Wind Turbine Aerodynamic Performance Calculation 9819935083, 9789819935086

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Tongguang Wang Wei Zhong Yaoru Qian Chengyong Zhu

Wind Turbine Aerodynamic Performance Calculation

Wind Turbine Aerodynamic Performance Calculation

Tongguang Wang · Wei Zhong · Yaoru Qian · Chengyong Zhu

Wind Turbine Aerodynamic Performance Calculation

Tongguang Wang College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

Wei Zhong College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

Yaoru Qian Nanjing Institute of Technology Nanjing, Jiangsu, China

Chengyong Zhu School of New Energy Nanjing University of Science and Technology Nanjing, Jiangsu, China

ISBN 978-981-99-3508-6 ISBN 978-981-99-3509-3 (eBook) https://doi.org/10.1007/978-981-99-3509-3 Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content. Translation from the Chinese language edition: “Wind Turbine Aerodynamic Performance Calculation” by Tongguang Wang et al., © Science Press 2019. Published by Science Press. All Rights Reserved. © Science Press 2023 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

Energy and the environment are two major factors that significantly affect the world’s politics, economy, and technology. Fossil energy sources such as oil and coal are becoming increasingly depleted, and these energy sources pollute the environment during utilization. As a renewable clean energy source, wind energy is demanded for large-scale development and commercialization in the era of increasing energy and environmental concerns. As an important aspect of future energy supply, its strategic position has been fully recognized by most countries worldwide. Wind power has developed rapidly over the past three decades and will continue to progress considerably in the future. A wind turbine is a device that converts the kinetic energy of wind into mechanical energy. In the past, they were known as windmills and were typically used for water irrigation and grain milling. Modern wind turbines are primarily used as core equipment in wind power plants for generating electricity. Wind turbines can be categorized into horizontal-axis wind turbines (HAWTs) and vertical-axis wind turbines (VAWTs), depending on the angle between the rotation axis and wind direction. The rotation axes of the HAWT and VAWT are parallel and perpendicular to the wind direction, respectively. Most wind turbines used in large-scale applications are HAWTs. The theories and methods introduced in this book are applicable to HAWTs. The main components of mainstream large-scale wind turbines include the rotor, nacelle, tower, main shaft, gearbox, generator, and controller. The rotor is a component that directly captures wind energy and is generally composed of three blades. The blade shape determines the wind energy extraction efficiency of a wind turbine. The blade-shape design is based on aerodynamic principles. Therefore, theories and methods associated with wind turbine aerodynamics are important to wind power technology. In academic studies and engineering applications, the blade element momentum method (BEM), vortex wake method (VWM), and computational fluid dynamics (CFD) method are three mainstream methods for investigating wind turbine aerodynamics, which are also the main focus of this book. Wind turbine blades typically feature large aspect ratios; thus, the flow around the blade, except the tip and root areas, exhibits a significant two-dimensional (2D) feature. As such, the blade performance can be computed easily using the v

vi

Preface

aerodynamic data of 2D airfoils from the blade profiles. The basic idea of the BEM is as follows: First, the blade is partitioned into several segments along its span, where each segment is known as a blade element. Aerodynamic interference is assumed to be absent between the blade elements. Second, based on momentum theory, the relationship between the local induced velocities and the aerodynamic force of the blade airfoils is established. Next, the induction factors and aerodynamic coefficients of each blade element are obtained by solving the equations corresponding to the relationship above. Subsequently, the thrust and torque contributed by each blade element are determined. Finally, the aerodynamic force of the entire blade is obtained by integrating along the blade span. The advantage of the BEM is that its computational model is relatively simple, and the amount of computation required is acceptable, thus resulting in computation results with reasonable accuracy. Therefore, it is suited for the significant amount of computations required in the design of wind turbines and has been used widely in engineering. In the VWM, the wind turbine velocity field is regarded as the result of the induction of blade-attached and wake vortices. The VWM solves the induced velocity based on the Biot–Savart law rather than the momentum theory. Its advantage over the BEM is that the 2D assumption of the blade element is disregarded, thus allowing three-dimensional (3D) effects to be included. The wake vortex shape is vital to the VWM. Models that effectively describe the wake vortex shape primarily include prescribed- and free-vortex wake models. The wake shape of the prescribed-vortex wake model is a predefined function of the local radius and the induced velocity at the blade. It is a semiempirical model based on experimental data that does not directly solve the effect of velocity induced outside the blade. The free-vortex wake model determines the wake shape by solving the induced velocity in the entire flow field. It provides a better theoretical basis and a more general applicability than the prescribed-vortex wake model. Furthermore, it can be used for computations in complex operating conditions where the prescribed-vortex wake model is incompetent, e.g., conditions with aeroelasticity effects. The BEM and VWM share some common shortcomings. For example, both rely on the aerodynamic data of airfoils, and their theoretical underpinnings prevent them from predicting the role of complex flow phenomena such as the 3D rotational effect and dynamic stall. CFD, which is an important branch of fluid mechanics, can overcome these disadvantages. It is based on numerical methods that solve the governing equations of fluid motion. Therefore, it is a general solution method for the spatial and temporal information of the flow field. CFD has received increasing attention among researchers of wind turbine aerodynamics because of its innate advantages. Almost all physical information from the wind turbine flow field can be simulated using CFD; thus, CFD provides incomparable advantages in comprehensively analyzing the flow characteristics of wind turbines. This book is organized into four parts, with a total of 12 chapters. Part I includes Chaps. 1–3, in which basic information regarding wind turbine aerodynamics is introduced, including the physical properties of air, mathematical description of air motion, and aerodynamics of airfoils. In Chaps. 4–6, steady and unsteady BEMs as well as the necessary correction models are discussed. Part III comprises Chaps. 7–

Preface

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9, in which the steady and unsteady VWMs are discussed based on the prescribedand free-vortex wake models. In part IV, which comprises Chaps. 10–12, the theory of CFD and its applications to wind turbines are discussed, including simulations pertaining to the aerodynamic performance of wind turbines based on Reynoldsaveraged equations and wind turbine wake simulations based on large-eddy simulations. The contents of part II, III and IV are relatively independent of each other, and readers can read any of the parts directly, in no specific order. The content of this book is based primarily on results obtained by the research team at the Jiangsu Key Laboratory of Hi-Tech Research for Wind Turbine Design, Nanjing University of Aeronautics and Astronautics, which is led by Prof. Wang Tongguang. Doctoral students Tang Hongwei, Yuan Yiping, and Gao Zhiteng compiled the results, whereas postgraduate students Chen Kai, Han Ran, Chen Jie, and Tang Zeling partially edited the manuscript. In writing this book, the authors received assistance from Tian Linlin, Wang Long, Xu Bofeng, Zhang Zhenyu, and Wu Jianghai. We would like to express our heartfelt gratitude to them. The publication of this book was funded by the National Basic Research Program of China (“973” Program) under grant number 2014CB046200. As this book includes various methods and was written in a relatively limited time, mistakes are inevitable. Readers are welcome to provide feedback such that revisions can be incorporated in the next edition. Nanjing, China

Tongguang Wang Wei Zhong Yaoru Qian Chengyong Zhu

Brief Introduction

Mainstream methods for computing the aerodynamic performance of wind turbines are systematically introduced in this book. The basic theory of aerodynamics related to wind turbines is first introduced. Subsequently, blade element momentum method, vortex wake method, and computational fluid dynamics method are discussed comprehensively, including the method introduction, formula derivation, computation process, and examples. For theoretical research and engineering applications, the three methods above are important for wind turbine aerodynamic performance analysis. Therefore, the discussion presented in this book provides both theoretical and practical significance. This book is suitable for professionals engaged in wind turbine aerodynamic design and simulation and can be used as a reference textbook for undergraduates and postgraduates studying wind energy and aerodynamics in colleges and universities.

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Part I

Fundamentals of Wind Turbine Aerodynamics

1

Physical Properties of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Continuum Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Pressure, Density, and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definitions of Pressure, Density, and Temperature . . . . . 1.2.2 Ideal Gas Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Compressibility, Viscosity, and Thermal Conductivity . . . . . . . . . 1.3.1 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Inviscid and Incompressible Assumptions . . . . . . . . . . . . . . . . . . . . 1.4.1 Inviscid Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Incompressible Assumption . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 4 4 5 5 6 8 9 9 9 10

2

Description of Air Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Motion of Fluid Microelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Analysis of Fluid Microelement Motion . . . . . . . . . . . . . . 2.1.2 Velocity Divergence and Its Physical Meaning . . . . . . . . 2.1.3 Curl and Velocity Potential Function . . . . . . . . . . . . . . . . . 2.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Governing Equations of Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Euler Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Governing Equations of Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . 2.5 Viscous Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Concept of the Boundary Layer . . . . . . . . . . . . . . . . . . . . . 2.5.2 Boundary-Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Pressure Characteristics in the Boundary Layer . . . . . . . 2.5.4 Boundary-Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Flow Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 14 15 16 18 18 21 22 24 24 25 27 28 29 xi

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2.6 2.7

3

Basic Concepts of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Wind in the Atmospheric Boundary Layer . . . . . . . . . . 2.7.1 Basic Characteristics of the Atmospheric Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Characteristics of the Mean Wind Speed . . . . . . . . . . . . . 2.7.3 Characteristics of Turbulent Wind . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 33

Fundamentals of Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Airfoil Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Geometric Parameters of Airfoil . . . . . . . . . . . . . . . . . . . . 3.1.2 Numbering of Typical Airfoils . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Parametric Description of Airfoil Geometry . . . . . . . . . . 3.2 Aerodynamics of Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Flow Around Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Aerodynamic Coefficients of Airfoil . . . . . . . . . . . . . . . . . 3.2.3 Aerodynamic Characteristics of Airfoil . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 39 41 42 44 44 44 46 48

Part II

33 34 35 36

Blade Element Momentum Method

4

Steady Blade Element Momentum Method . . . . . . . . . . . . . . . . . . . . . . 4.1 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 BEM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Blade Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effect of High Thrust Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Iterative Solution of BEM Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Calculation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 56 61 61 63 65 68

5

Correction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Tip-Loss Correction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Prandtl Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Glauert Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Goldstein Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Shen Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Zhong Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Blade-Root Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 3D Rotational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Category 1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Category 2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Dynamic Stall Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Beddoes–Leishman Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Øye Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 ONERA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Boeing–Vertol Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 73 73 74 75 76 77 79 80 81 91 92 93

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5.3.5

6

Coupling of Dynamic Stall Model and 3D Rotational Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94 97

Unsteady Blade Element Momentum Method . . . . . . . . . . . . . . . . . . . . 6.1 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calculation of induced Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Dynamic Inflow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Dynamic Wake Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Yaw/Tilt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Calculation Steps of Unsteady BEM Method . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 103 105 107 108 109 112

Part III Vortex Wake Method 7

Fundamentals of Vortex Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Vortex Lines, Vortex Tubes, and Vortex Strength . . . . . . . . . . . . . . 7.2 Velocity Circulation and Stokes Theorem . . . . . . . . . . . . . . . . . . . . 7.3 Biot–Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Vortex Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Model of Vortex Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Vortex Core Radius and Dissipation Model . . . . . . . . . . . 7.5 Helmholtz Vortex Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Kutta–Joukowski Lift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Flow Around a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Circulation and Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 118 122 123 123 125 127 129 129 131 132

8

Computational Models of Vortex Wake . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Definition of Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Models of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Models of Vortices Attached to Blades . . . . . . . . . . . . . . . 8.2.2 Models of Wake Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 PVW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 FVW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Governing Equations for Vortex Filaments . . . . . . . . . . . . 8.4.2 Description of Initial Wake . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Flow Field Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Wake Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Computation of Attached Vortex Circulation . . . . . . . . . . 8.5.3 Computation of Rotor Aerodynamic Performance . . . . . 8.5.4 Computation of Induced Velocity . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 134 135 135 138 140 142 142 143 143 143 145 146 148 151

