shell model includes the effect of transverse shear strains but does not account for variations of the thickness/neglects transverse normal strains. • Layer-wise and higher-order shell models • Continuum shell/degenerated solids: Order of dimensional reduction and discretization is switched, and the shell assumptions are applied to a 3D solid element. Note that the dimensional reduction introduces rotations as additional DoFs for some formulations. Those have to then be treated with additional boundary conditions.

Beam Elements Beam elements are used to model long and slender structures, where one dimension is significantly larger than the other two. The 3D continuum is reduced to a 1D topology. This introduces additional DoFs, e.g., for rotations. Also the application of boundary conditions changes due to the changed topology of the element. It is no longer possible to specify surface loads; only line or point loads are possible. Furthermore, the constitutive laws have to be adapted to the reduced dimensionality, and cross-sectional properties have to be provided. Mainly two different models are used to represent the transverse shear: The Euler-Bernoulli beam model (used, e.g., in Bauchau and Craig 2009) and the Timoshenko beam model (e.g., in Oˇnate 2013). Both models deal with slender beams under bending, torsional, and shear loads with the only difference that the Timoshenko beam model accounts for the shear deformations.

Application of High-Fidelity CSD FE Models to Wind Turbines The requested level of accuracy has a great influence on the computational cost, and the nonlinear 3D FEM can be considered as the computationally most expensive but

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advanced approach applied in wind turbine simulations. The approach to be used is selected based on the required accuracy/expected results; see Gasch and Twele (2010). If, for example, only the overall deformation behavior of the blade is of interest, then beam models are mostly sufficient. For capturing the deformation of the cross section, shell elements are needed. For a detailed stress analysis in the blade or when using an advanced damage model, then it might even be necessary to discretize the blade using several layers of solid elements across the thickness of the blade. Fig. 9 gives an overview of commonly used structural models for wind turbine blades. Due to the high computational cost and modeling effort, if the blades are represented by shell or solid elements, it is in several practical scenarios preferred to use beam elements in the aeroelastic modeling. A beam model provides less accurate but much faster results if combined with BEM (e.g., Malcolm and Laird 2003). Beam elements are a good choice for modeling modern large-scale wind turbine blades as they follow the slender beam theory having huge radius/blade length compared to the maximum chord. With Isogeometric Analysis (IGA) introduced by Austin et al. (2009), another possibility to model the parts of the turbine that are typically represented by a freeform surface description became available. The possible advantage of isogeometric structural analysis is twofold: firstly, one can have a more accurate description of the discretized geometry, and secondly – provided the same geometry description is used in CAD and as shape functions for the structural analysis – one can use the CAD geometry directly as basis for the structural simulation. Typical CAD models consist of trimmed and coupled NURBS patches which leads to the Isogeometric Boundary Representation Analysis (IBRA) (Breitenberger et al. 2015). Based on this CAD oriented extension of the original IGA, one can start the computation with the CAD parameterization (or the NURBS inherently contained refinements), and the structural model provides the exact geometry of the turbine part, e.g., the blade. Thus, no additional modeling error is introduced, and the solution quality and convergence rates can be increased. The use of the highly continuous surface description is especially beneficial for highly resolved flows close to the structure, where the small kinks in a “classical” FE mesh can influence/disturb the flow due to the discretization with standard loworder finite elements, as shown in Apostolatos et al. (2019). In a recent development, IBRA has been applied for modeling the blades of the NREL phase VI wind turbine (Simms et al. 2001) by Apostolatos (2019). The results are in agreement compared to “classical” shell elements, and it was possible to show that the model error can be decreased by using high-order NURBS shape functions as interface surface description. Solid elements are a suitable choice for the detailed analysis of local effects, one example being the aforementioned damage. Several layers of elements are needed across the thickness to accurately capture the behavior of the blade, which increases the number of elements significantly. Therefore, the computational cost is even higher than for shell elements. Peeters et al. (2018) compare shell and solid elements for a commercial blade of 43 m length and also give a good overview of similar work

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(a) CAD-Model

(c) Beam elements (with outline of Blade)

(b) “Classical” shell elements (unstructured mesh, triangle elements)

(d) IBRA shell elements (trimmed & non-matching NURBS-patches)

Fig. 9 Overview over different structural models for wind turbine blades for the NREL phase VI wind turbine (Simms et al. 2001). (Adapted from Apostolatos 2019). (a) CAD model (b) “Classical” shell elements (unstructured mesh, triangle elements) (c) Beam elements (with outline of blade) (d) IBRA shell elements (trimmed and nonmatching NURBS patches)

done in the past. Haselbach et al. (2016) use solids in a small part of the blade model to investigate the effect of delaminations on local buckling. The constitutive behavior of the turbine model can be represented with different material models; again it depends on the desired accuracy and the requested results. The simplest material law is isotropic linear elasticity. More advanced constitutive material laws can account for anisotropic materials such as composites or represent fatigue/damage that can occur over the lifetime of a turbine.

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Fig. 10 Shell and beam model representation of a wind turbine blade based on airfoil S809 (Somers 1997)

Another important aspect for the selection of the structural model and the element formulation is the meshing procedure. Meshes for beam elements are simple to create since it requires to mesh a line, which is little effort. Creating meshes for surfaces or even volumes is significantly more effort and also requires the availability of the 3D-CAD model which is not always the case. For IBRA shell elements, directly the NURBS-based CAD description is used for the analysis, which reduces the meshing effort compared to “classical” shell elements.

Obtaining Beam Sectional Data from Higher Dimensionality Models Beam elements require sectional data as input such as the bending stiffness (planar second moment of area), torsional stiffness (polar second moment of area), or the cross-sectional area:

List of Input Data Required by Beam Elements Young’s modulus Density Poisson’s ratio Cross-section area Bending stiffnesses Torsional stiffness Shear stiffnesses (only for Timoshenko beam)

• • • • • • •

Those can be derived from the cross section of the continuum; see Gross et al. (2018) and Giavotto et al. (1983). Fig. 10 illustrates the differences between a shell and a beam blade model. It is also possible to obtain those quantities from FE models of higher dimensionalities, shell, or solid models. Especially for shell models of wind turbines, there exist tools such as BEam Cross section Analysis Software (BECAS) (Blasques et al. 2013) or FOCUS6 (WMC 2019) that can read the input and calculate the beam properties from it.

Aspects of Computational Fluid-Structure Interaction Recently, multiphysics problems in many different fields became of great interest. FSI is one of these multiphysics problems, as the interaction of movable or deformable bodies with the effect of a fluid flow cannot be neglected. Traditionally, those problems are difficult to be solved as one system of equations and have

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to be split up. There are two main multiphysics solution approaches which are the monolithic (fully coupled and simultaneous) and the partitioned (staggered, iterative) approach. In the monolithic approach, the governing equations of the fluid and structure are solved simultaneously in one system of equations in one solver. In contrast, in the partitioned approach, stand-alone solvers can be used, and there is an exchange of interface data which requires more effort in the design of the coupling algorithm in order to stabilize the numerical solution of strongly coupled problems (Bungartz and Schäfer 2006). Concerning numerical stability, the monolithic approach can be utilized with coarser time step compared to the partitioned one which can lead to less computational costs. But solving the different fields in one single system of equations is rather complex and can cause numerical issues, because of the nonlinear nature of the coupling and the large total number of unknowns in the different fields. Nevertheless, it has the potential to provide more accurate results, as there is no time lag between the different solvers. In contrast to this, the partitioned approach is used where different fields are solved independently allowing that the two different solvers (fluid and structure) can be developed separately using their own specific and well-established numerical methods and discretization schemes. In addition, in the partitioned approach, different grids can be used for each solver, and the exchanged variables are mapped from one interface grid to the other. A potential time lag between the integration of the two solvers is a drawback, and a relatively small time step is needed to maintain the stability of the coupling scheme. The use of inner iterations at every time step can be a solution for the time lag. This has been done by Alonso and Jameson (1994), where the authors reduced the time lag by using an implicit algorithm for the time integration in the fluid and structure solver and the data has been exchanged at the coupling iteration level. Since modern wind turbines have large and flexible blades, this flexibility requires detailed investigation of the interaction between the deformed blades, as well as the other components, and the unsteady aerodynamic loads. This is a typical FSI problem which is solved by aeroelastic simulations of wind turbines with different levels of complexity. A comprehensive review of the different aeroelastic models and methods used in wind energy can be found in Hansen et al. (2006b) and Zhang and Huang (2011).

Nonmatching Grid Treatment Due to the different physical and technical requirements, the solvers for fluid and structure usually have different discretizations on the coupling interface, see Fig. 11. This means that some special treatment is required for exchanging the coupling data between the nonmatching grids. This is generally referred to as mapping. Mapping transfers quantities such as forces or displacements from one interface to another, considering that usually the meshes are not matching. Different techniques exist, ranging from simple and fast to complex and accurate. Interpolative mapping techniques such as nearest-neighbor or nearest-element (Gatzhammer

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(b) Structural: structured surface mesh discretized with quadrilaterals

Fig. 11 Nonmatching surface discretizations of a wind turbine blade. (a) Fluid: unstructured surface mesh discretized with triangles (b) Structural: structured surface mesh discretized with quadrilaterals

2014) are fast and can be implemented with little effort. The drawback of those basic methods is that they do not preserve the mapped quantities in the sense of an integral principle. More advanced methods such as the mortar method (Farhat et al. 1998a) are based on a weak form; hence, they aim to preserve the quantities on the interface. The drawback of such methods is that they are computationally more expensive and more difficult to implement. An overview including a comparison of different mapping techniques can be found in De Boer et al. (2007). Mapping in FSI can be done in a consistent or conservative way; Wang (2016) describes it as follows: Consistent mapping is a basic requirement for mapping techniques; it means that constant fields can be mapped exactly. Conservative mapping uses a special mapping operator aiming to conserve the overall energy while mapping tractions/forces (Farhat et al. 1998b). This operator can be the transpose of the mapping operator used for consistent mapping of displacements. It depends on the application whether to use consistent or conservative mapping for the forces in FSI; see de Boer et al. (2008) and Wang (2016) who did some comparisons consistent against conservative mapping. It shall be noted that the quantities to be mapped should be described by distributed fields, interpolated with the shape functions and the degrees of freedom of the underlying discretization. If they are available as concentrated nodal quantities, which can be the case for finite elements, then they have to be transformed into the aforementioned distributed fields before the mapping. After the mapping, they can then again be converted into concentrated nodal quantities, depending on the requirements of the corresponding solver. High-fidelity simulations are computationally expensive; therefore, in many cases, large computer systems are used, which employ distributed memory computing and hence require special programming techniques such as the message passing interface (MPI) (Forum 1994). Most fluid and structural solvers already adapted to those environments. The domains of the fluid and the structural solver have to

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(a) Blade partitioned into 5 partitions

(b) Blade partitioned into 14 partitions

Fig. 12 Wind turbine blade surface mesh partitioned. (a) Blade partitioned into five partitions (b) Blade partitioned into 14 partitions

be distributed/partitioned across the processors; an example of the partitioning of a turbine blade is shown in Fig. 12. In order to not slow down the computations and to avoid bottlenecks, also the mapping should be able to work in distributed memory environments. This means that the algorithms especially have to be able to deal with coupling interfaces that are distributed among many processors. Cotela et al. (2017) introduce a method with which the mapping can be done in distributed memory environments.

Special Aspects of Dimensionally Reduced Structural Models For 3D FSI problems, a coupling interface needs to be chosen for the exchange of data between the fluid and the structural solver. When the structural model bases on shell or solid elements, it is possible to use directly the wet surface of the structure (e.g., blade and/or tower) as interface. In this way, the topology of the turbine is preserved with respect to the spatial dimension without any dimensional reduction as mentioned in 4. If the structural model of the turbine is modeled with beam elements, then the dimensionalities of the coupling interfaces of the models are not matching; see Fig. 13. This means that some special treatment is required to transfer the quantities from fluid to structure and the other way round. A relation between the beam elements of the structural solver and the surface description of the coupling interface of the fluid solver has to be built. Wang (2016) describes a general algorithm with which those relations can be established.

Interface Coupling Conditions The coupling conditions on the interface of the partitioned coupling of solvers for FSI have to be fulfilled in order to obtain the same solution as with a monolithic approach (Mok 2001). Two sets of coupling conditions exist: firstly, the

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Fig. 13 CFD surface mesh and beam CSD mesh. (Adapted from Wang 2016)

kinematic coupling conditions which ensure that the displacements, velocities, and accelerations are conforming on both sides of the interface: f

u = us f

u˙ = u˙ s f

u¨ = u¨ s where u , u˙ , and u¨ are the positions, velocities, and accelerations at the flow (f ) and structure (s) side of the interface , respectively, and, secondly, the dynamic coupling conditions which ensure the dynamic equilibrium of forces across the interface: f

f = f s where f are the total forces at the flow (f ) and structure (s) side of the interface , respectively. When both the flow solver output (i.e., the forces) and the structural solver output (i.e., the displacements) are consistently transferred within the interface, then the simulation results of a partitioned approach can be the same as from a monolithic approach.

Coupling Schemes Depending on the level of interaction, different types of coupling are distinguished; see Fig. 14:

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(b) One-Way-Coupling: Rigid boundary for flow & considering transient characteristics of the structure

(c)Two-Way-Coupling: Moving boundary for flow & considering transient characteristics of the coupled system

Fig. 14 Overview of types of coupling. (Adapted from Wüchner and Péntek (2018). This is compared also in Andre (2018)). (a) Pure CFD: Rigid boundary for flow and fluctuating surface loads (b) One-way coupling: Rigid boundary for flow and considering transient characteristics of the structure (c) Two-way coupling: Moving boundary for flow and considering transient characteristics of the coupled system

• Pure fluid solution: Only the flow around the turbine is of interest, and no input or details of the structural response are of interest; see Fig. 15. • One-way coupling: This is used mainly when the details of the turbine are of interest, but the deformations are small and hence don’t influence the flow around the turbine. • Two-way coupling: The interaction between the turbine and the flow is strong, meaning that the deformations of the turbine are large and influence significantly the behavior of the flow, which in turn also changes the flow-based loading of the turbine. Coupling schemes are used to coordinate the execution of the participating solvers. Two-way coupling is generally referred to as computational FSI or aeroelasticity; hence, we will focus on this way of coupling in the following. Apart from the monolithic coupling strategy, for a partitioned coupling approach, a coupling scheme is used. In other definition, the solution scheme of the interaction between the different solvers needs to be defined. Let us assume that the fluid and the structure solvers are expressed as a “black boxes” whose inputs and outputs are defined at the coupling interfaces in Equation (11): f = F (d) and d = S (f )

(11)

In this equation, the fluid solver (F) receives the deformations d as an input and sends the forces f as an output where the structural solver (S) receives that as an input f and sends back the output d. The solution procedure for dealing

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Fig. 15 CFD flow simulation around a wind turbine

with equation (11) is defined as a coupling algorithm for the partitioned coupling methods. The specific aspects and methods involved in the partitioned coupling will be explained in the following. More mathematical details about all the coupling algorithms can be found in Mok (2001), Küttler (2009), and Wang (2016).

Communication Patterns The flow of data among the participating solvers in the FSI (or in any other partitioned coupling, respectively) is determined by the coupling scheme. Mainly two variants for the data exchange among the solvers are used, the Jacobi pattern and the Gauss-Seidel pattern, as sketched in Fig. 16. More information can be found in Sicklinger (2014) and Uekermann (2016). Jacobi Pattern The execution of solvers is always based on the data of the other solver only from the previous iteration or time step. This means that all solvers can be executed in parallel, since there are no data dependencies among them, see Fig. 16a. Because they do not use any updated values, the converence is slower than for the GaussSeidel pattern (Sicklinger 2014).

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(b) Gauss-Seidel Pattern (sequential due to data dependency within one time step or iteration between the solvers) with fluid solving before structure

Fig. 16 Communication patterns, including the mapping M. (a) Jacobi pattern (inherently parallel) (b) Gauss-Seidel pattern (sequential due to data dependency within one time step or iteration between the solvers) with fluid solving before structure

Gauss-Seidel Pattern Here, the solvers are executed sequentially within one time step or iteration; they always use the newest available data from the other solvers. This means that they cannot be executed in parallel, since they have data dependencies among each other; see Fig. 16b. Furthermore, there exists a time lag between the solvers. The advantage over the Jacobi pattern is the faster overall convergence. Weak and Strong Coupling Methods Two main coupling methods can be distinguished, namely, the weak and the strong coupling. The difference is whether more than one coupling iteration is performed during one time step, as shown in Fig. 17. There exist many variants of these solution procedures, but roughly one can categorize into these two main strategies for the consideration/treatment of the coupling conditions (5.2), whereas also different names for those methods are given. Weak/Explicit/Loose/Staggered Coupling Algorithm Loose coupling is a staggered coupling scheme where the data between the different solvers is transferred at every time step with only one coupling iteration executed. Since only one coupling iteration is performed, a time lag between the solvers exists if a Gauss-Seidel pattern is used (as shown in Fig. 17a). It is a relatively simple algorithm, and thus it is the most common coupling method for partitioned coupling of two systems. In such an algorithm, the computational cost is reduced since each connected subsystem needs to be solved only once per time step. It was frequently reported that the basic implementation of the loose coupling algorithm reduces the

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(b) Strong Coupling

Fig. 17 Weak and strong coupling with Gauss-Seidel pattern (basic scheme). (a) Weak coupling (b) Strong coupling

time accuracy of the coupled simulations at least by one order compared to the time accuracy of the original stand-alone solvers (Piperno and Farhat 2001). Moreover, the loose coupling schemes suffer from numerical stability problems when the interface residual is rather high or the physical coupling intensity of the underlying problem is strong. Thus, the staggering can induce loss of time accuracy and reduction of numerical stability and cause a lack of momentum and energy conservation at the coupling interface (Piperno and Farhat 2001). In addition, high temporal discretization might be needed to overcome the time-loss accuracy as reported by Richter and Wick (2015). A stability improvement for loose coupling schemes is to establish an improved prediction of the values for the next time step which is used as an initial guess for the computation of the values in the next time step; see Fig. 18. Different methods are proposed; see Mok (2001), Dettmer and Peri´c (2013), and Wang (2016).

Strong/Implicit/Iterative Coupling Algorithm In the strong/implicit/iterative coupling schemes, the variables at the coupling interface are exchanged several times per time step within coupling iterations which means that both solvers are required to run multiple times within the same time step as shown in Fig. 17b. Strong coupling schemes are generally a better choice for more accurate and stable coupled solutions, and they are frequently used where a weakly coupled scheme does not provide the desired results. Also it is used if the interaction between the fields is too strong to be solved by a weakly coupled scheme. In strongly coupled solution schemes, the interface residual is minimized by coupling iterations between the solvers within each time step. Hence, the interface compatibility and equilibrium conditions (5.2) are fulfilled up to a desired tolerance, which is checked with a convergence criteria.

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Fig. 18 Weakly coupled Gauss-Seidel with prediction P. Note that the order of execution of the prediction P depends on the specific coupling algorithm

Fig. 19 Strongly coupled Gauss-Seidel with convergence check C and acceleration A

In addition to the previously discussed prediction of the values for the next time step, relaxation techniques are often used to accelerate the convergence of the interface residual which reduces the number of coupling iterations; see Fig. 19. Degroote (2011) and Uekermann (2016) give an overview over the available methods for acceleration: fixed-point methods such as constant under-relaxation (Küttler and Wall 2008) or Aitken relaxation (Mok 2001) as well as quasi-Newton methods such as IQN-ILS (Degroote et al. 2009) or MVQN (Bogaers et al. 2014). In general, the strong coupling algorithm is computationally more expensive because several calls of the solvers per time step should be done but provides more robust and accurate solutions, especially for strongly coupled problems. Furthermore, it shall be noted that performing strongly coupled simulations requires the solvers to repeat the solution of a time step.

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Overall Procedures for High-Fidelity Wind-Structure Interaction Simulations This section describes a general workflow for wind-structure interaction simulation including all the previously mentioned aspects that can be part of FSI. The proposed workflow has proven to be efficient for many applications and flexible enough to deal with a variety of single field solution schemes and individual discretization approaches; see, e.g., Sicklinger (2014), Wang (2016), and Apostolatos (2019). It shall be noted that it might not be necessary for wind turbine FSI to use a strong coupling, even for large turbines; see Sayed (2018). However, it is definitely important to consider it for cases with very strong physical coupling. The choice of the suitable solution procedure in a specific problem to be analyzed depends finally on the physical properties of the considered coupled system, the available compute power, and the required accuracy of the parameters needed for the specific design task. R1: Also a physically significant modeling of the complex flow conditions within the atmospheric boundary layer is of utmost importance to eventually obtain expedient results out of the aeroelastic simulations of wind turbines. For more details on how to correctly model turbulence for wind turbine simulations, the reader should refer to  Chap. 26, “Turbulent Inflow Models”. Workflow for strongly coupled wind turbine FSI including wind as boundary condition (see also Fig. 20): • Initialization of the solvers. This can involve: — Reading the input files (e.g., mesh and solver configuration files) — Allocating memory used in the calculation • Initializing the coupling. This can involve:

Fig. 20 Strongly coupled Gauss-Seidel with prediction P, convergence check C, acceleration A, and wind boundary condition W

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— Connecting the coupling tool to the solvers — Initializing the grid mapping • Time loop: – Apply wind boundary condition to fluid. – Compute initial guess (prediction) for displacements on fluid. – Coupling iteration loop: Note: This is only executed once per time step for weak coupling. ∗ Solve fluid. ∗ Map forces from fluid to structure. ∗ Solve structure. ∗ Map displacements from structure to fluid. ∗ Only for strong coupling: Check convergence and repeat time step if no convergence is achieved. Optional: Apply relaxation/acceleration for displacements on fluid. • Finalizing the coupling. This can involve: — Disconnecting the coupling tool from the solvers. • Finalizing of the solvers. This can involve: – Deallocating memory used in the calculation. – Closing output files.

Simulation Results An example of CFD-based FSI simulations with the effect of the structure models on the aeroelastic response of the wind turbine is presented. In this example, two different structural elements are used which are 1D beam elements and 2D shell elements. An explicit CFD-CSD coupling approach was used to conduct aeroelastic simulations of the InnWind rotor (Bak et al. 2013b). Since the objective of this section is to investigate the effect of the structure models on the aeroelastic response of the wind turbine and to reduce the computational time, the coupled simulations were employed for a one-third model of the wind turbine. The wind turbine configuration is presented in the next subsection followed by a short description of the numerical models, and finally the results are presented.

Flow Solver FLOWer The CFD solver used in this section is FLOWer which is developed over the last years by a joined team from DLR, universities, Airbus Germany, and EADS-M (Kroll et al. 1998, 2000, and Kroll and Fassbender 2006). It is developed to simulate flows around complex aerodynamic bodies such as aircraft, helicopters, and wind turbines by solving the 3D RANS equations for the viscous flows and the Euler equations for the inviscid flows. FLOWer uses block-structured meshes to discretize

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the physical domain. For time integration, explicit multistage time stepping schemes as well as the dual time stepping approach, according to Jameson (1991), for implicit schemes are employed. Chimera overlapping technique is utilized in FLOWer to account for the overlapping blocks and also to enable better meshing control of complex geometries (Schwarz et al. 2010). There are different turbulence models implemented in FLOWer, and in this example, the Wilcox k − ω turbulence model is utilized, since it is suitable for low Reynolds number flows and free shear, and it has a superior performance for wall-bounded boundary layers.

Structure Solver Carat ++ The CSD solver used to solve the structure dynamics is Carat++ (Bletzinger et al. 2006 and Fischer et al. 2010). It is a general finite element solver developed over the last years at the Chair of Structural Analysis of Technical University of Munich (TUM). Different structural linear/nonlinear elements can be modeled in Carat++ including beam and shell elements. Besides linear and nonlinear as well as static and dynamic analysis, the special emphasis is node-based shape optimization as well as form finding of lightweight structures. Furthermore, it has an interface to the co-simulation environment Enhanced Multi-Physics Interface Research Engine (EMPIRE).

Co-Simulation Environment EMPIRE To establish a partitioned FSI, a co-simulation environment is needed to exchange, store, and control the interface variables between the parallel running simulation codes. The coupling framework used in this section is the EMPIRE developed at TUM (Sicklinger et al. 2014 and Wang 2016) to solve general multiphysics problems. It can be utilized for co-simulation with multiple codes which suit the multiphysics problems. EMPIRE is a model based on client-server method. This method has been widely used, where the solvers are represented by mean of clients connected to the main server that acts as a coupling supervisor. Multiple numbers of clients can be coupled via EMPIRE. These clients are then allowed to solve their physics independently and then exchange the data between them at the coupling interface. The data are transferred through the main supervisor who is called Emperor as shown in Fig. 21. During the coupling, all the functionalities including data exchanging, data extrapolation, mapping, and solving the coupled equations are implemented in the server. This server acts as an additional code that runs in parallel with the clients. This means that the communication between the different clients is only done via the server. The main advantage of this co-simulation environment is the capability of considering more than two clients in the coupling problem (Fig. 22).

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Fig. 21 Schematic description of EMPIRE flow work (Sayed 2018)

Fig. 22 Blade and nacelle/hub surface meshes as well as the assembly of the blade into the hub (Sayed 2018)

Wind Turbine Configuration The wind turbine examined in the present example is the generic reference turbine of the European InnWind project (Bak et al. 2013b). It is a state-of-the-art large HAWT with 178.3 m rotor diameter, a hub height of 119 m, and FFA-W3 airfoil

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series. The rotor is installed at 5° tilt angle and a cone angle of -2.5°. The wind turbine blade and nacelle/hub rotate clockwise as viewed from the inlet boundary (x-direction). Pre-bended blades were considered with a deformation of 3.73% R at the blade tip. The blade at the root region is equipped with wedge-shaped Gurney flaps to increase the aerodynamic performance (Bak et al. 2013b).

Blade Structural Properties The construction of the DTU wind turbine’s blade consists of reinforced composites fiberglass and balsa wood which was employed as a sandwich core material. This composite layup was defined regarding a stacking sequence of layers representing multidirectional plies. Two unidirectional laminae (Lamina 1 and Lamina 2) consisting of E-glass fibers in an epoxy matrix were considered to be the main building blocks of the multidirectional plies. The mechanical properties of the Eglass fibers, as well as the epoxy matrix, are given in Bak et al. (2013b). The blade beam properties were extracted out of the full FE model at DTU using BECAS which is a newly developed software used for structural design and analysis of slender beam-like structures (Blasques et al. 2013). Extracting the beam properties out of the full FE model might guarantee the consistency between the shell and the beam model used here. The beam model properties were calculated from the eight-node shell element modeled by ABAQUS.

CFD Setup The numerical domain is divided into subdomains to allow better mesh control. For each subdomain, an independent grid is created with adopted refinement. Finally, the volume domains are placed together and overlapped using Chimera overlapping mesh technique. The blade volume mesh (red grid in Fig. 23) is generated using a dedicated script at the Institute of Aerodynamics and Gas Dynamics (Schulz et al. 2014; Sayed et al. 2015). For the blade sectional mesh (shown in Fig. 24), 300 cells are used in chord-wise direction and 100 cells in (span-wise) radial direction. 35 cells are introduced to resolve the boundary layer with y + ≈ 1 for the first layer. The nacelle/hub mesh (green grid in Fig. 23) is also generated taking into account the y + ≤ 1 condition at the wall, and also the no-slip boundary condition is applied. The squared cell size at the surface is 0.3 m. The nacelle/hub is connected to the blade by means of a rotating blade-hub-connector mesh. The background domain (gray grid in Fig. 23) is created using a one third cylinder with a periodic boundary condition. The far field dimensions are defined in terms of the blade radius (R) as follows: 6R in upstream direction, 9R in downstream direction, and 6R in far field direction. These values are chosen based on studies done by Sayed et al. (2015). The grid summary of the different volume domains are shown in Table 2.

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Fig. 23 CFD computational domain

Fig. 24 Blade profile mesh

CSD Setup The blade was modeled structurally first by means of beam elements and second by means of shell elements in order to show the effect of the structural model on the aeroelastic response of the wind turbine blade.

Blade Modeled by Beam Elements The blade’s structure is modeled using 51 3D nonlinear corotational beam elements which allow large rotations. The beam formulation is based on the Timoshenko beam theory which takes into account the shear deformation. Each beam element has six DoFs per node (three translational and three rotational DoFs). In Bak et al. (2013b), the cross-sectional stiffness and mass properties of predefined sections along the radius of the blade are given. Fig. 25 shows the predefined cross sections (in green) together with the beam finite element mesh of the blade (in black).

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Table 2 Simulation parameters of the CFD-CSD coupled simulations of a one-third model of the wind turbine (Sayed 2018)

Blade Modeled by Shell Elements Carat++ was developed especially for the prediction of shell or membrane behavior. It was developed based on advanced solution strategies for form finding and nonlinear dynamic problems (De Nayer and Breuer 2014 and Bletzinger et al. 2005). The dynamic structure equilibrium was described by applying the momentum equation in a Lagrangian frame of reference. As described in Section “Structure Solver Carat ++”, a St. Venant-Kirchhoff material law expressed by a second Piola-Kirchhoff stress tensor was assumed to account for the stress-strain relation. The shell element used in the numerical structural model was a four-node shell element with six DoFs per node. The shell element formulation was based on the degeneration principle (Buechter and Ramm 1992). A linear shell formulation was applied with linear displacement variation across the thickness and with ReissnerMindlin kinematics. The enhanced assumed strain (EAS) and assumed natural strain methods are utilized to enhance the well-known locking phenomenon of displacement-based finite elements for shell elements. These locking phenomena in the shell elements are the transverse shear locking and the membrane locking. The six DoFs per node are three translations DoFs of the shell midplane and three deformation components of the nodal director. A seventh DoF was added via the EAS method which makes the element formulation called seven-parameter element formulation. No loads and no boundary conditions can be applied to this DoF since it was defined on the element level. The nodal director vectors are generated to

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Fig. 25 Predefined sections (in green) together with the CSD computational domain (in black) (Sayed et al. 2016)

Fig. 26 Eight-node solid element degeneration into four-node shell element (Sayed 2018)

be parallel to the radial vector of the node pointing from the inner surface of the element to the outer surface (Fig. 26). The blade was discretized in Carat++ (modeled at TUM) by means of highfidelity FE shell elements based on the material properties defined by 1100 regions over the blade surface. The complete FE shell model discretized in Carat++ was derived from the initial ABAQUS model proposed by Bak et al. (2013b). Out of this model, a Carat++ shell model was created with a total number of 34849 quadrilateral elements connecting 33271 nodes. Fig. 27 shows the final numerical model defined in Carat++ with internal details, such as the shear webs, presented in Fig. 28. Therefore, 27329 shell elements and 27374 nodes were used in the coupling interface surface and transferred to FLOWer to generate the surface to be deformed during the coupled simulations. Aerodynamic forces only were included in the coupled simulations, and no gravitational or centrifugal forces were used.

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Fig. 27 Structural model discretized by 34849 quadrilateral shell elements (Sayed 2018)

(a) Outer part

(b) Near root

(c) 3D sectional view

Fig. 28 Details of the full FE numerical Carat++ model (Sayed 2018) (a) Outer part (b) Near root (c) 3D sectional view

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Effects of the Structural Models CFD-CSD aeroelastic simulations were conducted at a time step corresponding to 1° azimuth angle to study the aeroelastic response and its impact on the wind turbine performance. Based on the results of the linear shell model and the linear and nonlinear beam models, the effect of the structural model on the aeroelastic results will be discussed. First, CFD rigid simulations were performed to achieve a steady-state solution including the complete developed wake. Afterward, three coupled revolutions were simulated achieving a convergence of the residual of 10−6 . The simulation parameters are summarized in Table 2, considering that an one-third model is an only blade model of the turbine and therefore no tower has been taken into consideration. The azimuthal variations of the flapwise and edgewise deformations at the blade tip are presented in Fig. 29. Larger linear beam model deformations were predicted in all directions with nearly 2.2 m larger flapwise deformation than the one predicted by the linear shell model. Compared to the shell model edgewise deformation at the tip, approximately 0.7 m larger edgewise deformation was also predicted. One reason behind the increase in the deformations predicted by the linear beam model compared to the shell model is the lower stiffness of the beam element as a consequence of the decoupling of the flapwise and edgewise deflection in the element stiffness matrix. The in-plane deformations are coupled with the out-ofplane deformations in the shell model so that the rigidity is increased and therefore smaller deflections are expected. Lower deflections were predicted using a shell model compared to a beam model, and the difference is attributed to a local buckling which is better predicted by the shell model (Piculin and Brank 2015). Research on the differences between the beam and shell models (column with I – section) was conducted by Sreenath et al. (2011), and it was concluded that stronger deformations were obtained from a beam element compared to a shell with the same load factor. The reason behind this is a local buckling, a phenomenon that the beam elements are unable to capture. From these examples, in addition to the stiffness of the shell model due to the coupling between the in-plane and out-of-plane deflections, the beam model’s stronger blade deformations were expected compared to the shell model. Overall, the differences in shell and beam element results performed by the same type of analysis (linear/nonlinear) are due to the different characteristics inherent in the different formulations of elements. Overestimation in the flapwise blade deflection was reported to be a primary indicator of the missing nonlinear effects in the blade dynamic response (Manolas et al. 2015). In addition, the geometrical nonlinearities must be included not only in the fatigue or buckling analysis (Castelos and Balzani 2016) but also in the performance analysis in the aeroelastic simulations for such large wind turbines. The blade tip flapwise deflection was significantly reduced compared to the linear model, including the geometric nonlinearities. For blade designers, this significant difference in deflections might be vital. Compared to the results of the nonlinear beam model, about 4.2 m larger flapwise deformation results from the linear model corresponding to ≈44% more flapwise deformation.

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(a) Flapwise [-]

(b) Edgewise [-]

Fig. 29 Azimuthal variations of the normalized blade tip deformations from beam and shell models (Sayed 2018). (a) Flapwise [-] (b) Edgewise [-]

The radial deformation that approaches ≈1.3 m at the blade tip from the linear shell model and ≈1.6 m from the linear beam model was a major cause of this massive increase in deformations. As a result of the large radial deformations (as shown in Fig. 30), the blade surface area was increased resulting in higher forces in the flapwise and edgewise directions, resulting in higher deformations in both directions. The linear models also overestimate tip deflections when large deflections occur because they do not capture the geometrical nonlinearities (Wang et al. 2014) and the deflection starts in the tangential direction (flapwise) and remains in the tangential direction (which is not in the x-direction due to the blade pre-bend). This does not happen in the nonlinear analysis as the beam cannot get longer and the nonlinear beam experiences increased vertical rigidity in bending while maintaining its overall length. Since the linear beam’s stiffness is independent of deflection and due to the increased stiffness in the nonlinear beam,

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Fig. 30 Surface deformation after three coupled revolutions from different structural models (Sayed 2018)

the nonlinear beam for a given applied force produces a smaller deflection than the linear beam. The effect of these changes in the structural models on the wind turbine performance is also subsequently discussed. The power was determined and normalized by the value of the rigid turbine blade. In Fig. 31, the azimuthal power variations over the last coupled revolution (after three coupled revolutions) are presented. It was found that the generated power from the flexible blade is higher than the one from the rigid blade. The normalized power was increased by ≈1%, ≈3.4%, and ≈4.3% from the nonlinear beam model, the linear shell model, and the linear beam model, respectively. Remarkable fluctuations are shown in the power output with respect to the azimuth angle. These low-frequency oscillations are because of the vortex shedding in the inner region of the blade, up to 20% of the blade radius, since the blade profile is almost circular in that region. In fact, the frequency of the power fluctuations is in the range of frequencies that would result for the typical Strouhal number of von Karman vortex sheets within the cylindrical part of the blade. Therefore, this is the cause for this power fluctuation. More detailed discussion about this behavior can be found in (Sayed 2018).

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Fig. 31 Normalized power result from CFD-CSD simulations by different structural models compared to the rigid result (Sayed 2018)

(a) Edgewise Force

(b) Radial Force

Fig. 32 The sectional load distributions in y−direction and z−direction over the blade radius at the position of 0◦ azimuth angle after 3 coupled revolutions (Sayed 2018). (a) Edgewise Force (b) Radial Force

As shown in Fig. 30, the large increase predicted from linear models was due to the increase in the rotor area since the blade was stretched in radial direction. In fact, the total torque is calculated by multiplying the radial distance and the edgewise deflections from the blade forces in the y− and z−direction, respectively. The blade sectional loads in the y− and z−direction are presented in Fig. 32. As shown in Fig. 32a, the distribution of tangential force is slightly different as a result of torsional deformation due to the change in angle of attack. This small change does not affect the power that much since the decrease predicted in the outer region was overcome by the increase in the radial distance and the total difference would be very small. By contrast, as shown in Fig. 32b, a complete change in radial force was predicted. For the rigid blade, a vertical upward radial force was predicted (positive) since the blade was pre-bended toward the flow. In the case of flexible blade, the blade was bent (due to deformations) to the other direction (negative, in the direction of the flow) which leads to an increase in the radial force over the blade. Therefore, the maximum radial force results from the linear beam model

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since it was the most bent one as shown in Fig. 30. If the edgewise deformations are small, changing the direction of the radial force does not contribute significantly to the torque. Since the edgewise deformations are ≈1.5 m, or more in case of the linear beam model, the contribution of the radial force increases such that more torque was predicted. The blade was represented by a discrete number of nodes along the blade radius in the simulations based on beams. These nodes are used to calculate the deformations and the rotations due to the blade forces. The torsional deformation is very sensitive to the location of the center of mass, the shear center, and the elastic center. This is not the case in the simulations based on the full FE shell model since the forces are applied over the whole surface instead of some radial positions. It is therefore to be expected that the shell model will yield better results as there is no need to calculate these centers as in the beam model and also full composite layup properties are considered rather than equivalent beam properties. Nevertheless, for such slender blades, the beam-based simulations give good results as shown in Sayed (2018), where further details of the computed torsion are prescribed. In the following, the impact of the wind speed on the differences resulting from the considered structural models shall be outlined. Five different wind speeds were analyzed ranging from 5 m/s up to above rated speed of 15 m/s. The flapwise deflection at the blade tip after three coupled revolutions compared to the results from literature (Bak et al. 2013b) is shown in Fig. 33. Nonlinear beam element was employed in aeroelastic simulations by Bak et al. (2013b). As shown, the results based on the nonlinear beam element are in good agreement with the results from literature. Moreover, for the small deformations resulting from the low wind speeds 5 and 7 m/s, the linear shell model results are close to the nonlinear beam models. The level of accuracy decreases with increased loading of the wind turbine blade since the deformation increases up to 14–16% of the blade radius and cannot be correctly modeled without including the geometrical nonlinearity.

Fig. 33 Blade tip flapwise deflection at different wind speeds and using different structural models compared to the results from three different aeroelastic tools (Sayed 2018)

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Conclusions and Recommendations Simulation results based on engineering models have good agreement on the structural side but at the same time are limited by the BEM aerodynamic predictions. As concluded from literature, RANS shows in general better predictions for the aerodynamic performance of the turbine blades compared to the BEM and VLM methods. It is noted that the BEM relies on experimental data and empirical correlations for the simulations as well as predicted polars. Therefore, CFD is a mature tool for accurate predictions of aerodynamic performance of wind turbines. For aeroelastic simulations, the time-accurate blade loads become important, and, hence, unsteady CFD can be used. In general, it can be said that the choice on the necessary fidelity level depends on the user’s purposes. High-speed calculations deliver, in general, results with low accuracy and the other way around. The accuracy of a RANS-CFD simulation relies on different delicate factors such as the chosen time step and the domain size. The first one depends on the unsteady effects arising at the wind turbine operating conditions and the desired degree of accuracy to solve them. The second one needs to be big enough to minimize the blockage effect and allow a complete wake development and, finally, to ensure grid independent results. Different techniques to handle with rotating components exist, and they differ from each other in terms of versatility and complexity. When using the Chimera technique, all the grids deform and move independently from each other, but it is the most time-consuming technique. MRF and mixing plane are more suited for steady-flow computations. The sliding mesh requires less effort for the user than the Chimera technique and is also suited for unsteady flows. Performing CFD-based aeroelastic simulations requires a volume mesh deformation technique to apply the deformations provided by the structural solver. A suitable mesh deformation approach is chosen according to its levels of robustness and accuracy. It is advised to use the mesh connectivity-based schemes for small deformations since it converges fast. Usually, the RBF delivers the highest quality of the deformed meshes. However, in some cases, the orthogonality near the deforming surface may deteriorate due to the fact that the displacements are not linked in different directions, such as the bending of a beam. In the absence of blade shape deformations, it is recommended to use beam elements including the geometrical nonlinearity for large deformations predicted from a large slender bladed wind turbine. In other complex simulations such as flutter analysis, airfoil decambering becomes very important, and shell elements should be used to capture such effects. If detailed analysis of specific parts of the blade regarding damage or other local phenomena is to be conducted, then this part can be modeled with solid elements. Moreover, at small blade loading (small wind speeds), linear models give reasonable results compared to the nonlinear ones. But with increasing the loading (at high wind speeds), nonlinear models should be used. Recently, the effect of the aerodynamic model fidelity on the aeroelastic response of a state-of-the-art wind turbine was studied by Sayed et al. (2019). CFD-based and BEM-based aeroelastic simulations were employed on the InnWind turbine with flexible rotor only. It was concluded that, for the above rated wind speeds, the

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impact of the model fidelity increases. Moreover, power and thrust reduction was predicted from the flexible rotor compared to the rigid one employing the BEMbased model. In contrast, they were increased utilizing a high-fidelity CFD-based model. The radial force was found to be the reason behind this increase in power from the CFD-based model. Since one of the main assumptions of the BEM theory is to exclude the radial force component, a reduction was predicted in contrast to the results from CFD model. This assumption could be used for small wind turbine simulations where the edgewise deformations are insignificant. But for such large wind turbines, the radial force distribution over the blade radius must be taken into account in the calculation of the power output due to the large edgewise deformations.

Cross-References  CFD for Wind Turbine Simulations  Pragmatic Models: BEM with Engineering Add-Ons

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Turbine Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helicopter Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instability or Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stall-Induced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aeroelastic Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearised Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Nonlinear Time Domain Simulation Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tool Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aeroelastic Design and Innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aeroelastic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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In this chapter aeroelastic stability for wind turbines is discussed. The complete wind turbine mode shapes, the harmonic modal components, and the main instabilities are explained, possible resonances addressed, and methods to analyze and improve the stability of a wind turbine design are discussed.

J. G. Holierhoek () JEHO BV, Rotterdam, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_23

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The main instabilities that current size wind turbines could suffer from are stall-induced vibrations (edgewise and flapwise, idling instabilities, and vortexinduced vibrations) and classical flutter. The stability of a design can be evaluated using linearized stability tools or nonlinear time domain tools. It is also possible to evaluate damping of some modes on an actual wind turbine. Current size wind turbine has become more flexible, and due to the large deformations, it is required to use advanced blade models when analyzing the stability of the turbine. Keywords

Wind turbine aeroelasticity · Instability · Resonance · Aeroelastic evaluations

Introduction Wind turbine aeroelasticity is probably best known in connection with the load calculations that are always performed to ensure that the designed wind turbine will not break before the end of the prescribed lifetime. However, there are many other aspects to this field of expertise, and in this chapter, the focus will be on instabilities and resonances, not on load calculations. The modes on a complete wind turbine will be addressed as well as the difference between resonance and instability. The most relevant aeroelastic instabilities are discussed, and some aspects of different types of aeroelastic tools are explained. The subjects addressed in this chapter are summarized in Fig. 1. In this first section, the basics of wind turbine aeroelasticity will be provided, starting with aeroelasticity in general followed by an introduction to the specific field of wind turbine aeroelasticity, discussing the modes of a wind turbine and the Campbell diagram.

Aeroelasticity What is aeroelasticity? It is the combination of three types of forces that have significant interaction: inertia forces, aerodynamic forces, and elastic forces. It is often depicted using Collar’s triangle (see Fig. 2). Aeroelasticity deals with

Fig. 1 An overview of the sections in this chapter and the subjects

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Fig. 2 Collar’s triangle (Collar 1946): interaction of three different forces defining aeroelasticity

dynamically deforming systems where aerodynamic forces also play an important role and are influenced by the dynamic deformations. The discipline is known from aeronautics and is important for fixed-wing aircraft as well as for helicopters. Wind turbines also need to be analyzed using aeroelasticity, and the importance of the field of aeroelasticity for wind turbines has increased over the years, due to the increased flexibility as a result of the upscaling of the turbine designs. If a structure is elastic, it will respond to periodic forcing functions by vibrating in discrete geometric patterns (Spera 1994). These geometric patterns are called the mode shapes of the structure. Each mode will have a corresponding mode frequency. If the system is excited (e.g., given an initial deformation) and free to vibrate, the vibration will be a summation of the different natural modes at these modal frequencies. Two common issues from an aeroelastic point of view are instabilities (including divergence) and resonances. These are not the same thing and should be distinguished from one another. A resonance is the situation where an external forcing frequency is close to or identical to a natural mode frequency of the system. An instability is a situation with negative damping, meaning that energy from aerodynamic forces is added to a natural mode during a cycle of that mode. The vibration then will occur at the natural frequency of that mode, while a forced vibration will result in a vibration at the forcing frequency. For resonance the damping of the mode is important; if there is a lot of damping present, the excitation at the natural frequency can be irrelevant, while for a mode with very little damping, a difference of 10% between excitation and natural frequency can still result in significantly larger amplitudes of vibration and therefore increased fatigue loads. This is illustrated by the amplification factor depicted

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Fig. 3 The frequency response of a forced system. The amplification ratios for different frequency ratios and different damping values are shown

in Fig. 3. These results are based on linear vibrations but are also indicative for nonlinear systems. Divergence, another possible issue in aeroelastic structures, is not related to a mode; it is only related to aerodynamic forces and elastic forces and concerns cases where the aerodynamic force results in a deformation which in turn increases the aerodynamic force (Bisplinghoff et al. 1996; Fung 1969). In Collar’s triangle in Fig. 2, it is named ‘static aeroelasticity’. A well-known example is torsion divergence where the increase in the aerodynamic moment about the shear center due to an increase in torsion deformation is larger than the restoring torque related to that torsion deformation. This results in a rapidly increasing torsion angle till the structure breaks. To gain insight and results in the field of aeroelasticity, models are used that take into account all three types of forces involved: aerodynamic, inertial, and elastic forces. When combined in, for example, a time domain simulation, it can provide the deformations of and the loads on the aeroelastic structure. This can be used to compare the calculated loads to the maximum allowable loads on the structure or check if the maximum deformations are within the allowable limits. It is also possible to calculate using linearized tools and find stability boundaries or mode shapes, natural frequencies, and damping values for the different modes. Strongly simplified models can be very instructive when trying to understand the reason for certain instabilities.

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There are many similarities when looking at aeroelastic issues on aircraft, helicopters, and wind turbines, but the differences between each of these disciplines make it into three separate fields of expertise. The basics that are needed to be able to discuss the aeroelastic stability aspects of wind turbines are therefore provided below.

Wind Turbine Aeroelasticity During the early development of modern wind turbines, aeroelasticity was not expected to become a relevant problem, as the stiffness of the blades and tower was relatively high. However, since then, the trend toward larger and larger wind turbines has resulted in a very different situation where aeroelasticity has become a vital subject that cannot be ignored. For most wind energy experts, the field of aeroelasticity is mainly related to the calculation of loads on the turbines. The simulation tools that are used to determine the loads on a wind turbine are also based on aeroelasticity, e.g., tools such as Bladed, FLEX5, PHATAS-FOCUS, HAWC2, and (OPEN)FAST. These tools are used to determine the response in time for different inflow conditions and situations a wind turbine will encounter during its lifetime. However, in this chapter the focus will be on the possible aeroelastic issues that can occur on wind turbines such as resonances and instabilities. To be able to discuss this, it is also important to explain the modes on a wind turbine. The Campbell diagram will also be explained, and then the modes will be readdressed showing that the complete turbine modes can be rather complicated.

Wind Turbine Modes As was described in section “Aeroelasticity,” a flexible system will have natural modes, where each mode has a specific natural frequency and a mode shape. The dynamic behavior of the system can be written as a summation of these natural mode shapes. For a continuous system such as a wind turbine rotor blade, there will be infinitely many different modes; however, only the lower modes are of interest due to the structural damping of the higher modes. Higher modes (meaning, for example, a fourth flapwise or third torsion mode) will have higher damping values than lower modes. Next to that higher modes also need a lot more energy to be excited due to the shape of higher modes and the corresponding necessary strain energy. Therefore the higher modes can be neglected and are irrelevant from an aeroelastic point of view. Note that a first torsion mode has a higher frequency than some of the flapwise and edgewise modes, but the damping will not be much higher than the damping of the first flapwise or the first edgewise mode. A wind turbine has different flexible structures that on its own will each present several significant natural modes. The main flexible structures will be the blades and the tower, but components such as the main shaft will also result in specific modes of that component. In the end the modes of the complete wind turbine will be combinations of deformations of the different components, but it is easiest to start

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a

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Fig. 4 Examples of mode shapes of a wind turbine blade, calculated using QBlade (QBlade). These modes are (a) first flapwise mode, (b) second flapwise mode, (c) first edgewise mode, (d) second edgewise mode, and (e) first torsion mode

the discussion of the turbine modes with the specific blade modes that will exist on a wind turbine blade. A blade will usually show modes that are mainly out-of-plane (flapwise, perpendicular to chord), mainly in-plane (edgewise, chordwise direction), or mainly in torsion. In Fig. 4, five modes are illustrated, the first two flapwise, the first two edgewise, and the first torsion mode. For the current size wind turbines, every mode will in fact be a combination of all three different directions, but there will be the first two or three modes in each direction that will have mode amplitudes in one direction that are clearly dominating. Multidirectional modes are due to the complexity of the blade shape, the aerodynamic twist, and the prebend that is included in the blade design resulting in coupling of flap, edge, and torsion, but especially the aerodynamic forces will result in rather complicated mode shapes. For many wind turbine blades, the relevant modes will be the first three or four flapwise modes, the first three edgewise modes and the first torsion mode. The higher modes will be difficult to be really identified as specific edgewise of flapwise modes due to the strong coupling that occurs, and they are irrelevant for the analysis of the wind turbine. As described by Hansen in 2007, the full turbine modes will result in combinations of blade modes that result in two whirling modes and one symmetric mode, for a three-bladed turbine. One flapwise mode with a frequency ω of the blade will result, on a three-bladed turbine, in three different modes on the full rotor. The resulting force (or moment) on the standstill structure of these mode combinations can be out-of-plane or in-plane. When the resulting force is in-plane, the rotation (with frequency Ω) gives rise to the whirling modes, one forward and one backward with frequencies ω + Ω and ω − Ω, respectively, in the standstill frame. The symmetric combinations of blade modes (all three blades flapping in

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Fig. 5 The first edgewise blade mode results in three different rotor modes, two asymmetric (b and c) that result in whirling and one symmetric mode (a)

the same direction or all three deforming collective with the rotor rotation and then against it) do not change in frequency from rotating frame to standstill frame. For the first edgewise blade mode, the three corresponding rotor modes in standstill are illustrated in Fig. 5. The symmetric mode in (a) results in an interaction with the drive train. The asymmetric modes in standstill (b) and (c) result in whirling modes on a rotating three-bladed turbine.

Campbell Diagram To ensure the design does not include any relevant resonance issues, usually a Campbell diagram is created (Fig. 6). In a Campbell diagram, the different excitation frequencies are depicted together with the natural frequencies, all as a function of the rotor speed. The excitation frequencies for a wind turbine are the nP values, so the rotor speed and multiples of the rotor speed. The sources for excitations are numerous. The most relevant that result in nP excitations will be the tower shadow and turbulence, but yaw error, tilt angle of the turbine, shear flow, and mass imbalance will also result in excitations at 1P. The tower will sense the effect of the tower shadow mainly at the number of blades times the rotor speed (BP), but at other nP values, there will also be an excitation. Turbulence will result in peaks at 1P, 2P, . . . nP values. This is due to the effect called rotational sampling (see Connell 1981; Kristensen and Frandsen 1982). The turbulence in the standstill frame will not show these excitation peaks, but the rotating blades will observe the turbulence while rotating resulting in a very different frequency spectrum. For an isotropic three-bladed rotor, the turbulence excitation, tower shadow, etc. will result in excitation in the fixed frame at 0P, 3P, 6P, 9P, etc. The natural frequencies of the complete wind turbine can be determined for different rotor speeds and included in the Campbell diagram. These modes should include the rotor whirling modes as well as the symmetric rotor modes and of course the tower modes. The Campbell diagram can give quick insight into possible ranges of rotor speed that need to be avoided, though as discussed earlier for the severity of a frequency closeness the damping of the mode is also very important (see Fig. 3).

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Fig. 6 Example of a Campbell diagram, based on work from Hansen (2016), extended with nP lines

This means that for each crossing of an excitation frequency and a natural frequency, one has to assess if the crossing is a problem or if it can be ignored. If it is known that a mode has high damping for the case where the crossing occurs, then the resonance is not an issue. However, if the damping is expected to be low, one needs to address the possible resonance. Often some of the nP values are ignored when evaluating possible resonance issues; however, this is generally incorrect. Even though for a perfectly isotropic three-bladed wind turbine the turbulence excitation, tower shadow, etc. will result in excitations in the fixed frame at 3P, 6P, 9P, etc., the turbine will not be perfectly isotropic. Therefore one cannot ignore a coincidence of 2P with the tower frequency on a three-bladed turbine, for example. The excitation at 2P is still significant and can result in increased fatigue loads, as was experienced on the WKA-60 turbine (Hau 2006). Therefore it is best to include all nP lines up to at least 12P in a Campbell diagram and then evaluate if the crossings could perhaps be ignored as the damping of the mode is expected to be high and/or the effect of the excitation on

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the mode is deemed to remain small. In case it is unclear if there is a resonance effect to be expected, one can also look at time simulations and evaluate the amplitudes of the vibrations at the resonance frequency.

Harmonic Components in Modes The short discussion in section “Wind Turbine Modes” concerning the changing of the modes in a forward and a backward whirling mode is not the complete story; it is a simplification of what actually happens, as is discussed in Skjoldan (2011). The actual mode shape contains harmonic components at different frequencies, but for three-bladed turbines, there will be a dominating component, and the other components are much less significant. However, note that for a two-bladed turbine, it is different. After all, there is only one asymmetric mode corresponding to one particular blade mode on a two-bladed turbine. Therefore it cannot become one forward and one backward mode, because that would mean that one mode changes into two modes. The forward whirling and backward whirling motions are both part of one combined mode (Hansen 2016). For a rotating rotor, the modes that actually occur on the turbine for an isotropic three-bladed turbine will include a forward whirl mode and a backward whirl mode. These modes can be illustrated using the mode shapes for standstill in Fig. 5 and decomposing the combination in circular components in the rotating frame (Petersen et al. 1998b). In Skjoldan and Hansen (2009), Skjoldan and Hansen derive modes for a very simple isotropic three-bladed model with flapping blades and a tilt and a yaw degree of freedom. These modes consist of different harmonic components where the blades all flap in backward whirling order or forward whirling order, due to phase differences between the mode amplitudes on the blades. The amplitudes are identical for each blade, similar to what was found in Petersen et al. (1998b); therefore this looks different than the standstill modes due to the phase differences that come into play. A complete wind turbine mode, taking into account the fact that it will never be completely isotropic, will contain different harmonic components, and therefore it is possible to excite these modes at different frequencies, not only the principal modal frequency. This is the case for three-bladed turbines, but the present frequencies are even more in number for two-bladed turbines. For example, Fig. 7 from Hansen (2016) illustrates the different modal components present in a two-bladed model.

This more complete description of the turbine modes illustrates that a natural mode on a rotating wind turbine can be excited at different frequencies. Therefore the mode could result in resonance issues when excited by any of these frequencies in the corresponding frame of reference. Of course, the amount of damping present in the mode is important, if there is little damping, possible resonance issues result in larger amplitudes. The amount (continued)

Fig. 7 Harmonic modal components for the two-bladed turbine mode dominated by antisymmetric deflection of the rotor in the first flapwise DOF across the rotor speed range. Top panel: modal amplitudes plotted versus rotor speed and frequency of the particular harmonic component. Lower left panel: projection onto the plane of modal amplitudes and rotor speed. Lower right panel: projection onto the plane of component frequencies and rotor speed (periodic Campbell diagram). Filled black circles show the principal modal frequencies in the frequencies and rotor speed planes. Only modal components with amplitudes larger than 10% of the overall maximum amplitude are plotted. (Reproduced from Hansen 2016)

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of energy present in the excitation affecting the mode amplitude also plays an important role, but this does illustrate that a simple Campbell diagram does not guarantee the design to be resonance-free. When an accurate enough tool is used to perform the load calculations, any increased amplitudes due to resonance issues should show up in the fatigue loads; therefore it should not result in loss of a turbine, but the design is perhaps not an optimal design as there is enough material in the structure to carry these increased fatigue loads, so without these loads, the design could have been lighter.

Instabilities This section will discuss the most important instabilities that we currently know of for wind turbines. Instabilities were first mainly an issue on research turbines where new concepts sometimes led to unexpected issues, but in the 1990s, it became a problem for commercial turbines as well. Since then, there has been a lot of effort put into the research of wind turbine aeroelasticity, not only looking at the correct way to predict loads but also trying to identify and prevent possible aeroelastic instabilities. In this section, first, the applicability of helicopter aeroelasticity research will be shortly discussed, followed by the difference between resonance and instability. The relevant instabilities currently known for wind turbines are discussed, starting with stall-induced vibrations including idling instabilities and vortex shedding, followed by classical flutter.

Helicopter Aeroelasticity Helicopter aeroelasticity is a field of research with a longer history than wind turbine aeroelasticity. Due to the similarity between these two rotating structures, the knowledge that was gained for helicopters can be a very valuable source for wind turbine aeroelasticity. However, there are also significant differences between these two, whereby the wind turbine aeroelasticity has become a special field of expertise. There are several models that are taken from the field of helicopters that have been very useful, for example, dynamic stall modelling or the multiblade coordinates. Differences are caused by the very different operating conditions that occur such as attained average angle of attack ranges, the difference in relative frequencies, as well as the shape of the blade. Therefore the instabilities for wind turbines will not correspond to those that are known for helicopters, and it is possible that new instabilities are encountered on wind turbines that were never an issue on helicopters. Experiences in helicopter aeroelasticity and the methods that have been derived in that field are often referenced to in wind turbine aeroelasticity and can be of great

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benefit for wind energy. Reference texts such as the books by Bielawa, Bramwell, and Johnson (Bielawa 1992; Bramwell et al. 2001; Johnson 1980) provide valuable background for all those involved in wind turbine aeroelasticity. However, one should not blindly assume that the results from helicopter aeroelastic research are also valid for wind turbines, the scale of the nonlinearities can be very different, and the aforementioned differences cannot guarantee the validity for wind turbines.

Instability or Resonance When discussing instabilities, there is often some confusion between instabilities and resonances. It is therefore important to distinguish between resonances and instabilities. Resonances are forced vibrations at a frequency that is equal or close to a natural frequency resulting in large vibrations. For resonance the damping of the mode is important; if there is a lot of damping present, the excitation at the natural frequency can be irrelevant, while for a mode with very little damping, a difference of 10% between excitation and natural frequency can still result in significantly larger amplitudes of vibrations and therefore increased fatigue loads. This was already illustrated in Fig. 3.

An instability is a case where a mode of the complete wind turbine has negative damping. This means that during one vibration in this particular mode, energy will be added to the vibration, resulting in increasing amplitudes of the mode vibration. The frequency of the mode is in case of an instability usually less important than the shape, and the shape for a large part determines the damping of the mode. The excitation frequencies on the turbine are not a cause of an instability.

Possible resonance issues on wind turbines will not specifically be discussed in this chapter. In the next sections, the different instabilities will be discussed which gives an indication of which mode can be negatively damped and some that will usually have very little to negative damping. Knowing about the amplification ratios, it is clear that for these low damped modes, it is more important to prevent instabilities, while for well-damped modes, the closeness of excitation frequency and natural frequency will be less of an issue. Looking at the low damped modes, one has to realize what happens with these modes when looking in rotating frame of reference and standstill frame of reference, as this can result in a shift of the natural frequency and therefore the actual resonance frequency. One should also take into account that the shift occurs both ways; a vibration of a whirling mode can be reflected again with an increase of the frequency by the rotor speed as well as decrease by the rotor speed. This is of course related to the harmonic modal components that are present in the turbine modes which can play a role, as there is not really just one natural frequency present in a mode. There will be one

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dominating principal modal frequency, but there can be significant contribution at other frequencies as well (see section “Harmonic Components in Modes”). Known resonance issues on wind turbines are related to tower frequencies being close to one of the excitation frequencies (usually 1P in current size turbines) or low damped rotor modes that are close to an nP frequency. The excitations in the standstill frame at 3P and 6P will be the most significant ones to avoid. This translates in the rotating frame to the same values for the symmetric modes, but, for example, a backward whirling mode that has a frequency in the rotating frame of 7P will be excited by 6P in the standstill frame. And if the damping of the mode is low, closeness of the standstill frequency of the whirling mode to the largest excitation frequencies should be avoided. The lowest damped blade modes will in general be the modes related to edgewise deformations. The flapwise mode will have a lot of damping except in stall. This is explained in the next section, where the different possible stall-induced instabilities are discussed. For the tower modes, the fore/aft will usually be well damped, except when the turbine is in stall. The sideways tower motion will, similar to edgewise blade modes, usually have little damping. There are several instabilities that are known to be a risk for current size wind turbines. These will be discussed in the next sections.

Stall-Induced Vibrations The instabilities for which the wind turbine is most at risk are usually addressed as stall-induced vibrations. There are several instabilities that are related to stall. These will be addressed in this section. First there are edgewise and flapwise instabilities. The cases that are related to idling or parked wind turbines are discussed next followed by a short explanation of vortex-induced vibrations.

Edgewise and Flapwise Instabilities During normal operation the edgewise mode will always have very little damping. This is due to the aerodynamic force changes in the in-plane direction that result in low or even negative damping, especially in and near stall as was shown using a very simple model by Petersen et al. in 1998a. To show this, a later model derived by Hansen in 2007 will be used here. This is also a simple model of a blade section. The model has one degree of freedom as illustrated in Fig. 8 the aerofoil can translate along a straight line. Assuming quasi-steady aerodynamics and linearizing about the steady state, the force in the direction of the motion can be expressed as the linearized expression Fx ≈ F0 − ηx. ˙ The damping coefficient η which is similar to a viscous damping coefficient can then be derived to be: η=

 1 cρW0 CD (3 + cos(2θ − 2φ0 )) + CLα (1 − cos(2θ − 2φ0 )) 2  + (CL + CDα ) sin(2θ − 2φ0 )

(1)

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Fig. 8 A model of an aerofoil, derived by Hansen in 2007 with one degree of freedom in x direction, defined by the angle θ. The gray shaded area indicates the directions for which positive values of CL and CDα reduce damping

In this equation W0 is the total relative inflow speed, φ0 the angle between inflow and rotor plane, ρ the air density, and c the chord. Readers will already be familiar with the meaning of the variables Cl , Clα , Cd , and Cdα . If the value for η is negative, this corresponds to negative damping; therefore energy is added to the vibration by the aerodynamic forces. A motion in the edgewise mode will be represented by a θ angle that is close to 0◦ , while for flapwise modes, it will be close to 90◦ . The equation shows that the drag coefficient always adds damping to the motion, as CD is always positive. The effect of CLα on the damping depends on the sign of CLα . As long as it is positive, it will add damping, but when it is negative, it will reduce the damping. This is therefore directly related to stall. The effect of this is much larger in the flapwise direction than in the edgewise direction. If the blade is not in stall, this term of the equation will result in large damping in flap direction, which is why flapwise modes usually are very well damped. In the flapwise direction, it dominates the result of the equation for the damping. The effect of the last term in the equation depends strongly on the direction of the vibration. Positive coefficients result in a decrease in damping for vibrations in directions for which θ − φ0 is between 0◦ and −90◦ , indicated by the gray area in Fig. 8. To further illustrate the results of Eq. 1, Fig. 9 illustrates the results for a NACA63418 aerofoil as well as the lift and drag coefficients for that aerofoil. This would be an aerofoil representing the outer part of the blade, and as the loads are largest in that part of the blade, this would be the most relevant part of the blade when analyzing damping. The results in Fig. 9 show that before stall only the exact in-plane motion is slightly negatively damped, but when stall is entered, a larger range of vibrations becomes negatively damped. First the range of directions close to edgewise for which the motion is unstable increases, but this quickly expands also to the region for flapwise motions. The graph also shows that for angles of attack before stall,

Fig. 9 The damping coefficients for different angles of attack α and as a function of the angle θ. The aerofoil data that was used is plotted on the right showing the stall behaviour of the aerofoil. The steady-state inflow angle that was used was φ0 = 9.5◦

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there is a lot of damping in the flapwise direction, while the positive damping in edgewise direction is always small at best. Of course the model discussed so far is an extreme simplification of the real situation. The natural frequency has no influence on the results, the model is based on quasi-steady aerodynamics, and a wind turbine mode will result in different situations along the blade radius. However, this simple model is able to explain the main background and provide the most effective solutions. It can obviously not be used to determine the actual damping of a complete mode on the blade. As is clearly visible in Fig. 9, ensuring that the edgewise mode has some out-of-plane motion adds significant damping to the mode, assuming the angle of attack is in the linear part of the lift curve. Recall that the edgewise mode on a blade translates to three modes on a complete turbine (see section “Wind Turbine Modes”). The symmetric mode will result in a torque on the rotor shaft and therefore can be damped by the generator. As long as an increase in the rotor speed results in an increase in the torque of the generator, this mode will get significant damping from the generator. The other two modes on the complete turbine, the whirling modes, will not get any damping from the generator; therefore these modes are most at risk of becoming unstable. This was the case in the reported instances of edgewise instabilities in the 1990s (Anderson et al. 1999; Stiesdal 1994; Møller 1997). Further studies showed that the backward whirling mode for an analyzed (stable) turbine was significantly less damped than the forward whirling mode (Thomsen et al. 2000), and this corresponded to the difference that was visible in the natural mode shapes of these two modes: the backward whirling mode was significantly more in-plane than the forward whirling mode (Hansen 2003). This again indicates similar to the model discussed here that out-of-plane motion adds damping to edgewise modes. Therefore when an increase in damping is required for the edgewise mode, it is possible to achieve this by changing the structural pitch angle, the angle that defines the principal stiffness directions.

Idling Instabilities During very high winds, a wind turbine will go to an idling setting; the pitch angle is 90◦ , and there is no generator torque. In case of grid loss, it might not be possible to yaw the turbine in the wind direction. Therefore it becomes possible that large yaw misalignment angles are attained in an idling situation. Due to the strange inflow angles that can occur during idling situations at high wind speeds and large yaw misalignments, there are possible instabilities that can occur during idling or parked situations. These instabilities are largely similar to the stall-induced instabilities, as they will mainly occur in the (positive and negative) post-stall region. The worst cases of this instability are usually found at yaw misalignment angles of around −30◦ and around 30◦ , but other regions of yaw errors can also result in instabilities. As was shown by Politis et al. (2009) using first an isolated blade model and steady aerodynamics, there are several ranges where blade modes can result in instabilities. The first flap frequency can become unstable in a small region, but for a large part, it concerns the first and second edgewise modes that result in negative aerodynamic

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damping. When angles of attack of just above the maximum lift angle of attack are attained, such stall-induced instabilities can occur, but also when the angle of attack reaches ± 90◦ , the situation can be unstable. Skrzypi´nski and Gaunaa in 2015 focus more on cases where the angle of attack attains much larger values and the aerofoil is in deep and post-stall. The unsteady aerodynamic effect was modelled using a temporal lag in the aerodynamic response, and using a 2-D model, it was found that the edgewise instability actually becomes less severe when unsteady aerodynamics are included. These results are mainly for angles of attack of 90◦ and −90◦ . The calculations also showed that an angle of attack of −180◦ results in an instability independently of the temporal lag in the aerodynamic model. This illustrates one difficulty with the idling instabilities: the uncertainty in the dynamic behavior in stall and deep stall. Depending on the azimuth angle and the yaw misalignment angle, angles of attack can be attained in the full 360◦ range, while measurements are available for only the range that is closer to normal operations, and the unsteady aerodynamic models are also tuned and derived for a limited range of angles of attack. In Wang et al. (2017) also conclude that the flapwise and edgewise damping is less negative when dynamic stall effects are included. In this article the complete wind turbine is addressed; therefore the effect of all three blades and the resulting rotor modes are evaluated. This is of course important as the flow conditions at one blade during idling and yaw misalignment will be very different than on the other two blades, or if one blade is in an unstable region, it does not need to result in an instability for the full rotor mode. For a yaw angle of ± 30◦ , the combination on the three blades can be such that one blade is in positive stall, while another blade is in negative stall, and the third blade is in the attached flow regime. This strongly depends on the azimuth angle, as can be seen in Fig. 10, taken from Wang et al. (2017).

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Fig. 10 Time series of edgewise bending moment at the blade root with uniform inflow of 42.5 m/s, yaw angle 30◦ , for a 10 MW design, including dynamic stall effects. (Reproduced from Wang et al. 2017)

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Fig. 11 Ultimate blade root edgewise bending moments for six different wind seeds using an average wind speed of 42.5 m/s and 11% turbulence intensity (Wang et al. 2017)

As illustrated in Fig. 11 taken from the same article (Wang et al. 2017), the largest vibrations occur for yaw misalignment angles of around 30◦ and −30◦ . This picture shows the results of several load simulations for high winds and at different yaw angles. The extreme attained edgewise bending moments in each simulation are shown, and the most extreme values are clearly in the aforementioned range of yaw misalignment angles. For turbulent wind the simulation will show results that go in and out of the instability, but it is known that these simulations generally show large load variations and can be a real problem when performing the load calculations of a wind turbine design. For the well-known NREL 5MW reference turbine design, instabilities during idling conditions were also encountered (Jonkman and Matha 2011). Studies performed during the AVATAR project also showed instabilities when using BEM (Stettner et al. 2016; Heinz et al. 2016a). The results were strongly dependent on the aerofoil data as well as on the dynamic stall model that is used (see, for example, Fig. 12 from Stettner et al. 2016). In a project report from AVATAR (Heinz et al. 2016a), simulations performed using HAWC2CFD and comparing results to BEM simulations showed that possibly BEM codes are too conservative for the deep stall cases. Based on the results discussed so far, there is still no certainty on the accuracy of the load calculations in idling at large yaw angles, but in case BEM is not too conservative, one can expect that the most effective solution is to include a system that can still yaw the turbine under extreme circumstances preventing large yaw alignment errors.

Vortex-Induced Vibrations Another instability that can possibly occur during standstill or idling situations is related to vortex shedding (VIV). Vortex shedding is well known to cause problems

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Fig. 12 Damping values of first edge mode for blade as wing versus pitch angle. No wake effects, different dynamic stall models (Stettner et al. 2016)

on, for example, tall chimneys and power transmission lines. Recently it has also become a focus of research for wind turbines as there are indications that even at low wind speeds, this can cause severe problems. This could, for example, be during the installation or maintenance of the turbine. The phenomenon is difficult to capture using simplified models; BEM is not able to provide any predictions concerning vortex-induced vibrations. There exist simplified models that predict vortex-induced vibrations on cylindrical structures and show reasonable results compared to FSI simulations (Chizfahm et al. 2018). In Zou et al. (2015) uses a double wake concept to simulate vortex shedding in a 2-D aerofoil model. FSI simulations of wind turbine blades have indicated that this instability can occur on wind turbine blades in specific situations (Heinz et al. 2016b, a). Vortex shedding vibrations will only occur if the shedding frequency is close to a natural frequency of the blade or tower and when the vortex shedding is spanwise correlated over a large part of the blade or tower. The vortex shedding can therefore occur if the shedding frequency is close to constant for a large part of the structure. The frequency can be estimated using ωs =

SV c

(2)

with ωs the shedding frequency, V the freestream velocity, and c the chord. S is the Strouhal number which can be estimated using CFD (Skrzypi´nski et al. 2014). In Heinz et al. (2016c) used HAWC2CFD to investigate the possibility of vortex shedding vibrations, where Eq. 2 can be used to estimate the flow for which the chance is greatest to find vortex shedding to occur. There were several cases of VIV found that occurred using realistic flow conditions for the wind turbine blade of a 10 MW reference design in parked or idling conditions and at yaw misalignments. This indicates that the possibility of VIV is something that should be addressed during the wind turbine design.

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Classical Flutter Classical flutter is an instability well known for fixed-wing aircraft as well as rotary wing aircraft and has recently become a possible issue for wind turbines as well. The theoretical background will be provided in this section, followed by a description of how to determine the flutter speed of a wind turbine design.

Theoretical Background Classical flutter is an instability that is known from fixed-wing aircraft (Bisplinghoff et al. 1996) as well as helicopters (Bramwell et al. 2001; Johnson 1980). For a fixedwing aircraft, there will be a maximum velocity which should never be exceeded due to the occurrence of classical flutter. In classical flutter a wing or blade will vibrate violently in torsion and flapwise direction resulting in the loss of the aircraft wing or the rotor blade. For wind turbines, if a turbine is allowed to increase the rotor speed beyond the overspeed limit, then at some rotor speed large vibrations will occur. These vibrations are often due to classical flutter occurring. It is not always classical flutter that occurs in cases with increasing rotor speed. For example, there have been measurements and simulations on turbines that resulted in one of the edgewise modes becoming unstable, in which case no classical flutter speed can be determined (Kallesøe and Kragh 2016; Stäblein et al. 2017; Pirrung et al. 2014). On wind turbines higher rotor speeds result in the flapwise frequency increasing due to centrifugal effects, while the torsion frequency will decrease due to the distance between aerodynamic center and shear center. This can result in a flapwise mode and a torsion mode almost coinciding in frequency and then splitting into two modes with flapping and torsion, one of which can show significant negative damping values. Classical flutter is an instability that can be very violent which will not be solved by the structural damping present in the blade (Hansen 2007). Due to the increase in size of wind turbines and the need to avoid the square-cube law, effectively wind turbine blades have become more and more flexible, especially in torsion direction. Therefore the occurrence of classical flutter is now deemed a realistic issue for current size wind turbines, and the designs should be checked to assure that the classical flutter speed is far enough above the operational speeds (Lobitz 2005). As stated by Hansen, classical flutter can occur when the blade stiffnesses are relatively low, the center of gravity is aft the aerodynamic center for the outer part of the blade, the rotor speed is high enough, and an increase in the angle of attack should result in an increase of the lift coefficient, so the flow must be attached (Hansen 2007). It has been shown that the phenomenon of dynamics stall will increase the actual flutter speed (Hansen 2007; Lobitz 2004), so not taking the hysteresis loops into account results in a conservative estimation of the classical flutter speed. Due to the mechanism that results in classical flutter, the position of the center of gravity has a significant effect on the classical flutter speed. In helicopter blades very often, nonload carrying mass is added to the leading edge of the blade to ensure the flutter

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speed to be high enough. For wind turbines this would also work, but it can be an expensive solution as the mass of the blade will increase; therefore the blade will require additional load carrying fibers to allow for the increased gravitational loads, and all other components of the turbine will need to be stronger to carry all increased loads. The classical flutter speed can also be increased by increasing the torsion stiffness of the blade; this is much more effective than increasing the flapwise stiffness. The actual flutter speed of the wind turbine will mainly depend on the position of the center of gravity and the torsion stiffness, but a possible sweep (or equivalently an edgewise deformation) of the blade will also influence the flutter speed (Hansen et al. 2011). As was already mentioned, the increase of the wind turbine size and beating the square-cube law has resulted in relatively lower torsion stiffness of the blades. This has resulted in classical flutter becoming a possible issue on current size wind turbines (Lobitz 2005). The increase in wind turbines being designed for offshore, which provides the option of operating at higher rotor speeds as the noise limits can be eased, means that the classical flutter speed of a wind turbine design is becoming more likely to be an issue. Therefore one has to check the design and ensure that the classical flutter speed is high enough to safely operate the wind turbine, even during extreme gusts.

Determine Flutter Speed A classical flutter speed can best be determined using a time simulation tool by running a stimulation with increasing rotor speed. There are also tools based on solving the linearized equations of motion that will directly provide the flutter speed as an output, but the time simulation is usually more accurate. As an example Fig. 13 shows a result from a classical flutter simulation. The turbine is free to spin up as the generator torque is set to zero. The wind speed slowly increases, and at some point, the unstable flutter mode appears; the torsion and flapwise vibrations rapidly increase. This is the typical behavior one will observe in a case of classical flutter. The slowly increasing uniform wind ensures that your turbine will increase its rotor speed while the angle of attack remains in the attached flow regime range. The actual slope of the wind speed will have only little influence on the flutter speed (Pirrung et al. 2014). The coupling between flapwise and torsion motion has to be with some phase difference. If the motions are exactly in phase, the torsion will not reduce the damping of the flapwise motion. So if one wants to ensure that the instability observed is indeed classical flutter, one can zoom in on the deformations and check if there is indeed a (often small) phase difference (see Fig. 14). A linearized aeroelastic tool (see also section “Linearised Analysis Tools”) can provide mode shapes and damping values and therefore provide the classical flutter speed by calculating the damping values for different rotor speeds until the flutter mode becomes unstable. As a phase difference between the motion in flapwise and torsion direction is the root cause of the instability, any tool that can be used must

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Fig. 13 Example of a classical flutter speed run, showing wind speed, rotor speed, flapwise tip deflection, and torsion tip deformation (Wiener-Milenkovi´c parameter (Wang et al. 2016))

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at least perform its calculations taking the phase differences into account. There are publications where linearized tools have been used to determine classical flutter, illustrating the mechanism causing the instability. It is also possible to predict the classical flutter speed for a single blade only, and as illustrated in Hansen (2007), the results for such an analysis are close to the results obtained using a

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full turbine model. Still, these kinds of calculations are linearizations, and there are often more simplifications of the turbine model included than will be the case for a time simulation; therefore it makes sense to always check the results using a time simulation tool. There are many different parameters that will influence your results. As was already mentioned, there is a clear effect of the dynamic stall model, despite the fact that the blade should not be stalling. The hysteresis effect in the attached flow regime adds damping to the classical flutter mode; therefore it becomes unstable at a higher rotor speed than when using steady aerodynamics (Lobitz 2004; Hansen 2007). Then the pitch angle will affect the flutter speed, and the results for higher air densities (equivalent with lower temperatures) will result in a lower flutter speed than lower air densities (Gao et al. 2018). Therefore when using a time simulation tool, one should run several simulations to ensure that the flutter speed is high enough for all cases. It has also been shown in Pirrung et al. (2014) that a near wake model can influence the flutter speed that is found. Note that determining an actual rotor speed as the flutter speed is actually not straightforward when the results look as those shown earlier in Fig. 13. These kinds of results do not clearly provide a particular rotor speed at which the damping has become negative. The vibrations start increasing in amplitude due to the increase in rotor speed and at some point due to the instability, but at the same time, the rotor speed continues to increase; therefore the actual limit is not clear. After the suggested simulation with increasing wind, it is possible to further identify the stability boundary by running simulations at constant wind speed and zero torque and the chosen pitch angle, where the wind speed is chosen such that the rotor speed will not increase any further or in other words when there is effectively no aerodynamic power generated by the turbine (Kallesøe and Kragh 2016). One could run simulations at the settings close to where the instability is expected to occur, and using a constant rotor speed, though this is not the exact same situation as free spinning, it can give a more accurate indication of the maximum allowable speed of the turbine than the initial overspeeding simulation will provide. The time simulation tool that is used for a classical flutter speed must include enough detail to provide accurate results. Obviously the torsion degree of freedom has to be included in the tool as the torsion mode plays a vital role in this instability. If a tool derives the mode shapes of the blades and uses this for the simulation, one can expect these results to be unreliable, as the modes will change significantly during the flutter speed simulation due to the change in aerodynamic forces and rotor speed. This could be solved by running several simulations at different rotor speeds. For a classical flutter speed, there must be sufficient detail in the turbine model, and it should be preferred to use the most accurate tools available and settings that are known to be valid for such a simulation. This will increase the computation time, but as one needs to perform only a few simulations, this should not be an issue. Using BEM with a dynamic stall model is not expected to be invalid for a classical flutter speed calculation, unless the tip speeds becomes so high that the aerofoil data and the assumption of incompressible flow are no longer valid.

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Aeroelastic Evaluations To assess if a design is aeroelastically stable, one can use specific stability or modal analysis tools based on linearizations of the equations in the time domain. These will provide mode shape, frequency, and aerodynamic damping for each of the relevant modes. Another option is to use nonlinear time domain simulation tools that are usually the same tools as those used for load set calculations. Then once there is a prototype, it is also possible to set up specific measurements to evaluate the total damping of some of the relevant modes on the turbine. This section will show some approaches to obtain linearized equations for a rotating wind turbine such that it becomes possible to perform stability calculations. The option of using a nonlinear time domain tool is also discussed. A short discussion of possible measurements is also provided, and finally the tool demands will be addressed.

Linearised Analysis Tools To calculate the aerodynamic damping of a wind turbine mode or of a blade mode, an aeroelastic stability tool can be used. This does require a linearization, as properties such as mode shapes and frequencies are linear properties of a system. Therefore aeroelastic stability tools will be less detailed than most time domain tools, but at the same time, the results will cost significantly less computation time. The nonlinear time domain tools such as FAST/OpenFAST or Bladed also allow for a linearization about an operating point, resulting in the state-space model for the operating point and small deviations from that point. For control design these tools are also required, as the controllers will be designed using linearized tools. To calculate mode shapes, damping values, and natural frequencies, one needs to set up a linearized model. For a wind turbine, there is always one big issue with setting up a linear model and that is the rotor rotation. This rotation results in an azimuth angle that reaches the complete range of 0 to 360◦ and therefore cannot be assumed small and linearized. There are several methods that can be used to set up a modal analysis of a periodic system, solving the issue with the azimuth angle. One rather simple and straightforward method is the Coleman transformation. When using the Coleman transformation, one assumes that the combination of deformations on the blades can be written in a different set of coordinates (Bir 2008; Hansen 2003). Assuming that the flapwise deformations of each blade are transformed, these are then represented using three new coordinates: one coinciding with a tilting motion of the rotor plane, one with a yawing motion of the rotor plane, and one that covers the symmetric fore/aft motion of the rotor plane. The coordinates that are used are illustrated in Fig. 15, and the equations to transform the flapwise deformations qj to these rotor coordinates are

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Fig. 15 Illustration of the three coordinates used in Coleman transformation for a flapwise deformation. One is for the collective motion (a), one for motion in a tilting plane (b) and one in a yawing plane (c)

a0 =

1 2 2 qj , a1 = cos(ψj )qj , b1 = sin(ψj )qj 3 3 3 3

3

3

j =1

j =1

j =1

(3)

The azimuth angle ψj will be ψj = ψ1 + 2π(j − 1)/3

(4)

The a0 coordinate in Eq. 3 is depicted in Fig. 15 (a), the a1 coordinate in (b), and the b1 coordinate in (c). The flapwise deformations of the blades can be written as a sum of these three coordinates instead of describing the deformation of each blade as a coordinate. The same equations can be used for the edgewise deformations of the blade, resulting in a collective coordinate, a horizontal coordinate, and a vertical coordinate.

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Using the Coleman transformation, the periodicity drops from the equations, and one can now linearize the equations. If any periodicity is still left in the equations, then this is removed by averaging. The resulting linear time-invariant model can then be solved to calculate Eigenvalues and with this find natural frequencies, damping values, and mode shapes. To ensure that the result is reasonably accurate, it is required that the steady-state deformation is first determined using the aerodynamic forces on the system. These forces and therefore the deformations will depend on the rotor speed, the pitch angle, and the wind speed that is analyzed. So for every combination that is to be analyzed and for which mode shape, frequency, and damping are to be calculated, the steady state should first be calculated. Due to its simplicity, the Coleman transformation is often the method used in stability tools. The Coleman transformation is in fact a special case of the Floquet analysis for an isotropic rotor (Skjoldan and Hansen 2009; Skjoldan 2011). For twobladed rotors and cases that cannot be assumed to be isotropic, it is however not possible to use the Coleman transformation, and one has to rely on more complex methods such as Floquet analysis or Hill’s method. In Floquet analysis, the linearized equations of motion are separated in a periodic part and an exponential part (Stol et al. 2002; Skjoldan 2011). So if the state-space form of the equations of motion is {y} ˙ = [A(t)]{y}

(5)

with {y} ˙ a column with the degrees of freedom qi and the corresponding time derivatives {q˙i }, then it is possible to set up a state transition matrix [(t, 0)] such that (Stol et al. 2002): {y(t)} = [(t, 0)]{y(0)}

(6)

This transition matrix can be decomposed [(t, 0)] = [P (t)]e[AL ]t where [P (t)] is a periodic matrix with period T = is [AL ] =

2π Ω

1 ln([(T , 0)]) T

(7) and the constant matrix [AL ]

(8)

The stability of {y} is determined by the eigenvalues of [AL ]. These eigenvalues are called the characteristic exponents of [A(t)]. A diagonal matrix [ ] with these characteristic exponents λi and the modal matrix [V ] define the eigen value problem: [AL ] = [V ][ ][V ]−1

(9)

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And similarly, [(t, 0)] = [V ][Σ][V ]−1

(10)

with [Σ] the diagonal matrix with the characteristic multipliers σi and the same modal matrix [V ] as before. From Eq. 8 one gets: [ ] =

1 ln [Σ] T

(11)

This indicates that one can determine the characteristic components using (Stol et al. 2002; Skjoldan and Hansen 2009) 1 1 λi = ln|σi | + j T T

  I m(σi ) arctan + 2π k = ζi + j ωi Re(σi )

(12)

with j 2 = −1 and k appearing in the equation due to the fact that a logarithm of a complex number will have infinitely many branches. The Floquet analysis for systems with a limited amount of degrees of freedom can be performed without any issues; however, when the number of degrees of freedom is above ≈ 50, then the calculation time becomes an issue in this analysis, specifically the calculation of the transition matrix [] (Stol et al. 2002). To solve this issue and allow for larger systems to be solved using Floquet theory, extensions to the Floquet theory have been suggested (see, for example, Peters 1994). In Hill’s method the separation that lays the basis for Floquet analysis is also used, so there is a periodic part and an exponential part. A selected number of terms in a Fourier series are then used to obtain eigenvalues (Skjoldan 2009). This method is also described and used in Hansen (2016) for two-bladed turbines. This is a short description of Coleman transformation and Floquet theory. The details of different methods can be found in the different cited references; specifically (Skjoldan 2011) provides an extensive overview of the different methods that are available. Once the transformation has been used and the Eigenvalues are determined, one knows the natural modes, frequencies, and damping values for the turbine. This provides very quick insight into possible instabilities and can be used to identify trends on how to increase damping. However, these models are usually much simpler than the models for nonlinear time domain tools; therefore the accuracy can also be less. As was mentioned at the start of this paragraph, nonlinear simulation tools such as Bladed and FAST or OpenFAST also allow for a linearization around an operating point. As the steady state of the rotating wind turbine will then be periodic, the coordinate transformations discussed above will also be used to transform these linear models.

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Fig. 16 Example of results obtained using linearized analysis, reproduced from Stäblein et al. (2017), showing frequencies and damping values for different modes and different blade designs, obtained using HAWCStab2

As an example of the kind of results that can be obtained using linearized aeroelastic stability tools, Fig. 16 shows results from Stäblein in Stäblein et al. (2017) for different blade designs. The blade designs included bend-twist coupling, and a linearized stability tool (in this case HAWSStab2) can very quickly illustrate the effect of the coupling on the natural frequencies and damping. Tools that only analyze the damping of a blade, such as Blademode (Lindenburg 2003), can directly solve the linearized equations of motion, and the transformations from rotating to standstill frame are then not required. Of course this also means that the effect of the tower and the other two blades is not taken into account and though it is possible to represent it for the collective/symmetric modes, there is no way to model the whirling effects.

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Using Nonlinear Time Domain Simulation Tools It is also possible to evaluate aeroelastic stability using time domain simulation tools. As the actual damping value is a linearized property, it is difficult to determine the damping in a nonlinear time domain simulation tool. After all, due to nonlinear effects, the damping will actually not be constant, so there is not one single value that can be determined. However, a nonlinear time simulation can provide qualitative information on the stability, and with some post-processing, it is also possible to evaluate damping in a quantitative way. The time domain simulation tools are generally used for the load calculations to determine the ultimate and fatigue loads that can be expected during the lifetime of the turbine. These tools can also be used with a different perspective, that of evaluating if there are any possible stability issues on the turbine. Of course in general, this is then already taken into account in the fatigue and ultimate loads that result from the full load set calculation; however, it should be preferred to perform additional analysis and use a more accurate tool or setting than used for the standard load cases. This is for several reasons. First, the amount of simulations needed is much smaller than a full load set calculation; therefore more accurate settings resulting in longer calculation times should not be too much of a problem. Also one does not need to include the aeroelastic evaluation in each loop of the design, at least one evaluation when the design is almost fixed as a final check could be enough, though performing a first scan earlier in the process can of course be beneficial. Second, the safety factor that is used in the load set calculations is a safety factor on the calculated loads only, but if something in the calculation is less accurate which might make the difference between badly damped and actually an unstable situation, then the safety factor will not cover the effect this will have on the loads. The classical flutter speed can be determined as was described in section “Classical Flutter.” Next to evaluating the flutter speed, one should analyze the damping mainly qualitatively by running simulations for a broad range of settings (wind speeds, pitch angles, rotor speeds). In these simulations an excitation should be included, such that one can evaluate the damping after the excitation. An excitation can be similar to hitting the system with a hammer, for example, a sudden jump in wind speed (so start at a lower or higher wind speed than the speed you want to evaluate, and then jump to the correct speed). Such an excitation will affect all modes on the turbine. It is also possible (if the tool allows you) to be more specific and include an excitation at the frequency to be analyzed and stop the excitation after a couple of cycles, similar to what was done on an actual turbine (Thomsen et al. 2000). There must be enough coupling between the effect on the loads due to the excitation and the mode that has to be analyzed. For example, collectively pitching the blades to try to stimulate a tower sideway mode will be less successful as there is very little interaction to be expected between the excitation and the mode one wants to excite. The simulations are preferably performed using the most accurate settings, but taking into account any uncertainties that might exist. For example, the dynamic

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stall model is known to have significant influence on stability, but it is also known that the accuracy of the dynamic stall models can still be improved. Therefore to ensure that there is some safety margin, evaluating without dynamic stall model or using different dynamic stall models for the most critical cases that are found would be preferred. The results of the simulations should be evaluated in detail, checking the output in time in detail to find out if there are any vibrations that are increasing in amplitude or if there are any vibrations that remain too long in the simulation indicating that the damping is low and the design could be improved by trying to increase the damping. Cases such as idling should also be assessed when analyzing the aeroelastic stability of a wind turbine design. As discussed in section “Stall-Induced Vibrations,” idling cases can result in instabilities occurring on a wind turbine. The results of a frequency tool can be used as a starting point for the time simulations; the frequency tool provides insight into when the turbine is expected to suffer from instabilities or low damping. The time simulation can then be used to verify these results and check the sensitivity of the issue to the different parameters to ensure that the turbine will not get into a situation that the load calculations predicted to be stable, but more accurate tools show to be possibly unstable. Clearly it is possible to analyze the aeroelastic stability using a time simulation tool; post-processing of the output can aid in assessing if there are any issues. Creating spectrograms of different output parameters can help identify low and negative damping value (see, for example, Fig. 17). This graph shows a spectrogram for a classical flutter speed calculation using the torsion deformation output of the

Fig. 17 Example of a spectrogram. This shows the spectrogram of the torsion degree of freedom during a classical flutter speed simulation. Such a simulation includes an increasing rotor speed. The time signal is shown in Fig. 13

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simulation. This clearly shows a frequency just below 6 Hz that becomes dominant in the last half of the simulation. That frequency corresponds to the classical flutter mode that is unstable for that part of the simulation. Using the same approach that is needed to create the spectrogram, it is also possible to create a graph using the moving block FFT (Holierhoek 2008; Hansen et al. 2006b). The color coding is then replaced by actual lines (3-D plot) or only the peaks (2-D plot) for the different FFT’s that are analyzed over different time segments. If the simulation is not too nonlinear, it can also be possible to use system identification and fit a linear function to the output and assess damping values this way (Holierhoek 2008). One can also use a band pass filter and try to fit an exponential function to the results (Thomsen et al. 2000). Methods that are used in the analysis of measurements are also possible. These are shortly addressed next. However, a complete discussion of the methods used in using measurement data as well as the aforementioned system identification and different filtering methods is outside the scope of this book.