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9

Contents

Solving Aerodynamic Performance of Wind Turbines . . . . . . . . . . . . . 9.1 Solution of Steady PVW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Solution of Steady FVW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Relaxation Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Unsteady PVW Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Calculation of Inflow Wind Speed . . . . . . . . . . . . . . . . . . . 9.3.2 Induced Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Coupling of Dynamic Stall Models . . . . . . . . . . . . . . . . . . 9.3.4 Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Unsteady FVW Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Time-Stepping Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Computation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 153 156 157 157 159 160 161 161 163 164 165 166 166 170 171 172

Part IV Computational Fluid Dynamics Method 10 Fundamentals of Computational Fluid Dynamics . . . . . . . . . . . . . . . . 10.1 Brief Introduction to CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Mathematical Description of Incompressible Viscous Flow . . . . . 10.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 RANS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Methods of Numerical Discretization . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 FDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 FVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Algorithms for Velocity–Pressure Coupling . . . . . . . . . . . . . . . . . . 10.5.1 SIMPLE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 PISO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Mesh Generation and Post-processing . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Applications of CFD in Wind Turbine Aerodynamics . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 177 178 179 179 180 181 182 182 183 185 185 187 187 187 189 189 190

11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Governing Equations and Discretization . . . . . . . . . . . . . . . . . . . . . 11.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 193 194

Contents

11.1.3 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 One-Equation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Two-Equation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Selection of Turbulence Models . . . . . . . . . . . . . . . . . . . . . 11.3 Transition Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Michel Transition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 γ –Reθ Transition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Inlet and Outlet Boundary Conditions . . . . . . . . . . . . . . . . 11.4.2 Rotational Periodic Boundary Condition . . . . . . . . . . . . . 11.4.3 Wall Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Mesh for Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 2D Mesh for Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Mesh for Wind Turbine Rotor . . . . . . . . . . . . . . . . . . . . . . 11.6 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Fully Turbulent Simulation Versus Transitional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Parameter Correction for SST Turbulence Model . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Large- and Detached-Eddy Simulation Methods for Wind Turbine Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 ALM for Wind Turbine Simulations . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Actuator Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Nacelle and Tower Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Large-Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Filtering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 SGS Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Turbulent Inflow Generation for LES . . . . . . . . . . . . . . . . 12.3 Detached-Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 S–A DES Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 SST–DES Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Unsteady Performance Simulations Using DES . . . . . . . 12.4.2 Wind Turbine Wake Simulated Using LES . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

195 195 196 197 199 200 200 201 204 205 205 206 206 206 208 209 209 210 211 213 214 214 215 216 218 219 222 225 225 226 227 227 230 236

Part I

Fundamentals of Wind Turbine Aerodynamics

Chapter 1

Physical Properties of Air

A wind turbine converts the kinetic energy of wind into mechanical energy via the relative motion between its blades and air. The aerodynamic performance of a wind turbine depends not only on the shape of its blades, but also the physical properties of air. This chapter introduces the physical properties of fluid media, in particular air.

1.1 Continuum Assumption Air is a fluid medium, and the mean-free path of its molecules is much larger than the molecules. Under standard conditions, the mean-free path of air molecules is approximately 6.9 × 10−8 m, the average diameter of air molecules is less than 3.5 × 10−10 m, and the ratio of the two is approximately 200:1. Therefore, microscopically, air is considered a discontinuous medium. However, in macroscopic aerodynamic studies, a detailed investigation of the microscopic motion of molecules results in laborious computations, which is neither practical nor necessary for most engineering problems. When we investigate the macroscopic motion of air, the continuum assumption can be adopted; i.e., air can be regarded as a continuous medium without gaps that fills the space it occupies. In most engineering analysis cases, the continuum assumption applies to gases and liquids. Based on the continuum assumption, a fluid microelement can be considered the object of analysis for analyzing fluid motion. The microelement contains numerous fluid molecules, and the characteristics of the microelement reflect the statistical properties of all the molecules. However, the microelement is infinitely small compared with macroscopic objects and can be approximated as a point. Based on the continuum assumption, the macroscopic physical properties of a fluid, such as its density, velocity, and pressure, can be regarded as continuous functions of space, thus providing conditions for the application of various mathematical tools.

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_1

3

4

1 Physical Properties of Air

The governing equations of fluids are derived by applying mathematical derivations to the dynamics of fluid microelements.

1.2 Pressure, Density, and Temperature 1.2.1 Definitions of Pressure, Density, and Temperature Consider a small surface in a fluid. It can be the surface of an actual object in direct contact with a fluid or an imaginary face in the fluid. The fluid pressure is the normal force on the unit area of the surface when the molecules of the moving fluid collide with the surface. Let dA be the area of the surface, and dF be the normal force; therefore, the pressure at a point can be defined as p = lim

dA→0

dF . dA

(1.1)

Fluid density is defined as the fluid mass per unit volume. Consider any point in the fluid, and let dV be a microvolume containing the point, and dm the fluid mass in the microvolume. Subsequently, the fluid density at the point can be defined as ρ = lim

dV →0

dm . dV

(1.2)

Additionally, temperature is an important property of fluids as it reflects the degree of coldness and heat of a fluid. Microscopically, it reflects the average kinetic energy of fluid molecules owing to their thermal motion. The relationship between the average kinetic energy of fluid molecules and the absolute temperature T of the fluid is E k = 23 kT , where k is the Boltzmann constant.

1.2.2 Ideal Gas Equation of State The ideal gas is a model used in the kinetic theory of gases [1]. Its molecules are regarded as wholly elastic microspheres. The intermolecular attractive forces can be ignored. The interaction between molecules only occurs when they collide. The total volume of molecules is negligible compared with the space occupied by gases. Gases far from the liquid state typically satisfy these assumptions. In addition, the air under normal conditions generally conforms to these assumptions. The functional relationship between the pressure, density, and temperature of an ideal gas is represented by the ideal gas equation of state, which was first proposed by Clapeyron [2]:

1.3 Compressibility, Viscosity, and Thermal Conductivity

p=

R ρT , m

5

(1.3)

where R is the universal gas constant with a value of 8.3145 J/(mol K), m is the molecular mass of the gas, and T is the thermodynamic temperature. If R/m is replaced by R, Eq. (1.3) can be expressed as follows: p = ρ RT,

(1.4)

where R is the gas constant, which varies for different gases. Air is a mixture composed of different components. Its gas constant is calculated as 287.053 J/(kg K) according to the mass proportions of its components.

1.3 Compressibility, Viscosity, and Thermal Conductivity 1.3.1 Compressibility Fluid compressibility is a characteristic in which the volume or density of a certain mass of gas changes with pressure. Compressibility can be measured based on the bulk modulus, which is defined as the pressure change required to yield a unit of relative volume change, as follows: E =−

dp , dV /V

(1.5)

where E is the bulk elastic modulus of a fluid, and V is the volume of a certain mass of fluid. For a certain mass of fluid, its volume is inversely proportional to its density. dV dρ =− . ρ V

(1.6)

Therefore, the bulk modulus of elasticity of a fluid can be written as E =ρ

dp . dρ

(1.7)

Different fluids have different bulk moduli and hence different compressibilities. For example, the bulk modulus of water at room temperature is approximately 2.1 × 109 N/m2 . When the pressure increases by 1 atm, the corresponding relative density change is /p /ρ = ≈ 0.5 × 10−4 . ρ E

(1.8)

6

1 Physical Properties of Air

In other words, the relative density change of water caused by a pressure change of 1 atm is only 0.5/10,000. Therefore, water is generally regarded as an incompressible fluid. Liquids are considered incompressible fluids in most engineering problems because of their relatively high bulk modulus values. The bulk modulus of air is much lower than that of liquids, i.e., approximately 1/20,000 that of water. Therefore, the density of air tends to change with pressure. However, if the relative change in air density is insignificant, then the airflow can be considered incompressible. According to the Bernoulli equation, p + 21 ρv 2 = const (v is the flow velocity), the pressure change /p caused by fluid motion can be regarded as exhibiting the same order of magnitude as the dynamic pressure q = 21 ρv 2 . Consequently, Eq. (1.8) yields /ρ ≈ Eq . Therefore, if q/E α1

,

(5.82)

where S 1 and S 2 define the static stall characteristics, and α 1 defines the break point corresponding to f = 0.7. S 1 , S 2 , and α 1 can be easily determined from the wind tunnel static lift data. A general expression for the pitching moment coefficient C m cannot be obtained using Kirchhoff flow theory. In the Beddoes–Leishman model, the variation in the ratio C m /C n is fitted using the least-squares method from the airfoil static data, as follows:

88

5 Correction Models

Cm = k0 + k1 (1 − f ) + k2 sin(π f μ ), Cn

(5.83)

where k 0 represents the offset of the aerodynamic center from the quarter-chord axis, k 1 imposes a direct effect on the center of pressure owing to the growth of the separated-flow region, and k 2 describes the shape of the moment break at stall. The values of k 0 , k 1 , k 2 , and μ can be adjusted accordingly for different airfoils to obtain the best reconstruction of C m /C n . For small AOAs, the tangential force coefficient can be approximated by C t = Cl α − C d .

(5.84)

Substituting the expressions derived using Kirchhoff theory, i.e., √ )2 1+ f 2 √ )2 ( 1− f C d = Clα α 2 , 2 (

Cl = Clα α

(5.85)

(5.86)

into Eq. (5.84) yields C t = Clα α 2

√

f

(5.87)

This equation above can be correlated with the normal force coefficient in a more generalized format, as follows: Ct = Cn α (α − α0 )

√

f sin α.

(5.88)

(2) Unsteady trailing-edge separation For unsteady conditions, a lag occurs in the leading-edge pressure responses with respect to the normal force, and the lagged normal force coefficient Cn' can be obtained using Eq. (5.77). The corresponding AOA becomes α ' (N ) =

Cn' (N ) + α0 . Cnα

(5.89)

Substituting Eq. (5.89) into Eq. (5.82) yields the trailing-edge separation point associated with the lag in the leading-edge pressure response, as follows: '

f (N ) =

⎧ ⎨

1 − 0.3e

α ' (N )−α1 S1

⎩ 0.04 + 0.66e

α1 −α ' (N ) S2

α ' (N ) ≤ α1 α ' (N ) > α1

.

(5.90)

5.3 Dynamic Stall Model

89

In addition to the unsteady aerofoil pressure response, an unsteady boundary-layer response exists. The effect on the trailing-edge separation point may be represented by applying a first-order lag to f ' . For a sampled system, f '' (N ) = f ' (N ) − D f (N ),

(5.91)

where − ΔS S

D f (N ) = D f (N − 1)e

f

[ ] − ΔS + f ' (N ) − f ' (N − 1) e 2S f .

(5.92)

The constant S f can be evaluated using unsteady airfoil test data. Thus, the value of f'' is used to obtain the nonlinear forces and pitching moments. 4. Modeling of dynamic stall Dynamic stall includes the formation of airfoil leading-edge vortices, vortex shedding from the airfoil surface, and rearward movement of the vortex. The vortex generates a vortex lift and the corresponding moment during the movement; therefore, when the dynamic stall begins, modeling the dynamic stall vortex is critical for calculating the aerodynamic load of the airfoil. The vortex normal force can be regarded as the accumulation of circulation near the airfoil until the critical condition is satisfied. For a sampled system, the increment in the vortex normal force C v can be determined by the difference between the linear value of the unsteady circulatory normal force and the nonlinear value of the unsteady normal force approximated by Kirchhoff’s theory, as follows: { Cv (N ) =

CnC (N )

[ 1−

1+

√

f '' (N ) 2

]2 } .

(5.93)

The total accumulated vortex normal force coefficient Cnv under unsteady conditions is allowed to decay exponentially with time and can be updated by a new increment. This can be formulated as follows: ΔS

ΔS

Cnv (N ) = Cnv (N − 1)e− Sv + [Cv (N ) − Cv (N − 1)]e− 2Sv .