Measurements Measurements can also be performed with the specific aim of trying to evaluate the aeroelastic stability of a certain wind turbine mode. Clearly the system will not behave linearly, and similar to the described results for nonlinear time simulations, additional post-processing is required to assess the damping. There have been specific measurement campaigns performed in research projects to asses damping values for some of the wind turbine modes (Thomsen et al. 2000; Hansen et al. 2006a). The possibilities one has to excite a specific mode or all modes on the turbine when performing measurements are of course different from the options one has in a simulation. You cannot control the wind speed in any way, so using this as a sudden excitation is out of the question. What can and has been done is exciting a particular mode using pitching actions at the mode’s frequency and stopping this excitation after a couple of cycles or letting an eccentric mass rotate in the hub again at a mode’s frequency and suddenly stopping the excitation. The generator has also been used as a means of excitation to evaluate the damping. In overspeed there will also be an instability occurring, as discussed in section “Classical Flutter.” This can be measured on a real turbine in the field as well and has been described in Kallesøe and Kragh (2016). To measure the overspeed limit, one can simply turn off the generator torque and let the turbine speed up. This type of measurement has of course significant risk of damaging the turbine and should therefore only be undertaken at relative low wind speeds, and one should be able to quickly shut down the turbine as soon as vibrations are increasing. To obtain frequency and damping values from measurements, one can analyze the time signal in case of a performed excitation and try to fit an exponential function through the peaks once the excitation was stopped. This was done by Thomsen et al. in 2000 as well as by Hansen et al. in 2006a. This method only provides results for modes that have been specifically excited; therefore the application is quite limited.

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Methods that try to identify the aeroelastic properties of a wind turbine based on measurement data of a wind turbine in turbulent wind are, for example, OMA, POMA, and PARMAX. Using OMA for such an analysis was done by Hansen et al. in 2006a and by Ozbek and Rixen in 2013. Riva et al. (2016) perform a comparison between POMA and PARMAX, concluding that PARMAX outperforms POMA. These methods could also be used to analyze results from nonlinear time domain simulations. A complete discussion of OMA, POMA, and PARMAX is outside the scope of this book.

Tool Demands To perform an aeroelastic analysis of a wind turbine design, one should check that the tool that is used suffices in accuracy. Over the years, the turbine design has changed significantly; therefore the demands on the codes have also increased. Where in the early years it was, for example, reasonable to neglect torsion and to model a structural twist angle with a single parameter, these kinds of simplifications are no longer valid. Here a short overview is provided of different aspects that should be included in the tools used. A linearized tool will always include more simplifications than the nonlinear time domain tools. The linearized tools can be valuable for a quick scan of possible instabilities, but its results should be verified using a more detailed nonlinear time domain simulation tool. The accuracy of the different linearized tools can also differ significantly, depending on the modelling assumptions that are used. The blade properties that are known to have significant impact on damping should be included in the model. This is, for example, the structural pitch distribution along the radius, due to the effect it can have on the low damping of the edgewise modes. But also the position of the CG should be provided along the radius due to the effect on the classical flutter speed. Clearly the mode shape is very important when looking at the damping of the mode, so any parameters that influence the mode shape significantly should be included in the tool. When Coleman transformations are used, the model is also less accurate than when Floquet analysis or Hill’s method are used. To be able to perform a realistic evaluation of the wind turbine design, the nonlinear time simulation tool that is used should be accurate enough to present reliable results. Due to the current size and flexibility of the wind turbine designs, unless it concerns a rather straight blade with relatively high torsional stiffness, the accuracy of simulations when using linearized blade modes really needs to be avoided, especially when the blade modes are determined for the undeformed state (Kallesøe 2011). The blades will have significant steady deformation during operation which has impact on the mode shapes. To give one example, an unloaded blade will have a coupling between edgewise and torsion due to the prebend of the blade, but when the blade is deformed, the flapwise deformation often results in a blade that is actually bent in the opposite direction, and the coupling between edge and torsion shifts in phase by 180◦ . The approach of evaluating the blade as different

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flexible (linearized) segments does provide results that are accurate enough, though there are more accurate nonlinear blade models that can be used. The model that is used for the tower will usually be simpler; based on the Craig-Bampton approach, only a couple of modes of the tower are included. There are no indications yet that this could be a problem for accurate evaluations, but it is known that not including torsion in the tower will affect the damping of whirling modes that would interact with the tower torsion. It is not difficult to imagine that the tower torsion will influence the turbine mode shape of especially the flapwise whirling modes, but small effects on edgewise whirling modes could result in significant differences in damping due to the out-of-plane blade motion that would be influenced by tower torsion. The frequency and time domain tools will include aerodynamic loads that are usually based on BEM. This is a valid approach for most cases, as long as the currently common engineering models are included to cover effects such as yawed flow, tip correction, dynamic stall, and dynamic inflow. The vortex-induced vibrations that were discussed in section “Vortex-Induced Vibrations” need to be evaluated using other tools, and this would require FSI based on CFD to be able to provide accurate results. In addition there are cases where it is not yet clear if the BEM approach is valid or needs to be further extended with additional engineering models, think of idling instabilities, for example. As wind turbines are still increasing in size, it can be expected that the validity of our tools will be challenged again and again. Research will continue to keep the tools up to date for the challenging designs that are produced.

Aeroelastic Design and Innovations How can we use all this knowledge on wind turbine aeroelasticity to further improve the analysis and the designs of wind turbines? For this goal, first the aspect of the aeroelastic design will be addressed, followed by some of the latest innovations that are used and can be deemed relevant from an aeroelastic point of view.

Aeroelastic Design When a turbine is designed, the design has to be evaluated from an aeroelastic point of view. In this chapter, issues such as resonances and instabilities have been discussed, and especially those instabilities that result in loss of the turbine must be prevented. Resonances and instabilities resulting in limit cycle effect can perhaps be allowed to occur on a design, but it is clear that the resulting design is then not the optimal design. The issues result in increased vibrations that increase the fatigue loads, and therefore the design would allow for further improvement in design by preventing the issue. It is of course not always that black and white, perhaps there is a small resonance effect for specific wind speeds that leads to an amplification factor of 1.1, and removing it from the design would reduce the

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fatigue loads only by a very small percentage and probably not be cost-effective to resolve the issue. It strongly depends on what design changes are needed to prevent an issue; some issues are solved by increasing stiffness, therefore increasing cost and mass of the blade or tower. Some issues are solved by changing the controller, for example, using different operating conditions, so an alternative rotor speed for certain wind speed ranges, which may reduce the power output. Every design change is a complex and careful weighing of the attainable decrease in loads, the change in power production, (especially in case of controller alterations) and the change in material costs. As a general guideline, it is recommended to ensure that the Campbell diagram is carefully checked. One should be able to evaluate if any closeness of an excitation frequency to a natural frequency is an issue as the damping of the mode is low, and changes must be made to prevent it or if it can be allowed. Changes can be alterations to the blade design resulting in more damping to the mode, a change in the natural frequency, or adjusting the rotor speed. Then one should also try to prevent aeroelastic stability problems, and even low positively damped modes can result in significant fatigue loads if they are excited regularly. Therefore the damping values of the modes on the turbine for the different operating conditions should be determined or at least qualitatively assessed. Instabilities should be prevented during all situations, and if possible increasing the damping of low damped modes can further improve the design as it can lead to a reduction of the loads. The instabilities that were discussed in section “Instabilities” all have clear ways to most effectively resolve them. So in case of negative damping of an edgewise or flapwise mode, the most effective solution will be to stay away from stall or, for the edgewise mode, increase the out-of-plane motion present in the mode. If the classical flutter speed is not high enough, increasing torsion stiffness and adding mass to the leading edge are very effective solutions. The aerofoils that are selected also have significant impact on the aeroelastic behavior of the wind turbine. A sudden and steep change in the lift coefficient when in stall will increase the risk of negative damping occurring for the edgewise and the flapwise modes. In case aerofoils with such behavior are selected, then the steadystate angle of attack during the power production cases should really be kept far away from the stall angle of attack, such that turbulence and gusts do not result in an unstable situation occurring. In the evaluation of the damping for different operating conditions, it is important that there is also a safety margin taken into account. At this moment we cannot perform simulations that are completely matching the real-life situation. Therefore during the load calculations that are performed to compare calculated ultimate and fatigue loads to the maximum allowable values for the design, a safety factor is used. However, thinking of an inaccuracy in a simulation that results in a small difference in the damping value, but it changes it from slightly positive to slightly negative, the loads can become infinitely larger than the calculated loads. Therefore a safety factor on the loads will not cover such a difference. One should take into account that, for example, the effect of a dynamic stall model on the damping can be significant,

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but the phenomenon of dynamic stall is still not completely captured in the models. Therefore the load calculations or the stability analysis should be performed with this in mind. Checks can be made by asking the following questions: How much effect does a certain engineering model have on the damping situations? How much effect do different properties and/or settings have on the results? Is this within an allowable limit? Structural damping has a significant influence on the modes with the least damping; however, one should be aware if the input data is accurate and test what would happen if the input is actually slightly too optimistic. So, would the turbine still be stable for the entire operating range if the structural damping is actually less than was assumed? To summarize, one should assure that even small inaccuracies in the input parameters and in the models used do not result in large differences in the calculated loads. And when there are any stability issues found or possible resonances identified, it is possible to change the design to prevent the issues. Knowledge of the background of instabilities is then vital, such that it becomes relatively easy to change the design in an effective way to increase damping in a certain mode.

Innovations Currently innovations from an aeroelastic design point of view will be mostly focusing on the blade design. One aeroelastic innovation that is being looked at is the possible fatigue load reduction that can be obtained by including bend-twist coupling. This coupling can be structural coupling, using an asymmetric lay-up of the fibers, but it can also be caused by including sweep in the blade. A sweep in the blade will make the production process more complicated and expensive due to the shape of the mold becoming even more complicated. Changing the fiber lay-up to achieve the structural coupling therefore has a larger cost reduction potential. The coupling between bending and torsion can be designed such that a sudden gust that increases the angle of attack is counteracted by a torsion deformation that reduces the angle of attack. This means that an increase in flapwise bending should result in a torsion deformation toward feather. As an example, Fig. 18 shows the results for the flapwise blade root bending moments. This illustrates that the frequency response for wind speed variations is reduced when there is a coupling of flap to feather. This confirms earlier results from, for example, Lobitz and Veers (2003) and Bottasso et al. (2013). Then there are many aerodynamic enhancements available that can change the aeroelastic loads on the turbine. For example, vortex generators (VGs) can be used on the wind turbine blade to improve power production. They will delay the onset of stall; therefore they also influence the aeroelastic stability of the blade. The difference between steady-state angle of attack and the stall angle can be increased. However, the VGs will also increase the drag at low angles of attack, and this can affect damping of modes, for example, if CDα increases, this will decrease the damping of the edgewise mode (see Eq. 1). Possible aerodynamic enhancements are discussed in other chapters of this book.

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Fig. 18 Frequency response of the flapwise blade root bending moment to mean wind speed variation between 0.0 and 2.0 Hz for steady-state operation at mean wind speeds of 5, 10, 15, and 20 ms−1 . (Taken from Stäblein et al. 2017)

Finally controls can be used to reduce the fatigue and extreme loads. Individual pitch control (IPC) has been specifically developed to reduce the fatigue loads on the wind turbine. By pitching the three blades on the turbine differently at a 1P and/or higher nP frequencies, some of the loads due to wind shear and other harmonic excitations can be reduced. The way controllers handle sudden gusts or direction changes can also further improve the turbine design; if extreme event control is implemented, the extreme loads can also be reduced (Kanev and van Engelen 2010). It is foreseen that the possibilities of further load reductions will be further extended due to improvement in sensors that are being developed. If it is possible to measure more and more accurately during operation, the possibilities for controllers and perhaps for including other controlling devices (e.g. flaps, tabs, etc.) will be further extended.

Conclusions Wind turbine aeroelasticity is a complex field of expertise, but it is of great importance. Designs must be free of instabilities and relevant resonances. A resonance is a case where an excitation frequency is close to a natural frequency, while an instability is a case where during one cycle of a mode energy is added by the aerodynamic forces to that mode vibration. Linearized tools can be used to quickly gain insight into the damping of different modes and for different operating conditions. Due to the simplifications that are used in linearized analysis, it is

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advisable to further investigate the design using nonlinear time domain tools to ensure that the design will not show any instabilities. The blade model needs to be accurate for these kinds of analyses, using modes for the blades can result in significant inaccuracies, and therefore in these kinds of analyses, one should use the more advanced blade modelling options. Knowing the cause of the currently relevant instabilities also provides the most effective solutions to these issues. The most relevant instabilities on current wind turbines are stall-induced vibrations, classical flutter, and idling instabilities. There are also indications that vortex shedding vibrations could occur on wind turbines in very specific situations. Resonances can also cause additional fatigue loads on a wind turbine. In this aspect one also has to be aware of the real frequencies of a natural mode. The natural modes have one principal (dominating) frequency; however, due to the rotor rotation, there are also other harmonic components in the mode that could also result in resonances. Resonance is only an issue if there is little damping present in a mode, and using the knowledge from the instabilities, one can increase damping in low damped modes effectively and reduce the effects of the resonance.

Cross-References  Pragmatic Models: BEM with Engineering Add-ons

References Anderson C, Heerkes H, Yemm R (1999) The use of blade-mounted dampers to eliminate edgewise stall vibration. In EWEA 1999, Nice Bielawa RL (1992) Rotary wing structural dynamics and aeroelasticity. AIAA Education Series, Washington, DC Bir G (2008) Multiblade coordinate transformation and its application to wind turbine analysis. In: Proceedings of the 2008 ASME Wind Energy Symposium Bisplinghoff RL, Ashley H, Halfman RL (1996) Aeroelasticity. Dover Publications, Inc., New York Bottasso CL, Campagnolo F, Croce A, Tibaldi C (2013) Optimization-based study of bend-twist coupled rotor blades for passive and integrated passive/active load alleviation. Wind Energy 16:1149–1166 Bramwell ARS, Done G, Balmford D (2001) Bramwell’s helicopter dynamics, 2nd edn. Butterworth-Heinemann, Oxford Chizfahm A, Yazdi EA, Eghtesad M (2018) Dynamic modeling of vortex induced vibration wind turbines. Renew Energy 121(C):632–643 Collar AR (1946) The expanding domain of aeroelasticity. J R Aeronaut Soc L:613–636 Connell JR (1981) The spectrum of wind speed fluctuations encountered by a rotating blade of a wind energy conversion system: qObservations and theory. Technical Report PNL-4083 UC-60, Pacific Northwest Laboratory, Battelle Fung YC (1969) An introduction to the theory of aeroelasticity. Dover Publications, Inc., New York Gao Q, Cai X, Guo X-W, Meng R (2018) Parameter sensitivities analysis for classical flutter speed of a horizontal axis wind turbine blade. J Cent South Univ 25(7):1746–1754

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Hansen MH (2003) Improved modal dynamics of wind turbines to avoid stall-induced vibrations. Wind Energy 6:179–195 Hansen MH (2007) Aeroelastic instability problems for wind turbines. Wind Energy 10:551–577 Hansen MH (2016) Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors. Wind Energ Sci 1:271–296 Hansen MH, Thomsen K, Fuglsang P, Knudsen T (2006a) Two methods for estimating aeroelastic damping of operational wind turbine modes from experiments. Wind Energy 9:179–191 Hansen MOL, Sørensen JN, Voutsinas S, Sørensen N, Madsen HA (2006b) State of the art in wind turbine aerodynamics and aeroelasticity. Prog Aerosp Sci 42(4):285–330 Hansen M (2011) Aeroelastic properties of backward swept blades. In 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition Hau E (2006) Wind turbines: fundamentals, technologies, application, economics, 2nd edn. Springer, Berlin/Heidelberg Heinz J, Sørensen NN, Riziotis V, Chassapoyannis P, Schwarz CM, Iradi SG, Stettner M (2016a) Stand-still operation. Technical Report Deliverable 4.5 AVATAR Heinz JC, Sørensen NN, Zahle F (2016b) Fluid-structure interaction computations for geometrically resolved rotor simulations using CFD. Wind Energy 19(12):2205–2221 Heinz JC, Sørensen NN, Zahle F, Skrzypi´nski W (2016c) Vortex-induced vibrations on a modern wind turbine blade. Wind Energy 19(11):2041–2051 Holierhoek JG (2008) Aeroelasticity of large wind turbines. Ph.D. thesis, Delft University of Technology Johnson W (1980) Helicopter theory. Princeton University Press, Princeton Jonkman JM, Matha D (2011) Dynamics of offshore floating wind turbines-analysis of three concepts. Wind Energy 14(4):557–569 Kallesøe BS (2011) Effect of steady deflections on the aeroelastic stability of a turbine blade. Wind Energy 14:209–224 Kallesøe BS, Kragh KA (2016) Field validation of the stability limit of a multi mw turbine. J Phys Conf Ser 753:1–7 Kanev S, van Engelen T (2010) Wind turbine extreme gust control. Wind Energy 13(1):18–35 Kristensen L, Frandsen S (1982) Model for power spectra of the blade of a wind turbine measured from the moving frame of reference. J Wind Eng Ind Aerodyn 10:249–262 Lindenburg C (2003) Bladmode; program for rotor blade mode analysis. Technical Report ECNC–02-050, ECN, Petten Lobitz DW (2004) Aeroelastic stability predictions for a MW-sized blade. Wind Energy 7:211–224 Lobitz D (2005) Parameter sensitivities affecting the flutter speed of a MW-sized blade. In 43rd AIAA Aerospace Sciences Meeting and Exhibit Lobitz DW, Veers PS (2003) Load mitigation with bending/twist-coupled blades on rotors using modern control strategies. Wind Energy 6:105–117 Møller T (1997) Blade cracks signal new stress problem. WindPower Monthly Ozbek M, Rixen DJ (2013) Operational modal analysis of a 2.5 MW wind turbine using optical measurement techniques and strain gauges. Wind Energy 16(3):367–381 Peters DA (1994) Fast floquet theory and trim for multibladed rotorcraft. J Am Helicopter Soc 39(4):82–89 Petersen JT, Madsen HA, Björck A, Enevoldsen P, Øye S, Ganander H, Winkelaar D (1998a) Prediction of dynamic loads and induced vibrations in stall. Technical Report Technical Report Risø-R-1045(EN), Risø National Laboratory, Roskilde Petersen JT, Thomson K, Madsen HA (1998b) Local blade whirl and global rotor whirl interaction. Technical Report Technical Report Risø-R-1067(EN), Risø National Laboratory, Roskilde Pirrung GR, Madsen H, Kim T (2014) The influence of trailed vorticity on flutter speed estimations. J Phys Conf Ser 524:012048 Politis E, Chaviaropoulos P, Riziotis V, Voutsinas S, Romero-Sanz I (2009) Stability analysis of parked wind turbine blades. In: Proceedings of the EWEC, pp 16–19 QBlade, http://www.q-blade.org

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Riva R, Cacciola S, Bottasso CL (2016) Periodic stability analysis of wind turbines operating in turbulent wind conditions. Wind Energy Sci 1(2):177–203 Skjoldan P (2009) Modal dynamics of wind turbines with anisotropic rotors. In: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition Skjoldan PF (2011) Aeroelastic modal dynamics of wind turbines including anisotropic effects. Ph.D. thesis, Risø DTU Skjoldan P, Hansen M (2009) On the similarity of the Coleman and Lyapunov-Floquet transformations for modal analysis of bladed rotor structures. J Sound Vib 327(3):424–439 Skrzypi´nski W, Gaunaa M (2015) Wind turbine blade vibration at standstill conditions – the effect of imposing lag on the aerodynamic response of an elastically mounted airfoil. Wind Energy 18(3):515–527 Skrzypi´nski W, Gaunaa M, Sørensen N, Zahle F, Heinz J (2014) Vortex-induced vibrations of a du96-w-180 airfoil at 90 degree angle of attack. Wind Energy 17(10):1495–1514 Spera DA (1994) Wind turbine technology: fundamental concepts of wind turbine engineering. ASME, New York Stäblein AR, Hansen MH, Verelst DR (2017) Modal properties and stability of bend–twist coupled wind turbine blades. Wind Energy Sci 2(1):343–360 Stettner M, Reijerkerk MJ, Lünenschloß A, Riziotis V, Croce A, Sartori L, Riva R, Peeringa JM (2016) Stall-induced vibrations of the AVATAR rotor blade. J Phys Conf Ser 753:042019 Stiesdal H (1994) Extreme wind loads on stall regulated wind turbines. In: BWEA 16, Stirling, UK. Mechanical Engineering Publications Ltd Stol K, Balas M, Bir G (2002) Floquet modal analysis of a teetered-rotor wind turbine. J Sol Energy Eng 124(4):364–371 Thomsen K, Petersen JT, Nim E, Øye S, Petersen B (2000) A method for determination of damping for edgewise blade vibrations. Wind Energy 3(4):233–246 Wang Q, Jonkman J, Sprague M, Jonkman B (2016) Beamdyn user’s guide and theory manual. Technical report, NREL Wang K, Riziotis VA, Voutsinas SG (2017) Aeroelastic stability of idling wind turbines. Wind Energy Sci 2(2):415–437 Zou F, Riziotis VA, Voutsinas SG, Wang J (2015) Analysis of vortex-induced and stallinduced vibrations at standstill conditions using a free wake aerodynamic code. Wind Energy 18(12):2145–2169

Part III Experimental Approaches to Wind Turbine Aerodynamics

Wind Tunnel Wall Corrections for Two-Dimensional Testing up to Large Angles of Attack

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W. A. Timmer

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blockage in Attached Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid and Wake Blockage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lift Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Corrections on Coefficients for Streamlined Flow . . . . . . . . . . . . . . . . . . . . . . Correction of the Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correction of Measurements in the Deep-Stall Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maskell’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corrections on Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Wall Pressure Signature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Source-Source-Sink Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Matrix Version of the Wall Signature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

602 603 603 605 609 610 611 612 615 616 616 622 624 626 627 628 628

Abstract

An accurate representation of two-dimensional airfoil characteristics measured in a wind tunnel generally requires the inclusion of corrections for interference effects that exist due to the presence of the wind tunnel walls. This chapter discusses the most commonly used correction schemes both for streamlined and separated flow regimes. The classical correction method based on small velocity perturbations gives very good results up to angles of attack of about 20 degrees

W. A. Timmer () Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_27

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for chord-to-tunnel height ratios c/h up to 0.36. Even with separation of the boundary layer at a chord location of 30% the corrected pressure distribution matches that of a much smaller model with c/h = 0.15. In the deep-stall range of angles of attack, where the flow separates from the leading edge, the method based on the wake analysis by Maskell with a blockage factor of 0.96 seems to give good results for two-dimensional models up to c/h values of 0.27. A comparison with measurements corrected with the matrix version of the pressure signature method, which uses the pressure distribution on the tunnel walls, shows that the latter leads to slightly larger corrections. Maskell’s method, for which the blockage parameter of 0.96 apparently is based on a single measurement of a two-dimensional flat plate, seems to give better results when a value of 1.03 is used. Keywords

Wall corrections · Wake blockage · Solid blockage · Wake buoyancy · Lift interference · Deep stall · Maskell method · Pressure signature method · Source-source-sink method

Introduction The two-dimensional aerodynamic characteristics of dedicated airfoils play a prominent role in the design of wind turbine blades. As long as the flow is attached and the airfoil surface is smooth, numerical codes are able to predict the airfoil force and moment coefficients to a satisfying degree of accuracy. However, when the angle of attack increases beyond the stall angle and boundary layer separation starts to move forward, or when the leading edge area of the airfoil is contaminated or degraded, or when flow dynamics play a significant role, these predictions increasingly lose their accuracy. At this point, wind tunnel measurements are still indispensable to determine the performance. Due to considerations of energy consumption and flow quality and stability the great majority of wind tunnel tests is performed in closed test sections. When a wind tunnel model is placed inside a closed test section the presence of the wind tunnel walls alters the flow field around the model. The model and its wake partly block the passage leading to a local increase of the velocity and the expanding wake induces a pressure gradient in flow direction referred to as wake buoyancy. In addition, the curvature of the streamlines associated with lift generated by the model will be affected by the straight walls of the tunnel. With increasing rotor diameter, blade design requires airfoil performance data at higher Reynolds numbers. This leads to existing wind tunnel facilities being used with increasingly larger model chords for two-dimensional testing, giving larger blockages in the same angle-of-attack range. In the following paragraphs examples and the applicability of the most common correction schemes for streamlined flow and for separated flow will be explored, each in their own angle-of-attack range.

17 Wind Tunnel Wall Corrections for Two-Dimensional Testing up to . . .

603

Blockage in Attached Flow Interference effects in wind tunnel testing have been the subject of a large number of publications, starting as early as the late 1920s with the work of Lock (1929) and Glauert (1933). The classical correction equations most commonly used in twodimensional sub-sonic wind tunnel testing find their origin in the assumption of linearized potential flow between the model and the walls. With a (limited) number of singularities, such as vortices to represent the lift, sources for the wake and source-sink doublets to represent the model volume, a theoretical model of the object and its wake is made. The method of images is then used to calculate the interference effects at the model location. The corrections refer to a situation in which the thickness and camber of the airfoil are small, the chord is small with respect to the tunnel height, and the induced velocities everywhere in the test section are small compared to the undisturbed flow velocity. This justifies the neglect of higher powers and products of the interference factors and enables the superposition of interference effects and consequently makes it possible to consider the influence of camber and thickness and of model and wake blockage separately.

General Form In a closed test section the presence of the model and its wake gives rise to an increase at the model location of the undisturbed (apparent) velocity U : U = (1 + εb )U 

(1)

where b is the total blockage factor. The prime denotes the uncorrected value. In its simplest form an arbitrary nondimensional force coefficient Ca in an incompressible flow can be corrected for blockage according to Ca =

Ca (1 + εb )2

(2)

For small blockage factors (2) may be written as Ca = Ca (1 − 2εb )

(3)

However, when the effect of compressibility in the correction is included, the equations take a slightly different form. The corrected-uncorrected dynamic pressure ratio is written as: q ρ =  q ρ



U U

2 (4)

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The true density at the model is related to the apparent density ρ by the isentropic relation   1   γ −1 U 2 ρ γ − 1 2 = 1 − − 1 (5) M ρ 2 U where M is the uncorrected Mach number. With equation 1 and the ratio of specific heats for air γ = 1.4 we find: 

2.5 ρ 2 2 + ε = 1 − 0.2M − 1 (1 ) b ρ

(6)

Now the combination of equations (1), (4), and (6) is sufficient to correct an arbitrary force coefficient ca for blockage including the effect of compressibility using   q  (7) ca = ca q Equation (6) can be further simplified by application of the binomial theorem (a + b)n = a n + na (n−1) b +

n(n − 1) n−2 2 a b ................. 2!

(8)

Neglect of higher powers of εb then gives ρ = 1 − M 2 εb ρ

(9)

Combining (1), (4), and (9) we find

q = 1 − M 2 εb (1 + εb )2  q

(10)

and written to the first order of b we arrive at

q 2 = 1 + (2 − M )ε b q

(11)

The effect of neglecting higher order terms of the blockage factor on the lift and drag coefficients for moderate angles of attack and reasonably sized models stays well below 0.5%. In the same manner the impact of blockage on the Reynolds number can be found using     ρ μ U Re = Re (12)  ρ μ U

17 Wind Tunnel Wall Corrections for Two-Dimensional Testing up to . . .

605

According to von Kármán and Tsien (Allen and Vincenti 1947) the ratio of the coefficients of viscosity is related to the ratio of temperatures by 

μ μ



 =

T T

0.76 (13)

The isentropic relation for the temperature is     U 2 T γ − 1 2 M = 1− −1 T 2 U

(14)

To the first order this leads to

Re = Re 1 + (1 − 0.7M 2 )εb

(15)

Likewise the correction of the Mach number is determined, writing M=M





U U



a a

 (16)

where a is the uncorrected speed of sound. In an ideal gas the speed of sound is only proportional to the square root of the temperature, which gives M=M





U U



T T

(17)

Written to the first order, this leads to

M = M  1 + (1 + 0.2M 2 )εb

(18)

Solid and Wake Blockage If it is assumed that the model is small compared to the tunnel test section and that the lift is not too large, the blockage due to the model (solid blockage) and that due to the wake (wake blockage) can be treated separately: εb = εs + εw

(19)

Solid Blockage Solid blockage is the result of the displacement of streamlines in the tunnel due to the volume of a non-lifting model. In an analysis by Lock the base profile in the center of the tunnel at zero incidence is modeled by a single doublet. The effect

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of the walls in two dimensions is replaced by an infinite system of doublet images extending on both sides of the model spaced at a distance equal to the test section height. By symmetry the velocity component normal to the walls is zero. The net effect of the tunnel walls upon the flow at the base profile is an increase in effective axial velocity of magnitude

εs =

  π 2 c 2 t 2 1 λ2 12 h c β3

(20)

where λ2 is a parameter related to the airfoil thickness t, and the airfoil surface pressure distribution. β is the Prandtl-Glauert compressibility correction factor  1 − M  2 , c is the airfoil chord, and h is the effective tunnel height. An approximation for λ2 was given bij Glauert, λ2 = 2A/(πt2 ), which effectively turns Eq. 20 into εs =

π A 6 β 3 h2

(21)

where A is the cross-section area of the airfoil. Thompson (Garner et al. 1966) suggests a relation in which the thickness of the airfoil is more prominently accounted for:    t A π 1 + 1.2β εs = 3 6 c β h2

(22)

The airfoil cross-section area A can be written as a combination of a factor and the product of maximum thickness and chord. For airfoil families resulting from analytical descriptions of the shape this factor is often a constant. For the old 4digit NACA series of airfoils based on the 00xx thickness form, for example, A= 0.69∗t∗c. The cross-section area of the 6-digit NACA 63-series, though strictly speaking not the result of a prescribed shape but rather of a systematically prescribed pressure distribution, can be approximated using a factor of 0.62. Allen and Vincenti found a similar equation as (20) but they use the expression  2 t =4 λ2 c

(23)

They write: εs =

σ β3

(24)

17 Wind Tunnel Wall Corrections for Two-Dimensional Testing up to . . .

607

with the tunnel blockage factor σ =

π 2 c 2 48 h

(25)

The body-shape factor is defined as 16 = π

1

    2  y x 1 − C  1 + dy d p c dx c

(26)

0

in which Cp is the inviscid zero-incidence pressure coefficient at the chord-wise station x and y is the ordinate of the symmetric (base) profile. Allen and Vincenti give values for a number of base profiles. With relation (26) the shape factor for the NACA 63-0xx series of airfoils has been calculated using the inviscid pressure coefficients at 300 chord stations. The present -values for the NACA 63 series in the thickness range of interest to wind turbine blades are represented by equation 27: N ACA63 = 1.4890

 2   t t + 0.0023 + 1.6143 c c

(27)

The parameter t/c is the maximum relative thickness of the airfoil in fractions of the chord. Figure 1 gives a comparison with the values presented by Allen and Vincenti. Their lower values can be attributed to the much lower number of pressure stations taken for the integration. 0.5

Λ

Allen & Vincenti

0.4

Present calculations 0.3

0.2

0.1

0 0

0.05

0.1

0.15

Fig. 1 The body shape factor for the Naca 63 series of airfoils

0.2

t/c

0.25

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Equations (22) and (24) do not differ very much in their calculation of the solid blockage factor. For an 18% thick laminar airfoil like NACA 633 -418 and a typical Mach number of 0.2 the factor by Allen and Vincenti is 1% lower. When using a 30% thick airfoil such as DU 97-W-300 the difference increases to -1.6%. The differences go up with decreasing Mach number and are around zero close to M = 0.4. The solid blockage equations were essentially derived with the assumption that the blockage is independent of lift. Based on the work of Batchelor (Garner et al. 1966), it is suggested that the simple theory expressed in equations (22) or (24) can be adapted to deal with the solid blockage of an airfoil at an angle of attack α by writing:   1.1 2 α εs (α) = εs 1 + β (t/c)

(28)

with α expressed in radians.

Wake Blockage The wake effect is simulated by a system of source images. Source strength is related to the measured drag coefficient using conservation laws with the boundary condition that the flow field far upstream remains unchanged. Under these assumptions the wake blockage at the tunnel center can be determined (Garner et al. 1966) from: εw =

1 c 1 + 0.4M 2  Cd ( ) 4 h β2

(29)

The Total Blockage Factor The total blockage factor can now be composed from the contributions of the solid and the wake blockages. Using the notation of Allen and Vincenti this gives:   α2 1 c 1 + 0.4M 2  σ εb = 3 1 + 1.1β + ( ) Cd 4 h β β2 (t/c)

(30)

The two-dimensional approach of the derivations essentially implies a rectangular test section. However, many wind tunnels have working sections with corner fillets, which raises the question what value of the tunnel height h should be used. From considerations of continuity an effective height he can be derived from the test section area divided by the span of the model. Insertion of Eq. (30) into Eq. (10) or (11) yields the change of the dynamic pressure due to the blockage inside the test section, which can be used with Eq. (7) to correct the force and moment coefficients for blockage. For small values of the blockage factor we may use

17 Wind Tunnel Wall Corrections for Two-Dimensional Testing up to . . .

1 q  ≈ 1 − (2 − M 2 )εb = q 1 + (2 − M 2 )εb

609

(31)

To correct forces and moments for blockage using Eq. 31 we may write Ca = Ca



q = Ca 1 − (2 − M 2 )εb q

(32)

Note that for incompressible flow (M = 0) this equation equals (3).

Wake Buoyancy Apart from a blockage effect, the developing wake induces a velocity increase in flow direction and consequently, applying Bernoulli’s equation, also gives rise to a pressure gradient along the model which would not exist in free air. This pressure gradient is felt by the model as buoyancy and the associated increase in drag follows from a derivation by Allen and Vincenti based on the work of Glauert: D = D 



1 + 0.4M 2 σ β3

 (33)

The true drag in free air is given by   1 + 0.4M 2 D = D  − D = D  1 − σ β3

(34)

With the definition of the drag coefficients we find   1 + 0.4M 2 σ D = cd qc = cd q  c 1 − β3

(35)

The drag coefficient in free air, with reference to the true dynamic pressure follows from    1 + 0.4M 2 q  cd = cd 1 − (36) σ q β3 Combined with Eq. (31) and written to the first order the corrected drag coefficient is given by cd =

cd

with εb given by Eq. (30).



 1 + 0.4M 2 2 1− σ − 2 − M εb β3

(37)

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There is some debate about the applicability of the buoyancy correction if the drag is derived from the pressures in the wake, reduced in the usual way. Allen and Vincenti state that the correction equations are primarily derived for drag measured with a balance, but that for normal chord-to-height ratio’s values differ by less than 0.5%. Rogers (Garner et al. 1966) argues that in particular the wake buoyancy equation should not be applied when the drag is measured with a wake rake. It is indeed questionable whether the impact of the drag increase due to the wake-wall interference is captured by the wake rake data reduction, since the speed-up due to the expanding wake for reasons of continuity mainly takes place outside the wake, which is discarded if the wake survey method is used in the conventional way. It can be argued that the wake buoyancy correction should only be applied if the drag is derived from balance measurements or from the model pressure distribution.

Lift Interference A straight walled closed test section prevents the normal curvature of the flow around an airfoil producing lift since the streamlines along the walls are straight. As a result the model appears to have more camber showing increased lift and moment coefficients and an induced upwash, changing the angle of attack. This problem of lift interference (streamline curvature) is evaluated o.a. by Allen and Vincenti for a thin airfoil with its chord on the tunnel center line. They approximate the load on the airfoil by distributed vorticity along the chord. Vortex theory is used on a system of images with alternating signs to mimic the tunnel walls. With the requirement that the distribution of lift along the chord and especially the magnitude of the lift component near the leading edge of the airfoil shall be the same both in free air and in the tunnel, their evaluation leads to a set of relations in the form of α = α  + α   q cl = cl + cl q

(38)

 q   + cm cm = cm q where the increments quantify the effect of the lift interference to the order (c/h)2 and are defined as  σ    Cl + 4Cm α = 2πβ Cl = −Cl

σ β2

(39)

17 Wind Tunnel Wall Corrections for Two-Dimensional Testing up to . . .

Cm = Cl

611

σ 4β 2

and q /q can be derived from Eq. (11). The angle of attack is in radians. According to Garner et al. (1966) the applicability of these relations is restricted to c r (Peinke et al. 2019; Behnken et al. 2020). Thus, the multipoint and multiscale probabilities of Eq. (35) can be determined by p(ur |u(x); ur  ). Furthermore, it is possible to express the p(ur |u(x); ur  ) by stochastic equations. How well this works for cases of ideal turbulence, wind turbulence, and wave turbulence is detailed in Peinke et al. (2019) and Behnken et al. (2020). Finally, we want to note that the central probability p(ur |u(x); ur  ) for a general N -point statistics is given by the velocity values at u(x), u(x + r), and u(x + r  ) and this is a statistical three-point quantity. In this way, it seems that turbulence can be described in a comprehensive way by the knowledge of three-point statistics. This shows that even the elaborated multifractal models of section “Multifractal Models” are missing statistically something, as these only consider two-point features to turbulence.8 Finally, several important features of turbulence on the background of homogeneous isotropic turbulence have been presented. Aspects of atmospheric turbulence like non-stationarity, directional dependencies (non-isotropic), and non-homogeneity are relevant in the context of atmospheric turbulence. Their understanding is crucial for quantifying their impact on wind energy systems. Nevertheless, the discussed basic features of HIT need still to be taken into account and may become modified like shown in Behnken et al. (2020).

Discussion For the practical application in the analysis of different turbulent situations, as it is necessary for the planning of new sites for wind turbines, it is important to consider in the so-called site assessment in which statistical way the wind situations are classified. If a classification is not statistically complete, one can expect surprises or unexpected phenomena. An example would be when the wind conditions at two different sites are the same after the incomplete characterization, the real wind conditions may differ. For the significance of such differences, it is the question whether the higher-order statistical features – multiple points or higher-

8 An

open-source software package is available to perform such analysis with given data (github.com/andre-fuchs-uni-oldenburg/).

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order moments – will have an impact on a wind turbine. In this way, it is interesting to understand whether, for example, intermittency, as the low order statistical feature of turbulence, which is not considered in current standards, already has an impact on wind turbines. This effect of intermittency is discussed controversially (Schwarz et al. 2019; Berg et al. 2016), which may be a consequence of the properties of intermittent signals discussed in our chapter. Note that very small and very large fluctuations are amplified for the intermittent case, so that larger and smaller loads may be the result, depending on the system behavior. In this chapter, it is shown that it is possible to parameterize the intermittency of wind conditions, which could be used in the future to better capture this aspect. Multipoint statistics have not previously been discussed as reasons for uncertainty in turbulent wind conditions. Here, it is suggested that such multipoint statistics can be approximated via a three-point closure, allowing new time series for wind turbulence to be generated (Behnken et al. 2020) and short-term forecasts to be made (Hadjihosseini 2016). In principle, the multipoint approaches should also make clear what wind gusts are statistically. In addition, the multipoint statistics should provide access to clustering of wind gusts, which is potentially important for wind turbine operations. If a gust hits a wind turbine and causes a vibration, then the question of whether a second gust occurred may cause additional loads that cannot be understood if the complexity of the wind turbulence is not properly accounted for. Finally, basic features of homogeneous isotropic turbulence and their relevance for wind energy applications have been discussed. Further effects caused by turbulence are present in the atmospheric boundary layer as wind situations are often nonstationary, non-isotropic, and nonhomogeneous. The importance of all these must be quantified with respect to their impact on energy systems (i.e., loads and power production).

Cross-References  Aeroelastic Stability Models  CFD for Wind Turbine Simulations  CFD-Type Wake Models  Turbulence of Wakes  Turbulent Inflow Models

References IEC 61400-1:2019 ©IEC 2019 INTERNATIONAL ELECTROTECHNICAL COMMISSION, Geneva Arneodo A et al (1996) Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity. EPL (Europhysics Letters) 34(6):411 Babiano A, Dubrulle B, Frick P (1997) Some properties of two-dimensional inverse energy cascade dynamics. Phys Rev E 55(3):2693

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Behnken C, Wächter M, Peinke J (2020) Multipoint reconstruction of wind speeds. Wind Energy Sci 5:1211–1223 Benzi R, Ciliberto S, Tripiccione R et al (1993) Extended self-similarity in turbulent flows. Phys Rev E 48(1):R29 Berg J, Natarajan A, Mann J, Patton,G (2016) Gaussian vs non- Gaussian turbulence: impact on wind turbine loads. Wind Energy 19:1975–1989. https://doi.org/10.1002/we.1963 Briscolini M, Santangelo P, Succi S et al (1994) Extended self-similarity in the numerical simulation of three-dimensional homogeneous flows. Phys Rev E 50(3):R1745 Carlson JA, Jaffe A, Wiles A (2006) The millennium prize problems. American Mathematical Society Castaing B, Gagne, Y, Hopfinger EJ (1990) Velocity probability density functions of high Reynolds number turbulence. Phys D Nonlinear Phenom 46(2):177–200 Castaing B (1996) The temperature of turbulent flows. J Physique II EDP Sci 6(1):105–114 Chilla F, Peinke J, Castaing B (1996) Multiplicative process in turbulent velocity statistics: a simplified analysis. J Phys II 6(4):455–460 Chevillard L, Castaing, B, Lévêque E, Arneodo A (2006) Unified multifractal description of velocity increments statistics in turbulence: intermittency and skewness. Phys D Nonlinear Phenom 218(1):77–82 Davidson PA, Kaneda Y, Moffatt K, Sreenivasan KR (eds) (2011) A voyage through turbulence. Cambridge University Press Davidson PA (2004) Turbulence. Cambridge University Press Dubrulle B (1994) Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance. Phys Rev Lett 73(7):959 Forschungs- und Entwicklungszentrum Fachhochschule Kiel GmbH (2020) https://www.fino1.de/ en/, visited 26 Nov 2020 Frisch U, Kolmogorov AN (1995) Turbulence: the legacy of AN Kolmogorov. Cambridge University Press Frisch U, Sulem P-L, Nelkin M (1978) A simple dynamical model of intermittent fully developed turbulence. J Fluid Mech 87:719–736 Gagne Y, Marchand M, Castaing B (1994) Conditional velocity pdf in 3-D turbulence. J Phys II 4(1):1–8 Gaudin E, Protas B, Goujon-Durand S et al (1998) Spatial properties of velocity structure functions in turbulent wake flows. Phys Rev E 57(1):R9 github.com/andre-fuchs-uni-oldenburg/ OPEN_FPE_IFT Hadjihosseini A, Wächter M, Hoffmann NP, Peinke J (2016) Capturing rogue waves by multi-point statistics. New J Phys 18:013017 IEC Standard 61400-1 Ed. 4 (2019) Wind turbines, Part 1: design requirements, EN 61400-1:2018 Khintchin A (1934) Korrelationstheorie der stationären stochastischen Prozesse. Mathematische Annalen 109:604–615. https://doi.org/10.1007/BF01449156 Kolmogorov AN (1941a) Local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Proc USSR Acad Sci 30:301–305 (reprint Proc R Soc Lond A 434:9– 13 (1991)) Kolmogorov AN (1941b) Dissipation of energy in locally isotropic turbulence. Proc USSR Acad Sci 32:16–18 (reprint Proc R Soc Lond A 434:15–17 (1991)) Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82–85 L’vov VS, Procaccia I (2000) Analytic calculation of the anomalous exponents in turbulence: using the fusion rules to flush out a small parameter. Phys Rev E 62:8037 Mann J (1994) The spatial structure of neutral atmospheric surface-layer turbulence. J Fluid Mech 273:141–168 Meneveau C, Sreenivasan KR (1991) The multifractal nature of turbulent energy dissipation. J Fluid Mech 224:429–484 Morales A, Wächter M, Peinke J (2012) Characterization of wind turbulence by higher-order statistics. Wind Energy 15:391–406

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Naert A, Castaing B, Chabaud B, Hebral B, Peinke J (1998) Conditional statistics of velocity fluctuations in turbulence. Phys D Nonlinear Phenom 113(1):73–78 Peinke J, Tabar MRR, Wächter M (2019) The Fokker-Planck approach to complex disordered systems. Ann Rev Condensed Matter Phys 10:107–132 Polyakov AM (1995) Phys Rev E 52(6):6183 St. B. Pope (2000) Turbulent flows. Cambridge University Press Renner C, Peinke J (2012) A generalization of scaling models of turbulence. J Stat Phys 146:25–32. https://doi.org/10.1007/s10955-011-0345-1 Renner C, Peinke J, Friedrich R (2001) Experimental indications for Markov properties of smallscale turbulence. J Fluid Mech 433:383–409 Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press Schwarz CM, Ehrich S, Martín R et al (2018) Fatigue load estimations of intermittent wind dynamics based on a Blade Element Momentum method. J Phys Conf Ser. https://doi.org/10. 1088/1742-6596/1037/7/072040 Schwarz CM, Ehrich S, Peinke J (2019) Wind turbine load dynamics in the context of turbulence intermittency. Wind Energy Sci 4:581–594 She ZS, Leveque E (1994) Universal scaling laws in fully developed turbulence. Phys Rev Lett 72(3):336–339 Siefert M, Peinke J (2004) Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence. Phys Rev E 70:015302 Vassilicos JC (2001) Intermittency in turbulent flows. Cambridge University Press Veers PS (1988) Three-dimensional wind simulation, Technical Report SAND88-0152, Sandia National Laboratories Veers PS et al (2019) Grand challenges in the science of wind energy. Science 366. https://doi.org/ 10.1126/science.aau2027

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recycling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthetic Coherent Eddy and Stochastic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Filter Based Wind Field Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Random Spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Fields Based on Continuous-Time Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sandia Method and Rotational Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Mann Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

834 835 836 837 838 839 841 844 847 847 852 855 856 856

Abstract

This chapter gives a short overview of different methods used for turbulence generation in the field of wind energy. The wind fields can be used as an inflow for computational fluid dynamics or blade element momentum-based simulations. For all presented models, the mathematical background is given, and it is discussed which advantages and drawbacks they have. The main focus lies on statistical properties in terms of one- and two-point statistics. This includes variance, autocorrelations, cross correlations, and spectral properties. First different recycling methods are explained, namely, the weak and the strong

S. Ehrich () Department of Physics, University of Oldenburg, Oldenburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_42

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recycling methods. In the following sections, synthetic coherent eddy methods are shown which approximate the turbulent properties well. Those are the digital filtering method and the random spots method. Also an inflow model based on continuous-time random walks is demonstrated which considers higher-order statistics, the increment statistics. In the last section, two spectral methods are in the focus which are used in a wide range in the field of wind energy, the Sandia method, and the Mann model.

Introduction The nonlinearity of the Navier-Stokes equation system results in turbulent structures which are at the same time random and coherent. Simulating those motions accurately with resolving computational fluid dynamics (CFD) methods like direct numerical simulation (DNS) or large eddy simulation (LES) leads to several challenges which are to be solved. Important for the physical evolution of the turbulence are numerical correctness but also inflow boundary conditions which represent the spatial and temporal properties in a proper way. Otherwise the turbulent fields can decay instantaneously (Jarrin et al. 2006), and prescribed turbulence intensities are not advected through the domain. The evolution of turbulent vortices in the flow is strongly dependent on their initial state. Therefore it is obvious that much effort has to be put into the development of an inlet boundary condition which is able to generate realistic turbulent structures which also persist over large distances downstream. Creating such a boundary condition is in general not an easy task because several statistical properties of turbulence like temporal and spatial correlations, length scales, spectra, or even higher-order characteristics have to be considered. For the application on bridges, towers, or wind turbines, the generated turbulent structures are very important because the energy content at different frequencies usually has very different effects on loads. This is obvious if eigenfrequencies of the rotor blades or the tower and vortex shedding are taken into account. In general two different groups of methods of generating turbulent velocity data exist. In the first group, the recycling methods, the equations of fluid motion are solved in an auxiliary CFD simulation where the inflow data is extracted from the interior of the domain. The methods of the second group generate artificial inflow data by using mathematical models and are therefore an approximation to the recycling methods and not a solution to the Navier-Stokes equations. These methods normally have some shortcomings in the physicality and correctness compared to the first group, but they are much more simple and less costly, and the characteristics of the created velocity fields are known much better. Linked to these points is the high flexibility of those models which is needed for setups where several simulations with different inflow conditions have to be performed, without the costs of undergoing additional auxiliary simulations.

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Recycling Methods For the CFD simulation of turbulent flow fields with small scales of motion, a highresolution domain is needed and a simulation method which makes high-quality results available. One of those methods can be DNS, where all turbulent scales are resolved if the domain consists of cells which are able to capture those scales. But the drawback is the high computational costs involved, which makes this procedure in most cases unusable. If compromises between resolution and computational costs have to be made, LES is the way to achieve this goal. According to the usage of a coarser grid, not all scales are resolved, and the smaller scales have to be modelled by LES subgrid-scale models. In order to simulate objects in turbulent flow by means of CFD, usually the simulation domain has to be very big to achieve fully developed turbulence close to the area of interest. The additional computational cost due to the region, where the turbulent builds up, plays a big role for DNS and LES and should be minimized. One solution of this problem is to use periodic boundary conditions for the inflow and outflow which allows the turbulence to build up in a much smaller domain. The resulting turbulence can then be introduced to blade element momentum (BEM) or other CFD domains. But this approach is only fruitful for simple geometries which normally are also very long, e.g., a channel, and where fully developed turbulence is expected. Therefore the method of using periodic boundary conditions will fail for more complex objects or regions where it is not expected to achieve fully developed turbulence, e.g., for hills or forests. However, for fully turbulent cases, auxiliary simulations in much smaller domains can be used to reach the fully turbulent state in a fast way and extract the velocity field which is then introduced in the actual simulation at the inlet boundary. Different approaches of using a strictly periodic auxiliary simulation are discussed by Chung and Sung (1997) and will be shown in the next section. Spalart (1988) extended the periodic boundary approach by adding a coordinate transformation to the formalismF. But even if some new applications became accessible, the approach from Spalart is restricted to flows whose transverse variation is much bigger than the mean streamwise variation and it is also very complex. Thus this idea was extended by Lund et al. (1998) who also used an auxiliary simulation which takes the wind field from a downstream location, rescales it, and reintroduces it at the inlet. Wind fields can then be extracted and introduced at the actual simulation. Of course all those simulations which use periodic boundary conditions have the drawback that big structures in the flow are disturbed and do not behave as they actually would in a much bigger domain as is shown by Lygren and Andersson (1999). Even if those simulations can be very costly, their advantage compared to synthetic turbulence methods is that they can also reproduce higherorder statistics like the velocity derivative skewness which is highly important for nonlinear energy transfer processes. Another advantage is the realistic transient in the section where the turbulence develops. This behavior and the evolution of properties like skin friction or momentum thickness can be described very well by standard empirical relations, but fitting both properties at the same time is very difficult (Akselvoll and Moin 1993, 1996).

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Even though recycling methods are not very popular for simulating wind turbines, they are used very often for the generation and study of atmospheric boundary layers (Mayor et al. 2002; Han et al. 2016) and for the study of the wake of wind turbines (Dörenkämper et al. 2015). There is a lot of potential in those methods especially with an increase of computational power. This is the reason why this topic is given attention in the next sections where examples for strong and weak recycling methods are given. With strong recycling, we mean here that an auxiliary simulation is used with periodic boundary conditions at the inflow and outflow patches. For weak recycling methods, the periodicity is also present but in the sense that also a rescaling of the wind fields at the boundary is possible. For best results, the mesh should be of same resolution and cross-sectional area as in the simulation of interest.

Strong Recycling One way of generating inflow turbulence by means of CFD is to perform a spatially homogenous auxiliary simulation of a channel-shaped domain with periodic inflowoutflow boundaries and noisy initial conditions. After several simulation cycles, the whole system converges to a fully developed turbulent channel or boundary layer flow, and an upstream sweeping through the domain with a constant velocity is done. At the different passed positions, the velocity data is then extracted. However, this approach, which is based on Taylor’s frozen turbulence hypothesis, would need an extremely long domain size for the auxiliary simulation to realize inflow data for a long time interval. This would lead very quickly to issues in CPU time and storage capacity. However, a simplification can be made by reducing the domain size of the auxiliary simulation and repeating the sweeping procedure several times. But this will lead to the unavoidable problem of periodicity which has to be overcome. Two different ways of modifying the inflow data were shown by Chung and Sung (1997). They relaxed the periodicity by use of phase jittering and amplitude jittering in Fourier space for each sweeping cycle. The former method adds a random phase to the phase of the flow field for each data point, while the second one multiplies a random factor to the amplitude, i.e., the energy spectrum is modified. The advantage of phase jittering is that the variance of the flow field is kept constant, but the spatial structures are partly destroyed. In contrast to this, amplitude jittering modifies the variance, but the turbulent structures are kept. This is especially important for wallbounded flows which usually consist of several kinds of eddy structures. Another advantage of amplitude jittering is the small adjustment zone which is needed in the main simulation to build up real turbulence again. But still there is space for improvement left. In addition to the method described to achieve spatial flow data, it is also possible to use a temporal method. This temporal method extracts the flow field at a fixed cross section far off from the boundaries in the auxiliary simulation while this simulation runs. In this way, the extracted fields will not show periodicities, but the additional simulation time has to be accepted. Chung et al. have shown that the temporal method and the spatial method with amplitude jittering are able to catch

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the statistics of the channel flow. Also a mixture of both methods performs well where the spatial and the temporal approaches alternate. Sweeping takes place at a time instant followed by extraction of the flow field at a specific position over a fixed time interval and so on. Compared to those methods, adding white noise to the mean flow field leads to big discrepancies to measurement data and the other methods in terms of skin friction and turbulent structures in general. The flow tends to be laminar after a small distance from the inflow plane. Overall it can be said that in terms of non¯/periodicity and physicality, the temporal method performs best followed by the mixture of spatial and temporal approaches and amplitude/phase jittering. In terms of computational costs, the temporal method performs worst. It is case dependent if periodicities are a big issue or not.

Weak Recycling One very often used example for weak recycling in evolving turbulent boundary layers is described by Lund et al. (1998) and is a simplification of an approach presented by Spalart (1988), where the inhomogeneity associated with the boundary layer growth in streamwise direction is minimized by defining a new coordinate system for the whole domain. Lund’s implementation is fast because a development region is not needed and comparably easy to implement. Properties like the momentum thickness or skin friction can also be controlled very well. The simplification proposed by Lund is to not transform the coordinate system of the whole domain but only the boundary conditions in the streamwise direction. Figure 1 shows the simulation setup of the method. In an auxiliary simulation of a flat plate boundary layer (dashed line), the velocity field is extracted from a plane close to the outlet (dotted line). It is then rescaled and reintroduced at the inlet of this domain, such that a spatially evolving flow is achieved where the simulation is generating its own inflow condition. The outlet condition is a convective boundary condition in contrast to a periodic one used by Spalart. After several time steps, the velocity field in the middle of this domain is saved to disk and introduced as an inlet boundary condition in the actual simulation (solid line). The rescaling is done for the mean velocity by using the law of the wall for the inner region and the defect law for the outer region and assuming universal laws for the vertical velocity component and neglecting spanwise velocity components which are zero in the mean. Additionally the velocity fluctuations are assumed to be linearly dependent on the friction velocity where the streamwise inhomogeneity comes from. Weighted averages are used for interpolating between the inner and outer regions of the flow. More detailed information can be found in the work of Lund et al. (1998). Lund showed that this method has outstanding results compared to random fluctuation simulations with prescribed Reynolds stresses or simulations with streamwise periodic boundary conditions. It produces the most accurate inflow data, and it does not need a development section because the transient near the inlet boundary is negligible. This also leads to the conclusion that this method also is the

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Fig. 1 The computational domains used by Lund et al. The inflow-outflow simulation is sketched by the solid lines, while the dashed lines illustrate the auxiliary simulation where the recycling method is used. The dotted line denotes the patch where the wind field is extracted for the inflow calculation (Lund et al. 1998)

most efficient, because for the final simulation no development section is needed. Additionally the skin friction and momentum thickness can be controlled very well.

Synthetic Coherent Eddy and Stochastic Methods A different way of generating turbulence without recycling the flow and without the need of CFD is to generate synthetic turbulence by usage of mathematical models which are able to prescribe several turbulence properties of the velocity field to an inflow boundary condition for BEM or CFD simulations, where the turbulence is convected by the mean flow through the domain. The usual way is to define a time-averaged mean profile and superimpose time-varying fluctuations onto this field which should have characteristics as close as possible to the expected turbulence from experiments. For CFD domains, the flow field is strongly influenced by the velocity field at the inlet of the simulation domain as shown by Lund et al. (1998), Klein et al. (2001a, b), and Stanley and Sarkar (2000). It also has to be considered that for CFD simulations, the object of interest has to be far away from the inlet if there should be fully developed turbulence close to this object. BEM simulations generally do not underlie the problem of decaying or unevolved turbulence. Thus, they sample the wind fields as they are introduced to the simulation. In comparison to recycling methods, the flow might not be as correct for simple geometry flows like channel flows. However, for several applications which are in general inhomogeneous, nonperiodic, and much more complex, the correct characteristics of the flow are not known, and a synthetic model can be used to obtain a first hint on the evolution of turbulence. The main advantage of synthetic methods is the big variability in their properties and the potential focus on specific characteristics of turbulence rather than on the whole picture. In conjunction with BEM, synthetic models have the big advantage that the user does not have to worry about complex CFD simulations and the turbulence can directly be introduced to the simulation object without changing its characteristics too much. Additionally, in contrast to recycling methods, synthetic methods benefit from their low computational costs and their simplicity in terms of mathematical complexity.

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Digital Filter Based Wind Field Generation A usual approach of generating correlated inflow data is to generate a threedimensional signal with specific temporal and spatial correlations for all three velocity components independently. Cross correlations can then be achieved by using the Cholesky decomposition of the Reynolds stress tensor which is then multiplied with the velocity field. The first step can be done by using an inverse Fourier transform method, where the three-dimensional energy spectrum tensor has to be used. But because of the complexity and the implicated periodicity of the flow field, which is rarely wanted, this approach is not often used. Additionally the random phase angle in wave number space is responsible for a slow convergence of simulations to the experimental results (Le et al. 1997). To overcome those problems, Klein et al. (2003) introduce the concept of digital filtering to synthetic turbulence, which is based on a book from Nobach (1998). This method can be used to generate artificial velocity data with prescribed firstand second-order one-point statistics as well as the autocorrelation function for each inflow cell. To be precise, a digital linear and non-recursive filter with filter coefficients bn has been used on uncorrelated random numbers rm with zero mean and unit variance to create velocity data um with certain correlations. For simplicity this can be written in one dimension as N 

um =

(1)

bn rm+n

n=−N

which is nothing else than a discrete convolution or filtering. The covariance of those numbers is then

um um+k =

N N  

bi bj rm+i rm+k+j

i=−N j =−N N 

=

(2)

bi bi−k

i=−N +k

where the only nonzero terms appear for j = i − k if the fact is used that ri rj = 0 for i = j and ri ri = 1 as mentioned before. If the covariance is normalized by the variance, with the grid spacing x, the correlation Ruu (kx) = =

um um+k um um N  i=−N +k

bi bi−k



N 

i=−N

bi2

(3)

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is achieved which only depends on the filter coefficients. This equation also shows a drawback of this method. Normally the autocorrelation is a given property, and the filter coefficients have to be calculated which is in general not possible analytically. A multidimensional Newton method has to be used for solving this problem numerically. For homogeneous turbulence, Klein derives with the simplified correlation function   πr2 Ruu (r) = exp − 2 4L

(4)

and the prescribed length scale L = nx the approximated filter coefficient   

  N   π k2 π i2  bk ≈ exp − 2 exp − 2 . 2n n

(5)

i=−N

This approach can be easily extended to three-dimensional fields, with two space dimensions y and z in the inflow plane and the space dimension x in the main flow direction which can be converted to the time dimension by using the Taylor’s hypothesis. The velocity field is found by the three-dimensional convolution Ny Nz Nx      ˜ k) = b i  , j  , k  r i  , j  + j, k  + k u(j, i

j

k

(6) of the filter coefficients b (i, j, k) = bi bj bk

(7)

with the random field r. The next step is to introduce the mean velocity u and Reynolds stresses rij between the different velocity components by means of the Cholesky decomposition. This is done by a linear combination of velocity components which is added to the mean velocity, i.e., the new velocity is vi (x) = ui (x) + aij (x)u˜ j (x).

(8)

with the coefficients ⎛ √

r11

 ⎜ r /a aij = ⎜ ⎝ 21 11

 0 2 r22 − a21

r31 /a11 r32 − a21 a31 /a22

0

⎞

⎟ ⎟ 0 ⎠  2 − a2 r33 − a31 32

(9)

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841

Further information on how the exact generation procedure is performed can be looked up in the work of Klein et al. (2003). Finally it has to be mentioned that introducing the cross correlations as it is done here can distort the shape of the autocorrelation function, but this is assumed to be a small change. A better way to handle this problem will be shown in section “Method of Random Spots.” It also has to be remarked that this model is not able to generate divergence-free wind fields. The method of digital filtering found its way into a wide range of applications also in the area of wind energy where, e.g., Breuer (2018) studied the flow past airfoils and the transition to turbulence in the airfoil boundary layer. For this approach, they introduced the turbulence by source terms in cells close to the airfoil to overcome the issue of too fast-decaying turbulence. Another work of Szasz and Fuchs (2010) studied with this inflow model the influence on the wake of a turbine.

Method of Random Spots Kornev and Hassel (2007) developed a method based on randomly distributed turbulent spots which are superposed to achieve prescribed turbulent characteristics in three dimensions. The inner distribution of velocity is strongly connected to the autocorrelation functions in space and time. Thus integral length and time scales, as well as one-point cross correlations between velocity components, can be provided. An advantage of this method compared to the work of Klein et al. is the independence of a uniform grid and time step but also a higher physicality and an exact analytical solution for correlation functions of arbitrary shapes for the homogenous case. It also has to be emphasized that the cross correlations resulting from the proposed algorithm are also preserving the autocorrelations, while the method provided by Klein has small errors. Also Kornev provides the two-point cross correlations which are not part of Klein’s method. However, the current approach shows a high complexity and is more difficult to implement. The goal of the method of random spots is to generate velocity fields at each point x of the inlet plane for all velocity components. For the generated fields, those components are assumed to have the same autocorrelation and cross correlation functions everywhere. The model is based on several calculation steps. ∗ In the first step, Equations 8 and 9 are used to derive auxiliary correlations Rjj which are related to the target autocorrelation functions Rjj and Reynolds stresses rij by ∗ (x, x + η) = R11 (x, x + η) R11   ∗ ∗ ∗ R22 (x, x + η) = r22 (x)R22 (x, x + η) − a21 R11 (x, x + η) /a22

(10) (11)

842

S. Ehrich ∗ R33 (x, x

 a∗ + η) = r33 (x)R33 (x, x + η) − 32 ∗ r22 (x)R22 (x, x + η) a22   ∗ a∗  a21 32 ∗ R11 (x, x + η) − a31 − ∗ a22

(12)

with aij∗ = aij (x)aij (x + η). In the next step for each velocity component uj with j = 1, 2, 3, the velocity u distribution fk j (x ri , xk , ρik ) of each random spot i = 1, . . . , M in the direction k = 1, 2, 3 is calculated. Those spots are centered at x ri and have the size ρik such u ˆ ˆ that fk j (x (n) ri , xk , ρik ) = 0, if |xk − x ri · k| > ρik . Here k is the unit vector in k u direction. Kornev has shown that the relation between the velocity distribution fk j ∗ and the auxiliary correlations Rjj is 

ˆ ik (x ri ) x ri ·k+ρ

u

ˆ ˆ x ri ·k+η−ρ ik (x ri +kη)

fk j (x ri , xk , ρik (x ri ))

∗ ˆ xk , ρik (x ri + kη))dx ˆ ˆ · fk j (x ri + kη, k − Rjj (x ri , x ri + kη) u

 ·

ˆ ri (x ri ) x ri ·k+ρ ˆ ri (x ri ) x ri ·k−ρ

u

(fk j (x ri , xk , ρik (x ri )))2 dxs

=0

(13) u

which can be numerically minimized to obtain the distribution fk j . The velocity (n) components at each time step uj (x) can be computed for each time instant n by (n) uj (x)

=

3 M  

u

(n)

fk j (x ri , xk , ρik )

(14)

i=1 k=1

· sign(di(n) − 0.5)

(15)

where di are uniformly distributed random numbers in the interval d ∈ [0, 1] and sign is the signum function. The sum is over a total amount of M turbulent spots, and the product is a simplification to split the three-dimensional problems into three independent one-dimensional problems. After that procedure, the velocities have to be normalized by their standard deviation such that the variance is unity. At this point, the velocity field has a diagonal Reynolds stress tensor, because each velocity component is generated ∗ . separately and satisfies the autocorrelation function Rjj Equation 8 now has to be used to introduce Reynolds stresses rij and the (n) correct autocorrelations Rkk . The resulting velocities vi (x) can then be stored and introduced to BEM or CFD simulations.

26 Turbulent Inflow Models

843

As Kornev has shown, his method generates velocity fields with a fairly good agreement of all statistical characteristics with prescribed fields. The main advantage is that arbitrary correlation functions gained from experiments can be used as an input to this method. Furthermore it was shown that for homogeneous velocity fields (n) uj (x)

=

M 

(n)

(n)

ri f uj (x − x ri )

(16)

i=1 (n)

with r ∈ [−1, 1], analytical solutions for the inner distribution functions f (x) = 3 i k=1 fk (xk ) can be found. The product used here is again a simplification to split the three-dimensional problem into three independent one-dimensional problems. This separation makes it possible to derive the two-point autocorrelations in each direction as ˆ = R(ηk)

ˆ u(x)u(x + ηk)

u(x)u(x)  ∞ fk2 (xk )dxk = (fk  fk ) −∞

(17) (18)

and by application of the convolution theorem the inner distribution function as f (x) = C

3 

F

−1

  1/2 ˆ F R(ηk)

(19)

k=1

where F and F −1 denote the Fourier transform and its inverse and C a normalization constant. If the one-dimensional energy spectra 1 (kk ) = 2π



∞ −∞

ˆ exp{−ikk η}dη R(ηk)

(20)

with the wave vector k = (k1 , k2 , k3 ) are given, the inner distribution function has the form  ∞ f (x) = C (21) [(k1 )(k2 )(k3 )]1/2 exp{−ikx}dk. −∞

The advantage compared to the inhomogeneous cases discussed before is that here a numerical solution for fk is avoided if the Fourier transform and its inverse in Equation 19 can be found or the integral in Equation 21 can be solved analytically. This leads to higher accuracy and a faster field generation procedure. Reynolds

844

S. Ehrich

stresses can again be introduced by the algorithm described for the inhomogenous case. Kornev reports that for high separation of turbulent spots, it is also possible to introduce intermittency to the system. But further publications with more insight are missing. Kornev’s general approach has a physical clear background, it is flexible with respect to BEM and CFD grid resolution and time stepping, and analytical solutions for specific autocorrelations can be found for homogenous turbulence. Additionally the length scales, which are given by fk , can be dependent on the location and time. This allows to simulate wall flows or atmospheric boundary layers for CFD or BEM. A drawback of this method for CFD is that it does not support divergencefree velocity fields, which has to be corrected by algorithms for the velocity-pressure coupling (e.g., the PISO algorithm). This method equally suits CFD and BEM simulations with the abovementioned divergence issue for CFD. As a CFD example, Sale and Aliseda (2016) simulated a two-bladed wind turbine by means of the actuator line method and an inflow turbulence based on the method of random spots for comparison of the wake structure with particle image velocimetry. Other applications of Kornev’s method for environmental flows are shown in the work of Bazdidi-Tehrani et al. (2017) and Kubilay et al. (2016).

Wind Fields Based on Continuous-Time Random Walks The previous methods are able to create wind fields with statistically correct oneand two-point statistics up to second order, i.e., mean, variance, autocorrelation, cross correlation, and one-dimensional energy spectra. However, higher-order turbulent statistics from the atmosphere like the increment statistics (Mücke et al. 2011; Morales et al. 2012) are not taken into account. Here, the increments are defined as uτ (t) = u(t + τ ) − u(t). The following model generates wind fields with prescribed one- and two-point statistics as well as fourth-order structure functions S4 (τ ) = uτ (t)4 , which are strongly related to the kurtosis of increments by K = S4 /S22 . The kurtosis is usually larger than 3 in atmospheric boundary flows, which follows, according to Kolmogorove (1991), from the irregular dissipation of energy. This irregularity is often called intermittency and is an essential part of homogeneous isotropic turbulence. For the other inflow models discussed here, intermittency is not included, and the increment statistics follow a Gaussian shape K = 3. Based on coupled stochastic differential equations, the continuous-time random walk (CTRW) model, introduced to wind energy applications by Kleinhans and Mücke et al. (2011), generates time-varying correlated velocity fields on an inflow plane. The velocity vector on each grid point i on this plane is chosen to be ui = (ui x , ui y , ui z ). The simulation of each velocity component ui (k) with k = x, y, z is splitted up into three parts which are briefly described in the following:

26 Turbulent Inflow Models

845

• A reference velocity is calculated which can be interpreted as a varying average over time corresponding to changing weather conditions. • The velocity at each grid point is calculated introducing also spatial correlations. • A time transformation has to be done for introducing intermittency into the field. The reference velocity vector ur , which describes the slowly moving average of the calculated velocity, is grid independent and written as a stochastic differential equation in the model intrinsic time s: ! dur (k) (s) = −γr (ur (k) (s) − u0 (k) ) + Dr Γr (k) (s) ds

(22)

This is an Ornstein-Uhlenbeck process with γr being a damping constant which drives the system back to the long time mean velocity u0 . The second term consists  r with zero mean of the diffusion constant Dr and delta-correlated Gaussian noise  and variance equal to two for all three components. So the system would approach exponentially u0 proportional to e−γr s if there would be no stochastic term in this equation. But because the second term on the right-hand side, the diffusion term, is stochastic, the overall process for the reference velocity is also stochastic and does not reach an equilibrium. Based on the reference velocity, it is now possible to simulate velocity components ui (k) at all grid points i = 1, 2, . . . , N by another Ornstein-Uhlenbeck model. N " # ! (k)  dui (k) (s) = − γ ui (k) (s) − ξi ur (k) (s) + Di Hij Γj (k) (s) ds

(23)

j

This equation is very similar to the equation for the reference velocity but with a different damping constant γ , mean value ξi ur k , and diffusion term. The autocorrelation function of the system of Equations 22 and 23 has the form of an exponential decay (Kleinhans). As described before, the mean value of this process is the reference velocity but multiplied by a factor ξi which is connected to the wind profile. This factor depends only on the height zi of the grid point and has to be seen relative to the reference height zr , which is the height, where the mean velocity u0 is set. If a logarithmic profile is used, then ξi =

log(zi /z0 ) log(zr /z0 )

(24)

where z0 is the roughness height. The sum over all grid points in the last term of Equation 23 introduces spatial correlations into the field. Γj is again Gaussian noise with the same properties as the noise for the reference velocity. The matrix H is built from a given two-point correlation matrix by using the Cholesky decomposition C = HH , such that for each component the right twopoint correlation can be achieved. In this model, an exponential decay of correlation

846

S. Ehrich

is assumed such that $ Cij = exp −

rij2

%

2lf2

(25)

2  2  with the distance square rij2 = yi − yj + zi − zj . For simplicity it is supposed that the correlation matrix is direction independent. The correlation in time joins by Equations 22 and 23. The diffusion constant Di is defined with the standard deviation σ as (k)

Di

= γ σ 2 − ξi 2 Dr

γ2 γr (γr + γ )

(26)

and is important for the variance of the field. The Equations 22 and 23 define a system of two coupled Ornstein-Uhlenbeck processes with statistical properties which can be easily derived. The most essential and at the same time difficult part of the CTRW model is a time transformation, which slightly changes those statistical properties. This time transformation is needed to achieve intermittency which can have an effect on wind turbines loads. The time transformation has the form dt (s) = τα,C (s) ds

(27)

which is a stochastic process between the model intrinsic time s, which is also used in Equations 22 and 23, and the physical time t, which is the time of the model output. The type of the random variable on the right-hand side is important for the time evolution of the overall process and the type of intermittency one would achieve out of the model. In this case, τα,C is chosen to be a realization of a fully skewed alpha-stable évy distribution. This distribution was taken because it is the most general stable distribution only defined for positive values which goes hand in hand with the monotony of time. The two parameters α and C prescribe the shape of the Lévy distribution and the degree of intermittency to obtain. Generally α is chosen between 0 and 1, but the smaller α is, the higher the intermittency and the kurtosis of the increment distribution. α = 1 means τα,C is delta distributed around the value 1, which corresponds to a constant step size for t and with it to a normal discrete random walk for the whole CTRW process, resulting in Gaussian increment statistics. α = 0 leads to a delta distribution around 0. Of course picking such a distribution is not allowed, because the time step t is also 0, meaning there are two times sn and sn+1 projected onto the same time t. Because the Lévy distribution has a divergent mean value, the time step t could be infinitely large. To prevent such behavior, a cutoff value C was introduced for the Lévy distribution which only allows random variables for τα,C < C. Small values for C have a damping effect on intermittency, while high values increase the

26 Turbulent Inflow Models

847

intermittency. Thus the degree of intermittency is controlled by the interplay of C and α. This model has been used in combination with BEM in some works dealing with load calculations on wind turbines (Mücke et al. 2011; Schwarz et al. 2018; Gontier et al. 2007), but a final conclusion about the effects of intermittency is not found yet, because it is difficult to make intermittent and non-intermittent fields comparable in all other statistical properties. However, in the work of Schwarz et al. (2018), a good comparison for wind turbine simulations with BEM has been made by fixing the one-point one-time statistics and spectral densities for the CTRW model. Another promising way to compare intermittent and non-intermittent fields is presented by Berg et al. (2016) who combined CFD fields with BEM turbine simulations. The authors performed an LES simulation of an atmospheric boundary layer which naturally shows intermittency. Those fields are then extracted in the middle of the domain and introduced to a BEM simulation of a wind turbine. Nonintermittent fields, which were used for comparison, have been constructed by a phase randomization after a proper orthogonal decomposition. This work gave a hint that intermittency might increase the loads on the tower, but those effects could be insignificant if a safety factor is taken into account. In general the inflow generation of the CTRW model is very fast because no full wind velocity fields have to be created at once and Fourier transforms are not involved. The bottleneck clearly is the Cholesky decomposition for inflow patches with several thousand cells. However for BEM simulations, such a fine resolution is not used, and the computational costs should not be an issue. In terms of statistical properties, one- and two-point statistics can be defined within the limits of this model. For the spatial correlations, arbitrary functions can be assumed, but the correlations in time are exponentially decaying by definition. Also divergence-free fields are not supported, and cross spectra are not part of the generation process which makes this model not very physical. But in principle, cross spectra can also be added to the matrix C involved in the Cholesky decomposition. Nevertheless the main usage of this model is the generation of intermittency which is an important part of turbulence. This is why it is still interesting for wind field .

Spectral Methods The Sandia Method and Rotational Sampling Veers (1984) developed a method, also known as Sandia method, to generate threedimensional wind fields by use of single point spectral densities at all inlet cells and a coherence function. The assumptions here are that the coherence is isotropic in the plane perpendicular to mean wind speed and the cross-spectral densities are real valued. In particular, time series at grid points on a plane perpendicular to the mean wind direction are produced and propagated into the mean wind direction with the mean wind speed. This method is used mainly for the analysis of aerodynamic and structural analysis of horizontal axis wind turbines. Even if the generation

848

S. Ehrich

process is very fast, the full wind field has to be stored, and consequently the storage requirement is the limiting factor. This can be improved in simulation codes where only the wind field at the blade points is needed for the actual simulation like in BEM simulations. This improvement is based on rotational sampling which makes this method very effective in terms of computational costs and storage (Veers 1988). The main functionality is to create N correlated time series, where N is the number of grid points of the inflow plane. The correlations correspond to a discrete spectral matrix E in the Fourier domain, where the diagonal components are the power spectral densities and the off-diagonal components the cross-spectral densities. The diagonal components Ejj (fm ) at the mean frequency fm of a frequency band f are obtained from the continuous one-sided power spectral density (PSD) Gjj (f ) as Ejj (fm ) = Gjj (fm )

f 2

(28)

and the magnitude of cross-spectral components is |Ej k (fm )| =Cohj k (fm , rj k , Uj k )  · Ejj (fm )Ekk (fm )

(29)

where the coherence function Cohj k (fm , rj k , Uj k ) depends on the frequency fm , the Euclidic distance rj k of points j and k, and the average wind speed Uj k . In the next step, N correlated time series are generated by use of the Cholesky decomposition as the following: E(fm ) = H(fm )HT (fm )

(30)

with the real-valued transformation matrix H with entries Hj k =

Ej k −

k−1 

 Hj l Hkl

Hkk

(31)

l=1

Hkk = Ekk −

k−1 

1/2 Hkl2

.

(32)

l=1

A vector of the Fourier coefficients V of correlated wind velocities is achieved by the matrix multiplication V = HX1 of the transformation matrix H with an N × N diagonal matrix

(33)

26 Turbulent Inflow Models

849

Xj k (fm ) =

$ eiθkm 0

for j = k for j = k

(34)

of independent white noise with unit variance in Fourier space. 1 is here an N × 1 vector of ones. In summation notation, this is equal to Vj (fm ) =

j 

Hj k (fm )Xkk (fm ) =

k=1

j 

Hj k (fm )eiθkm

(35)

k=1

which is nothing else than a multiplication of each column of H with a random phase. In the last step, an inverse Fourier transform is used on V j (fm ) for each position j to find the time series of velocities for each grid point. In Veers (1988) is given an example calculation which is easy to follow. Because a full wind field is simulated, the approach shown here is very costly in terms of computational power which is unavoidable for CFD simulations. But for BEM simulations of horizontal axis wind turbines, where not the whole flow field is needed but only the velocities at the position of the blades, a simplification can be used. Thus the method of rotational sampling is used where the time series seen by a rotating wind turbine rotor blade is sampled at different radial positions and transformed to a PSD by use of the Fourier transform. The speed up of the method is achieved by introducing a constant time lag between each point in space such that each simulated time point of the time series refers to the time when the rotor blade passes this point. In contrast to the work of Bierbooms and Dragt (1996), where also variable speed wind turbines can be simulated, here the assumption is used that the rotor frequency is constant and all space points are equidistantly distributed. The implementation is done by phase shifting of the frequency component of the PSD before using the inverse Fourier transform. The Fourier coefficient vector has then the form Vj (fm ) =

j 

Hj k (fm )ei (θkm −φj m )

(36)

k=1

where the phase lag is defined as  φj m = lj

2π NP r



fm Ω

 (37)

with the location index lj = 0, . . . , NP r /NB − 1, the number of blades NB , the number of sampling points per revolution NP r per blade, and the rotation frequency Ω. A sketch of the distribution of locations lj is given in Fig. 2. It has to be augmented that the sampling points have to be equidistant. The difference in the simulation method compared to Equation 35 is that Vj in Equation 36 is only

850

S. Ehrich

Fig. 2 Simple sketch of the sampling locations for a two-bladed wind turbine from Veers et al. (1988)

calculated once per revolution when one rotor blade passes the location lj = 0, and not for each time step. Each simulated data point belongs to the sampled time series of one of the rotor blades. Therefore no information is unused. Because the amount of frequencies to be considered is much less, also the time for the inverse Fourier transform is much less than for the simulation of a full three-dimensional wind field. Furthermore the amount of storage needed drops drastically. Veers method is widely used and part of the turbulent wind field generator TurbSim, which is often the basis for turbulent wind fields for blade element momentum codes like FAST (Jonkman and Jonkman 2016). The method proposed here can be used with arbitrary spectra, and the rotational sampling proposed here can also be combined to other models for a speed up. Classical spectra used here are the Kaimal and the von Kármán spectra. The Kaimal spectra (Kaimal et al. 1972) are based on empirical measurements over flat and homogeneous terrain and are assumed to be approximately

k1 E11 (k1 ) 52.5k1 z = 2 u∗ (1 + 33k1 z)5/3

(38)

26 Turbulent Inflow Models

851

k1 E22 (k1 ) 8.5k1 z = u2∗ (1 + 9.5k1 z)5/3

(39)

1.05k1 z k1 E33 (k1 ) = 2 u∗ (1 + 5.3k1 z)5/3

(40)

with k1 = 2πf U , height z, and the friction velocity u∗ . The Kaimal model is part of the IEC 61400-3 with the recommended coherence function ⎧ ⎫ )  2  2 ⎬ ⎨ f rj k 0.12rj k Cohj k (f, rj k , UHub ) = exp −12 + (41) ⎩ ⎭ UHub Lc where UHub is the wind speed at hub height of a wind turbine and Lc = 8.1Λ1 is a length scale with $ Λ1 =

0.7z

z ≤ 60m

42m

z > 60m

(42)

known as the longitudinal turbulence scale parameter. The Kaimal model is a standard turbulence generator in TurbSim which is used together with the blade element momentum solver FAST. It has been used for load calculations of wind turbines (Tabrizi et al. 2017; Gontier et al. 2007) but also for wake studies (Reiso and Muskulus 2014; Reinwardt et al. 2018). For isotropic turbulence, Von Karman (1948) proposed an energy spectrum function and the corresponding one-dimensional energy spectra. His assumption is based on a real and symmetric velocity spectrum tensor Φ(k) without shear. For incompressible fluids, the isotropic velocity spectrum tensor has in general the form ij (k) =

# E(k) " 2 δ k − k k ij i j 4π k 4

(43)

where k is the wave vector with magnitude k. E(k) is the energy spectrum function which von Kármán suggested to be L4 k 4 E(k) = αε2/3 L5/3  17/6 1 + L2 k 2

(44)

with the dissipation rate ε, a length scale L, and the empirical parameter α = 1.7. The velocity spectrum tensor can be used to derive the one-dimensional spectra  Eii (k1 ) = 2

∞



∞

−∞ −∞

ii (k)dk2 dk3

(45)

852

S. Ehrich

as E11 (k1 ) =

1 9 2/3 αε  5/6 55 −2 L + k2

(46)

1

E22 (k1 ) = E33 (k1 ) =

3L−2 + 8k12 3 αε2/3  11/6 110 L−2 + k 2

(47)

1

and Eij (k1 ) = 0

for i = j .

(48)

The length scale L can be achieved by finding the wave number kmax,i at the maximum of the measured modified one-point spectra k1 Eii (k1 ) which is related to the length scale as √ (3/2) kmax,1 ! √ 6+3 5 L= 2kmax,2

L=

(49)

(50)

for both spectra. The variances are σ12 = σ22 = σ32 =

√ 9 2/3 2/3 π(1/3) αε L 55 (5/6)

(51)

where  is the Gamma function. This model works well for high frequencies, i.e., small scales, compared to the turbulence length scale. However, atmospheric turbulence is not isotropic, and an improvement is needed. In particular, variances for the different velocity components are different, and cross spectra are not zero. Additionally shear plays a role for real cases, which is not supported, and divergence-free fields are also not created.

The Mann Model A big improvement to the spectral methods above was made by Mann (1994, 1998) who developed a turbulence model for the horizontally homogeneous neutral atmospheric boundary layer with a comprehensive description by means of the spectral velocity tensor. This model based on the rapid distortion theory and consideration of eddy lifetimes is able to reproduce velocity component spectra as well as coherences and cross spectra of all velocity components at two separated

26 Turbulent Inflow Models

853

points by adjusting three parameters in total. One parameter is connected to the size of largest eddies, the second one to the lifetime of eddies, and the third one to the dissipation of energy. The velocity profile is assumed to be linear, and the gravitation and the rotation of earth are neglected. The basis of this model is the isotropic von Kármán tensor, defined by Equations 43 and 44 which is used as an initial condition for the time evolution of the spectral tensor given by rapid distortion theory. The spectral tensor becomes more and more anisotropic with time until the eddies break up and a stationary state is reached. It is assumed that the destruction of an eddy with size k −1 is due to eddies comparable to or smaller than k −1 , where k = |k|. The lifetime of eddies is  τ (k) = γ

dU dz

−1

2 F1

"

(kL)−2/3 1 17 4 −2 3 , 6 ; 3 ; −(kL)

#

(52)

with the uniform shear dU/dz, the hypergeometric function 2 F1 and the unknown eddy lifetime constant or shear parameter γ . Finally Mann showed that the velocity spectrum tensor has, under the assumptions mentioned before, the form 11 (k) = 22 (k) = 33 (k) = 12 (k) = 13 (k) = 23 (k) =

# # " E(k0 ) " 2 k0 − k12 − 2k1 k30 ζ1 + k12 + k22 ζ12 4 4π k0 # # " E(k0 ) " 2 k0 − k22 − 2k2 k30 ζ2 + k12 + k22 ζ22 4 4π k0 # E(k0 ) " 2 k1 − k22 4 4π k # # " E(k0 ) " 2 2 ζ · −k k − k k ζ − k k ζ + k + k ζ 1 2 1 30 2 2 30 1 1 2 1 2 4π k04 # # " E(k0 ) " 2 2 −k ζ1 k + k + k 1 30 1 2 4π k02 k 2 # # " E(k0 ) " 2 2 −k ζ2 k + k + k 2 30 1 2 4π k02 k 2

(53) (54) (55) (56) (57) (58)

2 and k = k + βk , where β = (dU/dz)t. E refers again with k02 = k12 + k22 + k30 30 3 1 to the von Kármán energy spectrum function. The values ζ1 and ζ2 are

  k2 ζ1 = C1 − C2 k1   k2 ζ2 = C2 + C1 k1

(59) (60)

854

S. Ehrich

with C1 =

C2 =

2 + βk k ) βk12 (k02 − 2k30 1 30

k 2 (k12 + k22 ) k2 k02 (k12 + k22 )3/2

(61)

 ⎤ βk1 k12 + k22 ⎦ . arctan ⎣ 2 k0 − k30 k1 β ⎡

(62)

For the stationary state, t in the definition of β can be substituted by the eddy lifetime τ in Equation 52, and the time dependence vanishes. The variance σii and the covariance uw have to be calculated numerically and are dependent on γ as shown in Fig. 3 where they are normalized by the variance of the isotropic von Kármán model in Equation 51. It is obvious that γ is responsible for the degree of anisotropy. The other two free parameters are αε2/3 and L. It should be mentioned that because a linear and not a logarithmic velocity profile is considered, the symmetries used for this model are only approximately valid. Further approximations are done by linearizing the Navier-Stokes equations by means of rapid distortion theory and neglecting viscosity. Thus the spectral tensor is only able to model eddies close to the ground where the shear does not change too much. Nevertheless the performance concerning second-order two-point statistics is very good. The numerical generation of wind fields is done in terms of the Fourier series ui (x) =



exp(ikx)Cij (k)nj (k)

(63)

k

Fig. 3 Dependence of 2 normalized variances σii2 /σiso on the shear parameter γ for the Mann model (Mann 1994)

4

s 2/s 2iso

3

s

2

2 1 1

s 222 2 s 33

1

0

〈u1 u 〉 3 –1

1

2

γ

3

4

26 Turbulent Inflow Models

855

where the sum is over all wave vectors k with components ki = 2π m/Li , integers m = −N/2, . . . , N/2, and simulation box dimensions Li . The complex random variables nj (k) are independently and Gaussian distributed, and Cij are unknown Fourier coefficients needed for solving this equation. It can be shown that they are Cij (k) =

!

(64)

k1 k2 k3 Aij (k)

where the wave vector separations are given by ki = 2π/Li and Aij are obtained by decomposition of the spectral tensor Φij = A∗ik Aj k . This is only valid for a large domain compared to the length scale of turbulence, i.e., L1 , L2 , L3  L. If only the streamwise dimension L1 fulfills this condition, Equation 64 is approximately valid for k > 3/L. Otherwise Mann suggests ∗ (k)Cj k (k) = Cik

2π k ×



∞



∞

−∞ −∞

3  m=2

 sinc

2

Φij (k1 , k2 , k3 )  )L (km − km m 2

 dk2 dk3

(65)

which has to be solved numerically for the Fourier coefficients. Another problem concerning the periodicity of the wind field is addressed by increasing the size of the turbulent box to Li > 2L. The divergence problem is also not solved as for the other synthetic models and might be problematic for CFD simulations. This model has very broad applications in wind energy by means of CFD and BEM from wake studies of an actuator line (Troldborg et al. 2007), actuator disk (van der Laan and Andersen 2018; Eriksson et al. 2014), full rotor (Kim et al. 2016), and wind turbine performance (Paulsen 2018) to fatigue analysis (Gontier et al. 2007; Dimitrov et al. 2018), where the fatigue load were compared with the Kaimal, von Kármán, and CTRW model. The Mann model already has been implemented in the blade element momentum code HAWC2 (Larsen and Hansen 2007) and is together with the Kaimal model part of the IEC 61400-1 (IEC61400 IEC 2005) where they are proposed for fatigue simulations. It has also been used for extreme load analysis as shown by Bos (2017). Because of the decaying turbulence in CFD domains without shear, the turbulence intensity normally is increased by an adequate amount to achieve the correct turbulence intensities at the rotor.

Conclusion In this chapter, different methods of generating turbulence as an inflow for wind energy applications were shown. Very promising for future simulations are the recycling methods introduced by Lund because they are able to give very accurate inlet conditions for atmospheric boundary layers in terms of low- and higherorder statistics. Increasing computational power will make recycling methods more

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feasible and open a big field of applications in wind energy. But as of today, those approaches are not really usable because of the big computational effort needed. As an alternative synthetic methods can be used which are a simplification of reality and therefore lead to a loss of information. Most models try to capture oneand two-point statistics up to the second order like Klein and Kornev have shown. Other models try to consider higher-order two-point statistics like the CTRW model developed by Kleinhans. Even if it was not proven that higher-order statistics of atmospheric turbulence have a significant impact on wind turbines, those might become important because they are strongly connected to gust events. In general it can be concluded that the synthetic methods shown here are very fast with the drawback of strong simplifications and missing divergence-freeness. The last group of turbulence models are based on a spectral description in Fourier space. The models belonging to this group have a well-defined spectrum and correlation functions as well as mean and variance. They can be divergencefree and fast if small fields are generated, but they quickly run into memory and storage problems for larger wind fields. For BEM simulations, those problems can be reduced by using the rotational sampling method of Veers, but they still exist for CFD cases. As a very simple spectral model, the Kaimal model can be used which reproduces accurately atmospheric wind data. As an improvement, the Mann model can be used which is able to account for shear and the connected distortion of eddies. However the latter one is much more complex and difficult to implement, but it already has a broad field of applications.