(5.94)

The center of pressure on the airfoil varies with the chordwise position of the vortex shedding and achieves a maximum value when the vortex reaches the trailing edge after a non-dimensional time period τvl . A general representation of the behavior at the center of pressure (aft of quarter-chord) due to the dynamic stall vortex can be formulated empirically as follows: ) ( π τv , (C P)v = 0.25 1 − cos τvl

(5.95)

90

5 Correction Models

where τv is the non-dimensional vortex time, 0 ≤ τv ≤ 2τvl , where τv = 0 at the onset of dynamic stall, and τv = τvl when the vortex reaches the trailing edge. Both the vortex decay non-dimensional time constant S v and the non-dimensional time for the vortex to traverse chord τvl can be obtained statistically from different dynamic stall test data. Thus, the increment in the pitching moment about the quarter-chord owing to dynamic stall is expressed as Cmv = −(C P)v Cnv .

(5.96)

Abrupt airloading changes occur when the critical condition for leading-edge separation is satisfied, i.e., Cn' > Cn1 . At this point, the accumulated vortex lift is assumed to begin to convect over the chord. During vortex convection, the behavior of the vortex force is assumed to reflect Eqs. (5.93) and (5.94); however, the accumulation is terminated when the vortex reaches the trailing edge at τv = τvl . Simultaneously, the ongoing pressure changes because the vortex shedding process is sufficient to accelerate the forward movement of the trailing-edge separation, which is accomplished by halving the S f non-dimensional time constant. After the vortex passes the airfoil trailing edge, the vortex-induced lift on the airfoil decays rapidly because the non-dimensional vortex decay time constant S v is halved for the period τvl ≤ τv ≤ 2τvl . 5. Unsteady aerodynamic loads (1) Normal force coefficient The nonlinear term associated with the trailing-edge separation is ( Cnf

=

CnC

1+

√

f ''

)2

2

,

(5.97)

where CnC is obtained using Eq. (5.64), and f '' using Eq. (5.91). The total unsteady normal force coefficient is Cn = Cnf + CnI + Cnv ,

(5.98)

where CnI is obtained using Eq. (5.68). In the dynamic stall case, Cnv is obtained using Eq. (5.94). (2) Chordwise force coefficient After the onset of gross separation, the Kirchhoff modification of the chordwise force [as expressed in Eq. (5.88)] becomes invalid, and an alternative procedure is adopted. In the model, the chord force is correlated with the separation point by introducing an additional term in the expression, i.e.,

5.3 Dynamic Stall Model

91 f

Ct = kt ηt CnC

√

f '' sin αe ,

(5.99)

where k t is the chordwise force efficiency, which can be estimated using airfoil test data and whose value is typically approximately unity. { ηt =

'

( f '' ) K f (Cn −Cn1 ) for Cn' > Cn1 . 1 for Cn' ≤ Cn1

(5.100)

The total chordwise force coefficient is expressed as Ct = Ct + (CnI + Cnv ) sin α. f

(5.101)

(3) Pitching moment coefficient Based on Eq. (5.83), the contribution of the movement of the center of pressure due to the trailing-edge separation can be written as follows: ( ) ]} { [ Cmf = Cm 0 + k0 + k1 1 − f '' + k2 sin π( f '' )μ Cnf ,

(5.102)

where Cm 0 denotes the zero-lift pitching moment coefficient. Subsequently, the total pitching moment coefficient is expressed as Cm = CmC + Cmf + CmI + Cmv ,

(5.103)

where CmC and CmI are obtained using Eqs. (5.65) and (5.73), respectively. In the dynamic stall case, Cmv is obtained using Eq. (5.96).

5.3.2 Øye Model The Øye model [23] is used for trailing-edge stall, where flow separation begins at the trailing edge, and as the AOA increases, the separation point shifts forward gradually. The lift coefficient is obtained by interpolating between the aerodynamic data in two extreme cases, i.e., one where the flow is fully separated, and another where the flow is completely inviscid. The corresponding expressions are as follows: Cl = f s Cl,inv (α) + (1 − f s )Cl, f s (α),

(5.104)

where α is the local AOA, Cl,inv the lift coefficient of the inviscid flow without separation, and Cl, f s the lift coefficient of the completely separated flow. Cl,inv is typically obtained by interpolating the static airfoil lift curve. According to Hansen et al. [13], the coefficient Cl, f s can be calculated as follows:

92

5 Correction Models

Cl, f s =

Cl,st − Cl,α (α − α0 ) f sst , 1 − f sst

(5.105)

where Cl,st represents the static lift coefficient, Cl,α the slope of the lift line in the linear region of the attached flow, α0 the zero-lift AOA, and f sst the static value of the separation function f S. Here, Cl,st and f sst can be calculated as follows: ( Cl,st = Cl,α (/ f sst =

1+

√

f sst (α) 2

)2

Cl,st (α) −1 Cl,α (α − α0 )

(5.106) )2 .

(5.107)

Typically, f s is assumed to converge to the static value f sst , where f st − f s d fs = s . dt τ

(5.108)

Integration can yield Δt

f s (t + Δt) = f sst + [ f s (t) − f sst ]e− τ ,

(5.109)

where τ is a time constant approximately equal to A · Wc , c the local chord length, and W the resultant velocity at the blade section. Here, A is a constant that is generally specified as 4.

5.3.3 ONERA Model The ONERA model [24] utilizes the properties of differential equations to directly simulate the aerodynamic response of wind turbines in the time domain. Before stalling, the airfoil aerodynamic loads can be represented by the first-order differentials of the corresponding variables. After stalling, the loads are represented by the second-order differential of the corresponding variable, S. Let function F represent the total aerodynamic load, which can be written as F = F1 + F2 ∂ 2α ∂ F1 ∂α + λF1 = λF1 + (λs + σ ) +s 2 ∂t ∂t ∂t ) ( ∂ 2 F2 ∂ F2 ∂Δ + r F . + a = − r Δ + e 2 ∂t 2 ∂t ∂t

(5.110)

(5.111)

(5.112)

5.3 Dynamic Stall Model

93

The coefficients λ, s, σ, a, and r are only functions of the airfoil instantaneous AOA α. F 1 represents the value obtained by linearly extrapolating the static load curves, and Δ represents the difference between the results calculated using the extrapolation method and the actual static data. Similarly, F 1 and Δ are functions of α and are entirely determined by the static characteristics of the airfoil. When the oscillation frequency approaches zero, Eqs. (5.110) to (5.112) degenerate into lim F = F1 − Δ = Fs ,

τ →0

(5.113)

where F s denotes the static load response. If the airfoil motion is unsteady but remains within the linear range, then Δ = 0, and the airfoil load is determined by F 1 in Eq. (5.111).

5.3.4 Boeing–Vertol Model The Boeing–Vertol model [25] assumes that a certain relationship exists between the static and dynamic stall AOAs. Gross and Harris [26] define this relationship as follows: / c|α| ˙ , (5.114) αds − αs = A1 2V where α ds and α s are the dynamic and static stall AOAs, respectively; α˙ is the first derivative of the AOA with respect to time; c is the airfoil chord length; V is the freestream velocity. The relationship between the static and dynamic AOAs, i.e., α and α d , respectively, can be formulated as / αd = α − A1

c|α| ˙ α˙ . 2V |α| ˙

(5.115)

Subsequently, the lift coefficient can be expressed as follows: Cl = Cl (0) +

Cl (αd ) − Cl (0) α. αd

(5.116)

94

5 Correction Models

5.3.5 Coupling of Dynamic Stall Model and 3D Rotational Effects Both the dynamic stall and 3D rotational effects cause stall delay; they do not satisfy the linear superposition relationship and exhibit a certain coupling effect [27, 28]. In this section, the Beddoes–Leishman model is used as an example to discuss the coupling of 3D rotational effects and the dynamic stall model. 1. Normal force coefficient If the Kirchhoff formula is assumed to be applicable to 3D cases, then the 3D trailingedge separation point can be correlated with the 3D normal force coefficient by rewriting Eq. (5.81), as follows: [/ f 3D = 4

1 (Cn )3D − Cn α (α − α0 ) 2

]2 .

(5.117)

For a specified tip-speed ratio, f 3D is a function of not only the AOA, as in the 2D case, but also the local radius r. f 3D = g(α, r ).

(5.118)

Under unsteady conditions, a lag occurs in the leading-edge pressure response with respect to the normal force. Applying this lag to (Cn )3D [as in Eq. (5.77)] yields a substitutive normal force coefficient (Cn' )3D . Correspondingly, the AOA [Eq. (5.89)] becomes α' =

(Cn' )3D + α0 . Cnα

(5.119)

Substituting α ' into Eq. (5.118) yields the trailing-edge separation point associated with the lag in the leading-edge pressure response, as follows: ' f 3D = g(α ' , r ).

(5.120)

' The separation point f 3D depends on the AOA, radial position of the blade element, and tip-speed ratio. Therefore, the variation in the 3D separation point cannot be generalized in a simple explicit algebraic form as shown in Eq. (5.82). In addition to the airfoil unsteady pressure response, an unsteady boundary-layer response exists, the effects of which on the trailing-edge separation point may be ' . represented by applying a first-order type lag to f 3D '' ' f 3D = f 3D − Df,

(5.121)

5.3 Dynamic Stall Model

95

where the compensation function Df , which is similar to that of the 2D case for a sampled system, is expressed as − ΔS S

D f (N ) = D f (N − 1)e

f

[ ' ] − ΔS ' + f 3D (N ) − f 3D (N − 1) e 2S f .

(5.122)

Similarly, for 3D dynamic stall, Eq. (5.93) is written as ⎧ ⎨

[

Cv (N ) = CnC (N ) 1 − ⎩

1+

√

'' f 3D (N ) 2

]2 ⎫ ⎬ ⎭

.

(5.123)

Finally, the unsteady 3D normal force coefficient can be expressed as ( Cn = Cnα

1+

√

'' f 3D

)2

2

(αe − α0 ) + CnI + Cnv .

(5.124)

2. Chordwise force coefficient The 3D chordwise force coefficient associated with the nonlinear separation may be expressed as f

C t = k t ηt

/

'' f 3D Cnα (αe − α0 ) sin αe .

(5.125)

For a given airfoil, the chordwise force efficiency k t for the 2D case is fixed and its value is typically approximately unity owing to the significant peak suction in the leading-edge area. In rotational cases, however, where the 3D relational effects occur, the airfoil loading depends on both the AOA and the distance of the blade section from the center of rotation. Consequently, the 3D k t will be different for the same airfoil section at different spanwise positions and will be difficult to determine in advance as an input to the calculation. Therefore, determining the chordwise thrust for 3D cases using this method is difficult; hence, an alternative procedure has been developed. If the change in shear stress on the aerofoil surface is negligible, then the increment in the chordwise thrust coefficient owing to 3D effects can be expressed as ∫ ΔCt = −

ΔC p d

(y) c

,

(5.126)

where ΔC p denotes the increment in the pressure coefficient on the airfoil, and y is the coordinate perpendicular to the chord. Since the rotational effects are minimal in the attached flow regime and occur only in the separated-flow region, as stated previously, the increase in the surface pressure may be considered to be primarily restricted to the separation region (Fig. 5.8). Furthermore, if the change in pressure is assumed

96

5 Correction Models

Fig. 5.8 Illustration showing assumed change in pressure due to 3D effects

to be constant throughout the separated-flow region as a first-order approximation, then it can be correlated with the change in the normal force as follows: Δc p = −

ΔCn , 1− f

(5.127)

where ⎡( ΔCn = Cn α ⎣

1+

√

'' f 3D

2

(

)2 −

1+

√

'' f 2D

)2 ⎤

2

⎦(αe − α0 ).

(5.128)

Thus, the change in the chordwise force coefficient can be approximated as follows: ΔCt = −

ΔCn y f , '' 1 − f 3D c

(5.129)

where yf is the vertical distance between the separation point f and chord line. Finally, the total unsteady 3D chordwise force coefficient can be calculated as Ct = (Ct )2D + ΔCt ,

(5.130)

where the 3D chordwise force coefficient (Ct )2D can be obtained using Eq. (5.101). 3. Pitching moment coefficient The increment in the quarter-chord pitching moment coefficient ΔC m owing to 3D effects is expected to be due to the change in the normal force coefficient ΔC n and the separation location f . Additionally, a positive ΔC n results in a negative ΔC m if the change in pressure that causes an increase in the normal force occurs primarily in the separated-flow region. Hence, the increment in the pitching moment can be

References

97

expressed as (

∫1 ΔCm =

Δc p

) ∫0 (y) (y) x 1 (x ) + d . Δc p − d c 4 c c c

(5.131)

y f /c

f

If the change in pressure is assumed to be approximately constant throughout the separated-flow region, then Eq. (5.131) can be written as ⎡ ⎤ ) ( ) ∫0 ( ) ( ) ∫1 ( x 1 x y ⎥ y ΔCn ⎢ − d + d ΔCm = − ⎣ ⎦. 1− f c 4 c c c

(5.132)

y f /c

f

For 3D unsteady cases, the expression above reduces to ] [ '' y 2f ΔCn 1 + 2 f 3D ) . − 2( ΔCm = − '' 2 2 c 1 − f 3D

(5.133)

Finally, the total 3D pitching moment coefficient is expressed as Cm = CmC + Cmf + Cmv + CmI ,

(5.134)

where the contribution of the movement of the center of pressure owing to the trailingedge separation is written as '' '' μ Cmf = Cm 0 + {k0 + k1 (1 − f 3D ) + k2 sin[π( f 3D ) ]}(Cnf )2D + ΔCm ,

(5.135)

with ( (Cnf )2D

= Cnα

1+

√ 2

'' f 2D

)2 (αe − α0 ).

(5.136)

References 1. Breton, S. P., Coton, F. N., & Moe, G. (2008). A study on rotational effects and different stall delay models using a prescribed wake vortex scheme and NREL phase VI experiment data. Wind Energy, 11(5), 459–482. 2. Sørensen, J. N. (2016). General momentum theory for horizontal axis wind turbines. Springer International Publishing. 3. Sørensen, J. N. (2012). Aerodynamic analysis of wind turbines. In A. Sayigh (Ed.), Comprehensive renewable energy (pp. 225–241). Elsevier.

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4. Brandlard, E. (2011). Wind turbine tip-loss corrections: Review, implementation and investigation of new models. Technical University of Denmark. 5. Glauert, H. (1963). Airplane Propellers. Dover Publications. 6. Wilson, R. E., & Lissaman, P. B. S. (1974). Applied aerodynamics of wind power machines (p. 75). NASA STI/Recon Technical Report N. 7. de Vries, O. (1979). Fluid dynamic aspects of wind energy conversion. AGARD. 8. de Vries, O. (1979). Wind tunnel tests on a model of a two bladed horizontal-axis wind turbine and evaluation of an aerodynamic performance calculation method. NLRTR. 9. Wilson, R. E., Lissaman, P. B. S., & Walker, S. N. (1976). Aerodynamic performance of wind turbines (p. 77). NASA STI/Recon Technical Report N. 10. Shen, W. Z., Mikkelsen, R., & Sørensen, J. N. (2005). Tip loss corrections for wind turbine computations. Wind Energy, 8(4), 457–475. 11. Zhong, W., Shen, W. Z., Wang, T., & Li, Y. (2020). A tip loss correction model for wind turbine aerodynamic performance prediction. Renewable Energy, 147(1), 223–238. 12. Snel, H., Houwink, R., Van Bussel, G. J. W., et al. (1993). Sectional prediction of 3D effects for stalled flow on rotating blades and comparison with measurements. In European Community Wind Energy Conference Proceedings (pp. 395–399). Lübeck Travemünde. 13. Chaviaropoulos, P. K., & Hansen, M. O. L. (2000). Investigating three-dimensional and rotational effects on wind turbine blades by means of a quasi-3D Navier-Stokes solver. Journal of Fluids Engineering, 122(2), 330–336. 14. Du, Z. H., & Selig, M. (1998). A 3-D stall-delay model for horizontal axis wind turbine performance prediction. In 1998 ASME Wind Energy Symposium. 15. Banks, W. H. H., & Gadd, G. E. (1963). Delaying effect of rotation on laminar separation. AIAA Journal, 1, 941–941. 16. Raj, N. V. (2000). An improved semi-empirical model for 3-D post-stall effects in horizontal axis wind turbines. University of Illinois Urbana-Champaign. 17. Corrigan, J. J., & Schillings, J. J. (1994). Empirical model for stall delay due to rotation. In American Helicopter Society Aeromechanics Specialists Conference Proceedings (pp. 1–15). San Francisco. 18. Lindenburg, C. (2003). Investigation into rotor blade aerodynamics. Energy Research Centre of the Netherlands (ECN), ECN-C-03-025. 19. Bak, C., Johansen, J., & Andersen, P. B. (2006). Three-dimensional corrections of airfoil characteristics based on pressure distributions. In Proceedings of the European Wind Energy Conference (pp. 1–10). Greece. 20. Larsen, J. W., Nielsen, S. R. K., & Krenk, S. (2007). Dynamic stall model for wind turbine airfoils. Journal of Fluids and Structures, 23(7), 959–982. 21. Leishman, J. G., & Beddoes, T. S. (1989). A semi-empirical model for dynamic stall. Journal of the American Helicopter Society, 34(3), 3–17. 22. Leishman, J. G., & Beddoes, T. S. (1986). A generalised model for airfoil unsteady aerodynamic behaviour and dynamic stall using the indicial method. In Proceedings of the 42nd Annual Forum of the American Helicopter Society. Washington, DC, USA. 23. Øye, S. (1991). Dynamic stall simulated as time lag of separation. In Proceedings of the 4th IEA Symposium on the Aerodynamics of Wind Turbines. 24. Petot, D. (1983). Progress in the semi-empirical prediction of the aerodynamic forces due to large amplitude oscillations of an airfoil in attached or separated flow. In 9th European Rotorcraft Forum. Italy. 25. Harris, F. D. (1966). Preliminary study of radial flow effects on rotor blades. Journal of the American Helicopter Society, 11(3), 1–21. 26. Gross, D. W., & Harris, F. D. (1969). Prediction of in-flight stalled airloads from oscillating airfoil data. In 25th Annual Forum of the American Helicopter Society. Washington, DC. 27. Guntur, S., Sorensen, N. N., Schreck, S., et al. (2016). Modeling dynamic stall on wind turbine blades under rotationally augmented flow fields. Wind Energy, 19(3), 383–397. 28. Elgammi, M., & Sant, T. (2017). A new stall delay algorithm for predicting the aerodynamics loads on wind turbine blades for axial and yawed conditions. Wind Energy, 20(9), 1645–1663.

Chapter 6

Unsteady Blade Element Momentum Method

In the preliminary design of wind turbines and the estimation of annual power generation, the steady blade element momentum (BEM) method is typically used. However, in engineering practice, atmospheric turbulence, wind shear, and tower shadow effects enhance the non-uniformity and instability of wind turbine flow. Additionally, a yawed inflow causes the wind turbine to bear asymmetric unsteady loads, thus affecting its flow structure and aerodynamic performance. Therefore, the steady BEM results in significant errors. Variable unsteady sources must be considered by modifying the aerodynamic model and using the unsteady BEM method to predict the unsteady aerodynamic loads of wind turbines.

6.1 Coordinate Transformation A wind turbine is a multibody dynamic system comprising multiple components. To determine the distribution of vectors, such as load, displacement, velocity, and acceleration in space and time, and to facilitate the conversion between various components of the wind turbine, multiple orthogonal coordinate systems must be established for the equation descriptions. In this section, eight coordinate systems are presented [1], as shown in Fig. 6.1. • Coordinate System 0 is the wind-direction coordinate system—its origin is at the bottom of the tower, and its x-axis is consistent with the wind direction. • Coordinate System 1 is the tower-bottom coordinate system—its origin coincides with that of Coordinate System 0, and its x-axis points to the south; additionally, it differs from Coordinate System 0 by wind-direction angle θ wind . • Coordinate System 2 is the tower-top coordinate system—its origin is located at the intersection of the tower axis and tower-top plane, and its x-axis is consistent with that of Coordinate System 1.

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_6

99

100

6 Unsteady Blade Element Momentum Method

Fig. 6.1 Coordinate systems of horizontal-axis wind turbine

• Coordinate System 3 is the yaw coordinate system—its origin and z-axis coincide with those of Coordinate System 2, whereas its xy plane differs from that of Coordinate System 2 by a yaw angle θ yaw . • Coordinate System 4 is the hub coordinate system—its origin is located at the center of the wind rotor, its y-axis is consistent with that of Coordinate System 3, and its x-axis points downstream along the axial direction; additionally, it is positioned at a pitch angle θ tilt from the horizontal plane. • Coordinate System 5 is the wind rotor coordinate system—its origin and x-axis coincide with those of Coordinate System 4 and rotate with the wind rotor; its yz plane differs from that of Coordinate System 4 by an azimuthal angle θ wing .

6.1 Coordinate Transformation

101

• Coordinate System 6 is the blade-root coordinate system, which is located at the center of the blade root—its y-axis is in the same direction as that of Coordinate System 5 and its xz plane differs from that of Coordinate System 5 by a cone angle θ cone . • Coordinate System 7 is a cross-sectional coordinate system—its origin is at the intersection of the pitch axis and chord line of the blade section. Meanwhile, its z-axis is consistent with that of Coordinate System 6 and points from the blade root to the blade tip, although its direction differs from that of Coordinate System 6 by a twist angle θ twist . Parameters such as scalars and vectors defined in different coordinate systems require coordinate transformation between the coordinate systems. The coordinate of each physical quantity can be transformed using a coordinate transformation matrix. Every two-connected coordinate transformation matrix is derived based on the relationship between different coordinate systems. For vectors such as velocity and force vectors, the following formula can be used to transform their coordinates: X→ B = a→ B A X→ A ,

(6.1)

where X→ A = (x A , y A , z A ) and X→ B = (x B , y B , z B ) represent the vectors in different coordinate systems, and a→ B A is the transformation matrix from coordinate system A to coordinate system B. For the coordinate point position vector, the following formula can be used to transform its coordinates: X→ B = a→ B A X→ A + r→B A ,

(6.2)

where r→B A represents the vector from the origin of coordinate system A to that of coordinate system B in coordinate system A. Meanwhile, the coordinates of the moment can be transformed as follows: → B = a→ B A M → A + r→B A × F→A , M

(6.3)

→ A and M → B represent the moment vectors in different coordinate systems, where M → and FA is the resultant force vector simplified to the origin of the coordinate system A. The transformation matrix between the wind-direction coordinate system and tower-bottom coordinate system is written as ⎡

a→10

⎤ cos θwind − sin θwind 0 = ⎣ sin θwind cos θwind 0 ⎦, 0 0 1

(6.4)

102

6 Unsteady Blade Element Momentum Method

where θ wind denotes the wind-direction angle. The transformation matrix between the tower-bottom and tower-top coordinate systems is written as ⎡

a→21

⎤ 100 = ⎣ 0 1 0 ⎦, 001

(6.5)

T where a→12 = a→21 , r→21 = (0, 0, H ), and H denotes the tower height. The transformation matrix between the tower-top and yaw coordinate systems is written as ⎡ ⎤ cos θyaw sin θyaw 0 a→32 = ⎣ − sin θyaw cos θyaw 0 ⎦, (6.6) 0 0 1

where θ yaw denotes the yaw angle. The transformation matrix between the yaw and hub coordinate systems is written as ⎡ ⎤ cos θtilt 0 − sin θtilt ⎦, a→43 = ⎣ 0 1 (6.7) 0 sin θtilt 0 cos θtilt where θ tilt denotes the pitch angle of the rotation axis, r→43 = (−xov , 0, z h ), x ov denotes the extension distance, and zh denotes the difference between the hub and tower heights. The transformation matrix between the hub and rotor coordinate systems is written as ⎡ ⎤ 1 0 0 a→54 = ⎣ 0 cos θwing sin θwing ⎦, (6.8) 0 − sin θwing cos θwing where θ wing denotes the azimuthal angle. The transformation matrix between the rotor and blade-root coordinate systems is written as ⎡ ⎤ cos θcone 0 − sin θcone ⎦, a→65 = ⎣ (6.9) 0 1 0 sin θcone 0 cos θcone where θ cone denotes the cone angle, r→65 = (0, 0, Rhub ), and Rhub denotes the hub radius.