Cross-References  Pragmatic Models: BEM with Engineering Add-Ons

References Akselvoll K, Moin P (1993) Application of the dynamic localization model to large-eddy simulation of turbulent flow over a backward facing step. ASME-PUBLICATIONS-FED 162:1–1 Akselvoll K, Moin P (1996) Large-eddy simulation of turbulent confined coannular jets. J Fluid Mech 315:387-411. https://doi.org/10.1017/S0022112096002479 Bazdidi-Tehrani F, Badaghi D, Kiamansouri M, Jadidi M (2017) Analysis of various inflow turbulence generation methods in large eddy simulation approach for prediction of pollutant dispersion around model buildings. J Comput Methods Eng 35:85–112 Berg J, Natarajan A, Mann J, Patton EG (2016) Gaussian vs non-Gaussian turbulence: impact on wind turbine loads. Wind Energy 19(11):1975–1989 Bierbooms WAAM, Dragt JB (1996) SWING 4: a stochastic 3D wind field generator for design calculations. In: Proceedings of European Union Wind Energy Conference, EUWEC 1996, pp 942–945 Bos R (2017) Extreme gusts and their role in wind turbine design. Ph.D. Thesis

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Breuer M (2018) Effect of inflow turbulence on an airfoil flow with laminar separation bubble: an LES study. Flow Turbul Combust 101(2):433–456 Chung YM, Sung HJ (1997) Comparative study of inflow conditions for spatially evolving simulation. AIAA J 35:269–274 Dörenkämper M, Witha B, Steinfeld G, Heinemann D, Kühn M (2015) The impact of stable atmospheric boundary layers on wind-turbine wakes within offshore wind farms. J Wind Eng Ind Aerodyn 144:146–153 Dimitrov N, Kelly MC, Vignaroli A, Berg J (2018) From wind to loads: wind turbine site-specific load estimation with surrogate models trained on high-fidelity load databases. Wind Energy Sci 3(2):767–790 Eriksson O, Nilsson K, Breton S-P, Ivanell S (2014) The Science of Making Torque from Wind. Analysis of long distance wakes behind a row of turbines–a parameter study. J Phys Conf Ser 524(1):012152. The Science of Making Torque from Wind 2014 (TORQUE 2014) 18–20 June 2014, Copenhagen Gontier H, Schaffarczyk AP, Kleinhans D, Friedrich R (2007) A comparison of fatigue loads of wind turbine resulting from a non-Gaussian turbulence model vs. standard ones. J Phys Conf Ser 75(1):012070 Han Y, Stoellinger M, Naughton J (2016) Large eddy simulation for atmospheric boundary layer flow over flat and complex terrains. J Phys Conf Ser 753:032044 IEC61400 IEC (2005) 61400-1: wind turbines part 1: design requirements. Int Electrotechnical Commission 177:68–73 Jarrin N, Benhamadouche S, Laurence D, Prosser R (2006) A synthetic-eddy-method for generating inflow conditions for large-eddy simulations. Int J Heat Fluid Flow 27(4):585–593 Jonkman B, Jonkman J (2016) FAST v8.16.00a-bjj. NREL. https://wind.nrel.gov/nwtc/docs/ README_FAST8.pdf Kaimal JC, Wyngaard JCJ, Izumi Y, Cote OR (1972) Spectral characteristics of surface-layer turbulence. Q J R Meteorol Soc 98(417), 563–589 Kim Y, Jost E, Bangga G, Weihing P, Lutz T (2016) Effects of ambient turbulence on the near wake of a wind turbine. J Phys Conf Ser 753:032047 Klein M, Sadiki A, Janicka J (2001a) Influence of the boundary conditions on the direct numerical simulation of a plane turbulent jet. TSFP digital library online. Begel House Inc. Klein M, Sadiki A, Janicka J (2001b) Influence of the inflow conditions on the direct numerical simulation of primary breakup of liquid jets. In: Proceedings of ILASS Europe, 17. Annual Conference on Liquid Atomization and Spray Systems, pp 475–480 Klein M, Sadiki A, Janicka J (2003) A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. J Comput Phys 186(2):652–665 Kleinhans D, Stochastische Modellierung komplexer Systeme. Ph.D. Thesis Kolmogorov AN (1991) Dissipation of energy in the locally isotropic turbulence. Proc R Soc Lond Ser A Math Phys Sci 434(1890):15–17 Kornev N, Hassel E (2007) Method of random spots for generation of synthetic inhomogeneous turbulent fields with prescribed autocorrelation functions. Commun Numer Methods Eng 23(1):35–43 Kubilay A, Derome D, Carmeliet J (2016) Analysis of time-resolved wind-driven rain on an array of low-rise cubic buildings using large eddy simulation and an Eulerian multiphase model. Build Environ 114:68–81 Larsen TJ, Hansen AM (2007) How 2 HAWC2, the user’s manual. Risø National Laboratory Le H, Moin P, Kim J (1997) Direct numerical simulation of turbulent flow over a backward-facing step. J Fluid Mech 330:349–374 Lund TS, Wu X, Kyle D (1998)Squires: generation of turbulent inflow data for spatially-developing boundary layer simulations. J Comput Phys 140(2):233–258 Lygren M, Andersson HI (1999) Influence of boundary conditions on the large-scale structures in turbulent plane couette flow. In: TSFP digital library online. Begel House Inc. Mücke T, Kleinhans D, Peinke J (2011) Atmospheric turbulence and its influence on the alternating loads on wind turbines. Wind Energy 14(2):301–316

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Mann J (1994) The spatial structure of neutral atmospheric surface-layer turbulence. J Fluid Mech 273:141–168 Mann J (1998) Wind field simulation. Probab Eng Mech 13(4):269–282 Mayor SD, Spalart PR, Tripoli GJ (2002) Application of a perturbation recycling method in the large-eddy simulation of a mesoscale convective internal boundary layer. J Atmos Sci 59(15):2385–2395 Morales A, Wächter M, Peinke J (2012) Characterization of wind turbulence by higher-order statistics. Wind Energy 15(3):391–406 Nobach H (1998) Verarbeitung stochastisch abgetasteter Signale: Anwendung in der LaserDoppler-Anemometrie. Shaker, Aachen Paulsen S (2018) Uwe: simulation of shear and turbulence impact on wind turbine performance Reinwardt I, Gerke N, Dalhoff P, Steudel D, Moser W (2018) Validation of wind turbine wake models with focus on the dynamic wake meandering model. J Phys Conf Ser 1037:072028 Reiso M, Muskulus M (2014) Resolution of tower shadow models for downwind mounted rotors and its effects on the blade fatigue. J Phys Conf Ser 555:012084 Sale D, Aliseda A (2016) The flow field of a two-bladed horizontal axis turbine via comparison of RANS and LES simulations against experimental PIV flume measurements. In: Proceedings of the 4th Marine Energy Technology, at Washington, DC Schwarz CM, Ehrich S, Martin R, Peinke J (2018) Fatigue load estimations of intermittent wind dynamics based on a Blade Element Momentum method. J Phys Conf Ser 1037:072040 Spalart PR (1988) Direct simulation of a turbulent boundary layer up to R = 1410. J Fluid Mech 187:61–98 Stanley SA, Sarkar S (2000) Influence of nozzle conditions and discrete forcing on turbulent planar jets. AIAA J 38(9):1615–1623 Szasz RZ, Fuchs L (2010) Computations of the flow around a wind turbine: grid sensitivity study and the influence of inlet conditions. Notes Numer Fluid Mech 110:345–352 Tabrizi AB, Whale J, Lyons T, Urmee T, Peinke J (2017) Modelling the structural loading of a small wind turbine at a highly turbulent site via modifications to the Kaimal turbulence spectra. Renew Energy 105:288–300 Troldborg N, Sørensen JN, Mikkelsen R (2007) Actuator line simulation of wake of wind turbine operating in turbulent inflow. J Phys Conf Ser 75(1):012063. The Science of Making Torque from Wind 28–31, Technical University of Denmark van der Laan MP, Andersen S (2018) The turbulence scales of a wind turbine wake: a revisit of extended k-epsilon models. J Phys Conf Ser 1037:072001 Veers P (1984) Modeling stochastic wind loads on vertical axis wind turbines. In: 25th Structures, Structural Dynamics and Materials Conference, p 910 Veers PS (1988) Three-dimensional wind simulation. Sandia National Labs, Albuquerque Von Karman T (1948) Progress in the statistical theory of turbulence. Proc Natl Acad Sci USA 34(11):530 Wagner R, Courtney M, Larsen TJ, Paulsen US (2010) Simulation of shear and turbulence impact on wind turbine performance. Danmarks Tekniske Universitet, Risø Nationallaboratoriet for Bæredygtig Energi

Wind Shear and Wind Veer Effects on Wind Turbines

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Contents Introduction and Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variability of Shear and Veer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations of Wind Shear and Wind Veer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Site-Specific Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of Wind Shear and Wind Veer on Wind Turbine Power Production . . . . . . . . . . Influence of Wind Shear and Wind Veer on Wind Turbine Wakes . . . . . . . . . . . . . . . . . . . . Influence of Wind Shear and Wind Veer on Wind Turbine Loads . . . . . . . . . . . . . . . . . . . . Summary and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter highlights key contributions to the scientific literature on the sources of wind shear and wind veer in the atmospheric boundary layer, observations of shear and veer, and the effects of shear and veer on wind turbine power production, wind turbine wake evolution, and wind turbine loads. As wind turbines have grown larger, they encounter deeper and more complicated regions of the atmosphere. Over this height, profiles of wind speed shear and wind direction veer play a quantifiable role. Changes in the wind speed and wind direction across the vertical extent of a wind turbine rotor disk modify the inflow vector on the blades of the turbine, thereby affecting the magnitude and orientation of the lift and drag forces of the blade’s airfoil. These changes can affect the power production and loads on large modern turbines, as well as the evolution of the wake that could affect a downwind turbine.

J. K. Lundquist () ATOC, University of Colorado Boulder, Boulder, CO, USA e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_44

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Keywords

Atmospheric stability · Wind shear · Wind veer · Wind turbine wakes · Wind turbine loads · Wind turbine power production · Coriolis force · Frictional force · Pressure gradient force

Introduction and Definition of Terms As wind turbine sizes increase, their interactions with the atmosphere become more complex, thereby complicating predictions of power production and loads. In 1998– 1999, average turbine hub-heights in the United States were shorter than 60 m above the surface, with average rotor diameters approximately 50 m. As a result, turbines were constrained primarily to the lowest 85 m of the atmospheric boundary layer (Bolinger and Wiser 2020). In contrast, in 2019, the average turbine hub height was approximately 90 m with a 120-m rotor diameter, nearly doubling the uppermost height that turbine blades sweep to 150 m. At the time of this writing, the largest turbines in production extend to 250 m (GE Haliade X, https://www.ge.com/ renewableenergy/wind-energy/offshore-wind/haliade-x-offshore-turbine). As wind turbines have grown larger, they encounter deeper and more complicated regions of the atmospheric boundary layer (ABL). Some features of the atmospheric profile become more important when profile becomes deeper and extends beyond the atmospheric surface layer into the ABL. Specifically, the wind speed profile and the wind direction profile become more consequential when a wind turbine blade samples a depth of 220 m in one rotation. Changes in the wind speed and wind direction across the vertical extent of a wind turbine rotor disk will also modify the inflow vector on the blades of the turbine, thereby modifying the magnitude and orientation of the lift and drag forces of the blade’s airfoil. These changes can affect the power production and the loads on large modern turbines, as well as influence the evolution of a wind turbine wake and therefore the effect on downwind turbines. The wind shear profile is simply the change in wind speed with height. We expect the wind profile to increase with height because of the balance of forces in the ABL: near the surface, frictional forces are stronger, and therefore wind speeds are generally slower, increasing aloft. If the surface layer (the bottom 50 to 100 m of the ABL) is statically neutral – one in which the potential temperature is constant with height, with no surface heating or cooling – wind speed increases approximately logarithmically with height (Stull 2017). The exact shape of this profile is a function of the surface roughness z0 and the surface friction velocity u∗ : U (z) =

  z u∗ ln k z0

(1)

where k is the von Kármán constant, 0.4. In the wind engineering community, or when the characterization of the surface roughness z0 and friction velocity u∗ are

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not available, wind shear is often described by a power law profile: 

z U (z) = U (zr ) zr

α (2)

where the larger the wind shear exponent α, the greater the change in average horizontal wind speed U with height z, as compared to a reference height zr (Brower 2012). The International Electrotechnical Commission (IEC) standard for the normal wind profile assumes that α = 0.2 (IEC 61400-1 2019), and extreme wind shear is considered in the IEC standard to occur for short periods of time (on the order of seconds). When the atmosphere is not statically neutral (i.e., during daytime convective conditions or during nighttime stably stratified conditions), adjustments to the loglaw profile of equation (1) must be made. During nighttime conditions over land, the cooling surface causes the atmosphere to become stably stratified, and the height of the ABL collapses to a shallow region. Correspondingly, frictional forces are constrained to the near-surface region, so wind speed is slower near the ground but faster aloft than would be predicted by the log-law profile. The profile in the stable surface layer is empirically described by a log-linear profile formula that combines the logarithmic profile of (1) with a linear term in z defined by the Businger-Dyer relations:     u∗ z z U (z) = +6 ln (3) k z0 L where L is the Obukhov length (Stull 2017). This length scale can be considered to be the height in the stable surface layer below which the mechanical shear production of turbulence exceeds the buoyant consumption of turbulence. As the height increases beyond the Obukhov length, the linear term dominates the logarithmic term in equation (3), as shown in the schematic diagram of Fig. 1. The change in wind direction with height, “wind veer,” also derives from the balance of forces in the ABL. Veer is often introduced by the change in the balance of forces rising through the ABL and into the free troposphere. In the free troposphere, typically 1–3 km above the surface, winds evolve as a balance between the pressure gradient force (P in Fig. 2, directed from high-pressure regions toward low-pressure regions) and the Coriolis (C in Fig. 2) force, so that geostrophic winds (Vg in Fig. 2) flow parallel to isobars or lines of constant pressure on a weather map. Near the surface, however, the frictional force (Fs in Fig. 2) is large, and so the surface winds Vs are directed at angle ψ across isobars towards low pressure. The stronger the frictional force Fs , the larger the angle ψ between Vg and Vs and the slower (or more subgeostrophic) the surface wind speed Vs . Svensson and Holtslag (2009) suggest that the angle ψ between the surface wind and the geostrophic wind (aligned with the x-axis) is a function of the latitude through the Coriolis parameter f , the surface stress through the friction velocity u∗ , the ABL height h, and the averaged cross-wind component in the ABL v:

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Fig. 1 Typical wind speed (M) profiles in the surface layer, or bottom 5%–10% of the ABL, for different static stabilities: neutral, stable, and unstable. Note the different heights of the surface layer (zSL) for varying stabilities, with the deepest surface layer in convective conditions. The very shallow height of the surface layer in stable conditions is not marked. (Modified from Stull (2017), https://www.eoas.ubc.ca/ books/Practical_Meteorology/, Fig 18.10, Creative Commons license)

cos ψ =

f hv u2∗

(4)

As the frictional force weakens at higher altitudes, the angle ψ decreases and the winds accelerate. This pattern manifests as winds rotating clockwise (in the Northern Hemisphere) as one moves up in the atmosphere – that is, southerly flow near the surface and southwesterly flow aloft – and it is defined to be “veering.” The opposite pattern, “backing,” which can occur with cold air advection, has winds rotating counterclockwise with height in the Northern Hemisphere. Because of the change in the direction of the Coriolis force C in the Southern Hemisphere, winds will typically rotate counterclockwise with height as one moves up in the atmosphere in the Southern Hemisphere. The convention for defining veering and backing changes depending on the hemisphere. For the wind engineering community, focused on building construction, “the veering angle is clockwise in the Northern hemisphere and counterclockwise in the Southern hemisphere” (Yeo and Simiu 2010). However, while the American Meteorological Society glossary (https://glossary.ametsoc.org/wiki/Veering) states that while veering and backing should be considered the opposite of each other in either hemisphere, two definitions for veering are permitted. Veering can be either (1) the clockwise turning of the winds with height regardless of hemisphere, or (2) the clockwise turning in the Northern Hemisphere and counterclockwise turning in the Southern Hemisphere to maintain consistency with the relationship with warmair advection. Here, we use the latter definition, so “veering” describes the turning from a surface influenced by friction to a flow aloft aligned with the geostrophic flow (clockwise turning in the Northern Hemisphere and counterclockwise turning in the Southern Hemisphere).

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Fig. 2 The balance of forces and resulting winds above and below the height of the ABL (HABL ) in relation to isobars or geostrophic height contours (horizontal solid lines) in the Northern Hemisphere. In (a), the balance in the free troposphere, between pressure gradient force P and Coriolis force C leads to the geostrophic wind Vg , oriented parallel to isobars, westerly in this case. In (b), the balance within the ABL is between pressure gradient force P, Coriolis force C, and friction at the surface Fs . resulting in slower surface winds Vs which are directed across the isobars towards lower pressure at an angle ψ, here suggested to be ~45º. In (c), the resulting wind speed profile appears, while in (d), the resulting wind direction profile appears. Solid lines in (a) and (b) represent isobars or geopotential height contours on a weather chart. At the surface where Fs is non-zero, this balance results in winds Vs , which are slower than geostrophic. In the free troposphere, where Fs = 0, this balance results in the geostrophic winds Vg

This description emphasizes the role of the frictional force in determining wind shear and wind veer. Two other mechanisms can also enhance or suppress shear and veer. One mechanism is the effect of horizontal temperature gradients on wind flows. These gradients, known as baroclinicity or the thermal wind, can be created by sloping terrain (such as over the North American Great Plains), the passage of frontal systems, the contrast between land and sea interfaces, large weather patterns (Stull 1988), and drainage flows and cold-pool events (McCaffrey et al. 2019) in complex terrain. Warm-air advection tends to enhance veering in the Northern Hemisphere, whereas cold-air advection induces backing in the Northern

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Hemisphere. The other mechanism is the inertial oscillation in the ABL introduced by changes in stability, which will be discussed next.

Variability of Shear and Veer Because of its coupling to the surface, the ABL continually evolves, affecting the wind profiles that influence wind turbines. Over land, the ABL experiences strong cycles in atmospheric stability, with a well-mixed convective boundary layer during daytime and a stably stratified boundary layer at night. This diurnal cycle of atmospheric stability introduces cycles of wind shear and wind veer, as suggested in Fig. 1 and shown explicitly in Fig. 3, where “direction shear” refers to wind veer. Smaller values of wind shear and wind veer tend to occur during the day, when the frictional forces are rapidly distributed throughout a deeper boundary layer caused by convection rising from a warm surface. At night, this convection ceases, and frictional forces are constrained to a shallow layer near the surface: stability increases the decoupling between the surface and flows aloft in the residual boundary layer. As a result of these factors (this suppression of convective turbulence, boundary-layer decoupling, and the decrease in the height of the boundary layer), veer becomes more pronounced at night than during the day. Wind shear and veer during stably stratified conditions can be enhanced by the inertial oscillation mechanism (Blackadar 1957). This oscillation in the horizontal wind is generated by the release of turbulent stresses during the evening transition’s decay of convective turbulence. The inertial oscillation is superimposed on the geostrophic wind. According to the Blackadar model, inertial oscillations com-

Fig. 3 Diurnal cycle of mean wind speed shear (dotted line) and wind veer or “directional shear” (dash-dotted line) from lidar measurements in Iowa throughout 3 summer months, calculated between 40 and 120 m above the surface for times with hub-height wind speeds between 3 and 25 m s−1 . The dashed vertical lines indicate sunrise and sunset times for 1 August 2013, the midpoint of the dataset. The gray shaded region indicates the morning transition period (0600-0900 LT). (From Sanchez Gomez and Lundquist (2020a). This work is distributed under the Creative Commons Attribution 4.0 License)

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mence at sunset and occur throughout the residual boundary layer with amplitudes that vary with height. The largest amplitudes occur at heights where the turbulent stress divergence is largest during the daytime convective mixing. Overnight, the inertial oscillation rotates, eventually aligning with the mean wind to form a jet structure in the wind profile. This jet structure is known as the nocturnal low-level jet (LLJ), which leads to strong winds that can provide significant wind resources (Vanderwende et al. 2015). The LLJ, and its accompanying wind shear and wind veer, occurs globally (Rife et al. 2010), with seasonal variations. While the LLJ provides strong winds that enhance the wind resource, the accompanying shear and veer can adversely affect turbine power collection and loads (Eggers et al. 2003; Kelley et al. 2006), as will be discussed in sections “Influence of Wind Shear and Wind Veer on Wind Turbine Power Production” and “Influence of Wind Shear and Wind Veer on Wind Turbine Loads” below. Numerous studies have investigated the nocturnal LLJ (Stensrud 1996) in the US Great Plains (Vanderwende et al. 2015, Smith et al. 2019, and references cited therein), but LLJs have also been studied in the North Sea (Dörenkämper et al. 2015; Kalverla et al. 2019), Finland (Tuononen et al. 2017), Russia (Kallistratova and Kouznetsov 2012; Kallistratova et al. 2013), and the Baltic Sea (Smedman et al. 1993, 1996), among other locations. In addition to the classical inertial oscillation mechanism, other physical mechanisms can enhance the occurrences of shear and veer in the ABL. First, flow over complex terrain can induce internal boundary layers with step-change functions in the wind speed and wind direction profiles (Panofsky and Townsend 1964; Taylor 1969; Savelyev and Taylor 2005). Further, baroclinicity such as that induced by the diurnal heating and cooling of sloping terrain plays a significant role in enhancing the LLJ (Whiteman et al. 1997; Pan et al. 2004; Shapiro and Fedorovich 2009; Parish and Oolman 2010; Parish 2017; Fedorovich et al. 2017). Similar to the diurnal mechanism for generating LLJs at night over land, an analogous mechanism is induced by stable stratification when warm air flows over a colder sea (Smedman et al. 1993), leading to frequent occurrences of LLJs in coastal regions (Smedman et al. 1996). This stable stratification varies diurnally (occurring more frequently during the day when air over land is warmer). Dörenkämper et al. (2015) find this pattern is most pronounced in spring when the temperature difference between land and sea temperature is largest. Additionally, flow accelerates when flowing from large surface roughness (over land) to small surface roughness (over water). Although LLJs are common phenomena, generating strong shear and veer at altitudes relevant for wind turbines, their effects are not always considered in the design and operation of wind turbines. For example, IEC 61400-1 (2019) specifically does not consider strong shear. Specifically, the standard states: “Shear values with high variability have been reported for certain areas in connection with highly stratified flow, complex terrain or severe roughness changes. In this case using the average wind shear may not be sufficient.” Because “highly stratified flow” occurs nightly in many locations, this omission seems like a high priority for reconsideration. The IEC standard is also silent on wind veer.

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Observations of Wind Shear and Wind Veer Wind shear and wind veer occur globally. In this section, we highlight several key papers (among a large body of literature) that characterize shear and veer in the ABL, at altitudes relevant for wind energy.

Global Assessments Global assessments of occurrences of veer are available from global databases of radiosonde launches and from reanalysis products. Based on one such global radiosonde database from 1981 to 2005, Lindvall and Svensson (2019) calculate veer between the surface and the top of the ABL. They find that, on average, the wind veers (turns clockwise with height) in the Northern Hemisphere and turns counterclockwise with height in the Southern Hemisphere. The top of the boundary layer can range from 50 to 2500 m in their database, and veer across the ABL ranges from 0 to 45 deg. They find that the strongest veer occurs in stable boundary layers, with an average veer of 35 deg for stable boundary layer heights ranging from 350 m up to 1200 m (0.1 to 0.03 deg m−1 ) (see their Fig. 7e). They also find that terrain influences veer, with weaker veer in coastal or island sites. Lindvall and Svensson (2019) also compare the radiosonde data to a reanalysis product, ERAInterim (Dee et al. 2011), and find that ERA-Interim underestimates the magnitude of veer, typically by a factor of two (see their Fig. 6). As latitude increases, veer also tends to increase in the Integrated Global Radiosonde Archive dataset. The radiosonde estimates of veer over Europe tend to be large: nighttime values over Europe range from 25 deg in the summer to 30 deg in the winter (Lindvall and Svensson 2019, Fig. 6b). When veer is assessed by latitude, the maximum value of 35 deg occurs at a latitude of 53 deg, although the authors note that this latitude dependence can be offset by local topography. For example, sites poleward of 53 deg tend to be coastal sites, and coastal sites tend to manifest smaller amounts of veer. The reanalysis product, ERA-Interim, finds a maximum value of veer of only 15 deg, and exhibits a muted seasonable variability in veer.

Site-Specific Observations Specific locations with tall towers have also provided insights into the occurrences of wind veer. The Cabauw Observatory in the western Netherlands, 50 km from the North Sea in flat terrain, has collected wind and temperature profiles for decades. In unstable and near-neutral conditions, wind veer between 20 and 200 m is small (9–12 deg, or 0.05–0.067 deg m−1 ), but in stable conditions, veer of up to 40 degrees is observed (0.22 deg m−1 ) (Van Ulden and Holtslag 1985). When these measurements are compared to a reanalysis product, ERA-40, the reanalysis data again underestimates the magnitude of veer (Baas et al. 2009), as also shown in Lindvall and Svensson (2019).

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Data from the Texas Tech Tall Tower near Lubbock, Texas, in the Southern Great Plains of North America, also demonstrate the prevalence of shear and veer. A detailed study of nearly 1 year of data demonstrates the likelihood of strong wind shear and strong wind veer events (Fig. 4) during nighttime stably stratified conditions. A narrow distribution of small values of power law coefficient α emerges during daytime/convective conditions, whereas nighttime/stable conditions range from α values of 0 to 0.6, far exceeding the IEC standard case of α = 0.2 (IEC 61400-1 2019). Additionally, a narrow distribution of veer values emerges only for daytime/convective conditions, whereas nighttime stably stratified conditions show a broad distribution including large values exceeding 0.2 deg m−1 . Using both lidar measurements and the tall-tower measurements from the Høvsøre tower, Peña et al. (2014a, b) document cases of strong shear and veer, and assess the ability of mesoscale modeling tools to capture observed cases of strong veer. Their case studies include an example of 43 deg of veer over the lowest 100 m of the Høvsøre tower (and 66 deg over the lowest 200 m), strongly influenced by the thermal wind. Overall, their observations confirm that wind shear and wind veer are stronger in stable conditions than in unstable conditions. In a lidar-based study an onshore wind plant, Sanchez Gomez and Lundquist (2020a) also find a strong diurnal cycle in shear and veer (Fig. 3). They find a predominance of strong shear and veer during nighttime stably stratified conditions. Further, shear values increase with hub-height wind speed (Fig. 4), whereas veer values decrease with hub-height wind speed. The specific effects of shear and veer during morning and evening transition periods are addressed by (Sanchez Gomez and Lundquist 2020a and 2020b). Most backing conditions occur during the evening transition of the boundary layer while the nocturnal inversion rises through the lowest levels of the boundary layer. Furthermore, the dependence of strong veer and shear on wind speed changes between the morning and evening transition. Almost half of the morning transition observations with extreme shear and veer (α = 0.3

Fig. 4 Probability distributions of wind speed shear (power law exponent α), left, and wind direction veer, right, from the Texas Tech tower. Stability classes are defined by the hourly mean change in virtual potential temperature from 10 to 116 m, which correlates well with time of day. (From Walter et al. 2009)

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Fig. 5 Variability of veer (solid line, labeled as “Direction shear”) and shear (dotted line, labeled as “Speed shear”) calculated between 40 and 120 m with hub-height wind speed and with time of day as determined by sunrise and sunset. (From Sanchez Gomez and Lundquist (2020a). This work is distributed under the Creative Commons Attribution 4.0 license)

and veer of 0.2 deg m−1 ) report wind speeds faster than 6 m s−1 . In contrast, only 25% of the evening transition cases with extreme shear and veer ( α = 0.3 and veer of 0.1 deg m−1 ) are for wind speeds faster than 6 m s−1 (Fig. 5). The offshore lidar measurements of the Air-Sea Interaction Tower (ASIT) indicate the typical shear and veer conditions that can be experienced in the US East Coast wind energy lease areas (Bodini et al. 2019, 2020). The ASIT, south of the island of Martha’s Vineyard and near the Massachusetts/Rhode Island wind energy lease areas, experiences a large predominance of stable conditions, as measured by lidar assessments of turbulence and turbulence kinetic energy dissipation rate. An annual cycle emerges, with larger values of turbulence in the winter and smaller values in the summer. Correspondingly, the largest average values of veer occur in the summer (0.10 deg m−1 as measured between 40 and 200 m) with smaller values in winter (0.04 deg m−1 ). Large values of veer – exceeding 0.3 deg m−1 – occur throughout the year, with more such cases at slower wind speeds (Bodini et al. 2020). The diurnal cycle is very muted at this offshore location. Five years of data from a wind-profiling lidar near the Eolos wind turbine in Minnesota, United States, highlight the variability of wind veer (Gao et al. 2021). In this location, 58% of cases show backing between 30 and 130 m (with 42% veering), although the mean wind veer is positive because of the shape of the distribution, with a narrow peak at a small negative (backing) angle and a broader right tail (veering). This site also experiences a predominance of unstable conditions throughout the year – with 24% stable conditions, 21% neutral conditions, and 55% unstable conditions. Values of veer can exceed 0.5 deg m−1 at this site, and more veering conditions occur during nighttime conditions with more backing conditions occurring during daytime conditions. Analysis of wind profiles from a 444-m WKY TV tower in central Oklahoma over the course of a year also indicate a strong diurnal pattern of veering, with

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large values of veer occurring primarily during nighttime stably stratified conditions (Crawford and Hudson 1973). During the daytime, minimal veering occurs because of well-mixed profiles. Overnight, significant veering with height in the tower layer generally occurs between 1700 and 1000 local time; an average maximum veering of 30 deg (0.07 deg m−1 ) occurs at 0600.

Influences of Topography on Shear and Veer Although atmospheric stability tends to promote strong cases of shear and veer, this effect can be suppressed or even reversed by topographic influences. For example, Wharton and Lundquist (2012a, b) analyze measurements in a Northern Hemisphere wind power plant, finding very little veer in their stable conditions because of channeled flow through a valley. Even in stable conditions, because of topographical features around their study region, veer was not possible. Similarly, lidar measurements in the Columbia River Gorge region, collected as part of the Wind Forecast Improvement Project 2 (WFIP2, Wilczak et al. 2019) suggest very little veer with height because of channeling and gap flow through the Columbia River Gorge (Sharp and Mass 2002). Finally, at a wind power plant in Northern Colorado, a surprising prevalence of backing conditions occur, thought to be introduced by the complex terrain in the local environment. Although an extreme case of 60 deg of veer occurred between the 40 and 120 m levels (0.75 deg m−1 ), overall measurements at this site during summertime conditions found veering 35% of the time and backing 65% of the time (Murphy et al. 2020). At a site with two flow regimes (flow from open water and flow from hilly terrain), Shu et al. (2020) confirm the importance of terrain forcing and in terraindriven stability changes. Using lidar measurements of winds near Hong Kong, they show the characteristics of the wind direction profile change depending on whether the flow came from open water or from hilly terrain (Fig. 6). Wind veer angles for hilly terrain are generally larger than those for open water. Furthermore, as the wind speed increases, the veering angle decreases.

Influence of Wind Shear and Wind Veer on Wind Turbine Power Production Even before turbines extended beyond 100 m above the surface, some researchers pointed out the effects of shear on the shape of wind speed profiles and therefore turbine power production. Because of shear and veer, hub-height wind speeds alone may not be representative of the flow over the entire rotor disk. Wind speed and direction across the entire rotor disk can impact power production because wind turbines integrate the momentum flux normal to the rotor disk. Log-law estimates of hub-height wind speeds based on measurements at 40-, 50-, or 60-m above the surface introduce large wind speed errors at three offshore and one onshore (measured by towers extending only up to 40 or 50 m) site under very stable conditions (Motta et al. 2005). Analyzing data over 1 year for turbines with a hub height of 32 m and a rotor diameter of 34.8 m, Sumner and Masson (2006)

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Fig. 6 Composite wind veer profiles (relative to the surface measurements) for open-sea upwind fetch (left) and for hilly terrain upwind fetch (right). Note the maximum veer decreases as wind speed increases. (From Shu et al. (2020). This work is distributed under the Creative Commons Attribution 4.0 license)

also find that wind shear across the turbine rotor disk affects power production. Using measurements from the Cabauw tower, van den Berg (2008) point out that a logarithmic extrapolation of wind speeds from the surface to hub height would result in overestimation of daytime power production and underestimation of nighttime power production. The observed effects of wind shear on power production tend to vary with location, and are sensitive to the heights across which the wind shear is defined. Analysis of data from a US Great Plains wind plant suggests that wind shear defined by a power law coefficient α greater than 0.2 reduces annual energy production (AEP) by approximately 1.1–1.2%, depending on wind speed (Rareshide et al. (2009). Wharton and Lundquist (2012a) also find that larger values of shear induce turbine overperformance, by up to 9% in specific wind speed bins, although the shear parameter in the bottom half of the turbine rotor disk exerts a stronger influence than that in the top half of the rotor disk. Furthermore, the study of Wharton and Lundquist (2012a) does not consider the related effects of wind veer since that location experiences channeled flow and so veer cannot occur. In a location where veer could occur but was unfortunately not measured, stable conditions with values of α greater than 0.2 (measured between 10 and 60 m) are associated with reduced power performance of up to 6% (Vanderwende and Lundquist 2012). Incorporating wind shear (and turbulence intensity) into a Random Forest machine learning model reduces prediction errors by a factor of three compared to a traditional power curve method (Clifton et al. 2013). Using a

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variety of machine-learning models, Bulaevskaya et al. (2015) find that including wind shear into power estimation reduces error (compared to relying only on hubheight wind speed) and including veer information reduces error by another 26%. At a wind power plant on the central Norwegian coast, Bardal et al. (2015) find that high shear (which they defined as α >0.15) reduces power production slightly in the middle of region II (7–9.5 m s−1 ) of the power curve. They also find small negative effects of strong veer on power production in those same wind speed bins. In a study of power production at an onshore wind power plant with a strong diurnal cycle, Sanchez Gomez and Lundquist (2020a) find that a combination of shear and veer best explains power variability. Large wind veering occurring with small speed shear results in wind turbine underperformance up to 13%. However, overperformance occurs for large speed shear and small values of wind veer, up to 8% of the expected power performance. Overall, large values of wind veer undermine turbine operation. Based on 5 years of data collection at the Eolos facility in Minnesota, veering and backing in the wind profile influence the power production of the wind turbine (Gao et al. 2021). The average power deviation from the power expected based on hub-height wind speed for conditions with veering throughout the wind turbine rotor disk (occurring primarily in nighttime stable conditions) is – 6.5%. Furthermore, in extreme wind veer cases (veer between 0.6 and 0.7 deg m−1 ), the power deviation declines to −18%: larger values of wind veer tend to result in more dramatic power loss. Further, the shape of the wind veer profile – specifically, whether veering or backing occurs in the bottom half, the top half, or over the entire rotor disk–also makes a difference in the power performance. Simulation-based studies have also assessed the effects of shear and veer. Very high positive wind shear decreases power by 26%, according to a simulation-based study based on turbines on flat Danish terrain (Wagner et al. 2009). Simulations with extreme wind shear ( α = 0.35) and backing (−0.47 deg m−1 ) suggest underperformance of 6% (Walter et al. 2009). Walter et al. also find a slight benefit when small but positive veer occurs simultaneously with very small or very large shear magnitudes. As a result of these varying influences of shear profiles, the concept of the rotorequivalent wind speed (REWS) was introduced (Wagner et al. 2008, 2011; Antoniou et al. 2009). Instead of relating the hub-height wind speed to power production, the total momentum through the rotor disk better predicts power production (Wagner et al. 2011, 2014). This modification was incorporated into the second edition of IEC 61400-12-1 (2017). Later work incorporated the effects of turbulence and veer on calculating the REWS (Clack et al. 2016; Choukulkar et al. 2016). The REWS calculation has been implemented into mesoscale models predicting wind plant power production and wake effects (Redfern et al. 2019). However, theoretical arguments suggest that the REWS is not necessary unless the power law coefficient α exceeds 0.4 or if the wind shear profile is such that a single coefficient cannot describe the wind profile (Sark et al. 2019). By considering several formulations of REWS, Murphy et al. (2020) assess the impact of shear and veer at a complex terrain site where shear and veer

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correlate more with wind direction than with atmospheric stability. When REWS is significantly faster than the hub-height wind speed, the turbine tended to produce 4%–5% more power. In contrast, when REWS is slower than hub-height wind speed, the turbine tended to produce 1%–2% less power. For the same dataset, turbine power production does not vary clearly as a function of α, although the low-α case suggests higher power production than the high-α case).

Influence of Wind Shear and Wind Veer on Wind Turbine Wakes Wind turbine wakes respond to wind veer. Large-eddy simulations of wakes in stably stratified flow (Bhaganagar and Debnath 2015; Lundquist et al. 2015; Mirocha et al. 2015; Vollmer et al. 2016; Abkar and Porté-Agel 2015; Abkar et al. 2016, 2018; Bromm et al. 2017; Xie and Archer 2017; Churchfield and Sirnivas 2018) demonstrate that a wake will stretch from a circular shape into an ellipse in the veering flow typical of stably stratified conditions. Scanning lidar measurements of wakes later confirmed that the wake will veer (Bodini et al. 2017), albeit not to the extent of the inflow veer. Nacelle lidar measurements (Brugger et al. 2019) also show this skewed structure of wakes. The shape and magnitude of the veering profile exerts a strong influence on the evolution of the wake (Englberger and Lundquist 2020; Englberger et al. 2020). In a detailed parameter study using large-eddy simulations in which the shape and magnitude of veer is varied from 4 to 20 deg across the turbine rotor disk, Englberger et al. (2020) show that increasing veer enhances the erosion of the wind turbine wake (Fig. 7), both for clockwise-rotating and counterclockwise-rotating turbines. Further, when the shape of the veer profile is changed, with more veer in the bottom or in the top half of the wind turbine rotor disk, the veer of the wake also changes (Englberger et al. 2020). For veer only in the lower half, the wake veers less than the inflow veer. When veer is present only in the upper half of the wind turbine rotor disk, the ratio of wake veer to inflow veer approaches 1. Wake steering experiments indicate that wake steering exerts a stronger impact in stable conditions (Fleming et al. 2019, 2020) which are accompanied by more shear and veer. As a result, understanding the impacts of shear and veer on wake structure is critical (Howland et al. 2020). Further, wakes of entire wind plants can extend for tens of kilometers downwind (Platis et al. 2018), influencing the production of neighboring plants (Lundquist et al. 2019), and these wind power plant wakes are also influenced by veer (Fitch et al. 2013; Gadde and Stevens 2019). The 40-deg veer across the rotor disk region in the stable portion of the mesoscale wind power plant simulations of Fitch et al. (2013) is reduced downwind by the wind plant wake.

Influence of Wind Shear and Wind Veer on Wind Turbine Loads With the large-scale deployment of wind energy, correctly assessing loads and therefore operations and maintenance (O&M) costs becomes critical. The assess-

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Fig. 7 Contours of the streamwise velocity in meters per second for different directional shears at turbine hub height z = 100 m for a clockwise-rotating rotor disk (a, c, e, g, i) and a counterclockwise rotating rotor disk (b, d, f, h, j) for veers of 4 deg across a 100-m rotor disk (a, b), 8 deg (c, d), 12 deg (e, f), 16 deg (g, h), and 20 deg (i, j), each averaged over 30 min. (From Englberger et al. (2020). This work is distributed under the Creative Commons Attribution 4.0 license.)

ment of wind shear and wind veer effects on wind turbine loads is just starting to gain in importance, although the importance of wind shear and veer on loads were investigated numerically as early as 2003 (Eggers et al. 2003), motivated by the pioneering work of Kelley (1999), who pointed out the diurnal variability of turbulence stresses and resulting loads on wind turbines. Sathe et al. (2013) use aeroelastic simulations to assess the impact of atmospheric stability on wind turbine loads; in high-shear environments, rotor loads increase (with some influence on

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blade loads as well). Using load surrogate models calibrated with a database of high-fidelity load simulations, Dimitrov et al. (2018) assess the influence of wind shear and wind veer and other environmental variables. They find that wind shear exerts a pronounced influence on blade root loads while the effect of veer appears to be minimal for fatigue loads. The aeroelastic simulations of Churchfield and Sirnivas (2018) assess loads on a downwind turbine from a veered and a non-veered wake. The considered loads include blade-root out-of-plane bending moment, tower-top yaw moment, and tower-top pitch moment, as well as the resulting damage-equivalent loads (DELs). For cases with a downwind turbine directly downwind, each of the DELs is larger in the veered case than in a non-veered case. Both the normalized mean blade-root outof-plane bending moment DEL and the normalized tower-top yaw moment DEL are nearly 20% larger in the veer case as compared to the no-veer case. The normalized mean tower-top pitch moment DEL is less than 5% larger in the veer case as compared to the non-veer case when the downwind turbine is directly downwind. Finally, a sensitivity analysis explores the effects on wind turbine loads of a wide range of wind characteristics, including wind shear and wind veer (Robertson et al. 2019). Of the 18 parameters tested, wind shear and turbulence intensity exert the largest influence on loads, with wind shear exerting the largest influence on bladeroot out-of-plane pitching moment and blade shaft bending moments. To a lesser extent, wind veer (tested in a range of −25 deg to 50 deg across the 5-MW turbine’s rotor disk) influences the main shaft bending moments.

Summary and Recommendations We have highlighted key contributions to the scientific literature on the sources of wind shear and wind veer in the atmospheric boundary layer, observations of shear and veer, and the effects of shear and veer on wind turbine power production, wind turbine wake evolution, and wind turbine loads. As wind turbines have grown larger, they encounter deeper and more complicated regions of the atmosphere such that profiles of wind speed shear and wind direction veer play a quantifiable role. Changes in the wind speed and wind direction across the vertical extent of a wind turbine rotor disk modify the inflow vector on the blades of the turbine, thereby affecting the magnitude and orientation of the lift and drag forces of the blade’s airfoil. These changes can affect the power production and loads on large modern turbines, as well as the evolution of the wake that could affect a downwind turbine. Numerous avenues of important research remain regarding the effects of wind shear and wind veer. Assessments of the occurrences of shear and veer in locations with significant wind energy deployment are still required, considering that wind turbine design standards do not reflect the frequent occurrences of shear and veer although shear and veer do occur regularly at wind-turbine-rotor altitudes. While wind shear and wind veer clearly affect power production, the variability in these effects suggests that interactions between the exact shape of the shear and veer profiles and the operating regime of the wind turbine can manipulate that effect.

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Therefore, it may be possible for turbines to maximize power production despite shear and veer by operating with detailed knowledge of inflow profiles via a feedforward nacelle-mounted (Simley et al. 2016) or blade-mounted (Ungurán et al. 2019) lidars. Similarly, because shear and veer affect the evolution of wind turbine wakes, wind plant power optimization via yaw steering (Fleming et al. 2019, for example) will need to accommodate understanding of the shear and veer of the inflow profiles. Finally, our current understanding of the effects of shear and veer on turbine loads is grounded primarily in simulations: detailed measurements of loads, in conjunction with characterization of shear and veer, are required. Current versions of wind turbine design standards do not account for the variability of turbine inflow conditions that have been observed. Although 30-min averaged values of power law shear coefficient α across wind turbine rotor disks have been observed up to 0.6 and larger, the IEC standard for the normal wind profile assumes that α = 0.2 (IEC 61400-1 2019). Furthermore, the IEC standard considers extreme wind shear to occur only for short periods of time (on the order of seconds), but larger values persist for much longer periods of time. The IEC standard is also silent on wind veer. We suggest incorporating the possibility of extreme wind shear and wind veer into turbine design standards and into wind resource assessment strategies. The rotor-equivalent-wind-speed concept of the IEC standard for power performance (IEC 61400-12-1 2017) includes the effect of wind speed shear, but at this writing does not account for wind veer. Acknowledgments The author expresses appreciation to Alex Rybchuk for reviewing a draft version of this chapter.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence Wake Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Turbulence in the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Velocity Components in the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Length Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Turbulent Kinetic Energy in the Wake of a Wind Turbine . . . . . . . . . . . . . . Energy Spectral Density in the Wake of a Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Turbulence Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-scale Properties of Turbulence in the Wake of a Wind Turbine . . . . . . . . . . . . . . . . . Intermittency in the Wake of a Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence in the Wake of a Yawed Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Actuator Disk and Wind Turbine Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The size of wind turbines has been increasing steadily over the past decades, and the majority of these turbines are built in wind farms. As downstream turbines will be operating in the turbulent wakes of the upstream turbines, it becomes more and more important to understand the turbulence evolution mechanisms

I. Neunaber () LHEEA (CNRS) – École Centrale de Nantes, Nantes, France Institute of Physics and ForWind – University of Oldenburg, Oldenburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_45

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within the wake of a wind turbine for improved load calculations, wind farm layout optimization, and wind farm control methods. In this chapter, the evolution of turbulence within the wake of a single turbine exposed to uniform and atmospheric boundary layer inflow is therefore discussed in detail with views on velocity components, turbulence intensity, length scales, Reynolds stress, energy spectra, and intermittency. Approaches to include turbulence in wake models are explained. Further, turbulence in wakes of yawed turbines will briefly be commented on, and a comparison of turbulence generated by an actuator disk and a wind turbine will be given since the actuator disk concept is an established concept to simplify simulations and experiments. Keywords

Wind turbine wake · Turbulence · Turbulence evolution · Wake – ABL interaction

Introduction Over the past decade, the globally installed wind power capacity grew steadily and with it the number of wind energy converters installed in wind farms. While this setup is profitable, wind turbines operating in wind farms face additional difficulties as compared to a single turbine, as illustrated in Fig. 1: A single turbine operates in the atmospheric boundary layer (ABL) that is characterized by a wind velocity varying with height and turbulence. In a wind farm, a wind turbine may – depending on its position within the wind farm – be exposed to the wake that evolves downstream of an upstream turbine. For the downstream turbine, this means a reduction in power production around 10% and up to 50% on the one hand due to the reduced mean velocity in the wake and an increase in fatigue loads due to the higher levels of turbulence in the wake on the other hand (see, e.g., Barthelemie et al. 2007, 2010 and Vermeer et al. 2003). Turbulent fluctuations have been shown to translate directly to the loads, torque, and power output (e.g., Mücke et al. 2011 and Schottler et al. 2017). To improve load calculations, wind farm layout optimization algorithms, and wind farm control strategies, it is therefore necessary to understand the complex flow created by an interaction of the inconstant atmospheric wind with large, rotating, and energy-converting mechanical devices (van Kuik et al. 2016). A first approach to gain understanding is the investigation of isolated aspects within this complex interaction. The structures within a single turbine’s wake have been studied intensively in the past with focus on different aspects, such as the mean velocity and the turbulent structures (see, e.g., Porté-Agel et al. 2019 for a summary). For practical applications, the most important information is how the mean velocity deficit downstream a turbine evolves. Different models describe this with varying complexity, and the transfer from the behavior of an individual turbine’s wake to the behavior of a wake within a wind farm is accomplished by superposition and interaction techniques for several modeled wakes. The inclusion of turbulence

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in wake models is, however, more difficult. On the one hand, atmospheric turbulence influences the evolution of the mean velocity deficit and thus complicates the implementation of wake models. On the other hand, turbulence within the wake and in a wind farm evolves, spanning a wide range of scales. The largest structures originate from the atmospheric boundary layer inflow and the interaction between the atmospheric boundary layer and the wake. One often discussed example of the interaction between wake and ambient flow is the meandering, i.e., oscillating of the wake. Structures in the order of magnitude of the turbine parts, i.e., the rotor blades, the nacelle, and the tower, are imprinted due to the interaction between turbine and ambient flow. Tip and root vortices that are shed from the blades are an example for this. Finally, the turbulence evolves and decays, leading to small-scale turbulence. Different approaches can be used to increase knowledge of the complex turbulence interactions in the wake. Investigations can be carried out experimentally in the field or downscaled but under controlled conditions in a wind tunnel or numerically. The behavior of a single turbine or a turbine placed in a wind farm can be studied. The inflow conditions can be laminar and uniform to investigate the structures the turbine imprints onto the wake or turbulent with or without an atmospheric boundary layer profile to investigate the interaction between wake structures and the ambient turbulence, often focusing on the mechanisms of energy re-entrainment. To simplify wind tunnel experiments and simulations, the turbine can also be substituted by a static rotor model, the so-called actuator disk. In conclusion, there are many studies that focus on different aspects of the wake. This chapter aims to summarize the turbulent structures present in the wake of a wind turbine. Different analysis methods will be used to discover different aspects of turbulence such as the creation of turbulent structures by the turbine and the interaction of the wake’s turbulence with the ambient turbulence. For this, in the beginning, a brief comment on some atmospheric characteristics will be given, followed by a short overview of wake models and how they include turbulence. It follows the description of turbulence structures within a wake. Afterward, the evolution of turbulence quantities such as the mean velocity, turbulence intensity, and energy spectra is discussed. In addition, a comparison between the turbulence in the wake of a wind turbine and the wake generated by an actuator disk will be included since it is a common substitute. The focus will be on results obtained in wind tunnel experiments and simulations. Since the next section of this book is dedicated to Wind farm aerodynamics with  Chaps. 30, “Industrial Wake Models”,  31, “Wake Meandering”,  32, “CFD-Type Wake Models”,  33, “Wind Farm Cluster Wakes”,  35, “Wake Measurements with Lidar”, and  37, “Met Mast Measurements of Wind Turbine Wakes”, those topics will be excluded or treated briefly.