6.2 Calculation of induced Velocity

103

The transformation matrix between the blade-root and blade-section coordinate systems is written as ⎡

a→76

⎤ cos θtwist sin θtwist 0 = ⎣ − sin θtwist cos θtwist 0 ⎦, 0 0 1

(6.10)

where θ twist denotes the twist angle, r→65 = (0, 0, r ), and r denotes the radial location of the blade section. In summary, the transformation matrix from the wind-direction coordinate system to the blade-section coordinate system is written as a→70 = a→76 · a→65 · a→54 · a→43 · a→32 · a→21 · a→10 .

(6.11)

6.2 Calculation of induced Velocity In the unsteady load calculation of wind turbines, physical variables are typically mapped to the blade-section coordinate system (Coordinate System 7) to obtain the magnitude and direction of the sectional inflow velocity; subsequently, the sectional angle of attack (AOA) and aerodynamic forces are calculated [2]. The relationship between the sectional velocities of the wind turbine, which was operated under a yawed inflow, is shown in Fig. 6.2. Based on the vector relationship of the velocity triangle, the sectional inflow velocity can be written as → = V→0 + V→rot + V→i = W

(

Vx Vy

)

( +

−Ωz cos θcone 0

)

( +

) Vi,x , Vi,y

Fig. 6.2 Schematic illustration showing relationship among the sectional velocities

(6.12)

104

6 Unsteady Blade Element Momentum Method

Fig. 6.3 Skewed wake behind rotor under yawed inflow

→ denotes the relative resultant velocity at the blade section, V→0 the wind where W speed in Coordinate System 7, V→rot the rotational velocity of the wind rotor, and V→i the induced velocity. By simplifying the rotor as an actuator disk, the airflow creates a discontinuous pressure drop at the rotor location under the yawed inflow. This pressure drop results in a thrust perpendicular to the rotor plane, which consequently generates an induced velocity in the direction of the thrust T. This induced velocity further causes a skewed rotor wake, as shown in Fig. 6.3. The relationship between the induced velocity and thrust is Vi.n = n→ · V→i =

T | |, | | 2ρ A|V→ ' |

(6.13)

| | | | | | | | where |V→ ' | = |V→0 + n→(→ n · V→i )|; n→ is a unit vector in the direction of the thrust, which is written as n→ = (−1, 0, 0) in Coordinate System 5. Assuming only the lift force L induces the induced velocity at the blade section, and that the direction of the induced velocity is opposite to the direction of the lift force, T = −L. The annular area with radius r and radial thickness dr can be expressed as dA = 2πr dr.

(6.14)

Let B denote the total number of blades. Therefore, the force-affected area of each blade element at this radial position can be expressed as dA = 2πr dr/B.

(6.15)

Based on Eq. (6.13), we can describe the effect of the induced velocity on to the blade section. The normal and tangential components of the induced velocity of each blade element can be deduced as follows:

6.3 Dynamic Inflow Model

105

Vi.n =

−B L cos φ | )| ( | |→ 4πρr F |V0 + f g n→ n→ · V→i |

Vi.t x =

−B L sin φ | )| , ( | |→ 4πρr F |V0 + f g n→ n→ · V→i |

(6.16)

where φ is the inflow angle, ρ the air density, r the radial location of the blade section, n→ the normal vector of the rotor plane, and F the Prandtl tip-loss factor. Meanwhile, f g is the correction of the high thrust under the turbulent wake state and typically refers to the empirical relationship between the thrust coefficient and axial induction factor. According to Glauert, when a > 0.4, the thrust and power coefficient expressions are no longer applicable in the steady state. Hence, based on experiments, scholars have proposed formulae for the thrust coefficients when the induced velocity is large, as described in Sect. 4.4. Using the correction of de Vries and den Blanken as an example, f g can be expressed as follows: { fg =

1 2.93a−0.53 4a 2

a ≤ 0.4 a > 0.4.

(6.17)

Thus far, the induced velocity V→i can be determined to obtain the sectional inflow → and the incident angle φ, which can be expressed as velocity W tan φ =

Wy . −Wx

(6.18)

Based on the expression α = φ − (β + θ p ), the sectional AOA can be obtained. Subsequently, the lift and drag coefficients can be obtained using the airfoil aerodynamic data, and the new induced velocity can be obtained through iterative calculation.

6.3 Dynamic Inflow Model The induced velocity varies with the transient load, which is often referred to as the equilibrium wake assumption. However, when the load changes abruptly due to abrupt changes in the pitch angle, wind speed, or rotor speed, the airflow in the flow field accelerates or decelerates, and the induced velocity will lag behind [3]. In this case, the wake behind the rotor does not reach a steady state until after a certain delay. This phenomenon is known as “dynamic inflow”. Figure 6.4 illustrates the dynamic inflow effect, where a trailed vortex forms at the blade, convects downstream at a local resultant velocity, and thereby changes the wake vortex. Because the wake vortex convects at a limited speed, the wake comprises both old and new vortices. After the old vortex propagates a distance of

106

6 Unsteady Blade Element Momentum Method

Fig. 6.4 Wake with mixed vorticity

two to four times the rotor diameter downstream, it hardly affects the rotor plane and thus attains a new equilibrium state. During this process, the induced velocity changes gradually from the original to a new equilibrium value, and this change is the basic characteristic of the dynamic inflow phenomenon. To establish a dynamic inflow model, the first-order time derivative term of the axial-induced velocity is typically incorporated into momentum theory. European scholars Snel and Schepers [4] discovered that the delay in induction causes a temporary enlargment of the blade force, particularly in the fast pitch stage. Additionally, they derived an expression for the induced velocity at radial location r, as follows: 4R f a

du i 2 + 4u i (Vw − u i ) = σ Veff Cn . dt

(6.19)

When the wind speed is constant, the expression can be written as a first-order differential equation for the axial induction factor, as follows: R da + a(1 − a) = dCd.ax /4, fa Vw dt

(6.20)

where Cd.ax is the axial force coefficient at radial location r, and f a is expressed as ∮2π f a = 2π/ 0

[1 − (r/R) cos φr ] dφr . [1 + (r/R)2 − 2(r/R) cos φr ]3/2

(6.21)

Using the above first-order differential equation makes the induced velocity to respond gradually to the change in the axial force coefficient, which is recorded as a time constant τ = VRw f a (r ). This parameter decreases with an increase in the blade-tip distance, whereas it increases with the rotor diameter. Dynamic inflow occurs during transient changes in the pitch angle and is caused by changes in the axial force coefficients. Therefore, dynamic inflow effects are typically described in terms of pitch-angle changes. In addition, dynamic inflow effects appear when the wind or rotor speed changes. However, changes in the free-flow rate barely

6.4 Dynamic Wake Model

107

result in any dynamic inflow effects. This is because although the axial induction factor varies with the wind speed, the induced velocity is barely affected by this behavior. This was observed in Snel and Schepers’ linearization-based BEM model as well as in wind-speed variation experiments performed in open-jet facilities. The dynamic inflow effect becomes more significant as the rotor size increases, which can also be inferred from Eq. (6.20); i.e., the time constant increases with the rotor diameter. Therefore, dynamic inflow is not negligible when examining the aerodynamic performance of modern megawatt wind turbines.

6.4 Dynamic Wake Model As natural winds cannot be completely stable in terms of both speed and direction, the assumptions of momentum theory are rarely satisfied precisely. A certain amount of time is required for wind to flow from the upstream of the rotor to the wake far downstream of the rotor. During this time, the state of the wind changes and a steady flow cannot be maintained. Even if the average wind speed changes gradually, a persistent unsteady phenomenon occurs around the wind turbine blades owing to small-scale turbulent flows. Therefore, a dynamic wake model must be introduced to consider the time delay before the aerodynamic loads are in equilibrium. Øye [5] proposed a dynamic wake model, where a first-order filter was used for the induced velocity. The relevant expressions are as follows: Vi,temp + τ1 Vi + τ2

dVi,temp dVi,qs = Vi,qs + kτ1 dt dt

dVi = Vi,temp , dt

(6.22)

where Vi,qs is the quasi-static solution obtained using Eq. (6.16), Vi,temp a temporary intermediate variable, Vi the induced velocity obtained using the final filter, and k = 0.6. The two time constants τ 1 and τ 2 can be calibrated using the simple vortex theory as follows: 1.1R (1 − 1.3a)V0 [ ( r )2 ] τ1 , τ2 = 0.39 − 0.26 R τ1 =

(6.23)

where R is the rotor radius, and a is the axial induction factor, which is generally estimated as | | |→ | |V0 − V→ ' | | | . (6.24) a= |→ | |V0 |

108

6 Unsteady Blade Element Momentum Method

The time constant shown in Eq. (6.23) generally requires a < 0.5. Although Eq. (6.22) can be solved using different numerical methods, it is typically solved based on the backward difference with time steps. The specific steps for solving it are as follows: (1) Calculate Vi,qs using Eq. (6.16). dV (2) Estimate the right-hand side of Eq. (6.22) as H = Vi,qs + kτ1 dti,qs . (3) Solve Eq. (6.22) analytically, where Vi,int = H + (Wi−1,int − H )e −Δt Vi,int + (Vi−1 − Vi,int )e τ2 .

−Δt τ1

and Vi =

6.5 Yaw/Tilt Model In engineering practice, owing to the direction fluctuation of natural winds, a wind turbine inevitably undergoes a continuous yaw state. If the normal direction of the rotor plane is not consistent with the direction of the wind speed, then the speed perpendicular to the rotor plane is reduced to V∞ cos θyaw , where V∞ is the incoming wind speed, and θyaw is the yaw angle, as shown in Fig. 6.5. Yaw misalignment causes load variations with azimuthal angles during each revolution; therefore, the effect of yaw should be considered when determining extreme and fatigue loads in the preliminary design of wind turbines [6, 7]. Any yaw form generates a wind-speed component in the rotor plane V∞ sin θyaw , thus resulting in highly unsteady aerodynamic loads. Unsteady aerodynamic loads on rotating blades can degrade the system reliability and cause transient variations in the output power.

Fig. 6.5 Inflow model of horizontal-axis wind turbine under yawed inflow

6.6 Calculation Steps of Unsteady BEM Method

109

Previously, Scheper fitted measured data to obtain a new axial velocity engineering model. At a positive yaw angle, the second-order Fourier series of the model is expressed as follows: u i = u i,0 [1 − A1 cos(φr − ψ1 ) − A2 cos(2φr − ψ2 )].

(6.25)

Meanwhile, at a negative yaw angle, it is expression as follows, based on symmetry: u i = u i,0 [1 − A1 cos(2π − φr − ψ1 ) − A2 cos(2π − 2φr − ψ2 )],

(6.26)

where u i,0 is the average induced velocity obtained after considering the dynamic wake model, and the amplitude A and phase angle ψ are functions of the radial position (rrel = r/R) and yaw angle θyaw , respectively. Thus far, the sectional AOA can be obtained from the calculated induced velocity; subsequently, the lift and drag coefficients can be obtained by referring to the 2D airfoil data. The lift and drag forces can be obtained using the formulas shown in Eq. (6.27). Subsequently, the aerodynamic performance of the wind turbine can be solved iteratively. 1 || → ||2 ρ |W | cCl 2 1 || → ||2 D = ρ |W | cCd . 2 L=

(6.27)

6.6 Calculation Steps of Unsteady BEM Method Using the above equations of unsteady BEM theory and the corresponding correction models, a program can be written to solve the unsteady aerodynamic performance of the wind turbine by performing the following steps: Step 1: Read the blade geometry and operating parameters; initialize the blade position and rotor speed, and discretize them into N blade elements. Step 2: Initialize the induced velocity for each blade element. →. Step 3: Use Eq. (6.12) to calculate the sectional inflow velocity W Step 4: Calculate the incident angle φ using Eq. (6.18) and obtain the sectional AOA, α = φ − (β + θ p). Step 5: Using the result obtained from Step 4 and the dynamic stall model in Sect. 5.3, refer to the 2D airfoil data to obtain the aerodynamic coefficients C l and C d and use Eq. (6.27) to obtain the lift and drag forces. Step 6: Calculate the induced velocity using Eq. (6.16).

110

6 Unsteady Blade Element Momentum Method

Step 7: Use the dynamic wake model [Eq. (6.22)] and yaw model [Eqs. (6.25), and (6.26)] to correct the induced velocity. Step 8: If the difference between the induced velocity calculated for each blade element and the result of the previous iteration exceeds the set tolerance, then return to Step 3. Otherwise, proceed to the next step. Step 9: Perform Steps 2–8 for each blade element at the next azimuth position until the induced velocity converges at all azimuthal angles during each revolution, and then proceed to the next step. Step 10: Calculate the local loads of each blade element at different azimuthal angles and integrate along the blade to obtain the thrust, blade-root bending moment, and mechanical power at different azimuthal angles. The calculations are complete. The aerodynamic loads of the NREL Phase VI blade under unsteady inflow conditions were calculated by performing the calculation steps above. During the calculation, the Prandtl model was used for blade-tip correction, the Du–Selig model for considering three-dimensional rotational effects, and the Beddoes–Leishman model for considering dynamic stall. Figure 6.6 shows the calculated normal and tangential forces when the wind speed is 10 m/s and the yaw angle is 30°, along with the wind tunnel experimental data for comparison. Figure 6.7 shows the azimuthal variation of the normal force at different wind speeds and yaw angles at the 63% span. As shown in Fig. 6.6, the mean normal force at the blade tip, as calculated by performing the steps above, is not within the error band of the experimental value, possibly because the blade-tip correction is insufficient, and the calculated values at other radial positions are consistent with the experimental data, which suggests the reliability of the unsteady BEM method.

Fig. 6.6 Comparison of aerodynamic loads at 10 m/s wind speed and 30° yaw angle

6.6 Calculation Steps of Unsteady BEM Method

111

Fig. 6.7 Comparison between BEM results and experimental data under different wind conditions

(a) 10 m/s wind speed and 10° yaw angle

(b) 10 m/s wind speed and 30° yaw angle

(c) 13 m/s wind speed and 30° yaw angle.

112

6 Unsteady Blade Element Momentum Method

References 1. Wang, T. G. (1999). Unsteady areodynamic modelling of horizontal axis wind turbine performance. University of Glasgow. 2. Hansen, M. O. L. (2008). Aerodynamics of wind turbines (2nd ed.). Routledge. 3. Leishman, J. G. (2002). Challenges in modelling the unsteady aerodynamics of wind turbines. Wind Energy, 5(2–3), 85–132. 4. Schepers, J. G. (2012). Engineering models in wind energy aerodynamics: Development, implementation and analysis using dedicated aerodynamic measurements. Delft University of Technology. 5. Øye, S. (1991). Dynamic stall simulated as time lag of separation. In Proceeding of the 4th IEA Symposium on the Aerodynamics of Wind Turbines. 6. Sørensen, J. N. (2011). Aerodynamic aspects of wind energy conversion. Annual Review of Fluid Mechanics, 43, 427–448. 7. Sathyajith, M., & Philip, G. S. (2011). Advances in wind energy conversion technology. Springer.

Part III

Vortex Wake Method

The vortex wake method is based on vortex theory and is used for analyzing the aerodynamic performance of wind turbines. When air flows through a wind turbine blade at a positive angle of attack, the pressure on the lower surface (pressure surface) of the blade is higher than that on the upper surface (suction surface), resulting in a lift force perpendicular to the inflow. For a blade with a finite span, the pressure difference between the upper and lower surfaces causes spanwise flow at the blade ends (tip or root). On the lower surface, the spanwise flow propagates toward the ends. On the upper surface, another spanwise flow occurs in the opposite direction. When the two spanwise flows merge at the trailing edge of the blade, the velocity jump generates numerous vortices that propagate downstream with the airflow. The vortex filaments roll up gradually to form concentrated tip and root vortices. The wind turbine flow field can be regarded as a result of the induction of the vortex system, which can be solved using vortex theory. In this part, we first introduce the theoretical basis of vortex theory, followed by the prescribed- and free-vortex methods.

Chapter 7

Fundamentals of Vortex Theory

Vortex motion is a typical phenomenon in nature and engineering practice. Wind turbine wakes comprise numerous complex vortices; thus, the investigation of vortex motion is critical.

7.1 Vortex Lines, Vortex Tubes, and Vortex Strength A vortex field can be described by vortex lines, similarly as the flow field can be described by streamlines. At a certain instant, the rotational angular velocity vector of any fluid microelement on a vortex line is tangent to the vortex line, as shown in Fig. 7.1. The differential equation for a vortex line is expressed as dx dy dz = = , ωx ωy ωz

(7.1)

where ωx , ω y , and ωz are the angular velocity components of the fluid microelement, which are expressed as follows: ωx = ωy = ωz =

(

)

1 ∂w − ∂v 2 ∂y ∂z ( ) ∂w 1 ∂u − 2( ∂ z ∂x ) 1 ∂v − ∂u 2 ∂x ∂y

,

(7.2)

where u, v, and w are the three components of velocity V→ .

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_7

115

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7 Fundamentals of Vortex Theory

Fig. 7.1 Vortex line in flow field

In general, ω → is a function of spatial coordinates and time, and vortex lines change → is only a function of spatial coordinates, and vortex over time. If the flow is steady, ω lines remain unchanged with time. At a certain moment in the vortex field, by selecting any smooth and closed curve that is not a vortex line (the curve must not intersect the same vortex line at two points or more) and then generating vortex lines through each point of the curve, the tubular surface formed by these vortex lines is known as a vortex tube, as shown in Fig. 7.2. The size of the vortex tube is arbitrary, i.e., it can be thick, thin, or even infinitely small. A vortex line can be regarded as a vortex tube with a cross-sectional area approaching zero. Similar to the definition of flux, the amount of vortex passing through the crosssection of a vortex tube can be defined as ∫∫ ∫∫ → · n→dσ = ⃝ ωn dσ , (7.3) ⃝ ω σ

Fig. 7.2 Vortex tube in flow field

σ

7.1 Vortex Lines, Vortex Tubes, and Vortex Strength

117

where σ denotes the cross-sectional area, and n→ is the unit normal vector of the cross-section. The vortex cannot cross the surface of the vortex tube just as the flow cannot cross the surface of the flow tube. The vortex strength, or vorticity strength, is defined as ∫∫ κ = 2 ⃝ ω → n dσ .

(7.4)

σ

In fluid mechanics, fluid motion can be classified into two categories, swirling and non-swirling, depending on whether the fluid elements exhibit rotational motion. If ∇ × V→ = 0 holds throughout the flow field, then the flow is non-swirling. Otherwise, the flow swirls. The non-swirling flow results in only translational and deformation motions, whereas the swirling flow also results in rotational motion. For non-swirling flow, ∇ × V→ = 0.

(7.5)

∇ × (∇φ) = 0.

(7.6)

If φ is a scalar function, then

This implies that the curl of the gradient of a scalar function is equal to zero. Comparing Eqs. (7.5) and (7.6), we obtain V→ = ∇φ.

(7.7)

Equation (7.7) indicates that a scalar function φ exists for a non-swirling flow such that the gradient of φ is equal to the velocity. The scalar function φ is known as the velocity potential. The function φ can be expressed as φ = φ(x, y, z) in the Cartesian coordinate system and based on the definition of the gradient as well as Eq. (7.7), we obtain → ∂φ − → ∂φ − → − → − → ∂φ − i + j + k. u→i + v j + w k = ∂x ∂y ∂z

(7.8)

u = ∂φ ∂x v = ∂φ ∂y . w = ∂φ ∂z

(7.9)

Therefore,

Notably, vortex lines are different from streamlines, although their vortex field, vortex line, and vorticity are conceptually similar to the flow field, streamline, and

118

7 Fundamentals of Vortex Theory

flow rate, respectively. Streamlines exist in all flows, whereas vortex lines only exist in flows with vortices.

7.2 Velocity Circulation and Stokes Theorem The total effect of all fluid curls in a certain flow region is represented by the velocity circulation ┌. The line integral of the velocity vector in a flow field along a specified curve is ∫B

V→ · d→s .

(7.10)

A

The variables are defined as V→ = u→i + v →j + w k→ → d→s = dx →i + dy →j + dz k.

(7.11)

Subsequently, Eq. (7.10) can be written as ∫B (udx + vdy + wdz),

(7.12)

A

where A and B are the start and end points of the specified curve, respectively. The line-integral result is typically related to the integration path from A to B. However, in a non-swirling flow, owing to the presence of the velocity potential function, ∫B A

V→ · d→s =

∫B

∫B (udx + vdy + wdz) =

A

dφ = φ B − φ A .

(7.13)

A

Based on Eq. (7.13), when the velocity potential is present in the flow field, the line integral of the velocity vector along any curve depends only on the difference in velocity potential between points B and A, i.e., it is independent of the integral path. Therefore, the most convenient path can be determined by calculating the line integral of the velocity in a non-swirling flow field. If the integration path is a closed curve, then the value of the velocity line integral is defined as the velocity circulation, i.e.,

7.2 Velocity Circulation and Stokes Theorem

119

Fig. 7.3 Velocity circulation and integral path

∮ ┌=

V→ · d→s .

(7.14)

The velocity circulation is considered to be positive in the counterclockwise inte− → gral direction, as shown in Fig. 7.3, where Vs is the projection of V→ in the direction of s→. Based on Eq. (7.13), in a non-swirling flow field, if the velocity potential is a single-valued function, then ┌ should be equal to zero, i.e., ∫ ┌=

dφ = φ A − φ B = 0.

(7.15)

According to the definition of curl, ∇ × V→ =

(

) ) ) ( ( ∂w → ∂u → ∂w ∂v → ∂u ∂v − − − i+ j+ k, ∂y ∂z ∂z ∂x ∂x ∂y

(7.16)

and its components are ( ( (

∇ × V→ ∇ × V→ ∇ × V→

) )x )y z

=

∂w ∂y

−

∂v ∂z

=

∂u ∂z

−

∂w ∂x

=

∂v ∂x

−

∂u ∂y

.