Atmospheric Boundary Layer In the following, a brief overview of the main characteristics of the atmospheric boundary layer will be given, as wind turbines operate in this part of the atmosphere.

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For a detailed discussion, the reader is referred to textbooks as, for example, Stull (1988) and Schlichting and Gertsen (1997). The atmospheric boundary layer is the lowest part of the atmosphere, and due to the interaction with the earth’s surface, it is naturally turbulent. For the mean velocity profile of the lower part of the atmospheric boundary layer, a logarithmic profile can be derived u(z) ∝ ln(z/z0 ) + ψ,

(1)

where z0 is the surface roughness length that ranges from 0.001 m (rough sea) over 0.03 m (open farm land) to >2 m for city centers with varying buildings and ψ is a function accounting for the atmospheric stability. The height of the atmospheric boundary layer and the turbulence depend on the thermal stratification of the ABL. A near-neutral ABL often occurs in strong winds, on cloudy days, or during dusk and dawn. In case of a neutral atmosphere, ψ = 0. During daytime when surface heating fuels convection, an unstable, or convective, stratification is found. When convection is stopped by surface cooling, for example, during night time, the ABL is stable. The turbulence in the ABL is anisotropic and depends on the surface roughness, the atmospheric stability, and the distance from the ground. Above the ABL is the free atmosphere that is not influenced by the earth’s surface anymore. Here, the pressure gradient force sets air into motion, and this motion is balanced by the Coriolis force. The result is the geostrophic wind that, as will be discussed later, is also the source of energy that re-energized the flow within a wind farm.

Turbulence Wake Structures When looking at the turbulent structures of a wake, two effects have to be distinguished: On the one hand, the turbine itself is imprinting flow structures onto the wake due to interaction between the turbine parts with the inflow. On the other hand, the wake interacts with its surrounding flow that is naturally turbulent. Thus, in the following, both aspects are examined by first explaining flow phenomena from the perspective of a wind farm operating in the atmospheric boundary layer and afterward looking at the wake itself. For a first overview of the development of turbulence in the wake generated by a wind turbine, Fig. 1 illustrates the initial situation: As illustrated in Fig. 1a, wind turbines are usually clustered into wind farms. While the front row turbines are exposed to the inflow of the atmospheric boundary layer, the downstream turbines are exposed to the wakes of the upstream turbines. Inside a wind farm, the wakes of different turbines interact, and also, the wakes interact with the ambient flow (Fig. 1c). The velocity recovers as mean flow kinetic energy and momentum are vertically entrained from the high-energetic flow above the wind farm by turbulence and mixing. These transport mechanisms are crucial for the efficiency of a wind

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Fig. 1 (a) Illustration of different turbulence regions in a wind farm operating in the atmospheric boundary layer (after Stevens and Meneveau 2017): Turbines are exposed to the inflow of the atmospheric boundary layer (ABL) of height δ that is often described to follow a logarithmic velocity profile. Above the atmospheric boundary layer is the geostrophic wind. Downstream of the turbines of hub height zh and rotor diameter D, the wakes evolve and interact with each other and the ambient flow. Above a wind farm, an internal boundary layer evolves with height δBL (x). In sufficiently large wind farms, this internal boundary layer will eventually reach a fully developed state. (b) Turbulence structures in the wake of a wind turbine (top view): The turbine is exposed to turbulent inflow and imprints its own structures, in particular the tip and root vortices that are transported downstream on a helical path, to the flow while also generating a velocity deficit. The wake expands downstream. A shear layer evolves between the wake and the ambient flow. Due to larger flow variations in the inflow, the whole wake can also meander. (c) “iso-surface of vorticity magnitude equal to 0.5 1/s colored by streamwise velocity” (Jha et al. 2015) in a wind farm consisting of five NREL 5MW turbines exposed to a moderately convective boundary layer at u0 = 8 ms show the turbulence structures and interactions in a wind farm. (Sub-figure (c) is taken from Jha et al. (2015) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

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farm as without it, “analyses based on ideal flow would predict that successive streamtubes would slow the air down to asymptotically small hub height velocities, so that the power density of a very large wind farm would tend to zero” (Meneveau 2019). The big reservoir from which the energy is drawn thanks to the pressure gradient is the geostrophic wind (see section “Atmospheric Boundary Layer”), as, for example, Stevens and Meneveau (2017) and Meneveau (2019) write. Above the wind farm, an internal boundary layer evolves that grows downstream, and when its height δBL (x) has reached the height δ of the atmospheric boundary layer, the fully developed wind turbine array boundary layer regime is reached (Calaf et al. 2010). When looking at a single wake, the structures that are imprinted by the turbine onto the flow are brought into focus. This is illustrated in Figs. 1b and 2. Since these illustrations give a top view, the boundary layer profile of the flow field is not visible. The turbine interacts actively with the inflow, first by converting energy and thus reducing the velocity downstream of the turbine and second by imprinting flow structures onto the wake flow: Vortices are shed from the turbine’s tower and nacelle. The turbine rotor induces a spin that is due to conservation of momentum

Fig. 2 Illustration of the wake regions near wake, intermediate wake, and far wake with an indication of the pressure and mean velocity behavior in the wake (top) with an example of these regions in a simulated wake of the NREL 5MW turbine at an inflow velocity of u0 = 8 ms . The vorticity magnitude is indicated in an axial wake plane. (This figure is taken from Jha et al. (2015) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

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counter-rotating compared to the turbine rotation. Additionally, from the blades, tip and root vortices are shed that are, respectively, rotating in opposite direction due to conservation of momentum (i.e., the root vortices rotate with the turbine’s sense of rotation and the tip vortices against it) (e.g., Hau (2014) and Sanderse (2009)). The tip vortices are transported downstream with the pivoting wake on a helical trajectory. Downstream, the tip vortices will start to interact with each other, an effect that is called leapfrogging, which causes an instability and a breakdown of the “tip vortex sheet” surrounding the wake (Lignarolo et al. 2014). With that, the shear layer expands and the wake mixing process is started. Once the shear layer reaches the wake’s center, the turbulence decays. In addition, the wake expands due to turbulent diffusion. This illustrates that a transition takes place from the near wake that is determined by the presence of turbulent structures induced by the turbine to the far wake where turbulence is fully evolved and flow structures cannot be separated anymore. The above-described evolution process of the turbulent structures within the wake is highly dependent on the interaction with the surrounding flow, as already indicated above. Higher turbulence levels in the atmospheric boundary layer increase the turbulent mixing effects which lead to an earlier breakdown of the tip vortices and a faster recovery of the mean velocity in the wake. Another turbulence phenomenon that is present in the wake and that represents turbulence at larger scales is the so-called meandering, i.e., the slow oscillating of the wake. While it was, for example, shown by Espana et al. (2011) that meandering is linked to large-scale motions in the inflow (>Dw ), another possible explanation of this phenomenon arising from classical wake turbulence theory is that the “wake eddies generated by the shear behind the wind turbine” (Meneveau 2019) cause the oscillation. Currently, it is widely assumed that the predominant effect is the interaction between inflow and wake but that the shear-generated eddies will also have an effect (Meneveau 2019).

Modeling of Turbulence in the Wake Overall it can be seen that mechanisms determining the evolution of the wake in the atmospheric boundary layer are highly complex. This high complexity of the evolution of the wake flow and especially the turbulence is challenging for a mathematical description of the wake. In theory, the three-dimensional unsteady Navier-Stokes equations describe the problem; however, exact solutions only exist for a few problems with simple boundary conditions. The closest we come to this is by using computational fluid dynamics, more precisely direct numerical simulations, where the Navier-Stokes equations are solved for each cell and time step in a discretized mesh that encompasses the whole area of interest and is so fine that the turbulence can be resolved down to the smallest scales within the flow. While generally, the achieved resolutions and Reynolds numbers are continuously improving, unfortunately, for a full-scale turbine, this would take in the order of magnitude of 100 years with current computers which makes a direct simulation of

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a full-scale turbine too costly and time-consuming for applications. An alternative exists that combines the simulation of larger flow structures on a coarser mesh with a turbulence model for the small scales, the so-called large-eddy simulation (LES). The results of LES investigating turbulence structures in wakes are promising. However, while LES is used in some wind farm models, it is still sophisticated and time-consuming for the use of wind farm and control optimization (Boersma et al. 2017). Therefore, depending on the application, a simple modeling of single or few combined flow properties within the wake can be an alternative. When modeling a wake, different strategies with varying complexity are used depending on the accuracy needed. The underlying idea of many quasi-stationary analytical wake models is the treatment of the wake as a canonical wake of a bluff body exposed to a uniform laminar inflow: This wake is axisymmetric and self-similar, and it expands downstream according to a power law, Dw (x)/D ∝ (x/D)1/3 with a velocity recovery according to uw (x)/u0 ∝ (x/D)−2/3 (see, e.g., Townsend 1976 and George 1989). While there is evidence that these scalings hold for a wind turbine wake under uniform laminar inflow conditions (see Okulov et al. 2015), the mixing effects are altered in an atmospheric boundary layer inflow which alters the scaling, and the inflow is sheared in z-direction so that axisymmetry does not hold anymore. An analytical, quasi-static model that considers the turbine thrust is, for example, easy to use when the downstream turbine’s power estimated from the mean velocity deficit is of interest. The most prominent one is the model derived by Jensen (1983), where the wake is modeled by using a top hat-shaped velocity deficit and conservation of mass. If also the turbulence in the wake is wished for, a model derived by Frandsen (2003) is used, for example, in the IEC 61400-1 Ed. 4 norm that explains the design requirements for wind turbines (IEC Standard 2019). There, the concept of an effective turbulence intensity in the wake is introduced to estimate the loads acting on a turbine within a wind farm. The model depends on the inflow velocity, the atmospheric turbulence intensity, the turbine thrust, the turbine spacing, and the number of surrounding turbines. The turbulence intensity is assumed to be the same over the whole rotor-swept area. While the Frandsen model may deliver reasonably good results for modeling the far wake, it is not suitable for the near wake due to the complex flow structures. Recently, the importance of a more precise, dynamic approach to model wind turbine wakes including the small-scale turbulence has led to the inclusion of the dynamic wake meandering model (DWM, Larsen et al. (2008), see also section 5 –  Wake Meandering) to the IEC 61400-1 Ed. 4 norm (IEC Standard 2019). Within this model, different turbulence scales are included by splitting the wake into large scales imprinted by the atmospheric flow, intermediate scales of the size of the turbine diameter, and added small-scale turbulence. First, a velocity deficit at time t is modeled according to blade elementum theory (BEM). This initial deficit is transported downstream. The deficit expands due to turbulent diffusion. The expansion can be modeled using a thin shear layer approximation of the NavierStokes equation and assuming axisymmetry. The transport of this intermediate-scale deficit is modeled using the large turbulent scales within the incoming flow field,

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including the vertical and spanwise movement from the turbine center, thus also including meandering. The small-scale turbulence is believed to be homogeneous and following a Kaimal spectrum (see Kaimal et al. 1972), assuming ideally decaying turbulence. As compared to a simple engineering model like the Frandsen model, this approach leads to more precise wake deficit and turbulence intensity profiles and a better power and load estimation. It is suitable for modeling the intermediate to far wake. Using the presented methods, the evolution of the mean velocity deficit can be modeled accurately. However, when it comes to the modeling of the evolution of turbulence quantities such as the turbulence intensity, the deviations quickly become more significant (cf. Xie and Archer 2015 and Cabezón et al. 2011). A different approach to describe the wake turbulence is by using data to extract and model turbulent structures present in the wake. One method follows the approach of superpositioning turbulence structures of different sizes to model the wake, and it is pursued by stochastic models that use proper orthogonal decomposition (POD) to decompose the wake velocity field into spatial modes with time-dependent weighting coefficients (Bastine et al. 2018). For this, POD is applied to data with high spatial resolution that is usually obtained numerically from LES (in Bastine et al. 2018) or experimentally from particle image velocimetry (PIV) (e.g., Hamilton et al. 2016). Field measurements currently lack the required spatial resolution. To reduce the complexity of the POD, it can be truncated, and the temporal dynamics can be included by introducing stochastic weighting coefficients. The larger flow structures can be modeled using the first several modes, but in order to also model the small structures well, a high number of modes are needed. Whether small structures are needed in the model depends on the quantity that should be predicted with the modeled flow field. For example, it was shown in Bastine et al. (2018) that the torque acting on a turbine can be predicted well using the first 6 POD modes, while for thrust and yaw moment, more than 20 modes are needed. This improves significantly if a field of homogeneous turbulence is added to the truncated POD as now, six modes are sufficient to predict the thrust and yaw moments correctly. This shows the importance of the inclusion of smallscale turbulence. One drawback of this method is that the POD is not suitable to also include meandering. Overall, this illustration shows that even while power output and loads are often predicted reasonably well, a correct modeling of a rather simple quantity as the turbulence intensity within the wake of a wind turbine is still a challenge due to the complexity of the problem. When expanding the analysis to quantities such as the energy spectra or Reynolds stress tensor, modeling becomes even more challenging. The validation of the presented models is done with measurements, preferably by comparison with field data. Information on the mean flow profiles can be obtained using met masts or remote sensing techniques such as Lidar (light detection and ranging) and radar systems, and the latter two can also be used to investigate wake meandering (see, e.g., Schroeder et al. 2017 and Trujillo et al. 2011). While field measurements provide important insight, they are at the same time time-consuming and expensive. In addition, due to the permanently changing inflow conditions, the

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often limited number of downstream positions, and the low temporal resolution of the measurement data, field measurements are not well suitable to investigate the evolution of small-scale turbulence structures (< D) downstream of a wind turbine. The examination of the influence that different inflow conditions or operational modes of the turbine have on the wake is also challenging. For the future, another possibility lies in the use of drones equipped with five-hole probes that deliver high-frequency data, are flexible, and can capture flow structures such as the tip vortices, as presented in Mauz et al. (2019). Overall, field measurement techniques have become more advanced over the past years; however, the investigation of turbulence quantities with focus on small-scale turbulence remains today difficult. Simultaneously, CFD simulations become more and more accurate but still need validation. In Wang et al. (2019), wind tunnel measurements are therefore used to validate simulations. Thus, to gain knowledge about the evolution of turbulence downstream of a wind turbine, measurements can provide valuable insight. Wind tunnel experiments are a good option to study the turbulence evolution downstream of a wind turbine model under controlled and reproducible conditions despite the significant difference in Reynolds number that is in the order of magnitude of 1000 lower compared to full-scale turbines. In combination with a comparison of field data and results from simulations, a good picture of the turbulence evolution within the wake of a turbine can be drawn. In the following, the turbulence evolution in the wake of a wind turbine will be discussed by examples from different wind tunnel experiments and simulations.

Evolution of Velocity Components in the Wake When investigating flows, the first quantity that is investigated is the mean velocity. Therefore, for a first overview of the behavior of the flow within the wake, the evolution of the mean velocity with increasing downstream position is explained in the following. Starting from the evolution of the mean velocity in the wake in uniform inflow, a topic that is covered, for example, in Lignarolo et al. (2014), Maeda et al. (2011), and Neunaber (2019), the discussion will go to the influence of an atmospheric boundary layer inflow as covered in, e.g., Aubrun et al. (2013), Bastankhah and Porté-Agel (2014), Camp and Cal (2016), Wu and PortéAgel (2012), and Zhang (2012), and close with the review of the single-velocity components, which is, for example, discussed in Maeda et al. (2011) and Lignarolo et al. (2014). Since a wind turbine converts the kinetic energy from the wind into mechanical and then electrical energy, the velocity downstream of the turbine is reduced compared to the inflow velocity (cf. Hau 2014 and Sanderse 2009). This velocity deficit has a cross section of approximately the rotor-swept area directly downstream of the turbine in the near wake. The downstream evolution of the normalized mean velocity at the centerline is exemplarily plotted in Fig. 3. Here, the turbine was exposed to three uniform, differently turbulent inflows. The influence of the tower and the nacelle are present close to the turbine. The pressure that had increased

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Fig. 3 Evolution of the mean velocity u that is normalized by the inflow velocity u0 in the wake of a model wind turbine exposed to different inflow conditions at the centerline. As comparison, the wake evolution of an actuator disk is plotted (gray dashed line). (Taken from Neunaber 2019)

upstream the turbine above ambient pressure and abruptly dropped at the rotor plane below ambient pressure tends back to ambient pressure downstream (see also Fig. 2). While the pressure inside the wake is still increasing, the velocity is decreasing and the wake expands due to conservation of mass, and where the pressure just reached ambient pressure again, the velocity within the wake is minimum. The wake recovery starts from this point. Kinetic energy is re-entrained from the high-energetic free flow around the turbine, especially above the turbine, by turbulent mixing. The velocity deficit can be approximated by a Gaussian curve if uniform inflow is considered, and additionally, it is often described as self-similar. This shows that the velocity is smallest in the central region within the wake. Figure 4 shows the evolution of vertical and horizontal velocity deficit profiles in the wake of a single turbine exposed to different atmospheric boundary layer inflows as obtained by LES. With increasing downstream position, the velocity in the wake recovers and the mean velocity deficit decreases. The wake expands due to turbulent diffusion. While the horizontal velocity profiles are axisymmetric, the vertical shear in the inflow affects the axisymmetry in vertical direction. To account for a sheared boundary layer flow, the Gaussian profile can be corrected with a boundary layer profile as was shown in Bastankhah and Porté-Agel (2014). A more turbulent inflow is generated with increasing roughness length, and this also finds expression in the accelerated wake recovery that is especially pronounced between x/D = 3 and x/D = 7 in this example. As the stability state of the ABL is also related to the turbulence, a similar effect can be shown for different atmospheric stabilities, which was done, for example, by Abkar and Porté-Agel (2014). As mentioned above, the wake recovery is determined by the turbulence within the ambient flow since the breakdown of the tip vortices is accelerated and turbulent mixing is increased. A practical example of this effect is given also by the investigation of Platis et al. (2018) who measured the recovery of a wake of a whole wind farm in different atmospheric stabilities. They found that within a stable stratification that is usually characterized by low turbulence, the wake of the investigated wind farm is significantly lengthened as compared to an unstable stratification that is characteristically turbulent.

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Fig. 4 Influence of neutral ABL profiles with different surface roughness lengths on the downstream evolution of mean velocity deficit profiles of a single turbine in vertical (a) and horizontal (b) measurement direction. (This figure is taken from Wu and Porté-Agel (2012) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

To increase the understanding of innate mechanisms that fuel the wake recovery (i.e., without ambient turbulence), it can be useful to investigate the evolution of the velocity components in the near wake with the transition to the far wake. This is done using PIV in Lignarolo et al. (2014) downstream of a two-bladed rotor operating in two different tip-speed ratios both for the unconditioned velocity field to investigate the evolution of the shear layer between the wake and the ambient flow and the phase-averaged velocity field to visualize the tip vortex structures. In the near wake where the tip vortices are stable, the shear layer is thin, but it expands once the tip vortices become unstable. This effect is dependent on the tip-speed ratio. In Figs. 5 and 6, the phase-averaged spanwise and vertical velocity component are plotted for two different tip-speed ratios. From these indications and (more explicitly) from measurements as, for example, obtained in Zhang (2012), one can see that within the near wake, the spanwise velocity field is anti-symmetrically split by the horizontal rotor axis into a bottom part with positive velocity and a top part with negative velocity; downstream, this structure disbands. Similarly, the vertical

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Fig. 5 Phase-averaged spanwise velocity component of a horizontal measurement plane downstream of a two-bladed turbine that was operated with a tip-speed ratio of λdesign = 6 (a) and λ = 4.8 (b). Measurements were obtained using PIV. (Reprinted from Lignarolo et al. (2014), Copyright (2014), with permission from Elsevier)

Fig. 6 Phase-averaged vertical velocity component of a horizontal measurement plane downstream of a two-bladed turbine that was operated with a tip-speed ratio of λdesign = 6 (a) and λ = 4.8 (b). Measurements were obtained using PIV. (Reprinted from Lignarolo et al. (2014), Copyright (2014), with permission from Elsevier)

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velocity component is anti-symmetrically distributed around the vertical rotor axis with positive values at positive Y /D and negative values at negative Y /D due to rotation. Also, when looking at the conditionally averaged velocity components in Figs. 5 and 6, the tip vortices can be identified and it can be seen how they become unstable downstream and interact with each other. Overall, already the evolution of the mean velocity gives a quite detailed picture of flow structures within the wake.

Turbulence Intensity In the following, the investigation of turbulence in the wake of a wind turbine will be followed further by examining the behavior of the turbulence intensity I = σ/u¯ (σ being the standard deviation and u¯ being the mean velocity of the flow). In calculations, the turbulence intensity in the wake of a turbine is currently often taken into account using the concept of effective or added turbulence by superimposing the turbulence from the inflow with the modeled turbulence created by the turbine (Vermeer et al. 2003; IEC Standard 2019). However, by doing so, turbulence is reduced to a single value valid for the far wake, while the evolution of the turbulence from near wake to far wake is neglected. This section will explain the evolution of turbulence intensity downstream of a wind turbine similarly to the evolution of the mean velocity by first looking into uniform inflow and afterward discussing the influence of atmospheric boundary layer turbulence. As already indicated above, downstream of a turbine within the wake, the flow is more turbulent than the inflow, which is also mirrored in the turbulence intensity (see, e.g., Vermeer et al. 2003). Uniform inflow conditions enable to see the part of the turbulence intensity evolution that is imprinted by the turbine onto the flow. The evolution of the turbulence intensity at the centerline, which is presented in Fig. 7 and was investigated, for example, in Kermani et al. (2013) with LES, and in Neunaber et al. (2017) and Neunaber (2019) experimentally, already allows some conclusions about the evolution of the turbulent structures in the wake. First, the turbulence intensity drops while moving out of the nacelle’s lee. Then, the turbulence intensity builds as the shear layer expands, and when the shear layers surrounding the wake meet and the turbulence is fully developed, the turbulence intensity decays. It was shown in Neunaber et al. (2017) and Neunaber (2019) that this decay follows a power law. The position of the turbulence intensity peak at the centerline was shown to be related to the inflow turbulence, as a higher ambient turbulence increases the turbulence mixing and accelerates the whole turbulence evolution process. In Fig. 8, horizontal and vertical profiles of the added turbulence intensity evolution downstream of a turbine exposed to atmospheric boundary layer inflows with different turbulence levels are presented. The results are obtained by LES. Looking at the horizontal profiles, one can see again the axisymmetry of the wake, especially close to the rotor. Close to the rotor, the highest turbulence intensity

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Fig. 7 Evolution of the turbulence intensity T I that is normalized by the peak turbulence intensity T ICLpeak in the wake of a model wind turbine exposed to different inflow conditions at the centerline. As comparison, the wake evolution of an actuator disk is plotted (gray dashed line). (Taken from Neunaber 2019)

Fig. 8 Influence of neutral ABL profiles with different surface roughness lengths on the downstream evolution of turbulence intensity profiles of a single turbine in vertical (a) and horizontal (b) measurement direction. (This figure is taken from Wu and Porté-Agel (2012) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

values are found at the position of the tips where the shear layer evolves. The two peaks flatten and converge downstream. Looking at the vertical profiles of added turbulence intensity illustrates the influence of the sheared boundary layer as now, the profiles are asymmetric and exhibit the highest turbulence intensity values at the top tip of the rotor at x/D = 3. With increasing downstream position, the

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profiles flatten out. The influence of different inflow turbulence levels is in this example especially pronounced in the vertical profiles where the highest amount of turbulence is added in case of the highest turbulent inflow in the upper part of the wake. Interestingly, as compared to the inflow, a turbulence intensity reduction in the lower part of the wake occurs for the more turbulent inflows. With increasing downstream position, the influence of the ambient turbulence intensity decreases. Similar results to those found in Wu and Porté-Agel (2012) can, for example, be found in Chamorro and Porté-Agel (2010) and Abkar and Porté-Agel (2014). In Lignarolo et al. (2014), the turbulence intensity is presented downstream of a two-bladed rotor exposed to uniform laminar inflow both unconditioned and phase-averaged. A comparison of both representations shows that close to the rotor (x/D < 2.5), the contribution of the periodic structures (i.e., the tip vortices) is dominant, while with increasing downstream distance, the contribution of random structures becomes more and more prevalent.

Integral Length Scale When investigating turbulence for wind energy applications, the turbulent length scales present in a flow are an important parameter since they are included in different turbulence models. The integral length scale is the turbulent length scale that is discussed the most frequent in wind energy applications. In developed turbulent flows, it presents the energetic turbulent flow structure from which turbulence decays in a cascade and transports turbulence kinetic energy to smaller and smaller scales until dissipation becomes dominant. This means that turbulence is correlated within the span of the integral length while it is uncorrelated for larger structures. The integral length scale is therefore an estimated measure for the size of the high energetic structures in the flow. Considered with respect to turbine-related and turbine-relevant scales as the rotor diameter, these structures are important (e.g., Dimitrov et al. 2017). The behavior of the integral length scale in the wake of a turbine is compared in Aubrun et al. (2013), Chamorro et al. (2013), Jin et al. (2016), and Neunaber (2019) to the integral length scale in the inflow to understand the influence of the turbine on flow structures. Directly downstream of the turbine, the turbulence evolution and mixing processes have not yet started. Here, a comparison of the integral length scale in the wake with the integral length scale in the inflow shows that the integral length scale is smaller in the wake than in the inflow. This can be explained as the turbine acting as an active filter, thus dampening flow structures that are larger than the rotor (see, e.g., Chamorro et al. 2012). When comparing the integral length scale directly downstream of the rotor for laminar inflow and turbulent inflow, larger values are found in the case of turbulent inflow as compared to laminar inflow. This leads to the conclusion that, while large

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structures are dampened (or cut up) by the rotor, structures that are smaller than the turbine radius can pass through the rotor. With increasing downstream distance, when the turbulence evolves, the integral length scale increases again, depending on the inflow toward rotor-related scales or, if large-scale structures are present in the inflow, toward larger scales. Far downstream, it tends to the integral length scale in the inflow.

Evolution of the Turbulent Kinetic Energy in the Wake of a Wind Turbine To understand the physical mechanisms that create the turbulent mixing and are thus responsible for the energy entrainment and the wake recovery, the flux and the production of mean flow kinetic energy by turbulence are important. They are thus subject to several studies, for example, Chamorro and Porté-Agel (2010), Chamorro et al. (2012), Barlas et al. (2016), Hamilton et al. (2017), Lignarolo et al. (2014), or Wu and Porté-Agel (2012). According to Hamilton et al. (2017), the mean kinetic energy equation in a wind turbine boundary layer can be expressed by ∂ui uj ui ∂ 12 u2i ui ∂p ∂ui   uj =− + ui uj − − Fxi . ∂xj ρ ∂xi ∂xj ∂xj

(2)

On the left of the equation is the convection term. It equals the sum of the pressure gradient term, the production of TKE, the flux of TKE, and a forcing term originating from the thrust force acting on the flow while the turbine converts energy. As can be seen from this equation, the Reynolds shear stress tensor components τij = ui uj are directly related to the production and flux of the turbulent kinetic energy. Therefore, when discussing the turbulent kinetic energy in the wake of a wind turbine, often, the most dominating component of the Reynolds shear stress tensor is discussed, namely, the streamwise-vertical component τxz (see, e.g., Hamilton et al. (2017) who uses a wind turbine array boundary layer inflow). Exemplarily, in Fig. 9, the turbulence kinetic energy downstream of a wind turbine exposed to uniform inflow is plotted 6D downstream of the rotor. Highest values are found here in a ring slightly larger than the rotor. While this result indicates similarly to the turbulence intensity regions of strong fluctuations surrounding the wake, the more interesting part is, as mentioned above, the flux and production of the TKE. In Fig. 10, the downstream evolution of the kinematic shear stress τxz is plotted at several lateral cross-sectional planes for differently turbulent ABL profiles. It can be seen that τxz is maximum in the near wake at top tip where the shear is strongest. The highest values are found for the highest inflow turbulence (Case 1). In the upper half of the wake, the values are positive while they are negative on the lower half. This

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Fig. 9 Contour plot of the turbulent kinetic energy (T KE = 12 ui ui ) 6D downstream in the wake of a wind turbine. High levels of TKE are found in a ring surrounding the wake. (Taken from Schottler et al. (2018) under permission of the Creative Commons Attribution 4.0 License Creative Commons Attribution 4.0 License 2009)

Fig. 10 “Contours of the kinematic shear stress τxz (m2 s−2 ) at the lateral cross-sectional planes of x/D = 3, 5, 7, 10, and 15 for the turbines installed over flat surfaces with four different roughness lengths [Case 1, z0 = 0.5 m; Case 2, z0 = 0.05 m; Case 3, z0 = 0.005 m; Case 4, z0 = 0.00005 m]. The dotted line denotes the rotor region” (Wu and Porté-Agel 2012). (This figure is taken from Wu and Porté-Agel (2012) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

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Fig. 11 “Contours of the TKE budget terms (advection, shear production, turbulent transport, and residual) for the stand-alone turbines installed over flat surfaces with the two different roughness lengths: z0 = 0.5 m (a panel) and z0 = 0.00005 m (b panel). All the quantities are normalized by u3hub /δ = 1.458 m2 s−3 .” (Wu and Porté-Agel 2012). (This figure is taken from Wu and Porté-Agel (2012) under permission of the Creative Commons Attribution 4.0 License (Creative Commons Attribution 4.0 License 2009) without alterations)

indicates the entrainment of the surrounding flow into the wake. An investigation of the energy entrainment process from Hamilton et al. (2017) shows similarly that an inward flux of kinetic energy that is strongest in the near wake is responsible for the wake recovery. To further investigate the processes that drive the production and transport of turbulence in the wake, Fig. 11 shows the contributions of the production, convection and transport of TKE, and a residual term that accounts for “contribution of viscous dissipation and pressure-correlation effect as well as some contributions of the SGS turbulence to the production, advection and transport of TKE that cannot be computed from LES data” (Wu and Porté-Agel 2012). The highest values of all TKE budget terms, in particular the shear production term, can be found in the near wake at top-tip level where the production of TKE is high. A comparison of the two inflow conditions shows that higher turbulence levels in the inflow lead to higher values of all budget terms. Also, according to Wu and Porté-Agel (2012), the transport of turbulence in the wake away from the shear layer is important for the redistribution of the TKE and contributes to the wake expansion by turbulent diffusion. In Fig. 12 (obtained from Lignarolo et al. 2014), the phase-locked average of the τxy shear stress component is plotted in a horizontal plane at hub height. It is attributed to a random share of turbulence, and it can be seen how in the near wake where the tip vortices are present, ambient flow is entrained between the tip vortices. When the tip vortices break down, within the expansion of the shear layer the turbulent mixing marked by the Reynolds shear stress is strong, and the

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  Fig. 12 “Phase-locked average x − y component of Reynolds stresses u v  /u20 at λ = 6 (a) and λ = 4.8 (b)” (Lignarolo et al. 2014). (Reprinted from Lignarolo et al. (2014), Copyright (2014), with permission from Elsevier)

energy entrainment is inward. As further investigated in Lignarolo et al. (2014), the unconditioned averaged Reynolds shear stress is assigned to periodic turbulent motions. Highly negative values can be identified in the shear layer in the near wake. In addition, the expansion of the shear layer can be seen.

Energy Spectral Density in the Wake of a Turbine The evolution of the energy spectral density E(f ) represents the distribution of energy E over frequency f and is thus another measure of turbulent and periodic structures in a flow. Downstream of a wind turbine, the evolution of energy spectra can be used to track the tip and root vortices and their breakdown in dependence of the turbulent structures in the inflow. Additionally, with a view from classical turbulence research, the decay of energy within the inertial subrange of the energy spectrum can be used to analyze the turbulence evolution, also with regard to the interaction between inflow and turbine. Energy spectra have, for example, been investigated in Aubrun et al. (2013), Al-Abadi et al. (2016), Barlas et al. (2016), Eriksen and Krogstad (2017), Maeda et al. (2011), Neunaber (2019), and Zhang (2012) with regard to the evolution of the tip vortices. The latter are captured at the outer radial positions of the wake

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at R/D ≈ 0.5 (R being the radius of the turbine) and they can be tracked up to 2-3 diameters downstream the turbine depending on the inflow. This is exemplarily shown in Fig. 16 where the downstream evolution of pre-multiplied energy spectra f · E(f ) is plotted in comparison to the inflow generated by a regular grid. To validate that certain effects are related to the turbulence generated by the turbine and not to the evolution of the inflow turbulence, the spectrum of the inflow is plotted at the rotor position and at the last measurement position.1 Generally, the investigation of energy spectra confirms that tip vortices break down faster in higher ambient turbulence which accelerates the wake recovery due to the stronger interaction between the wake and its surroundings. To understand how a turbine interacts with the inflow, Bastine et al. (2015), Chamorro et al. (2012, 2013), Jin et al. (2016), Neunaber (2019), and Singh et al. (2014) compare the turbulence spectra of the inflow and the wake at different downstream positions. As an example, Fig. 13 shows the evolution of f · E(f ) with increasing downstream position with regard to the inflow at the centerline. In the near wake, the root vortex is still present. From the near wake to the transition region, the spectra evolve and tend toward one universal spectrum in the far wake that shows an inertial subrange with a decay of E(f ) ∝ f −5/3 . Compared to the inflow, the turbine introduces energy across all scales. Overall, summarizing the results from different studies, it can be concluded that the evolution of the energy spectrum downstream of a turbine within the central region of the wake is dominated by the turbulence generated by the turbine. Depending on the different inflow conditions, the energy distribution may therefore be enhanced or reduced across certain scales within the wake as compared to the inflow which makes the turbine an active filter of turbulent structures. In the far wake where the turbulence has evolved, the inertial subrange of the spectrum follows a decay of E(f ) ∝ f −5/3 and the spectra collapse to one universal spectrum which indicates that the turbulence imprinted by the turbine onto the flow is dominating and tending toward an equilibrium state of turbulence.

Further Turbulence Quantities In the following, some turbulence quantities that are used to investigate the turbulence in the wake of a wind turbine by higher-order statistics are briefly introduced.

Fig. 16, a peak can be seen around f · D/u0 ≈ 0.2 both in the far wake of the turbine and the evolved inflow at x/D = 4.69. Since the experiments have been carried out in an open test section, this peak is assigned to vortices shedding from the wind tunnel inlet, and it is not related to the wake evolution.

1 In

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Fig. 13 Evolution of the pre-multiplied energy spectra f · E(f ) over f · D/u with increasing distance downstream at the centerline position in the wake of a turbine exposed to inflow generated by a regular grid: The downstream position is color-coded, in blue is a decay of E(f ) ∝ f −5/3 indicated and the red dashed lines represent the spectra of the inflow at the rotor plane x/D = 0 (dark red) and its evolution without turbine at x/D = 4.69 (red). (Adapted from Neunaber 2019)

Multi-scale Properties of Turbulence in the Wake of a Wind Turbine In some investigations, the turbulence within the wake of a turbine is also examined with additional methods from turbulence research to gain understanding of the underlying turbulence processes. For example, the multi-scale nature of turbulence has been investigated by means of structure functions and the existence of Markov properties in Melius et al. (2014) and Ali et al. (2019). When Markov properties are present, the respective quantity can be modeled with a stochastic process generated by a Langevin equation. The authors show that within the near wake region, the flow does not follow a universal behavior and shows strong variations depending on the measurement position. However, in the far wake, Markov properties down to scales in the order of magnitude of the Taylor length are exhibited and a universality of the scaling behavior of turbulent structures is present. As a consequence, the turbulent wake flow can be modeled using the Langevin equation.

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Intermittency in the Wake of a Turbine Another quantity that can be examined is the intermittency, i.e., gustiness, within a flow. It can be identified by means of two-point statistics: Given the flow velocity at two times t and t + τ , the velocity increment with respect to the time scale τ is defined as the difference between the two velocities, δuτ = u(t + τ ) − u(t). If the flow is with respect to the time scale τ intermittent, the probability density function of the velocity increments p(δuτ ) has a significantly higher probability of large velocity fluctuations than a Gaussian distribution would predict. It has been shown that the flow within the atmospheric boundary layer is intermittent (Wächter et al. 2012). For a downstream turbine, the presence or absence of intermittency can be of interest for several reasons. First of all, as Schwarz et al. (2018) showed, intermittency in the flow on scales related to the rotor diameter can increase fatigue loads. Also, intermittent fluctuations are transferred from the inflow to the turbine and its power output, and this can have an influence on the grid stability. Thus, the question arises whether and to what extend intermittency is present in the complex wake of a wind turbine since the ABL inflow is on the one side already intermittent but the turbine cuts up larger structures and acts as a filter for certain structures on the other side. Also, the downstream evolution of intermittency in the wake of a turbine is of interest with regard to the question whether regions with high intermittency levels exist. Different studies obtained by Ali et al. (2017), Bastine et al. (2015), Neunaber (2019), Singh et al. (2014), and Wessel (2008) show that the intermittency at scales corresponding to the rotor diameter is reduced in the far wake of a wind turbine as compared to the inflow. In particular, when looking at the downstream evolution of the intermittency at scales related to the rotor diameter, one finds that the evolution of intermittency in the wake does not depend on the presence of intermittency in the inflow. Close to the rotor, intermittency is not present in the wake, it increases where the turbulence builds up in the transition region and decreases toward a non-intermittent state in the far wake. For small scales (i.e., < D), the flow is intermittent, and strong indications of homogeneous isotropic turbulence in the center of a wind turbine far wake are present for these structures. Although intermittency is reduced in the center of the far wake by the turbine as compared to the inflow, it was additionally found that a ring surrounding the wake exists within the wake’s shear layer that is highly intermittent at scales related to the rotor diameter (cf. Schottler et al. 2018 and Neunaber 2019). This is shown in Fig. 14 where the shape parameter λ2 (τ ) = 0.25·ln(F(δuτ )/3)), which characterizes the shape of the probability density function of the velocity increments p(δuτ ), is plotted for a time scale τ that corresponds to the rotor diameter. Here, F denotes the flatness. λ2 approaches 0 for flows with Gaussian distributed fluctuations on a respective time scale. For a detailed explanation of λ2 , the reader is referred to the chapter on description methods for wind turbulence characteristic in Section 4 of this book and to Morales et al. (2011). A consequence of this result is that a

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Fig. 14 Contour plot of the shape parameter λ2 6D downstream in the wake of a wind turbine for a scale related to the rotor diameter. High levels of intermittency are found in a ring surrounding the wake. (Taken from Schottler et al. (2018) under permission of the Creative Commons Attribution 4.0 License Creative Commons Attribution 4.0 License 2009)

downstream turbine exposed to a half-wake experiences higher loads due to the high intermittency within the flow.

Turbulence in the Wake of a Yawed Turbine For an increase of the power generation and a decrease of loads, within wind farm control applications, methods to deflect the wake are of interest. Wake deflection can be achieved by yawing the turbine with regard to the inflow direction. The classical shape of the wake is then altered to a curled or kidney shape which is illustrated in Fig. 15 by means of the mean velocity (cf. Howl et al. 2016, Bastankhah and PortéAgel 2016, Schottler et al. 2018, Lin and Porté-Agel 2019, and Porté-Agel et al. 2019). The reason for the curled shape is that a counter-rotating vortex pair is shed from the upper and lower part of the rotor. In the framework of turbulence in wind turbine wakes, this is of interest as, depending on the yaw angle, the counter-rotating vortex pair alters the flow field and adds an asymmetry. The turbulent kinetic energy also plotted in Fig. 15 shows the highest values at the border of the wake, and its shape is similarly altered. The previously mentioned intermittency ring is also found for yawed turbines (see Fig. 15), but it, too, is altered to a curled shape rather than a ring (Schottler et al. 2018). This graphic also emphasizes that the intermittency ring surrounds the wake and its region with high TKE.

Comparison of Actuator Disk and Wind Turbine Wakes When investigating wind turbine wakes both by using simulations and wind tunnel experiments, the modeling of a turbine is complex and costly. The use of a static

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Fig. 15 Color-coded contour plots of the to the inflow velocity uref normalized mean velocity u (left), turbulent kinetic energy (middle), and shape parameter λ2 calculated for a time scale τ corresponding to the rotor diameter. The measurements were carried out in turbulence generated by a passive grid and the turbine was yawed by an angle γ = −30◦ . (Taken from Schottler et al. (2018) under permission of the Creative Commons Attribution 4.0 License)

rotor model, the so-called actuator disk, is a simple and established alternative that has been proven to provide good results regarding the modeling of the mean velocity profile in the far wake. However, for simple yet accurate calculations, it is of interest to what extent the turbulence evolution downstream of an actuator disk and a wind turbine are similar since it was shown before that the breakdown of the tip vortices adds structures to the wake and influences the turbulence mixing process in the near wake. In the following, results from different experimental studies that compare wind turbine and actuator disk wakes are summarized (cf. Aubrun et al. 2013, Camp and Cal 2016, Lignarolo et al. 2015, 2016, and Neunaber 2019). One question that arises when using an actuator disk is how to model it. In Aubrun et al. (2013), the disk is designed from a mesh, and the nacelle is modeled by using a different mesh as compared to the outer rotor area. The mesh was chosen to generate a similar velocity deficit as that of a model wind turbine. In Lignarolo et al. (2015, 2016), a combination of three mesh layers has been used with the aim of creating a disk with uniform blockage and a thrust coefficient that matches the turbine’s thrust coefficient. Camp and Cal (2016) and Neunaber (2019) follow a different approach, using coarser models with nonuniform, radially changing blockage, where a nacelle is modeled and the thrust coefficient matches the one of the turbine. From the different studies, it can be concluded that although the very near wake shows differences, within the intermediate and far wake • The mean velocity is modeled well by an actuator disk. • The wake expansion is similar downstream of both objects. • A similar turbulence intensity profile is found despite the different turbulence evolution and a different shear layer. • Energy spectra evolve similarly downstream of a turbine and an actuator disk even in the outer wake regions once the tip vortices broke down. This is also

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Fig. 16 Evolution of the pre-multiplied energy spectra f · E(f ) over f · D/u with increasing downstream distance at the blade tip position (Y = −0.51 D) in the wake of an actuator disk (left) and a turbine (right) exposed to inflow generated by a regular grid: The downstream position is color-coded, in blue is a decay of E(f ) ∝ f −5/3 indicated and the red dashed lines represent the spectra of the inflow at the rotor plane x/D = 0 (dark red) and its evolution without turbine at x/D = 4.69 (red). (Adapted from Neunaber 2019)

• • • •

indicated in Fig. 16 where the evolution of the energy spectral density with increasing distance from the model is plotted for the actuator disk (left) and the turbine (right), also with regard to the inflow. It can be seen that while the spectra evolve differently in the near wake, the spectra tend toward a universal spectrum in the far wake that follows a decay E(f ) ∝ f −5/3 . Reynolds stresses and the mean kinetic energy budget evolve similarly. There are strong signs that homogeneous isotropic turbulence can be found in the central wake of both objects independently on the inflow. Both objects reduce intermittency as compared to the inflow. A disk can also be used to investigate yaw effects (Howl et al. 2016).

Since experiments have both been conducted in laminar and different turbulent inflows, it is also possible to comment on the importance of the tip vortex evolution for the wake recovery. It was found that the turbulence mixing downstream of a turbine is not enhanced by the strong fluctuations within the near wake as compared to the actuator disk. In conclusion, it can be confirmed that even with regard to turbulence quantities, the wakes generated by different actuator disk and a wind turbine are despite the fundamentally different turbulence generation mechanisms astonishingly similar.

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Conclusion While the starting point of the description of wind turbine wakes (partially) originates from classical bluff body wake analysis, this chapter summarizes the progress that has been made over the past decades in the description and analysis of turbulence phenomena characteristic of a wind turbine exposed to atmospheric boundary layer inflow. For this, both the structures imprinted by the turbine onto the flow and the structures from the interaction of the wake with the ABL have been discussed. Close to the rotor, the structures imprinted by the turbine, above all the tip vortices, characterize the wake, but with the transition from near wake to far wake, they break down while simultaneously, the shear layer between ambient flow and wake expands. Energy is re-entrained to the wake and the wake recovers. The axisymmetry innate to the classical wake is broken in vertical direction due to the sheared inflow of the ABL, and the turbulence in the ABL accelerates the turbulence evolution processes and thus the wake recovery. Also, higher turbulence in the inflow leads to higher turbulence levels in the wake. Turbulence quantities and the energy entrainment have shown to be most distinctive at the top-tip level of the rotor in the ABL. In the far wake, the turbulence is fully developed and decays according to classical turbulence scalings. While the intermittency inside the wake is reduced compared to the inflow, a ring of high values of intermittency at scales related to the rotor diameter is found to surround the wake. Since intermittency on these scales was shown to increase fatigue loads, this is important for load calculations and wind farm layouts. Overall, an overview over turbulence structures in the wake of a wind turbine and the interaction with the atmospheric boundary layer was given. In the next section, wind farm aerodynamics will be discussed, focusing on the influence of rotor aerodynamics on the wind farm.