(7.17)

In Fig. 7.4, the velocity line integral for the fluid microelement abcd is defined as ∮ lim

ˆ Δ S→0

(udx + vdy) Δ┌ = lim , ˆ ˆ ˆ Δ S→0 Δ S ΔS

(7.18)

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7 Fundamentals of Vortex Theory

Fig. 7.4 Velocity components of fluid microelement boundaries

where Δ Sˆ is the area enclosed by the integral path. For any plane, Eq. (7.18) can be written as ( ) ωn = ∇ × V→ = lim n

ˆ Δ S→0

{∮

} V→ · d→s , Δ Sˆ

(7.19)

ˆ where ωn is the component of the curl perpendicular to the plane of Δ S. When segmenting any continuous surface in a flow field into small pieces, as shown in Fig. 7.5, if the flow is known, then the curl component perpendicular to a small piece can be calculated using Eq. (7.19). Applying Eq. (7.19) to the first small surface in Fig. 7.5 yields . ˆ 1= (ωn Δ S)

(∫

) → V · d→s 1

Fig. 7.5 Stokes theorem

(7.20)

7.2 Velocity Circulation and Stokes Theorem

121

ˆ i yields Summing over all the small surfaces on (ωn Δ S) k ∑

ˆ i= (ωn Δ S)

i=1

k (∫ ∑ i=1

) → V · d→s

(7.21)

i

By summing the right-hand side of Eq. (7.21), the integrals over the shared boundary of adjacent small surfaces cancel each other. Therefore, the sum of the line integrals over all small surfaces is exactly equal to the line integral around the boundary of the entire surface S. Equation (7.21) is an exact expression for each small surface when Δ Sˆ → 0, i.e., ∫ ∫∫ (7.22) ⃝ ωn dˆs = V→ · s→ = ┌. Sˆ

Equation (7.22) shows the mathematical expression for the famous Stokes theorem. Stokes’ theorem states that the circulation along any closed curve L in space is equal to the area integral of the curl on any surface bounded by the curve. According to this theorem, the vortex strength of a vortex tube can be replaced by the circulation value along a closed boundary curve of the vortex tube; thus, the circulation reflects the vortex strength. If no vorticity flux exists in the area bounded by the curve, then the circulation along the curve is zero. Equation (7.22) shows that if the velocity circulation along any closed curve in the flow field is zero, then the flow is non-swirling. For example, in point vortex motion where Vθ = k/r , the flow is non-swirling everywhere except at the origin r = 0. The integral show in Eq. (7.22) is zero only if the integral path does not contain the origin. In the point vortex motion above, if we create a velocity line integral along a streamline (r = constant) and considering that V→ · s→ = (k/s)r dθ , we obtain ∫ ┌=

V→ · s→ =

∫2π (k/r )r dθ = 2π k.

(7.23)

0

Substituting ┌/2π for k, an equivalent expression for the vortex motion can be ┌ . obtained, i.e., Vθ = 2πr The velocity potential and steam function of the vortex motion are expressed as follows: φ = (┌/2π )θ

ψ = −(┌/2π ) ln r.

(7.24)

The velocity circulation ┌ of a closed curve containing a point vortex is known as the point vortex strength.

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7 Fundamentals of Vortex Theory

7.3 Biot–Savart Law The velocity generated by vortices in a flow field is typically known as the induced velocity. For an inviscid and incompressible flow, the magnitude of the induced velocity can be determined using the Biot–Savart law, as follows: d→ v=

┌ dl→ × r→ , 4π |→ r |3

(7.25)

− → where d→ v is the induced velocity at any point in space P(x, y, z), d l is an infinitely short straight segment of a vortex element with a vortex strength of ┌, and r→ is a vector pointing from the straight segment to point P. Consider a straight vortex element with a finite length that begins and ends at A(x A , y A , z A ) and B(x B , y B , z B ), respectively, as shown in Fig. 7.6. The intensity of the vortex element is represented by the circulation ┌. Based on the Biot–Savart law, the integral form of the induced velocity generated by the vortex element at point P can be obtained as follows: ┌ V→ = 4π

∫B → dl × r→ . r3

(7.26)

A

Geometric relationships exist between r = sinh θ and dl = r dθ = sinh θ dθ . Substituting these relationships into the equation above yields the following expression for the induced velocity: ┌ V→ = (cos θ A − cos θ B )→e 4π h in which cos θ A = r B sin θ B ,

r→AB ·→ rA , r AB r A

Fig. 7.6 Diagram illustrating the Biot–Savart law

cos θ B =

r→AB ·→ rB , r AB r B

e→ =

r→AB ×→ rA , |→ r AB ×→ rA|

(7.27) and h = r A sin θ A =

7.4 Vortex Models

123

where

π ) 2

r→A = (x − x A )→i + (y − y A ) →j + (z − z A )k→

(7.28)

r→B = (x − x B )→i + (y − y B ) →j + (z − z B )k→

(7.29)

→ r→AB = (x B − x A )→i + (y B − y A ) →j + (z B − z A )k.

(7.30)

Using Eq. (7.27), the induced velocity for a semi-infinite vortex (θ B → π , θ A → can be expressed as ┌ e→ V→ = 4π h

(7.31)

If the vortex line extends to infinity at both ends (θ A → 0, θ B → π ), then the induced velocity becomes ┌ e→. V→ = 2π h

(7.32)

In this case, the flow is identical in all planes perpendicular to the vortex line. Therefore, this flow can be considered a plane flow.

7.4 Vortex Models 7.4.1 Model of Vortex Core Near the vortex line, the value of the induced velocity computed based on the Biot– Savart law increases rapidly. On the vortex line, the computational results show singular values. To avoid such unphysical results, the effect of air viscosity must be considered by introducing a vortex core model [1, 2]. The vortex core is defined as a small finite region, inside which the velocity distribution resembles the rotation of a rigid body. However, outside the vortex core, the velocity distribution is consistent with that of a point vortex flow. Lamb and Oseen established the classical Lamb–Oseen vortex model [3] by solving the one-dimensional Navier–Stokes equation. In this model, the circularinduced velocity of a vortex element is expressed as Vθ (r ) =

[ ( )] r2 ┌ 1−e − , 2πr 4νt

where ν denotes the kinematic viscosity coefficient.

(7.33)

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7 Fundamentals of Vortex Theory

Based on the equation above, the induced velocity reaches its maximum at the radius of the vortex core. In addition, it reflects the dissipation of velocity with time. Vatistas et al. [4] proposed the following algebraic expression for the circularinduced velocity of a concentrated vortex: Vθ (r ) =

r r ┌ ┌ , = ( ) 2π r 2n + r 2n 1/ n 2πrc (1 + r 2n )1/ n c

(7.34)

where, n is an integer parameter, rc the radius of the vortex core, r the radial distance / from the center of the vortex core to the calculation point, and r = r rc . Different values of n correspond to different vortex models. When n → ∞, the Rankine vortex model is represented, i.e., { Vθ (r ) =

┌ r 2πrc ┌ 1 2πrc r

0≤r ≤1 . r ≥1

(7.35)

When n = 1, the Scully vortex model [5] is represented, i.e., Vθ (r ) =

r ┌ . 2πrc 1 + r 2

(7.36)

When n = 2, the Leishman–Bagai vortex model [6] is represented, i.e., Vθ (r ) =

r ┌ √ . 2πrc 1 + r 4

(7.37)

The Rankine vortex model confines all vorticities to the interior of the vortex core; therefore, it is also known as the “concentrated” vortex model. The Scully and Leishman–Bagai vortex models are both “distributed” vortex models, where vorticity is distributed inside and outside the vortex core. The induced velocity distributions of the different vortex models mentioned above are shown in Fig. 7.7. Based on the results, the main disadvantage of the Rankine vortex model is the first-order discontinuity of its induced velocity distribution; meanwhile, the other two models achieve a smoother induced velocity distribution near the vortex core boundary. Experimental results pertaining to helicopter rotors [6] showed that the computationally induced velocity of the Leishman–Bagai vortex model agreed well with measured data. All the vortex models mentioned above are based on the assumption of laminar flow. Notably, the induced velocity distribution around a vortex in a turbulent flow differs significantly from that in a laminar flow. Vatista [7] modified the Leishman– Bagai vortex model and proposed the following turbulence-related vortex model: ) ( αT + 1 m ┌ , r Vθ (r ) = √ 2 2πrc αT + r 4

(7.38)

7.4 Vortex Models

125

Fig. 7.7 Distribution of circular-induced velocity of different vortex models

Fig. 7.8 Variation in αT with age angle of blade-tip vortex

where the parameters αT and m are turbulence scaling constants, and m = (αT + 1)/4. When αT = 1, the Leishman–Bagai model is represented. As the turbulence increases, αT decreases. Dobrev et al. [8] performed a detailed analysis of the wind turbine wake vortex structure using the Praticle Image Velocimetry (PIV) technique and obtained the variation in the parameter αT with the age angle of the blade-tip vortex, as shown in Fig. 7.8. As shown in the figure, as the age angle increases, the turbulence weakens, and αT increases from about 0.5 to about 0.8. The averaged experimental value of 0.653 was used as the value of αT and was substituted into Eq. (7.38). Figure 7.9 shows a comparison of the induced velocity distributions calculated using the laminar and turbulent flow models. As shown in the figure, after the same peak value near the vortex core boundary is attained, the induced velocity of turbulent flow decreases much slower than that of laminar flow.

7.4.2 Vortex Core Radius and Dissipation Model In a vortex model, the radius of the vortex core is an extremely important parameter. It is not only related to the specific distribution of the induced velocity in space but also to the convergence when solving the wake shape. Owing to the effect of viscosity, the vortex presents a dissipative effect. Two approaches are typically used to consider

126

7 Fundamentals of Vortex Theory

Fig. 7.9 Induced velocity distributions of laminar and turbulent models

the dissipation effects of the vortex. One is to fix the radius of the vortex core and attenuate the vortex intensity at an exponential rate. The other is to fix the vortex strength and increase the radius of the vortex core. The latter approach is introduced in this section. At the radius of the vortex core, the induced velocity of the vortex is the largest in the circumferential direction. Let the derivative of the radial distance in Eq. (7.33) be zero; therefore, the radius of the vortex core can be obtained as √

rc =

4α L νt,

(7.39)

where α L =1.25643 is the Lamb–Oseen constant, which characterizes the rate of increase of the vortex core; and ν is the laminar kinematic viscosity coefficient of the fluid. The growth rate of the vortex core in the classical Lamb–Oseen vortex model is slightly smaller than the measured data because the effect of vortex viscosity on the vortex size is disregarded. Bhagwat and Leishman [9] modified the model and derived the following equation for the vortex core radius as a function of the wake age angle: / rc (ζ ) =

r02 +

4α L δνζ , Ω

(7.40)

where ζ is the wake age angle, r0 the vortex core radius when the age angle is zero (i.e., the initial radius), ν the vortex viscosity parameter, and Ω the angular velocity of the blade. The initial radius of the vortex core r0 is generally in the same order of magnitude as the blade thickness. It can be set as 10–20% of the local chord length for the trailing and detached vortices, and 5–10% of the rotor radius for the blade-tip vortex. However, the effect of turbulence on the vortex core radius has not been elucidated. Vortex dissipation is typically determined by the viscous effect, and the growth / rate of the vortex core radius is related to the vortex Reynolds number Reν = ┌ ν. The turbulence effect on the vortex core is assumed to be evident only when the vortex Reynolds number exceeds 105 . Consequently, the vortex viscosity parameter δ is assumed to be:

7.5 Helmholtz Vortex Theorem

127

δ = 1 + a1 Rev ,

(7.41)

where a1 is the empirical constant obtained experimentally. For helicopter rotors, it is set as a1 = 2 × 104 . Experimental data are inadequate for wind turbines; therefore, the value stated above is also used for wind turbine rotors. Owing to the expansion of the wind turbine wake, vortex lines are stretched as they propagate downstream; thus, the length of the vortex elements increases. If the incompressibility assumption is adopted, then the stretching of the vortex element reduces the vortex core radius because of mass conservation. Let the vortex element length before stretching be l, the vortex core radius be rc , and the linear strain of the vortex element at Δt be ε=

Δl l

(7.42)

Subsequently, the change in the radius of the vortex core due to the stretching can be expressed as ( ) 1 Δrc = rc 1 − √ 1+ε

(7.43)

Combining Eqs. (7.40) and (7.43), the change in the effective vortex core radius with the wake age angle can be obtained as follows: ┌ | ∫ζ | | 2 4α L δνζ rc (ζ ) = √r0 + (1 + ε)−1 dζ Ω

(7.44)

0

7.5 Helmholtz Vortex Theorem Helmholtz theorems are applicable to vortex flow. Theorem I: The strength of a vortex filament is constant along its length. This theorem is known as Helmholtz’s first theorem. At a certain instant, a slotted cylinder encloses a vortex line in the flow field, as shown in Fig. 7.10. If no swirl exists everywhere except at the vortex line, then the curl at any point on the slotted cylinder is zero. Therefore, the velocity integral along the closed curve of the cylinder is zero. This requires that the integral along a → b to be equal in magnitude and opposite in sign to the integral along c → d. Hence, the vortex strengths of the vortex lines passing through the upper and lower faces of the cylinder should be exactly the same. Because the upper and lower faces of the cylinder are arbitrarily selected, the vortex strength along the vortex line is constant.