Cross-References  CFD-Type Wake Models  Doppler Lidar Inflow Measurements  Industrial Wake Models  Met Mast Measurements of Wind Turbine Wakes  Turbulent Inflow Models  Wake Meandering  Wake Measurements with Lidar  Wind Farm Cluster Wakes  Wind Shear and Wind Veer Effects on Wind Turbines

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Part V Wind Farm Aerodynamics

Wake Structures

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Features and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Near Wake Length and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . End Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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This chapter will concentrate on the near wake that also can be divided into very near and near wake. Over the past years, the length of the near wake has become a very important parameter with the appearance of increasingly larger wind farms. The focus of this chapter will be on the physics and possible ways of modeling or estimating the near wake length. The chapter is outlined with a background introducing the subject, a section with basic features and theorems introducing basic concepts, a section describing the wake structure followed by a description of influence of turbulence. The chapter will end with a “rough and ready model” as one way of estimating the length of the near wake followed by an end note with final comments and recommendations.

S. Ivanell () Section of Wind Energy, Department of Earth Sciences, Uppsala University, Campus Gotland, Visby, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_48

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Keywords

Wake · Wake Length · Tip spirals

Introduction The basic flow features in the wake of a wind turbine have been a topic of interest during the last decades, and even though intensive research has been performed in this area, basic understanding of the flow behind the turbine is still lacking. The interest in wake characteristics became more intense when turbines started to be placed in clusters, and therefore started to influence each other. Today, turbines are most commonly sited in clusters, that is, wind farms. The number of wind turbines in a wind farm is increasing, and thus, wind farms are growing. Entire farms are positioned in the near vicinity of each other which means that wakes from these farms need to be considered. When considering the farm wake, the interaction heavily depends on the large scales in the atmospheric boundary layer. This is, however, a topic far from the near wake region. In this chapter, we will focus on the near wake by emphasizing its physics and the possible ways of modeling or estimating its length. Figure 1 illustrates the complexity of the wake flow. Here one can identify how the vortex system, that is formed from circulation distributions on the blades, rolls up into a center vortex and distinct tip vortices. These tip vortices destabilize and break down into turbulence, where turbulent mixing finally results in what is referred to as the far wake. A distinct definition of the near and far wake was introduced by Vermeer et al. (2003), where the near wake was defined as the area just behind the rotor, where the properties of the rotor can be distinguished. The near wake is followed by the far wake where the focus is placed on the influence of wind turbines in a farm situation, and modeling of the actual rotor is less important. The number of blades, blade aerodynamics, and phenomena like stall, 3-D effects, and tip vortices therefore

Fig. 1 The figure illustrates an iso-surface of vorticity in the flow field from a large eddy numerical simulation

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influence the properties of the near wake. In the far wake, turbulence physics and wake interaction are the principal phenomena. Distinguishing between the near and far wake can be difficult in simulations since the near wake is the initial condition of the far wake. Typically, one uses a Gaussian velocity profile for the definition of the far wake. This is because the Gaussian velocity profile of the wake is a result of turbulence mixing after the tip vortex breakdown and when the wake recovery is in progress. One may also divide the near wake into a near wake and a very near wake. In the very near wake the tip spirals are still dominant while in the rest of the near wake these spirals are not dominant anymore but a Gaussian shaped velocity profile has not been reached yet. Wakes in general have been modeled by a large range of models, ranging from linear (e.g., the Jensen model (Jensen 1983)) to more complex modeling types. Göcmen et al. (2016) give an overview of wake models from The Technical University of Denmark (DTU). Sanderse makes a review of wake models, Sanderse et al. (2011), and Porté-Agel makes a review of farm flows in Porté-Agel et al. (2020). The stability of the tip spirals has been under investigation during the last decade, by for example, Ivanell et al. (2010) and Sarmast et al. (2014). A comparison between Large Eddy Simulation (LES) and experimental work has been conducted by, for example, Ivanell et al. (2015). For a more extensive review of experimental and modeling approaches for the near and far wake, the reader is referred to the classical work by Vermeer et al. (2003). There are many wake models based on analytical formulation that can be used independently or as a part of the post processing of flow field analyses. However, many of these models aim to model the far wake for estimation of losses due to neighboring turbines. These models do not aim to cover the detailed physics of the near wake where one can identify the origin of upstream turbines. Instead these models use a Gaussian velocity profile rather than the distinct tip and root vortices of the near wake. For additional far wake modeling studies the reader is referred to Crespo et al. (1998). To be able to simulate the distinct tip vortices existing in the near wake one has fewer options. The dominant approach, which will be addressed here, is numerical computational fluid dynamics (CFD). Alternate options, such as vortex methods and lifting line methods, are also possible but will not be addressed here. The numerical method used in relation with CFD will not be described in detail here as that would require a book on its own, the basic approaches linked to wake modeling will instead be briefly described. The most commonly used methodology in modeling the turbine wake system is to model the turbine with so-called actuator methods where the turbine itself is replaced by body forces given by tabulated data for the specific simulated turbine. This approach can be used either with the so-called actuator line or disc methodology. The method was introduced by Sørensen and Shen (2002). Using this

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method the physical properties of the simulated turbine are included but with the compromise of not resolving the blade geometry and its boundary layer. The total circulation distribution is well represented using the actuator line method while also saving significant amount of numerical effort in resolving the blade geometry which allows for a higher resolution of the wake. There is always a compromise between numerical effort and the possible number of cases that can be performed. The actuator line method is able to model the very near wake while the actuator disc model is not capable of modeling the tip spirals. This means the actuator disc model is capable of modeling the near wake but is unable to model the very near wake with the same degree of accuracy as the actuator line method. An alternative is to model the full rotor, that is, resolve the blade geometry and its boundary layer, see, for example, Schulz et al. (2017). Using this approach will, however, limit the possibilities to model clusters of turbines due to the high numerical cost.

Basic Features and Theorems A useful quantity in the context of induction theory is the circulation, . The circulation is defined as the vorticity integrated over an open surface bounded by S:   = v · dS (1) where v is the velocity along the curve S which encloses the vortex. The wake structure in the near wake is determined by the distribution of bound circulation,  = (r), along the blade. The bound circulation is determined according to the Joukowsky’s circulation theorem related to the distribution of lift:  = ρ  × Vrel L

(2)

 is the lift, Vrel is the relative velocity on the aerofoil section, and ρ is the where L density. The circulation can be computed from the lift coefficient:  = CL Vrel c

(3)

where CL is the lift coefficient and c is the chord length. Theoretically, the bound circulation on the blade is equal to the circulation behind the blade, that is, in the wake. For inviscid flows, the sum of the tip and root vortices should be zero, which is however not the case for viscous flows. The tip and root vortices do however, both for inviscid and viscous flows, have a different sense of rotation, that is, different signs of circulation, see Fig. 3. A steep decline of circulation toward the tip will lead to a rapid concentration of the vortex at the tip (occurring a few chords behind the tip). The sign of the circulation gradient, d dr , along the blade will also determine the sense of rotation of the vortex behind the

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Fig. 2 The figure illustrates a schematic picture of the circulation distribution at a radial position where 1 corresponds to the blade tip (R). The marked areas schematically illustrate where d dr is larger or smaller than 0. The dashed gray line illustrates a theoretical bad blade design

blade, see Fig. 2. This means that if there is more than one region with a negative gradient, there will be more vortices than the tip and the root vortex, see the dashed gray line in Fig. 2. This dashed gray line is of course of theoretical nature as such a design would be impractical, but possible. The vortex is formed during a short time period behind the blade. In inviscid theory the circulation for the tip and root vortices should both be equal to the maximum circulation along the blade. In reality, the viscosity affects the vortices differently depending on the flow field. Therefore, it is not possible to make any conclusions about the dependence of the tip and root vortices other than that when neglecting viscosity, one would expect the sum of the circulation of the tip and root vortices to be approximately zero. The tip vortices are shed into the wake in a continuous fashion and appear to emanate from a certain radius which is slightly smaller than the radius of the turbine blade tips. The vortices also concentrate to a root vortex. Therefore it is of great interest to simulate swirl behavior and to identify where the concentrations toward the outer and inner vortices takes place. The structure of the wake and the tip and root vortices has been studied in different investigations as earlier discussed in section “Introduction”. Generally, the root vortex structure has not been as relevant as the tip vortex structure since it is destroyed earlier due to its interaction with the induction from the hub. The three root vortices (or two for two-bladed turbines) are also much closer to each other when compared to the three tip vortices and the spiral structure is therefore destroyed much earlier than the tip spiral structures. The stability of the tip vortex thus sets the basic behavior of the wake. The basic mechanisms behind this instability have been quantified, the length of the wake can therefore be related to turbulence intensity and the frequency of wind fluctuations (Sarmast et al. 2014).

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r

z

Fig. 3 Since the blade tip follows a circular orbit it leaves a trailing vortex with a helical structure. The trailing tip vortex moves downstream

The wake length has become a very important parameter during the last years with the advent of increasingly larger wind farms. When a number of turbines are positioned in close proximity there will be wake interaction, at least for some wind directions. Wake interaction affects the power output and fatigue load, and therefore the lifetime of the turbines. Wake interaction also increases in off-shore wind farms where the turbulence intensity in the atmosphere is lower and the wakes are thus longer since there is less turbulence to mix the flow behind the turbines. Wake interaction could therefore be seen as less important when considering more complex or rough terrain sites. A site inside a forest, for example, could perhaps result in reduced wake interaction, but nevertheless one has to consider effects from a very complicated boundary layer. Today, we are far from understanding all features of such a complicated flow situation but are making progress in the field (Ivanell et al. 2018). It is, however, well known that turbulence generated from shear, buoyancy, or the turbulence generated from upstream turbines all results in a decrease of the wake length (Fig. 3).

Wake Structures Figure 4 shows the wake structure and the surrounding flow. The pressure and velocity are depicted on two perpendicular planes. Here, the tip and root vortices’ structures are identified by an iso-surface of vorticity. The figure is generated from LES modeling of one well-resolved single turbine in a polar mesh. (LES (Large Eddy Simulations) is one of a number of CFD approaches used in wind energy modeling which, however, will not be described in detail here.) The thrust coefficient is set to 0.8, the tip speed ratio to 7, wind speed 10 m/s and it is operating in optimum conditions according to Betz’s limit. The horizontal plane (x = 0) illustrates the velocity and here one may identify that the axial induction factor is 1/3 which

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Fig. 4 y = 0-plane: pressure distribution, x = 0-plane: streamwise velocity, red: path of tip and root vortices as iso-surface of constant vorticity

corresponds the Betz optimum. The vertical plane (y = 0) illustrates the pressure increase before the rotor and its corresponding decrease in the wake area. Figure 5 shows curves corresponding to Fig. 4 where the azimuth distribution of axial and azimuth velocity is depicted on the top and the axial and azimuth induction factors are depicted on the bottom. The four curves illustrate the wake distribution extracted at a rotor position (RP) and at three downstream positions 1–3 rotor radii downstream the rotor.

Influence of Turbulence The wake system is highly dependent on interactions with the atmospheric boundary layer. This becomes clear already in the very near wake, and since the spiral system largely dictates wake properties, this has a significant impact. Figure 6 illustrates this with an example of a 3 × 3 turbine farm layout. All figures show the result at an inflow angle of 30◦ with an atmospheric boundary

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〈 Uθ 〉

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Fig. 5 Velocity components (top) and induction factors (bottom) as functions of radius at different axial positions

layer profile with a power law exponent of 0.15. Figures (a) − (b) illustrate a case with 1% turbulence intensity. Figures (c)–(d) illustrate a case with 6% turbulence intensity. The left column, (a)–(c), shows the results of the 3D field. The wake structure is identified by an iso-surface of the vorticity. What appears to be a ground surface is the same iso-surface level illustrating the wake structures. The pressure is identified by color contours at the iso-surface of the vorticity. The right column, (b)–(d), shows the corresponding 2D plane with the color contours illustrating the velocity at hub height. One can clearly identify here the impact on the wake length as the turbulence increases. In the (a)–(b) cases significantly more dominant wake structures reach the downstream turbines. In the (c)–(d) cases the turbulence will affect the wake structure and decrease the length of the wakes. This is here identified by the shorter blue areas representing the level of deceleration compared with the undisturbed flow. To fully assess farm flows one need to consider ambient turbulence, atmospheric stability, roughness, shear, veer, expanding internal boundary layers to fully assess the farm flow physics.

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Fig. 6 Results of simulations using 9 turbines with 4 rotor diameters of separation

Near Wake Length and Stability Two questions always come up when talking about wakes. The first being – how long is the wake? The second – how does the wake impact downstream turbines? The answers to these questions are the goal of all the methods mentioned earlier. The industry uses a wide range of methodologies in combination with experience to conduct these complex estimations. Many studies have been conducted for the validation of different methods. All these studies will not be described or addressed here; however, a “rough-and-ready” equation which addresses the first question for near wakes will be supplied and discussed. The equation originates from a number of studies starting from a more theoretical understanding of the wake’s spiral system by Ivanell et al. (2010) and Sarmast et al. (2014). As always, these types of simplified models include constants that need to be set. The full derivation of equation 4 can be found in Sørensen et al. (2015). Remember that this equation estimates the near wake length. It does not directly indicate anything about any potential losses of downstream turbines. If the potential

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losses are of primary concern, other methods considering the velocity deficit in the downstream wake might then be more suitable. However, this equation gives an understanding of the near wake length and how it depends on the operation conditions of specific turbines in combination with turbulence levels based on a fundamental understanding of the turbine wake physics. The rough and ready equation is as follows:      1 16u˜ 3c ln(C1 Ti ) + C2 ln(Ti ) =− R nwl Nb λCT

(4)

where R is the rotor radius, and u˜ c is the convective propagation velocity of the wake system. Nb , λ, CT are turbine characteristics corresponding to number of blades, tip speed ratio, and thrust coefficient. Ti corresponds to the turbulence intensity. C1 and C2 are constants set to C1 = 0.3 and C2 = 5.5 according to Sørensen et al. (2015). The definition of the near wake length (nwl) resulting in the used coefficients is based on a 99% fit of an analytical Gaussian profile to the mean velocity distribution in the wake. Note that this 99% fit of an analytical Gaussian profile might be set too close to a theoretical, fully developed Gaussian velocity profile for industrial applications. However, this gives a simple and ready to use model for the wake length given these conditions. Figure 7 illustrates the near wake length dependence on the turbulence intensity based on equation 4. The vertical axis represents the wake length in number of radii,

Fig. 7 Results from equation 4 compared to data from simulation

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the horizontal axis represents the turbulence intensity. The example here illustrates a case with λ = 7.07, CT = 0.78, and u˜ c = 0.78. As expected, the wake length is highly dependent on the turbulence level. In Fig. 7, the near wake (modeled) curve illustrates the result of equation 4. The near wake (sim) diamonds show calibration data obtained in Sørensen et al. (2015). The very near wake (sim) dots illustrate the very near wake length based on results from Sarmast et al. (2014). The results show that the very near wake is in the order of 15% of the length of the near wake, using the 99% criterion. A standard approach would, as an alternative to the approach using equation 4, instead typically use an assumption of a shorter wake structure. Here, based on fundamental theories, the difference in length from the very near wake to the fully developed Gaussian shaped profiles stretches approximately from about 1–2 diameters to 10 diameters. Depending on which approach to use of the far wake, one will result in different assumptions of the wake length.

End Note This chapter has described wake physics with examples from very detailed modeling approaches to farm modeling as well as noting the influence of turbulence. The focus in this chapter has been on the near or even very near wake. One should remember that atmospheric conditions including shear, veer, turbulence, diurnal cycles, and so on have a significant impact on the wake length. The ready-to-use equation presented is based on a theoretical physical understanding and gives a solid foundation for determining wake length; however, the near wake length set to 99% of a Gaussian velocity profile needs to be balanced with other experiences when evaluating the influence of the wake in industrial applications. Optionally, other constants can be chosen to fulfill different applications.

Cross-References  Aerodynamics of Wake Steering  CFD for Wind Turbine Simulations  CFD-Type Wake Models  Turbulence of Wakes

References Crespo A, Hernandez J, Frandsen S (1998) Survey of modelling methods for wind turbine wakes and wind farms. J Wind Energy 2:1–24 Göcmen T et al (2016) Wind turbine wake models developed at the technical university of Denmark: a review. Renew Sustain Energy Rev 60:752–769

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Ivanell S et al (2010) Stability analysis of the tip vortices of a wind turbine. Wind Energy 12(8):705–715 Ivanell S et al (2015) Comparison between experiments and Large-Eddy Simulations of tip spiral structure and geometry. J Phys Conf Ser 625:012018 Ivanell S et al (2018) Micro-scale model comparison (benchmark) at the moderately complex forested site Ryningsnäs. Wind Energy Sci 3(2):929–946 Jensen NO (1983) A note on wind generator interaction. Technical report, Risoe-M-2411(EN). Risoe National Laboratory, Roskilde Porté-Agel F, Bastankhah M, Shamsoddin S (2020) Wind-turbine and wind-farm flows: a review. Bound-Layer Meteorol 174:1–59 Sanderse B, van der Pijl SP, Koren B (2011) Review of CFD for wind-turbine wake aerodynamics. Phys Rev Lett, Wind Energy 14(7):799–819. https://doi.org/10.1002/we.458 Sarmast S et al (2014) Mutual inductance instability of the tip vortices behind a wind turbine. J Fluid Mech 755:705–731 Schulz C et al (2017) CFD study on the impact of yawed inflow on loads, power and near wake of a generic wind turbine. Wind Energy 20:253–268 Sørensen JN, Shen WZ (2002) Numerical modeling of wind turbine wakes. J Fluid Eng 124:393 Sørensen JN et al (2015) Simulation of wind turbine wakes using the actuator line technique. Philos Trans R Soc A Math Phys Eng Sci 373(2035):20140071 Vermeer LJ, Sorensen JN, Crespo A (2003) Wind turbine wake aerodynamics. Prog Aerosp Sci 39:467–510. Division L

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytic Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top-Hat Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double-Gaussian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation System Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD Lookup-Table Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearized RANS Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eddy-Viscosity Wake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Farm Modelling and Wake Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Equivalent Flow Quantities and Partial Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Farm Calculation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter deals with the description of wind turbine wakes by means of reduced-complexity flow models. These models offer the appeal to conduct a vast number of simulations of the wake flow for different atmospheric boundary conditions in short time, thus, they are usually the models of choice in the wind energy industry for assessing and optimizing long-term energy production. The intent of the chapter is to provide an overview over a subset of the most prominent wake models, their physical approximations, and the resulting equations that the

J. Schmidt () · L. Vollmer Aerodynamics, CFD and Stochastic Dynamics, Fraunhofer IWES, Oldenburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_49

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models use to describe the flow. At the end, two algorithms are presented to derive a converged wind farm flow by superimposing the wake deficits derived from the presented single wake models. This offers the readers the opportunity to implement a wind farm flow model with their wake model of choice themselves. Keywords

Wake models · Wake TI models · Simplified flow modelling · Numeric implementation · Wind farm algorithm

Introduction From the modelling perspective, the wake effect can be interpreted as the difference between the flow field in the presence of a wind turbine and the flow field under identical ambient conditions without any disturbances induced by the rotor. In other words, the wake-corrected flow equals the background flow plus the wake effect. The first step for wind farm modelling is therefore the calculation of the aerodynamic background conditions, possibly including complex three-dimensional effects which stem from the local topography, forest canopy, thermal stratification and other site-specific details. Often, however, this complexity is vastly reduced, and accuracy is treated for efficiency and the benefit of high computational speed. In fact, in many applications, homogeneous wind and turbulence fields are applied, or simple vertical profiles are used. Note that this may or may not be a good assumption, depending on the case at hand. In general, the impact of the background flow is often decisive for the accuracy of the final answer of a wind farm modelling code, and its impact on the results should not be taken lightly. Wakes have an impact on all components of the flow description: the wind speed and the wind direction, pressure, temperature and turbulence. In principle, all these quantities are interdependent and connected via the Navier-Stokes equations. However, an often defining feature of modelling is the focus on a specific subset of the relevant physics, yielding a significant reduction of the complexity. This can be a single conservational law, an imposed symmetry, the isolation of a single component of the wind vector, the neglection of interactions, the simplification of turbulence, or other assumptions. As a consequence, wake models often describe the change of a single undisturbed quantity behind the wind turbine rotor and do not provide the complete aerodynamic answer to the full-fledged physics problem, as raised by nature. For example, often the wind speed deficit and the increase of turbulence intensity (TI) are obtained from two different independent models, even though physical principles connect the two. Another example for the reduction of the physical degrees of freedom is the restriction to a single wind vector component in many models, potentially violating the principle of local mass conservation.

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These are examples for the mental flexibility that is crucial when doing modelling work. It is important to realize that the purpose of models is to give fast results for a selected set of aerodynamic quantities, at the cost of consistency – and often consequently also of accuracy. The modeller’s point of view is that, in the end, validation justifies the means. And it has been shown in numerous studies that wake models can be very reliable when applied to the cases for which they have been designed and optimized. Hence, decades after the invention of the first and simplest wake models, they and their successors are still applied very frequently as core parts of many commercial software tools and in-house codes, for industrial and research projects, all over the world. In fact, sometimes even high-fidelity approaches like CFD are not guaranteed to give better results, depending on the setup and the modelling effort (cf. Moriarty et al. (2014)). In this chapter, we describe the basic principles of wind and turbulence intensity wake models and briefly sketch a few popular examples. The focus is on describing a consistent approach for modelling the wind field inside a wind farm without considering any restriction by boundary conditions on the flow. For a full representation of the wind conditions in a particular wind farm, these boundary conditions, e.g. manifested by constraints in vertical or horizontal momentum transfer, need to be carefully considered in the wake model input parameters or as a modification of the background flow. For recent more extensive reviews on wind farm flows and wake models, see, e.g. Gögmen et al. (2016) and Porte-Agel et al. (2020). All figures in this chapter were created by using the Farm Layout Program in Python (flappy) by Fraunhofer IWES (Schmidt et al. 2021; Centurelli et al. 2021; von Brandis et al. 2021).

Analytic Wake Models Analytic wake models describe the wind deficit or the wake-added turbulence intensity at any point behind the wake-causing rotor without invoking differential equations. Instead, the result is directly obtained by explicit functions of the turbine parameters and the coordinates at the evaluation points. The evaluation of such models is extremely fast, hence their continuous popularity in industrial and also scientific communities since almost 40 years. The derivation of analytic wake models in general starts from basic laws of physics, like the conservation of mass and momentum, combined with simplifying assumptions, like an assumed wake extension behaviour or a Gaussian wake profile shape. Due to the nature of the derivation of those models, they are also commonly referred to as kinematic wake models. Often the modelling involves the introduction of constants which are a priori unknown and then determined by fits to measurements or to high-fidelity wake simulations.

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Top-Hat Models Top-hat models assume a known relation between the width of the wake and the distance behind the rotor. In addition, the wind deficit vector inside the wake region is modelled by a single-component vector that is parallel to the ambient wind (which is assumed to be homogeneous), and the magnitude of this component is only a function of the distance. In other words, there is no dependency on the radial distance from the rotor centre within the wake, and the deficit drops to zero discontinuously at the border of the wake region. The resulting wake deficit field is therefore axially symmetric and has a top-hat-shaped form in every slice parallel to the rotor. Also the wake-added TI is often modelled by top-hat models. In this case, it is the increase of TI within the wake region that is described by a function of the distance to the rotor, i.e. independently of the radial coordinate.

The Jensen Deficit Model The most prominent example of this model class is the Jensen model. Without doubt, this can be called a classic wake model; maybe it is the classic wake model among them all. It is based on the works by Jensen (1983) and Katic (1987) from the 1980s and has been used in wind farm calculations and model studies ever since. The linear expansion of the wake radius is described by a model constant k, Dw (x) = D + 2kx ,

(1)

where Dw denotes the wake diameter at distance x from the rotor and D the rotor diameter. Typical considered values for k range between 0.04 and 0.075 for offshore and onshore applications, respectively. Once k is fixed, the shape of the wake region is determined. The requirement of mass conservation together with induction formulated in terms of the thrust coefficient ct then leads to the result u∞ − u(x) = u∞



D Dw (x)

2 (1 −



1 − ct ) ,

(2)

where u∞ is the undisturbed wind speed and u(x) the wind speed at distance x behind the rotor within the wake region. The Jensen model, or derived versions of the model, is frequently used, for example, in the framework of commercial software (e.g. WindPRO, WindSim, WindFarmer and OpenWind). The typical flow field result is visualized in Fig. 1.

The Frandsen Deficit Model The Frandsen model (Frandsen et al. 2006) is based on a control volume analysis for which the wake diameter Dw at position x behind the rotor is given by 

Dw (x) = β

k/2

x 1/k + 2α D D

√ 1 + 1 − ct β= . √ 2 1 − ct

(3)

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3

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Fig. 2 The Frandsen wind deficit model (Frandsen et al. 2006)

Here k = 2 is the shape parameter and α the expansion constant which should be calibrated. A starting point could be α ∼ 0.5, roughly O(10) times what might be chosen for the Jensen model wake expansion parameter (Andersen et al. 2014). The wake velocity then follows as  2  u∞ − u(x) D 1 1 1−2 = ± ct , u∞ 2 2 Dw (x)

(4)

where the positive sign before the√square root term applies to cases with a ≤ 0.5 for the induction factor a = 1 − 1 − ct , and the negative sign otherwise. A flow impression of the Frandsen model is shown in Fig. 2.

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3 0.6 2 0.5 1

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Fig. 3 The IEC 61400-1 TI wake model

The Frandsen (IEC) Turbulence Intensity Model The historic reference for modelling the wake-added TI is the Frandsen model as described in the IEC 61400-1 norm (IEC-61400-1 2019). In this model, the wake diameter follows a simple cone with opening angle 21.6◦ (cf. Fig. 3). The wakeadded TI does not explicitly depend on the wind speed in this model: I+ (x) =

1 √ . 1.5 + 0.8x/(D ct )

(5)

The Crespo-Hernandez Turbulence Intensity Model Another simple model for the increased TI in the wake region was proposed by Crespo and Hernandez in 1996 (Crespo and Hernández 1996). The model parameters were obtained by fitting to findings from CFD simulations, resulting in  I+ (x) =



√ 0.326 1 − 1 − ct for x < 3D 0.83  √ x −0.32 1− 1−ct −0.0325 I∞ for x ≥ 3D 0.73 2 D

(6)

where I+ (x) denotes the TI value at distance x behind the rotor that is to be added 2 +I 2 (x). Note quadratically to the undisturbed background value I0 , i.e. I 2 (x) = I∞ + that this deviates slightly from the original paper (Crespo and Hernández 1996). Instead, we are following (Ishihara and Qian 2018) (Eq. (46) therein), since this implies a smooth transition between near and far wake (cf. Fig. 4). The original reference (Crespo and Hernández 1996) states limits of validity of the model, x < 15D, 0.07 < I∞ < 0.14 and 0.1 < a < 0.4 for the induction factor a, but it is not uncommon to ignore those in modelling. The shape of the wake region on the other hand is not discussed in the original paper. Reference Niayifar and Porté-Agel (2015) suggested a top-hat shape with a wake diameter

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3 0.18 2 0.16 0.14

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0.08

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Fig. 4 The Crespo-Hernandez TI wake model (Crespo and Hernández 1996)

that coincides with four times the standard deviation of the Gaussian wake model from Bastankhah and Portè-Agel (2016) (cf. section “The BP Deficit Models”) and a specific parameter fit,    Dw (x) = 4 k  x + 0.2 βD ,

k  = 0.3837I∞ + 0.003678,

(7)

with β from Eq. (3). The corresponding TI wake is visualized in Fig. 4.

Gaussian Models The BP Deficit Models Two Gaussian wind deficit wake models with increasing popularity in recent years were proposed by Bastankhah and Porté-Agel. We denote them by BP 2014 (Bastankhah and Portè-Agel 2014) and BP 2016 (Bastankhah and Portè-Agel 2016) in the following. Using momentum and mass conservation and assuming a Gaussian shape of the wind deficit, 2 u∞ − u(x) − r = C(x)e 2σ 2 (x) , u∞

the authors find for the BP 2014 model the centreline deficit:  ct . C(x) = 1 − 1 − 8(σ (x)/D)2

(8)

(9)

The width of the wake is then modelled as a linear function of the distance to the rotor,

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10

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0.8

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y [D]

0.6 4

0.4

2

0 0.2 −2 0

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4

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8 x [D]

10

12

14

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Fig. 5 The BP 2016 wind deficit model (Bastankhah and Portè-Agel 2016) in yawed and unyawed situations

σ (x) = k  x + ε,

 ε = 0.25 βD,

(10)

with β from Eq. (3). The model parameter k  describes the wake growth and has to be calibrated. For the BP 2016 model, the authors modify their model by the introduction of near and far wake regions. Additionally, they introduce sensitivity to yawed situation and formulate a concept for modelling wake deflection. A flow impression of the model is shown in Fig. 5.

The Ishihara Deficit Model Another Gaussian wake model for the wind speed deficit with parameterized dependency on turbulence intensity has been proposed by Ishihara and Qian (2018). The model parameters were obtained by fits to LES simulations that were carried out for varying mean TI. For the detailed form, the reader is referred to the original paper (Table 2 therein gives a summary). The resulting wind deficit field is shown in Fig. 6.

Double-Gaussian Models The Double-Gaussian Deficit Model A double-Gaussian structure for the wind velocity deficit in the wake was proposed by Keane et al. in 2016 (Keane et al. 2016). Further improvements of this model

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were recently published by Schreiber et al. (2020). The latter model evaluates mass conservation at the end of a stream tube in the near-wake region, whose length x0 is an input parameter to the model, resulting in a ct -dependent initial wake expansion  at that point. Strictly speaking, the model only predicts data at locations further downstream than x0 . From that point onwards, the width σ of the two Gaussian functions grows linearly with the distance, as it is the case in many models. Here this is governed by the model parameter k  : σ (x) = k  (x − x0 ) + .

(11)

The third model parameter is the radial spacing of the origins of the two Gaussians kr . The model does not provide an explicit dependency on turbulence intensity, and therefore the case-specific calibration of parameters is recommended. Figure 7 shows the wind velocity deficit for the choice x0 = 4.55D, k  = 0.011 and kr = 0.535D.

The Ishihara Turbulence Intensity Model As discussed in Section “The Ishihara Deficit Model”, the Ishihara wake model (Ishihara and Qian 2018) describes the wind deficit by a Gaussian wake model. Additionally, it provides a wake model for turbulence intensity, as visualized in Fig. 8. The wake-added TI is determined by the addition of two Gaussian functions for each rotor, shifted by half the rotor radius from the centre. As stated earlier, the model parameterizes the effects of TI and ct variations on the wake effect as found in LES simulations.

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Equation System Wake Models The complete description of all flow phenomena behind a wind turbine (and elsewhere) is in general assumed to be given by the Navier-Stokes equations. However, for almost any application, be it an engineering problem or a research question (with few exceptions), it is unavoidable to simplify the latter. The reason is that turbulence covers a vast range of scales, all the way down to Kolmogorov microscales, and for practical applications those are unreachable by simulations. The solution of simplified Navier-Stokes equations in three-dimensional grids, where small-scale turbulence production and dissipation effects are calculated by models, is the subject of the field of computational fluid dynamics (CFD). Either

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results from large eddy simulations (LES) or Reynolds-averaged Navier-Stokes (RANS) equations can be used as the basis for industrial wake models, as we shall see below. However, the corresponding systems of differential equations are still rather complex. For example, RANS simulations solve three equations for the three wind velocity components plus one for pressure and often another two for turbulence model fields. This adds up to solving six coupled partial differential equations in a three-dimensional mesh, for many inflow conditions. Hence, further reductions of the complexity are a common goal in order to reduce the model calculation time. Two approaches will be briefly discussed below, the idea of linearization and the reduction to axial symmetry.

CFD Lookup-Table Wake Models Steady-state or time-averaged results from LES and RANS simulations of single rotors can be used in wind farm modelling by means of a lookup-table approach (Schmidt and Stoevesandt 2014, 2015). This can only be sketched very briefly in this section; for the details of CFD modelling, we refer the reader to other chapters in this book. In the context of steady-state RANS simulations, the single wind turbine rotor is often modelled by actuator discs (Réthoré et al. 2014). These identify cells inside the mesh that represent the rotor (cf. Fig. 9) and distribute the thrust force accordingly. This introduces local drag terms into the momentum equation, which are determined

Fig. 9 Mesh refinement near an actuator disc for RANS simulations

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by the ct value. It is common practice to include mesh refinement in the vicinity of the disc in order to increase the accuracy of the wake simulation. Both uniform and radially varying force distributions (for a recent analysis, see, e.g. Pirrung et al. 2020) are applied for wake simulations in RANS. Example results for the uniform case are shown in Figs. 10 and 11 for wind speed and turbulence intensity, respectively. Note that the latter is derived from the turbulence model fields, in this case from the turbulent kinetic energy k and the turbulent dissipation ε. The results naturally depend on the turbulence model choice, and the standard k-ε RANS turbulence model has been improved over the years for the purpose of increasing the accuracy of wake modelling by actuator discs (Kasmi and Masson 2008; Laan et al. 2015; Laan and Andersen 2018).

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As an alternative to simulating single turbine wakes by RANS-CFD, actuator disc or actuator line LES (cf. Witha et al. 2014; Churchfield et al. 2017) and subsequent time averaging is an option. Of course, this requires a lot more computational power and is therefore rarely used for the generation of lookup-type wake models. The process of creating an industrial wake model from steady-state or timeaveraged CFD simulations can be sketched as follows: • Create a mesh with good rotor and wake resolution. • Pick a suitable inflow wind speed. Vary the thrust coefficient ct of the disc and the inflow turbulence intensity I over the desired parameter range of the wake model. The TI variation can be either realized by changing the roughness length at the ground or by reducing the ground completely and inducing uniform TI. For each (ct , I ) combination, run CFD simulations until convergence. • Interpolate the results to a coarser, and for convenience, Cartesian spatial grid. This grid should not cover more space than necessary for the wake modelling. Rescale the spatial distances by the rotor diameter and translate the results to dimensionless wake deficit quantities. • Store all results as a table in ct and I , such that an interpolation to intermediate values is possible. Note that also spatial interpolation with respect to wake frame coordinates will be necessary. The speed of running such CFD lookup wake models then crucially depends on the performance of the interpolation method at all the parameters and points of interest. Note that the uniform actuator disc model in the above formulation has no turbine-specific features. However, discs with radial force distributions might reflect specific properties of the blades. Similarly, models that are derived from timeaveraged actuator line LES results intrinsically depend on blade section lift and drag coefficients as well as the blade geometry.

Linearized RANS Models A linearization of the RANS equations for the purpose of wake and wind farm modelling has been proposed by Ott et al. in 2011 (Ott et al. 2011), known under the name Fuga (see also Gögmen et al. 2016). It is derived by assuming that the forcing due to the presence of actuator discs can be treated as a small perturbation of the momentum equations. Consequently, a Taylor expansion is carried out, and all flow quantities are written as the sum of contributions of different orders with respect to the perturbation parameter. Together with a simple eddy-viscosity-type closure, the zeroth-order equation has no force contribution and is easy to solve. All other orders reduce to linear equations and can be solved subsequently. Fuga is not so much a single turbine wake model but a linearized CFD model that has to be solved in a discretized computational domain which contains the full wind farm.

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Eddy-Viscosity Wake Models The reduction of the above-mentioned six equations of RANS simulations to only two for modelling a single turbine’s wake was proposed by Ainslie in 1988 (Ainslie 1988). The key ingredient was the introduction of rotational symmetry, reducing the computational domain to the axial x and the radial r coordinates and the wind velocity to the axial and the radial components u(x, r) and v(x, r), respectively. Furthermore, Ainslie restricted the validity of his model equations to the far wake, such that pressure effects could be ignored. He ended up with two equations, the momentum equation for u and the continuity equation:     ∂u ∂u 1 ∂ ru v u +v =− , ∂x ∂r r ∂r ∂u 1 ∂ (rv) + = 0. ∂x r ∂r

(12) (13)

Here u v  denotes the relevant component of the Reynolds stress tensor. Ainslie realized turbulence closure by assuming u v  = −ε(x)

∂u(x, r) , ∂r

(14)

where ε(x) is the so-called eddy viscosity. He modelled the latter as the sum of a wake-induced part, which is assumed to be proportional to the wake deficit at the centre line at any distance x behind the rotor, and a constant part, which represents the contribution by the ambient turbulence intensity. Notice that Ainslie’s differential equations do not explicitly depend on turbine properties like ct or D. The dependency on turbine data is instead realized by imposing a specific near-wake solution for the field u(x, r) up to a certain distance x = x0 , which then defines a Dirichlet-type boundary condition for the momentum equation. Different proposals for the near-wake and the eddy viscosity formulation have been studied, e.g. Lange et al. (2003); Larsen et al. (2008); Beck et al. (2014), and for one of them the resulting axial wind component field is shown in Fig. 12. As for the CFD lookup-table models, the Ainslie model also provides results for the radial wind component. Furthermore, a parametrization-dependent derivation of a TI wake field has been proposed (cf. Lange et al. (2003)). An example result is shown in Fig. 13.

Wind Farm Modelling and Wake Interactions The main application of industrial wake models is the fast evaluation of wake losses and other wake-related quantities for a wind farm. In this section, we will discuss

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the commonly applied steps from single turbine wake models to the net results at the turbines of the wind farm.

Rotor Equivalent Flow Quantities and Partial Wakes All wake models that were described in this chapter depend on the turbine’s thrust coefficient ct . This quantity is usually provided by the manufacturer as a function of wind speed in terms of the turbine’s thrust curve, possibly additionally depending on air density or other quantities. Likewise, the produced electrical power P can be

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Point weight

deduced from the corresponding power curve, and those two inputs are in general crucial for any wake model and wind farm energy yield calculation. Consequently, the first question that arises is how the rotor equivalent wind speed (REWS) uREWS should be determined, which is then used during the calculation for looking up the values ct (uREWS ) and P (uREWS ). Considering rotor diameters of more than 100 m, the wind speed gradients over the rotor area are not always negligible. Also partial wakes, i.e. situations in which a part of the rotor is exposed to wake effects more severely than the rest, are common phenomena and need to be reflected in the modelling. The simplest approach for calculating the REWS is the direct evaluation of the wind speed at the rotor centre point, as sketched in Fig. 14 (left). Alternatively, more than one point can be used in order to probe a broader region of the rotor area (cf. the middle and right panels of Fig. 14). Notice that the statistical weights of the probe points are either identical, if each point represents a rotor area element of the same size, or varying. For grid-type models, for example, not all outer points represent a square area element that is fully covered by the rotor disc area, and therefore such points have reduced weight. Obviously, higher-point rotor models have a better capability to capture arbitrary spatial variations of the local background wind field. Alternatively, wind shear and wind veer effects can be tabulated and given to the rotor as input data for the evaluation of the gross REWS. In the context of wind farm wake flow, however, the calculation of the net REWS is required, i.e. wake effects need to be included. Figure 15 demonstrates the importance of the rotor model for the REWS result at a turbine that is fully or partially hit by the wake of another turbine. Clearly, the simple centre point model does not give satisfactory results. However, rotor models with a large number of evaluation points may slow down the overall calculation, depending on the implementation. Concerning the partial wake effects from Fig. 15, this can be avoided by taking into account the shape of the wake. Figure 16 sketches this idea for top-hat wake models (left) and Gaussian wake models (right). For top-hat models, the wake radius is well defined at any downstream distance x behind the wake-causing rotor. Hence, the intersection of the wake section and the rotor disc area is calculable, and the wake effect can then

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Fig. 15 Comparison of the REWS as seen by a turbine at downstream distance x = 6D behind the wake-causing turbine of the same hub height for different rotor models, as a function of the orthogonal horizontal coordinate y. Results for the Jensen (left) and the BP 2014 wake model (right) are shown. Here ‘partial’ denotes the wake overlap calculation that is sketched in Fig. 16

Fig. 16 Left: Intersection of a top-hat wake (red) and a rotor disc (blue). Right: Four intersection areas for the Gaussian wake model overlap integration

be weighted accordingly. This approach is well established for top-hat models and reproduces well the results of higher point rotors (cf. Fig. 15, left, line “partial”). The generalization to Gaussian (and double-Gaussian) wake models has been developed and applied recently to different scenarios (Schmidt et al. 2021; Centurelli et al. 2021; von Brandis et al. 2021). It is based on the idea of making use of the wellknown radial behaviour of the wakes when integrating over the rotor disc, after evaluating the magnitude of the wake effects only once at the centre line. Also this method reproduces well the results of higher rotor models (cf. Fig. 15, right, line “partial”), at lower computational costs. Note that similar arguments can be applied to the calculation of the rotor equivalent turbulence intensity (RETI) IRETI at the turbines. This quantity can be an input to some of the wake models. Often it is also required for the evaluation of tabulated turbine loads and other quantities.

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Wake Superposition In many situations, a turbine in a wind farm is subject to multiple wakes from many upstream wind turbines. Therefore, in the context of single-turbine wake modelling, a calculation method that can predict the overall wake effects needs to be specified. In the following, we first discuss the superposition of multiple wake effects for the wind velocity. Due to the vector nature of the latter, but also due to the fact that the dimensionless results from wind deficit wake models require rescaling, a few technical complications arise in this context, and we will have a closer look at them. Afterwards, we briefly address superposition models for turbulence intensity.

Homogeneous Background Flow In many applications of industrial wake models, the background wind conditions are assumed to be described by unidirectional flow without spatial variation, and this setup is combined with single-component wind deficit wake models. The background wind field can then be expressed by a uniform horizontal wind velocity vector with two components, u0 = (u0 , v0 ) in the global frame of reference, and the orientation of this vector is defined by the uniform wind direction. The background turbulence intensity is specified by a single scalar value I0 in this scenario. We begin by looking at the most upstream turbine of a wind farm, subject to the above homogeneous background conditions. Let ω0 = |u0 | denote the background wind speed in the following. For uniform flow, all rotor models then agree on the (0) (0) REWS and RETI results uREWS = ω0 and IRETI = I0 , respectively. Here, the superscript (0) indicates the index of considered turbine. Next, the wake model is evaluated in the wake frame of reference. This frame is defined by aligning the first axis with the background wind velocity vector u0 (and the other axes follow by the right-hand rule). Indicating the wake frame by the subscript ‘wf’, the background wind velocity vector reads u0 = (ω0 , 0)wf in the wake frame. Therefore, when evaluating the above-presented wake models, we can identify u∞ = ω0 for the most upstream turbine. The result of a single-component wake model for turbine 0 is then given by the dimensionless wind speed deficit δ (0) that is related to the wake-corrected wind velocity u(0) = (u(0) , 0)wf by

δ

(0)

u∞ − u(0) (x) (x) = u∞

= wf

ω0 − ω(0) (x) . ω0

(15)

Here ω(0) (x) = |u(0) (x)| is the wake-corrected wind speed behind turbine 0 at the point of interest x. The above relation gives   ω(0) (x) = ω0 1 − δ (0) (x) ,

(16)

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and the full wind velocity vector u(0) (x) behind turbine 0 in the global frame of reference can be deduced by combining the above wind speed with the uniform wind direction of the homogeneous background flow. Now let’s consider a second turbine (with index 1) somewhere in the downstream region of turbine 0, i.e. located at a position where the wake effects originating from turbine 0 are not zero. How can we calculate the wind speed at a point of interest x behind turbine 1, which possibly may also be located further downstream but still inside the wake region of turbine 0? The first step is the evaluation of the rotor model and the thrust curve for turbine 1 in the corrected wind field u(0) (x), yielding (1) (1) (1) uREWS , ct and IRETI . Secondly, the evaluation of the single-component wake model for turbine 1 results in the dimensionless wind deficit δ (1) (x), and we obtain relations similar to Eqs. (15) and (16):

δ (1) (x) =

  ω(1) (x) = u∞ 1 − δ (1) (x) .

u∞ − ω(1) (x) , u∞

(17)

Step number three is a decision: Which value do we identify as u∞ when interpreting the wake model of turbine 1? Three different choices arise, and all have been applied in the literature: (a) The background wind speed, u∞ = ω0 (b) The REWS at the wake-causing turbine u∞ = u(1) REWS , (c) The wind field that is corrected by upstream wake effects, u∞ = ω(0) (x) Approach (a) can be related to the idea that the uniform background wind speed sets the scale of the problem, and the whole calculation can be carried out in dimensionless form. The original works on the topic of wake superposition from Lissaman in 1979 (Lissaman 1979) and Katic in 1986 (Katic et al. 1987) were formulated based on approach (a), with different superposition rules: Linear (Lissaman): Quadratic (Katic):

  ω(1) (x) = ω0 1 − δ (0) (x) − δ (1) (x) ,    ω(1) (x) = ω0 1 − δ (0) (x)2 + δ (1) (x)2 .

(18) (19)

The quadratic superposition is also called root square sum (RSS) model. For turbine n in a set of N turbines, ordered according to their downstream positions, the above generalizes to

Linear (Lissaman):

ω

(n)

(x) = ω0 1 −

n  i=0

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δ (x) ,

(20)

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 ⎞  n   ω(n) (x) = ω0 ⎝1 −  δ (i) (x)2 ⎠ . ⎛

Quadratic (Katic):

(21)

i=0

The linear superposition is justified by suggesting momentum conservation, while the quadratic superposition is motivated by energy conservation, but strictly speaking this cannot be proven. The linear approach is flawed in the sense that it can lead to negative wind speeds for large wind farms. Due to the stronger suppression in the quadratic addition, this problem does not occur in the RSS model. (i) The idea behind approach (b), u∞ = uREWS when evaluating the wake deficit (i) δ (x) of turbine i, is that the local wind speed at the turbine can be interpreted as the undisturbed background wind speed during the wake calculation (see, e.g. Bastankhah et al. 2021 for a recent discussion). If turbine i itself is in a wake situation, then this will be reflected in its own wake. The linear and quadratic models become Linear:

ω(n) (x) = ω0 −

n 

(i) u(i) REWS δ (x),

(22)

i=0

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  n  2  (i) uREWS δ (i) (x) . ω(n) (x) = ω0 − 

(23)

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The argument behind approach (c) is that the background wind field for the evaluation of turbine 1 is given by the result from Eq. (16) for the wind field behind turbine 0. By replacing u0 → u(0) and, consequently, ω0 → ω(0) (x), this yields   ω(1) (x) = ω(0) (x) 1 − δ (1) (x) . For turbine n in the above set of N downstream ordered turbines, one obtains the closed expression (cf. Lanzilao and Meyers 2020) Product:

ω(n) (x) = ω0

n  

 1 − δ (i) (x) .

(24)

i=0

Note that this product includes terms with up to n factors of the wake deficit. Furthermore, it is based on the local rescaling of the dimensionless wind deficit by the local wind speed at the point of interest x, rather than by the REWS at (i) the turbine’s location. The thrust coefficient ct that enters the wake calculation of turbine i, however, is still based on the turbine’s REWS. All above approaches are rather simple models for a complex physics problem; the interaction of wind turbine wake effects. Notice that also other models than the ones presented here have been proposed (for recent model comparisons, see,

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e.g. Machefaux et al. 2015; Vogel and Willden 2020; Bastankhah et al. 2021). A part of the problem is also inherent in most wake models, since they often assume a homogeneous and unidirectional background wind field. However, in any multiple wake situation, the local background wind field has gradients in streamwise and (depending on the model) radial directions. The product superposition model explicitly violates the assumption of a uniform background wind field, while the linear and quadratic models ignore this spatial variation altogether.

Heterogeneous Background Flow In general flow situations, the background wind field is not homogeneous and not unidirectional, especially on larger scales like wind farm clusters. Furthermore, wake models may in general provide more than one wind deficit component, as, for example, the CFD lookup models or the Ainslie wake model from Sections “CFD Lookup-Table Wake Models” and “Eddy-Viscosity Wake Models”, respectively. We therefore have to address the vector field nature of the wind velocity and the wake deficit fields. Similar to the unidirectional case, the wake frame of reference is defined by the location of the origin in the rotor centre and alignment of the first coordinate axis with the background wind velocity field. The remaining two directions follow by application of the right-hand rule. In other words, the wake frame by definition follows a streamline that goes through the rotor centre point. Hence, the background wind vector in the wake frame (subscript ‘wf’) at any point behind the turbine has only a single component, ⎛

⎞ ω0 (x) u0 (x) = ⎝ 0 ⎠ , 0 wf

(25)

with background wind speed ω0 (x) = |u0 (x)|. Introducing a slightly generalized notation, we can now express Eq. (15) for turbine 0 to the vector notation, ⎛

u∞ − u(0) (x) δ (0) (x) = , u∞

u∞

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(26)

The approaches (a), (b) and (c) from the discussion of the unidirectional case can (0) now be adopted to the multidirectional situation, for example, by u∞ = uREWS . Notice that in general the REWS does not coincide with the background wind speed at the rotor centre in inhomogeneous wind fields, even for the upstream turbine 0. For single-component wake models, the vector nature of the wind deficit disappears. In that case, only the first component along the streamline direction is nonzero, i.e. δ (i) (x) = δ (i) (x)eflow (x), where eflow (x) denotes the unit vector that is aligned with the background wind velocity. The wake superposition models (18)– (24) can then be translated straightforwardly to the multidirectional case in the wake

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frame of reference. The only relevant additional effort is the execution of the back transformation to the global coordinate system. In the case of multicomponent wake models, it might still be the best option to drop the two additional components, since they hardly have any impact on further wind farm calculations. In that case, we are back in the convenient single-component situation from above. If the (y, z)-components in the wake frame are of explicit interest, then only the linear superposition models have simple generalizations, e.g. Eq. (22): Linear:

u(n) (x) = u0 (x) −

n 

(i) u(i) REWS δ (x).

(27)

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A visual example of curved wake frames in a heterogeneous background flow is shown in Fig. 17. Wakes that follow the convective flow have gained recent attention in the context of offshore wind farm modelling and cluster wakes (Lanzilao and Meyers 2020; von Brandis et al. 2021).

Superposition of Wake-Added Turbulence Intensity Turbulence intensity is defined as the standard deviation of the wind speed divided by the mean wind speed, both with respect to the same time averaging interval. In

Fig. 17 Illustration of the wake frame bending in heterogeneous background flow, for the BP 2016 wake model (Bastankhah and Portè-Agel 2016). In this case, the background wind velocity was interpolated from the boundary points of an irregular quadrangle

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the context of wake modelling, the latter is often assumed to be at least a period of 10 min. In wake situations, two sources of turbulence contribute to the total result, namely, the ambient turbulence intensity that is associated with the background flow I0 (x) and the contribution due to wake effects Iwake (x). It is generally assumed that these contributions add quadratically, following Frandsen et al. (2001) and Frandsen (2007), I (x) =



2 (x). I02 (x) + Iwake

(28)

The wake-added turbulence intensity for a set of N downstream ordered turbines (i) can then be calculated from the individual wake model results I+ (x) as follows: 2 (x) Iwake

Quadratic:

=

N 

I+(i) (x)2 .

(29)

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Note that similar to the wind deficit models above, also the turbulence intensity wake models are evaluated in the wake frame of reference. Due to the scalar nature of turbulence intensity, this only has an impact on the explicit coordinates of the evaluation point x, and all further complications related to rescaling the wake model results do not apply. In some applications, the total wake-added contribution is limited to the dominant wake, i.e. to the wake of the upstream turbine with the largest wake effect: Maximum:

  (i) Iwake (x) = maxi I+ (x) .

(30)

All wake models, i.e. wind deficit as well as turbulence intensity wake models, are sensitive to turbulence intensity since this quantity determines how quickly the disturbed flow recovers (in some models this is hidden in the parameters). In principle, this is an integrated rather than a local process, and the turbulence over the complete wake region determines the mixing that leads to wake decay. However, in the context of industrial wake modelling, this complexity cannot be addressed, and either the background or the wake-corrected turbulence intensity has to be chosen as a representative value.

Wind Farm Calculation Algorithm Above, we have discussed the individual ingredients that are needed for running wake model calculations: • The background flow field • The calculation of REWS, RETI and ct values

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• The wake models for wind deficits and added TI • The wake superposition models for wind velocity and TI In the remainder of this chapter, we shall now discuss how to bring these ingredients together. Two different algorithms for wind farm wake calculations will be presented, one based on turbine ordering and the other on iterative wake calculations. Both follow a grid-less paradigm, i.e. both are based on the model computations at arbitrary evaluation points. Other implementations are possible, and the following should be only understood as illustrative examples.

Downstream Turbine Evaluation In homogeneous background wind fields, the calculation of the downstream order of wind turbines is straightforward. If eflow denotes the uniform unit vector that is aligned with the flow direction, then the order simply follows from evaluating the projection of the rotor centre coordinate of each turbine in the global frame of reference onto eflow . For each turbine, this results in a scalar value, and the downstream order simply follows from ordering that respective list. For general heterogeneous background wind fields, the downstream order can be obtained by evaluating the matrix of wake effects. For N turbines, the latter has dimension N × N, where the (i, j )th entry contains the magnitude of the wake effect of turbine i onto turbine j . For the determination of the downstream order, it is generally sufficient to calculate this matrix once, based on background wind conditions only. Assuming that the turbine indices of a wind farm with N turbines follow the downstream order, such that turbine 0 is guaranteed to experience free stream conditions and each turbine is only affected by wakes from turbines with smaller indices, the wind farm calculation algorithm is sketched in Algorithm 1. Note that the calculation of the rotor equivalent quantities in this notation is based on a certain set of evaluation points for each rotor (cf. section “Rotor Equivalent Flow Quantities and Partial Wakes”). The mentioned calculation of other quantities of interest could, for example, refer to the evaluation of the power curve of the (j,i) (j,i) turbine. The quantities δpoints , I+ points denote entries of the wake matrices from i i wake-causing turbine j onto the evaluation points of turbine i for wind velocity and turbulence intensity, respectively. The sketched algorithm has complexity of O(N 2 ), where N denotes the number of wind turbines. The background wind and TI fields are evaluated in an initial step for the most upstream turbine with index 0. After that, the target turbines with index 1 or larger are evaluated. Firstly, in an inner loop over all upstream turbines, all required wake matrix elements are computed. Secondly, the wake effects are combined into the wake-corrected flow results at the target points. Finally, based on the latter, all relevant quantities of the target turbine are calculated. This includes

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Algorithm 1 Downstream turbine evaluation Require: Background wind field u0 Require: Background TI field I0 Require: Turbines in downstream order points0 = evaluation points for turbine 0 (0) (0) (0) calculate uREWS , IRETI , ct from u0 , I0 for the wake models calculate other derived quantities of interest for turbine 0 for i: 1 < i < N do pointsi = evaluation points for turbine i for j : j < i do (j,i) δpoints = wind deficit due to wake j at pointsi (j,i)

i

I+ points = TI contribution due to wake j at pointsi i end for (j,i) upointsi = wind deficit superposition using u0 and δpoints for all j : j < i i

(j,i)

Ipointsi = TI wake superposition using I0 and I+ points for all j : j < i i

(i) (i) calculate u(i) REWS , IRETI , ct from upointsi , Ipointsi for the wake models calculate other derived quantities of interest for turbine i end for

all data that are needed for the subsequent wake evaluations originating from the turbine in question.

Iterative Wake Calculation The iterative approach does not require any specific ordering of the turbines. Instead, it treats all of them equally (cf. Algorithm 2). First, starting from background flow conditions at all turbines, all wake matrix elements are computed. Then, in an iterative loop, the wake effects are applied, and all turbine quantities are updated accordingly, for example, the REWS and consequently the thrust coefficients of turbines in wake situations. The iteration proceeds until convergence is reached, for example, when none of the REWS and the RETI values have changed for any of the turbines. For a row of N turbines that is aligned with the background wind direction, this algorithm requires at worst N iterations, since each implies converged results at least at one additional turbine. This suggests that the algorithm has the order O(N 3 ). Note that the performance can be improved by avoiding the double calculation of wake matrix elements for already converged turbines. In the best case, i.e. for accidental downstream ordering or wind from the orthogonal direction, the complexity reduces to O(N 2 ). The benefit of the iterative approach lies in its ability to handle implicit dependencies of the involved quantities on one another. For example, control strategies might be included that influence ct depending on the current wind farm state, for example, depending on the overall produced power or other information that involves downstream turbines.

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Algorithm 2 Iterative wake calculation Require: Background wind field u0 Require: Background TI field I0 for i: i < N do pointsi = evaluation points for turbine i upointsi = u0 evaluated at pointsi Ipointsi = I0 evaluated at pointsi (i) (i) calculate u(i) REWS , IRETI , ct from upointsi , Ipointsi for the wake models calculate other derived quantities of interest for turbine i end for repeat for i: i < N do for j : j = i do pointsj = evaluation points for turbine j (i,j ) δpoints = wind deficit due to wake i at pointsj j

(i,j )

I+ points = TI contribution due to wake i at pointsj j end for end for for i: i < N do pointsi = evaluation points for turbine i (k,i) upointsi = wind deficit superposition using u0 and δpoints for all k: k = i i

Ipointsi = TI wake superposition using I0 and I+(k,i) points for all k: k = i (i)

(i)

(i)

i

calculate uREWS , IRETI , ct from upointsi , Ipointsi for the wake models calculate other derived quantities of interest for turbine i end for until convergence

Naturally, the benefits of the downstream order and the iterative algorithms can be combined. Note, however, that introducing an iterative loop around Algorithm 1 would increase its order of complexity to O(N 3 ) as well in worst-case scenarios. Nevertheless, the latter would be a consequence of the mentioned implicit dependencies only and can therefore be expected to appear less frequently than in the pure iterative approach. If available, it is therefore always recommendable to take advantage of the downstream order when evaluating wake effects.

Cross-References  Aerodynamics of Wake Steering  CFD-Type Wake Models  Optimizing Wind Farm Layouts  Turbulence of Wakes  Wake Meandering  Wake Measurements with Lidar  Wake Structures  Wind Farm Cluster Wakes

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References Ainslie JF (1988) Calculating the flowfield in the wake of wind turbines. J Wind Eng Indus Aerodyn 27(1):213–224 Andersen SJ, Sørensen JN, Ivanell S, Mikkelsen RF (2014) 524:012161 Bastankhah M, Portè-Agel F (2014) A new analytical model for wind-turbine wakes. Renew Energy 70:116–123 Bastankhah M, Portè-Agel F (2016) Experimental and theoretical study of wind turbine wakes in yawed conditions. J Fluid Mech 806:506–541 Bastankhah M, Welch BL, Martínez-Tossas LA, King J, Fleming P (2021) Analytical solution for the cumulative wake of wind turbines in wind farms. J Fluid Mech 911:A53 Beck H, Trabucchi D, Bitter M, Kühn M (2014) The Ainslie wake model – An update for multi megawatt turbines based on state-of-the-art wake scanning techniques. EWEA, Barcelona, Spain Centurelli G, Vollmer L, Schmidt J, Dörenkämper M, Schröder M, Lukassen L, Peinke J (2021) Evaluating Global Blockage engineering parametrizations with LES, Wake Conference (in review) Churchfield MJ, Schreck SJ, Martinez LA, Meneveau C, Spalart PR (2017) An advanced actuator line method for wind energy applications and beyond. In: 35th wind energy symposium Crespo A, Hernández J (1996) Turbulence characteristics in wind-turbine wakes. J Wind Eng Ind Aerodyn 61(1):71–85 Frandsen S (2007) Turbulence and turbulence-generated structural loading in wind turbine clusters Ph.D. Thesis (DTU, Risø National Laboratory) Frandsen S, Gravesen H, Jørgensen L, Eriksson C, Halling KM, Skjærbæk P, Jørgensen U, Werner NE, Lemming J, Bjerregaard E (2001) Recommendation for technical approval of offshore wind turbines (Danish Energy Agency) Frandsen S, Barthelmie R, Pryor S, Rathmann O, Larsen S, Højstrup J, Thøgersen M (2006) Analytical modelling of wind speed deficit in large offshore wind farms. Wind Energy 9(1– 2):39–53 Gögmen T, van der Laan P, Réthoré P-E, Pena Diaz A, Larsen GC, Ott S (2016) Wind turbine wake models developed at the Technical University of Denmark: a review. Renew Sustain Energy Rev 60:752–769 IEC-61400-1 (2019) Wind energy generation systems – Part 1: Design requirements Ishihara T, Qian G-W (2018) A new Gaussian-based analytical wake model for wind turbines considering ambient turbulence intensities and thrust coefficient effects. J Wind Eng Ind Aerodyn 177:275–292 Jensen NO (1983) A note on wind generator interaction (2411) El Kasmi A, Masson C (2008) An extended k-ε model for turbulent flow through horizontal-axis wind turbines. J Wind Eng Ind Aerodyn 96(1):103–122 Katic I, Højstrup J, Jensen NO (1987) A simple model for cluster efficiency. In: Palz W, Sesto E (eds) EWEC’86. Proceedings, vol 1, pp 407–410 (A. Raguzzi 1987) Keane A, Olmos Aguirre PE, Ferchland H, Clive P, Gallacher D (2016) An analytical model for a full wind turbine wake. J Phys Conf Ser 753:032039 van der Laan MP, Andersen SJ (2018) The turbulence scales of a wind turbine wake: A revisit of extended k-epsilon models. J Phys Conf Ser 1037:072001 Paul van der Laan M, Sørensen NN, Réthoré P-E, Mann J, Kelly MC, Troldborg N, Gerard Schepers J, Machefaux E (2015) An improved k-μ model applied to a wind turbine wake in atmospheric turbulence. Wind Energy 18(5):889–907 Lange B, Waldl H-P, Guerrero AG, Heinemann D, Barthelmi RJ (2003) Modelling of offshore wind turbine wakes with the wind farm program FLaP. Wind Energy 6(1): 87–104 Lanzilao L, Meyers J (2020) A new wake-merging method for wind-farm power prediction in presence of heterogeneous background velocity fields

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Larsen GC, Madsen Aagaard H, Larsen TJ, Troldborg N (2008) Wake modeling and simulation, Denmark. Forskningscenter Risoe. Risoe-R (Danmarks Tekniske Universitet, Risø Nationallaboratoriet for Bæredygtig Energi) Lissaman PBS (1979) Energy effectiveness of arbitrary arrays of wind turbines. J Energy 3(6): 323–328 Machefaux E, Larsen GC, Murcia Leon JP (2015) Engineering models for merging wakes in wind farm optimization applications. J Phys Conf Ser 625:012037 Moriarty P, Rodrigo J, Gancarski P, Chuchfield M, Naughton J, Hansen KS, Machefaux E, Maguire A, Castellani F, Terzi L, Breton SP, Ueda Y (2014) IEA-task 31 WAKEBENCH: Towards a protocol for wind farm flow model evaluation. Part 2: Wind farm wake models. J Phys Conf Ser 524:06 Niayifar A, Porté-Agel F (2015) A new analytical model for wind farm power prediction. J Phys Conf Ser 625:012039 Ott S, Berg J, Nielsen M (2011) Linearised CFD models for wakes, Denmark. Forskningscenter Risoe. Risoe-R (Danmarks Tekniske Universitet, Risø Nationallaboratoriet for Bæredygtig Energi Pirrung GR, van der Laan MP, Ramos-Garcìa N, Meyer Forsting AR (2020) A simple improvement of a tip loss model for actuator disc simulations. Wind Energy 23(4):1154–1163 Porte-Agel F, Bastankhah M, Shamsoddin S (2020) Wind-turbine and wind-farm flows : a review. Boundary-layer Meteorol 174:1–59 Réthoré P-E, van der Laan P, Troldborg N, Zahle F, Sørensen NN (2014) Verification and validation of an actuator disc model. Wind Energy 17(6):919–937 Schmidt J, Stoevesandt B (2014)) Wind farm layout optimisation using wakes from computational fluid dynamics simulations. In: EWEA conference proceedings Schmidt J, Stoevesandt B (2015) Wind farm layout optimisation in complex terrain with CFD wakes. In: EWEA conference proceedings Schmidt J, Requate N, Vollmer L (2021) Wind farm yield and lifetime optimization by smart steering of wakes, Wake Conference (in review) Schreiber J, Balbaa A, Bottasso CL (2020) Brief communication: a double-Gaussian wake model. Wind Energy Sci 5(1):237–244 Vogel CR, Willden RHJ (2020) Investigation of wind turbine wake superposition models using Reynolds-averaged Navier-Stokes simulations. Wind Energy 23(3):593–607 von Brandis A, Centurelli G, Dörenkämper M, Schmidt J, Vollmer L (2021) Investigation of mesoscale wind direction changes and their consideration in engineering models, in preparation Witha B, Steinfeld G, Heinemann D (2014) High-resolution offshore wake simulations with the LES Model PALM. In: Hölling M, Peinke J, Ivanell S (eds) Wind energy – impact of turbulence, pp. 175–181. Springer, Berlin/Heidelberg

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Physics Behind Wake Meandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Meander Modeling: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Medium-Fidelity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Meander Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example DWM Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The present chapter deals with wake meandering – its physics, its modeling, and its consequences for production and loading of wind turbines erected in wind farms. Wake meandering is the phenomenon describing the dynamics of wind turbine wakes. Nowadays there is almost unanimous agreement in the wind energy community that wake meandering is caused by large turbulent eddies in the atmospheric boundary layer. In the introductory part of this chapter, an accounting of the development leading to this conclusion will be given. This includes both full-scale experiments using advanced lidar technology, scaled wind tunnel experiments using both boundary layer wind tunnels and conventional wind tunnels, and last, but not least, detailed unsteady computational fluid dynamics large eddy simulations with wind turbines modeled as actuator lines. G. C. Larsen () DTU Wind Energy, DTU, Risø Campus, Roskilde, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2022 B. Stoevesandt et al. (eds.), Handbook of Wind Energy Aerodynamics, https://doi.org/10.1007/978-3-030-31307-4_50

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Recognizing the fundamental physics behind the wake meandering phenomenon, both high-fidelity and medium-fidelity modeling approaches are described. Being related to large-scale turbulence structures in the atmospheric boundary layer, impact from atmospheric boundary layer stability should be expected, and this important aspect is therefore also included in the modeling part. The chapter is concluded with various example applications ranging from wind turbine production prediction over wind turbine load prediction to optimal wind farm layout, for which both accurate production and load prediction are needed. Keywords

Wake meandering · Wake dynamics · Wake meander physics · Non-stationary flow fields · Wake meander modeling · Dynamic wake meandering model · Single wake model · Wake superposition · Large-scale ABL turbulence · ABL stability · Wind farm loading · Wind farm layout optimization · Full-scale lidar measurements · Wind tunnel measurements · Turbulence spectral tensor · Taylor advection

Introduction Cost of energy (COE) from wind farms is probably the most important single factor in deployment of wind in the energy system, although the functionality – and consequently the economic importance – of grid services is becoming increasingly more significant concurrently with the expansion of wind power, because more and more fluctuating power sources are supplying power grids. Since the development of wind energy on a large scale is most efficiently achieved by collecting wind turbines (WTs) in wind farms, there is a direct link from COE to wind turbine wakes; to wind farm wind fields; and to optimal layout and control of wind farms. Wind farm performance is intimately connected with wind farm production as well as loading of the individual WTs within the wind farm. The understanding and modeling of these wind farm design aspects are thus of major importance and therefore remain to be a hot topic in the wind energy research community. A WT wake is characterized by a downstream flow regime, in which the velocity is reduced and the turbulence is increased compared to the ambient inflow conditions, as can be observed from full-scale measurements using advanced lidar recording technology (Hansen and Larsen 2011; Herges et al. 2018). The reduction in wind speed is traditionally considered an organized flow structure denoted the wake deficit and is due to the extraction of kinetic energy by the WT. The increase in downstream turbulent energy relates both to wake-generated turbulence and to the lateral and vertical wake dynamics – the so-called wake meandering – of the wake deficit. Characteristics of two-dimensional instantaneous wake deficit and wake

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Fig. 1 Snapshots of instantaneous wake deficits (left) and instantaneous wake turbulence profiles (right) scanned at four different times. (Reproduced from Herges et al. 2018)

turbulence cross-sectional profiles are illustrated in Fig. 1, showing lidar scans of the flow field 3 rotor diameters downstream a Vestas V27 WT with a hub height of 32.1 m and a rotor diameter of 27 m. “Instantaneous” relates, in the present context, to a lidar cross-sectional scan frequency of 0.5 Hz. As a result of wake meandering, both wake deficit and small-scale wake generated turbulence structures are seen collectively displaced – in this case varying from approximately 0.5 rotor diameters to the right of the upstream rotor location to approximately 2 rotor diameters to the left of the upstream rotor location. A movie put together by such instantaneously recorded cross-sectional deficit and turbulence profiles reveals that these largely share a synchronous dynamic behavior. Similar results were obtained from a full-scale experiment performed at the DTU Risø campus, in which the same synchronous dynamics of the wake deficit and the wake generated turbulence were resolved behind a Tellus 95 kW WT using a horizontally downstream scanning lidar mounted on the hub of the wake generating WT (Larsen et al. 2010).

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The wake self-generated turbulence is mechanically generated turbulence caused by the wake shear, and with additional contributions from the blade trailed vorticity, consisting mainly of tip and root vortices. The tip vortices will initially take the form of organized coherent flow structures but later, due to instability phenomenon, gradually break down and approach the characteristics of conventional turbulence – although with modified turbulence characteristics compared to conventional atmospheric turbulence. Contrary to conventional atmospheric boundary layer (ABL) turbulence, wake generated turbulence is typically inhomogeneous and approximately isotropic over the wake regime and with length scales considerable less than those observed for atmospheric turbulence (i.e., typically of the order of the rotor length scale) (Hansen and Larsen 2011; Larsen et al. 2008b, 2012). With wake turbulence scales comparable to the scale of the organized wake deficit flow structure, the closely correlated dynamics of, respectively, wake deficit and wake generated turbulence is intuitively clear. Wake meandering appears as an intermittent phenomenon, where the flow at downstream positions may be undisturbed for part of the time but at other times interrupted by episodes of intense turbulence and reduced mean velocity, as the wake hits the observation point. The focus of the present chapter is on the wake meandering phenomenon.

The Physics Behind Wake Meandering The first to notice and investigate the meandering aspect of WT wakes was probably Baker and Walker (1984), who performed full-scale experiments to characterize the flow field behind a 2.5 MW WT and compared these measurements with predictions from simple wake models. They identified a downstream fluctuating movement of the wake deficit in the lateral direction – which they denoted wake meandering – and linked this phenomenon to wind direction variability quantified in terms of recorded wind direction standard deviation. They furthermore noted a relationship between ABL stability and the downstream wake flapping, with ABL stability quantified in terms of wind direction standard deviation. About the same time, Taylor et al. (1985) identified the same flow phenomenon as based on analysis of full-scale measurements behind the Danish Nibe 630 kW turbines. Ainslie (1988) was perhaps the first to incorporate the wake meandering phenomenon in a model context. Based on a thin shear-layer Navier-Stokes approach, he modeled the instantaneous wake deficit (i.e., excluding the “long-time-scale meandering effect”), which subsequently was extended to include wake meandering in a statistical sense by adding an empirical correction factor depending on the wind direction standard deviation. Although recognizing wake meandering as a dynamic phenomenon, the abovementioned pioneering contributions have a focus on steady wake performance, thus taking a statistical approach to estimate the steady wake equivalent of the observed dynamic phenomenon, which in turn potentially can be used to improve power production predictability of downstream-located WTs.

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In the following years, the research focus gradually changed to additionally include structural loading of WTs exposed to waked flows (Vermeer et al. 2003), whereby the non-stationary wind field effects – inextricably linked with wake dynamics – attracted more attention. Medici and Alfredsson (2006) conducted a series of wind tunnel experiments with scaled WT models. They observed periodically fluctuating helical movements of the wake, which, contrary to the wind direction variability hypothesis introduced in the pioneering works (Baker and Walker 1984; Taylor et al. 1985), made them conjecture that wake meandering was caused by a wake instability phenomenon similar to traditional vortex shedding behind a bluff body. This conjecture received quite some attention and led to a passionate debate on the fundamental mechanism driving the wake meandering within the wind energy community. However, as mentioned by Larsen et al. (2008a), there exist a number of notable differences between the experimental conditions obtained in the referred wind tunnel environment – being based on high-loaded rotors – and the conditions realized for a full-scale turbine located in the atmospheric boundary layer, where rhythmic vortex shedding has never been observed/reported. This position was later supported by both numerical studies (Lu and Porte-Agel 2011) and dedicated scaled studies in wind tunnels (España et al. 2011, 2012). Based on advanced computational fluid dynamics (CFD) large eddy simulations (LESs) of a stable boundary layer, using an actuator line representation of the WTs, Lu et al. concluded that for the modeled scenario, the observed wind turbine wake meandering is driven by large-scale turbulence. España et al. conducted a comprehensive series of wake experiments using two different types of wind tunnels – an ABL wind tunnel and a conventional wind tunnel with homogeneous and isotropic turbulence (HIT). In both cases, the WTs were modeled as static porous disks, with the solidity of the porous disk selected to reproduce a realistic velocity deficit and thrust coefficient compared to a full-scale WT rotor. The crucial difference between these two series of wind tunnel experiments is that the turbulence length scale of the ABL wind tunnel flow exceeded considerably the diameter of the rotor disk (i.e., approximately by a factor of 10), whereas the HIT flow was characterized by turbulence length scales between 3 and 10 times smaller than the disk diameter. Because random “flapping” (i.e., meandering) of the wake was observed only in the ABL wind tunnel flow, España et al. concluded that wake meandering is caused by large-scale turbulence structures. The ABL wind tunnel experiments also showed that the instantaneous wake width remains almost constant downstream the rotor disk, and that broadening of the mean wake therefore predominantly is due to the meandering stochastic process. Bingöl et al. (2010) and Trujillo et al. (2011) reached the same conclusion based on a series of full-scale experiments representing a variety of ambient meteorology conditions. For the first time, they applied a lidar-based remote sensing technology to characterize wake-affected flow fields. The lidar-recorded wake dynamics downstream a three-bladed Tellus 95 kW turbine was studied in detail and subsequently correlated with measurements of large-scale inflow turbulence

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Fig. 2 Lateral full-scale wake dynamics recorded three rotor diameters downwind of a turbine. Velocity contours obtained from lidar measurements in the field are shown in grayscale. The red line represents the temporal variation of the wake center predicted by a wake passive tracer approach

from a nearby upstream located reference mast, which enabled them to conclude that the wake deficit is laterally advected passively by the larger-than-rotor-size turbulent eddies in the inflow. Figure 2 below shows lateral full-scale temporal wake dynamics recorded three rotor diameters downwind this turbine for two different inflow turbulence intensities – 15.7% and 9.0%, respectively. Velocity contours obtained from lidar measurements in the wake-affected flow field are shown in grayscale, with “dark” flow regimes representing low wind speeds and “light” flow regimes representing high wind speeds. The fluctuating narrow dark flow regime across the 10-min period identifies the wake position, and the red line represents the temporal variation of the wake center predicted by a wake passive tracer approach. Consistent with the passive tracer conjecture, it is clear that the character of the inflow ABL turbulence has a significant effect on wake meandering. In conclusion, it is fair to state that today there is almost unanimous agreement in the wind energy community that wake meandering is predominantly caused by large turbulent eddies in the ABL, and this realization is perfectly in line with the basic conjecture put forward with the introduction of the dynamic wake meandering (DWM) model in 2008 (Larsen et al. 2008a). This model is now included in the IEC code as a recommended practice for computation of WT loads in wind farms and will be described in more detail in section “Medium-Fidelity Approach”.

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Fig. 3 ABL scales ranging from airfoil boundary layer thickness to weather system macro scales. (Reproduced from Porté-Agel et al. 2020)

Wake Meander Modeling: General Considerations Modeling of the atmospheric boundary layer is a complicated task for many reasons, among which the most important ones are the huge spectrum of turbulence flow length scales represented ranging from the Kolmogorov scale to big macro-scale weather systems with sizes of continents (Fig. 3); seasonal and diurnal variabilities; possible complicated terrain orography; plant canopies; and buoyancy effects related to temperature gradients. The modeling task is further complicated with introduction of wind farms in the ABL, where the individual WTs are interacting mutually through wakes and where the wind farm potentially interacts with the regional wind climate in a complicated two-way process coupling. Even with today’s largest supercomputers, a complete modeling of a wind farm flow field, including all relevant scales ranging from the boundary layer of the WT blade aerodynamic profiles to characteristic mesoscales of the regional wind climate, is impossible. This calls for simplifications. A first, often taken, simplification is to de-couple the local wind farm flow field from the regional wind climate by assuming a one-way interaction and thus introducing the regional wind climate in the modeling in terms of stationary boundary conditions – e.g., mean wind speed; mean wind direction; vertical temperature profiles; and air densities with only spatial dependencies. Whereas stationary meso-scale boundary conditions may suffice for prediction of wind farm performance, the non-stationary characteristics of the local wind farm

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wind climate are essential for prediction of wind farm WT fatigue loading. This is contrary to prediction of average wind farm production, where stationary flow field modeling often suffices. As the focus in this chapter is on wake dynamics, the models described in the following will be restricted to models accounting for the ABL non-stationarity. Even with the simplifications described above, the modeling task is still quite complex, and further simplifications are in general needed for practical applications as well as for scientific applications. The model description in the following is organized in one section (i.e., section “CFD-Based Approach”) dealing with a highfidelity/high-effort approach based on CFD, where the Navier-Stokes equations – or some simplified version of those – are solved numerically; and another section (i.e., section “Medium-Fidelity Approach”) with focus on a “medium-fidelity” approach dedicated design and optimization of wind farm performance in practice. The latter approach will be based on the DWM model (Larsen et al. 2008a).

CFD-Based Approach Given stationary boundary conditions, the ultimate CFD approach is a direct numerical simulation (DNS) of the Navier-Stokes equations. However, even with the described simplifications, this is not possible with today’s computer power due to the wide range of spatial scales/temporal involved. A popular further simplification is CFD LES, where the basic principle is to resolve the larger scales by direct solution of the Navier-Stokes equations and supplement this with models accounting for the sub-grid-scale turbulence regime. This type of wind farm flow simulation is typically organized in a sequence consisting of two steps: (1) modeling of the non-stationary ABL and (2) embedding a wind farm in the simulated flow field. Two different approaches are usually applied to simulate the ABL turbulence. The first approach is to take advantage of a consistent synthetic turbulence model as, e.g., the Mann spectral tensor (Mann 1994), and introduce such a field in the LES simulation by means of embedded time-varying inflow boundary conditions based on body forces immersed in an upstream fringe region (Troldborg et al. 2014). The same type of technique is used to establish the requested vertical and/or horizontal wind shear profile. A drawback with this approach is that equilibrium state of the flow is not assured, with the consequence that the intended turbulence and mean shear flow properties might change/evolve during the LES simulation (Troldborg et al. 2014). The advantage is considerable savings in time and computer power. The second approach – denoted the precursor approach – is considerably more elaborate but also more consistent. The strategy is to generate a fully developed turbulent ABL flow by means of a so-called spin-up simulation with periodic boundary conditions. The spin-up simulation is typically1 initialized with a suitable 1 It should be noted that other approaches exist to spin up the precursor simulation. Such alternatives

are usually associated with modeling of molecular viscosity.

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boundary layer shear profile, upon which random divergence-free perturbations are imposed. The flow is advanced for a time period of typically 24 h on a super computer, after which spurious effects of the random perturbations have disappeared, and the flow has reached a fully developed and statistically stationary state in the neutral ABL case. Finally, the obtained flow field is advanced in time, using a periodic precursor simulation, to generate statistically stationary time series of unsteady turbulent inflow conditions. The simplest way to generate a precursor ABL is to mimic these as canonical pressure-driven boundary layers, in which the governing Navier-Stokes equations are simplified by omitting terms associated with Coriolis forces and thermally driven buoyancy effects. This simplification is obviously on the cost of not being able to include ABL stability effects as well as Coriolis force-driven effects. Surface roughness is often derived from an assumed mean wind speed shear. A more elaborate and complete approach is to drive the spin-up process by the geostrophic wind. In contrast to the simplified pressure-driven approach, where the boundary layer grows naturally to the top of the defined domain, the height of the boundary layer will with the latter approach result from a balance between entrainment on the one hand and a stably stratified free atmosphere and capping inversion on the other. The relevant externally imposed parameters, defining the canonical ABL, are kept constant during the spin-up simulations. These include in general the surface heat flux (for non-neutral ABLs); the geostrophic wind speed and direction (which might be challenging to specify upfront); the Coriolis parameter at the relevant latitude; and the surface roughness. For non-neutral ABL conditions, the simulations are in addition geostrophic wind shear initialized with a constant potential temperature gradient throughout the domain. As time advances, boundary layer growth occurs as turbulent kinetic energy is produced by shear production at the surface and entrainment of potential temperature at the top of the domain. With the precursor field in place, the final step is to use this field as inflow field to a pre-defined wind farm. After a transient phase, where equilibrium of the wind farm flow field is obtained, the requested wind farm flow and WTs load sampling is performed. The WTs are typically embedded in ABL field – established as based on either of the approaches described above – as actuator disks (ADs), actuator sectors (ASs), or actuator lines (ALs). If the focus is on wind farm flow field analysis only, the structural dynamics of the WTs can be omitted. Further, it is worth noticing that LES AD and LES AL flow field simulations are almost identical for downstream distances of practical importance – only in the flow regime very close to the wake generating rotor differences can be observed (Troldborg et al. 2015). This is an important observation, because the time steps in the numerical schemes for the actuator disk solution can be increased compared to the AL schemes, thus typically resulting in a reduced computational time of the order of a factor of 5. If the focus, on the other hand, is on the (fatigue) loading of the wind farm WTs, then full aero-elastic modeling of each and every wind farm WT is required including modeling of the WT control system, and for such detailed WT load studies, the AL approach (Shen and Sørensen 2002) is often used. An example of a LES simulation of a WT wake-affected flow field, illustrating the apparent

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Fig. 4 Meandering wakes related to stable (left) and unstable (right) ABL conditions. As seen, the higher turbulence level associated with the convective ABL has a significant impact on wake meandering

wake meandering phenomenon, is shown in Fig. 4 for two different turbulence intensity/structure cases.

A Word on CFD and ABL Stability As illustrated in Fig. 4 and concluded in many CFD-based studies (e.g., Machefaux et al. 2016, Larsen et al. 2016, Abkar and Porte-Agel 2015, Sullivan and Patton 2011, and Sullivan et al. 2016), ABL stability conditions have a pronounced effect on wake-affected flow fields including wake meandering. As for neutral ABLs, such fields can be established either using a precursor approach or, alternatively, by introducing embedded time-varying inflow conditions in an upstream fringe region established based on a consistent kinematic turbulence model.The latter approach is sometimes referred to as the forced boundary layer (FBL) method. Regarding the FBL method, kinematic turbulence models accounting for the effect of non-neutral ABLs on turbulence are obviously needed, and a priori it is expected that synthetic turbulence models, providing the correct correlation structure of the requested turbulence, will facilitate “closer to CFD flow equilibrium” conditions in the fringe region. Only a few kinematic turbulence models offer this functionality Chougule et al. (2017, 2018) and Segalini and Arnqvist (2015). We will return to this issue in section “The ABL Stability Aspect” related to mediumfidelity modeling of wake meandering. An additional related issue is the associated modeling of the ABL wind shear. Conventionally, the Monin-Obukhov similarity theory (MOST) (Kaimal and Finnigan 1994) is used to prescribe the inflow wind shear. However, MOST is approximately valid in the surface layer, but, especially for stable stratification, large deviations between predicted and measured shears are reported outside the surface layer (cf. Larsen et al. 2016), where most modern WTs are operating. This is especially important for modeling of loads on the rotating parts of a WT. A modified formulation of MOST for stable stratification outside the surface layer is offered in Larsen et al. (2016). Regarding the spin-up approach, non-neutral ABL conditions are to be driven by the geostrophic wind, as mentioned above. In addition to the input needed for simulation of neutral precursor fields, the surface heat flux must be specified; fundamentally, the requested buoyancy effects can be accounted for by solving a transport equation for temperature as part of the spin-up simulation. Usually, periodic boundary conditions are applied to the inlet and outlet of the domain, and the simulation is set to run long enough for the generated turbulence to be

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fully developed and horizontally homogeneous. This type of LES flow predictions is generally considered to be the most accurate numerical representation of the nonneutral ABL that can be achieved today. A major disadvantage, in addition to its high CPU requirements, is that the stability characteristics change in time due to the stochastic nature of the LES and are therefore not straightforward to specify a priori. Thus, cases with requested specific stability conditions can be challenging and difficult to match observed surface fluxes and profiles of wind and temperature, due also to thermal inertia of the simulated ABL. Therefore, the influence of particular stability conditions on, e.g., wind turbine wakes needs to be addressed statistically. A pragmatic, but resource intensive, approach to address this issue is initially to simulate a diurnal cycle in the ABL using a series of unsteady RANS (URANS) simulations, representing relevant temporally varying surface heat fluxes; then subsequently determine the particular surface heat flux, which dictates the requested stability condition best possible; and use this as input for the LES simulation. It should, however, be stressed that the turbulence models used to close the URANS system equations are valid only as long as the time, over which changes in the mean occur, is large compared to the time scales of the turbulent motion containing most of the energy. Note moreover that due to different turbulence closures in, respectively, URANS and LES, an exact stability match should not be expected.

Concluding Remarks and Outlook Several integrated platforms exist, which facilitate advanced simulations of the performance of WTs embedded in a CFD-generated flow field – e.g., SP-Wind (KUL), Ellipsys3D (DTU), and SOWFA (NREL). Common for these platforms are that a CFD LES solver is coupled with an aero-elastic model, which in most cases is based on a multi-body finite element (FE) formulation. Although often using the terminology “high-fidelity” synonymously with LES simulations, this approach still faces unsolved challenges in addition to the excessive CPU demand. The most essential of these are grid convergence and the two-way coupling of LES with large-scale geophysical models. The considerable CPU demands so far render LES as impossible for wind farm design and optimization studies, where a huge amount of flow cases – preferable also including a variety of stochastic realizations for each flow case when analyzing WT fatigue loading – is to be investigated. Consequently, the LES approach is thus mostly applicable for scientific/phenomenological studies of fundamental wind farm flow physics. The latter two “issues” mentioned above relate to such basic investigations. The grid convergence challenge has been investigated in detail in Sullivan and Patton (2011) for convective ABLs, in Sullivan et al. (2016) for stable ABLs, and in Berg et al. for conditionally neutral ABLs. For all three canonical ABL types, it is concluded that second-order turbulence statistics can be strongly mesh dependent. The mesh dependence is usually largest in areas of the computational domain, where the spatial scales of turbulence are the smallest such as near the surface and at the top of the ABL in the entrainment zone – an issue which is generally magnified when associated with stable ABL stratification. Basically, the

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observed mesh dependence is an artifact of the sub-grid-scale turbulence modeling not perfectly reflecting reality. The two-way coupling of the local unsteady LES wind farm flow field with large-scale models of the (unsteady) regional wind climate is perhaps the largest unresolved issue. Here, a unified systematic procedure to transfer and enrich relevant information from large- to small-scale computational domains is still lacking.

Medium-Fidelity Approach From the full-scale experimental studies as well as from the detailed numerical CFD-based studies described in the previous sections of this chapter, the importance of unsteady wind farm flow fields – including wake dynamics – for reliable prediction of wind farm production and loading is obvious. However, it is at the same time clear that applying CFD LES, even coupled with the AL simplifying approach, is associated with a computational burden in terms of both computational time and CPU requirements, which is excessively high as seen from a practical wind farm design perspective. This is due to the required large number of design computations – each involving simulation of the full wind farm flow field – that have to be performed. These considerations emphasize the need for development of simpler and faster (kinematic) medium-fidelity models, which facilitate description of the fundamental features of dynamic wake flow fields with sufficient accuracy to be used for both reliable wind turbine load and production computations in wind farms. The development of the DWM model by Larsen et al. (2008a) is in a sense the breakthrough in understanding and modeling of wake dynamics called for in the concluding remarks of the review paper of Vermeer et al. (2003), and the model was subsequently included as a recommended practice in the recent update (i.e., Edition 4) of the IEC 64100-1 code for wind turbine design (IEC 2019). The core of this fast kinematic model is a binary split of scales in the wake flow field, with large turbulence scales being responsible for stochastic wake meandering and small turbulence scales being responsible for wake attenuation and expansion in the meandering frame of reference as caused by turbulent mixing. The DWM model thus essentially assumes that the downstream transport of wakes in the ABL can be modeled by considering the wakes to act as passive tracer plumes driven by a combination of large-scale ABL turbulence structures and a mean downstream advection velocity, adopting the Taylor hypothesis (Kaimal and Finnigan 1994). The wakes – including their dynamics as described above – are within the DWM framework treated as linear perturbations on the undisturbed ambient ABL wind field. Thus, combined with an aero-elastic model and a description of the ambient ABL flow field, the DWM model facilitates prediction of both production and loading of WTs operating in wake-affected flow conditions. The DWM model is fundamentally a single wake model consisting of three basic elements: a quasi-steady wake deficit formulated in the meandering frame of reference, a stochastic wake meandering process formulated in a fixed meteorological frame of reference, and a wake self-generated small-scale turbulence field

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Fig. 5 Illustration of the three basic components of the DWM model

formulated in the meandering frame of reference. The components of the model are illustrated in Fig. 5. Although the DWM model is formulated as a single wake model, it is in various implementations coupled with different wake superposition strategies to facilitate a description of complete wind farm flow fields. In the following, the different model components are described together with a selection of applied wake summation philosophies.

Quasi-steady Wake Deficit The quasi-steady mean wake deficit expands and attenuates with downstream distance from the wake generating rotor due to diffusion processes. The wake deficit can in principle be modeled as general 3D organized wake flow structures. However, the wake deficit is usually assumed rotational symmetric and modeled as based on the thin shear layer approximation of the Navier-Stokes (TL-NS) equations in their rotational symmetric form with the pressure term disregarded. This is convenient because of simplicity and justified because of gradients of the mean wake flow quantities being much bigger in the radial direction than in the downstream axial direction. Taking a turbulence eddy viscosity approach for the Reynolds stresses, the system of equations to be solved in the (meandering) polar frame of reference is thus (Madsen et al. 2010):  ∂ U¯ (r, x) 1 ∂  ¯ r V (r, x) + =0 r ∂r ∂x  ¯  ∂ U¯ (r, x) ∂ U¯ (r, x) νt ∂ ∂ U (r, x) ¯ ¯ U (r, x) + V (r, x) = r ∂x ∂r r ∂r ∂r

(1) (2)

where U¯ and V¯ denote the mean wake velocities in the axial and radial directions, respectively, and (r, x) are the radial and axial (i.e., downstream) coordinate pair.

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A further simplification of this system allows for an analytical solution, however, only for the flow regime far downstream the wake generation rotor (Larsen 2009). For the present application, we need a solution extending from the rotor plane and onward, because the meandering process, as well as the diffusion-driven expansion of the wake deficit, is initiated already with the creation of a wake deficit. Therefore, a numerical solution must be pursued. The eddy viscosity, νT , introduced by Madsen et al. in (2010) and subsequently re-calibrated in Larsen et al. (2013), includes contributions from both ambient turbulence and wake self-generated wake turbulence, and it is formulated as  x   σ 0.3 νt u = 0.023F1 RU0,h R U0,h x 1 x   x  1 1− + 0.008F2 Umin Rw R R R U0,h R

(3)

where U0,h is the ambient mean wind speed referring to hub height (i.e., rotor center), σu is the standard deviation of the ambient longitudinal turbulence component referring to hub height, R is the rotor radius, Rw is the wake radius at a specified downstream location, and Umin is the minimum axial wake velocity. The functions F1 and F2 are empirical filter functions, which depend on the axial coordinate. These filter functions account for the near field pressure influence on the wake deficit growth (i.e., F1 ) and for the lack of equilibrium between the mean shear flow field and the turbulence field in the near and intermediate wake regime (i.e., F2 ), respectively. The filter functions are defined as (Madsen et al. 2010)

F1

F2

x R

=

⎧ ⎨ ⎩ ⎧ ⎪ ⎨

 x 3/2 8R

−

  x 3/2 sin 2π ( 8R ) 2π

1;

 x

R

;0 ≤ ≥8

x R