128

7 Fundamentals of Vortex Theory

Fig. 7.10 Slotted cylinder enclosing a vortex line

Theorem II: A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path. This theorem is known as Helmholtz’s second theorem. For example, when conducting experiments in a two-dimensional wind tunnel, the spanwise vortex lines of the tested wing can be terminated on the sidewalls of the tunnel. In a threedimensional case, the vortex lines of the wing can only propagate downstream at the end of the wing and become wake vortices that extend to infinity. Theorem III: In the absence of rotational external forces, a fluid that is initially irrotational remains irrotational. In an inviscid flow, any fluid element is only subjected to the normal force perpendicular to its surface and not to the tangential force; therefore, no external force is available to rotate the element. In this case, the fluid will never become swirling if it is originally non-swirling, or the vortex strength will remain unchanged if the fluid is originally swirling. Actual fluids are viscous, and their vortex strength varies with time. However, the viscosity of air is low, and the decay of the vortex strength due to viscosity is insignificant. Therefore, air can be generally regarded as an ideal fluid with no decay in vortex strength.

7.6 Kutta–Joukowski Lift Theorem

129

7.6 Kutta–Joukowski Lift Theorem 7.6.1 Flow Around a Cylinder The flow formed by a uniform flow, dipole, and point vortex is shown in Fig. 7.11. The potential function φ and flow function ψ of such a flow are expressed as follows: ) ( 2 ┌ θ φ = v∞ r + ar cos θ − 2π ) ( 2 ┌ ψ = v∞ r − ar sin θ + 2π r

(7.45)

Based on Eq. (7.45), ψ is constant along a circle of radius a centered at the origin. Therefore, the circle is a streamline, as shown in Fig. 7.11. The velocity at any point in the flow field can be obtained via differentiation, which results in the following: ( vx = v∞ 1 − 2

a2 r2

) cos 2θ +

v y = −v∞ ar 2 sin 2θ −

┌ sin θ 2π r ┌ cos θ 2π r

.

(7.46)

On a cylindrical surface, r = a. Substituting this relation into Eq. (7.46) results in the following expressions for the velocity distribution on a cylindrical surface: ┌ vx = v∞ (1 − cos 2θ ) + 2π ┌ cos θ v y = −v∞ sin 2θ − 2π a

sin θ a

.

(7.47)

Therefore, on a cylindrical surface, the radial velocity vr , circumferential velocity vθ , and resultant velocity v are written as

Fig. 7.11 Flow around a cylinder

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7 Fundamentals of Vortex Theory

vr = 0 ┌ . vθ = −2v∞ sin θ − 2πa ┌ v = −2v∞ sin θ − 2π a

(7.48)

At the stagnation point, the resultant velocity v is equal to zero, and the angle θs is expressed as θs = sin

−1

(

) ┌ − . 4πav∞

(7.49)

/ Based on the equation above and sin θs = ys a, the stagnation point is located at √ xs = ± a 2 − ys2 ┌ ys = − 4πv ∞

(7.50)

Based on Eq. (7.50), two stagnation points propagate downward as the strength of the point vortex increases. As the strength of the point vortex continues to increase to ┌ = 4πv∞ a, the two stagnation points coincide at (0, −a) on the y-axis. As the strength of the point vortex increases further, Eq. (7.50) no longer holds, and the stagnation points depart from the cylindrical surface. Figure 7.12 presents a diagram that shows the locations of the stagnation points for various point vortex strengths. As shown in Fig. 7.12, the flow spectrum is symmetrical from left to right but not from top to bottom. Therefore, a force should be applied in the direction perpendicular to the incoming flow. This force is known as the lift force, which can be obtained by integrating the pressure along the surface of the cylinder.

Fig. 7.12 Stagnation points for various point vortex strengths (the point vortex is clockwise, and the incoming flow is from left to right)

7.6 Kutta–Joukowski Lift Theorem

131

7.6.2 Circulation and Lift Based on the origin as the center, a large control surface S with radius r1 can be constructed. Supposed that the entire control surface includes the cylinder surface S1 and two split lines (dotted lines in Fig. 7.11) connecting S and S1 . The changes in pressure or momentum on the two split lines cancel each other out; therefore, they do not affect the overall result and can be disregarded. The aerodynamic force exerting on S1 is the resultant force, which is symmetrical from left to right in the case investigated. Therefore, drag force is absent, and only a force exerting on the cylindrical surface (i.e., the lift force) exists, which is denoted as Y . Its value can be obtained based on momentum theorem, as follows: ∫ ∫ Y = − p cos(→ n , y)ds − pvn v y ds, (7.51) S

S

where vn is the partial velocity perpendicular to the control surface, and n→ is the unit vector in the normal direction of the control surface. The integration shown in Eq. (7.51) is performed along the circle with radius r1 , which is exactly the control surface S. Therefore, the following relation applies to the circle: cos(→ n , y) = sin θ . ds = r1 dθ

(7.52)

Consequently, π

π

∫2 Y = −2

∫2 r1 p sin θ dθ − 2

− π2

ρr1 vn v y dθ .

(7.53)

− π2

The pressure term in Eq. (7.53) can be replaced with a velocity term using the Bernoulli equation. Furthermore, using the equation obtained in Sect. 7.6.1 for the velocity around the cylinder, the first term on the right-hand side of Eq. (7.53) can be written as π

∫2 −2

r1 p sin θ dθ =

− π2

) ( 1 a2 ρv∞ ┌ 1 + 2 . 2 r1

(7.54)

Let r = r1 ; therefore, v y in the second term is written as vy = −

┌ cos θ v∞ a 2 sin 2θ − 2π r1 r12

(7.55)

132

7 Fundamentals of Vortex Theory

and the second term of integration is expressed as ∫ −2

π 2

− π2

) ( 1 a2 ρr1 vn v y dθ = ρv∞ ┌ 1 − 2 . 2 r1

(7.56)

Hence, the lift can be calculated as Y =

1 ρv∞ ┌ 2

) ( )] [( a2 a2 = ρv∞ ┌. 1+ 2 + 1− 2 r1 r1

(7.57)

Equation (7.57) expresses the lift force exerting on a unit-length cylinder in terms of the product of the fluid density, incoming velocity, and circulation. This result is known as the Kutta–Joukowski theorem.

References 1. Xu, B., Feng, J., Wang, T., et al. (2018). Application of a turbulent vortex core model in the free vortex wake scheme to predict wind turbine aerodynamics. Journal of Renewable and Sustainable Energy, 10(2), 023303. 2. Xu, B. F., Yuan, Y., Wang, T. G., et al. (2016). Comparison of two vortex models of wind turbines using a free vortex wake scheme. Journal of Physics: Conference Series, 753(2), 022059. 3. Lamb, H. (1993). Hydrodynamics. Cambridge University Press. 4. Vatistas, G. H., Kozel, V., & Mih, W. C. (1991). A simpler model for concentrated vortices. Experiments in Fluids, 11(1), 73–76. 5. Scully, M. P. (1967). A method of computing helicopter vortex wake distortion. In Massachusetts institute of technology Cambridge aeroelastic and structures research lab. 6. Bagai, A., & Leishman, J. G. (1993). Flow visualization of compressible vortex structures using density gradient techniques. Experiments in Fluids, 15(6), 431–442. 7. Vatistas, G. H. (2006). Simple model for turbulent tip vortices. Journal of Aircraft, 43(5), 1577– 1579. 8. Dobrev, I., Maalouf, B., Troldborg, N., et al. (2018). Investigation of the wind turbine vortex structure. In: Proceedings of the 14th international symposium on applications of laser techniques to fluid mechanics, Lisbon, Portugal (pp. 7–10). 9. Bhagwat, M. J., & Leishman, J. G. (2000). Correlation of helicopter rotor tip vortex measurements. AIAA Journal, 38(2), 301–308.

Chapter 8

Computational Models of Vortex Wake

This chapter primarily presents the models used in the vortex wake method, i.e., the prescribed-vortex wake (PVW) and free-vortex wake (FVW) models. In the PVW model, an empirical description function of the wake shape is established based on a significant amount of experimental data pertaining to the wake of wind turbines. The description function can be a function of the induced velocity factor or the blade circulation. During the solution process, the new induced velocity and circulation are computed based on the wake shape, and then the induced velocity and wake shape are iteratively solved until the flow field converges. The PVW model is applicable to various cases, such as for computing the steady aerodynamic performance of wind turbines [1, 2]; combining with a dynamic stall model to compute the aerodynamic performance of wind turbines under unsteady conditions, such as in the yaw state [3, 4]; analyzing the effect of tower shadows on downwind rotors [5, 6]; and predicting the stall-delay effect of a rotor using a three-dimensional (3D) rotational effect model [7, 8]. The PVW model offers the advantage of high computational efficiency. However, it cannot simulate the distortion of the wake shape or the rolling-up of tip vortices. These deficiencies have limited the development of PVW model. Compared with the PVW model, the FVW model does not require the predetermination of the vortex shape using experimental data. The vortex filaments in the wake can be deformed unrestrictedly under the effect of the local velocity. The FVW model can represent the distortion and rolling-up of the wake; thus, it has become an important method for analyzing the aerodynamic characteristics of wind turbines [9–14]. For analyzing the complex flow fields of wind turbines, the FVW model presents

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_8

133

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8 Computational Models of Vortex Wake

greater advantages than the PVW model [15]. However, it also incurs to a higher computational cost. Therefore, various simplified FVW models have been proposed to reduce the computational cost [9, 16–19]. Additionally, the FVW model has been investigated extensively, including the effects of flow separation [20], rolling-up of the wake, nonlinear aerodynamic force of the blade [21], and unsteady aerodynamic characteristics of wind turbines [22–24].

8.1 Definition of Coordinate Systems The wake shape of a rotor is constructed based on the Cartesian coordinate system (x, y, z) and cylindrical coordinate system (r, ψ, z) of the rotor, as shown in Fig. 8.1a. The origins of the coordinate systems are located at the rotation center in the rotor plane. In the Cartesian coordinate system, from the upwind side, the x-axis points horizontally to the right. The y-axis is perpendicular to the x-axis and points downward. The z-axis is determined by the right-hand rule and points downwind. In the cylindrical coordinate system, r is along the pitch axis of the blade, pointing outside the blade span. The clockwise direction is defined as positive for ψ, and ψ = 0 when the r-axis coincides with the x-axis. The z-axis coincides with that of the Cartesian coordinate system. Because the wake rotates around the z-axis, it is more convenient to use the rotor cylindrical coordinate system to represent the position of the wake nodes. When solving for the entire flow field, the coordinates of the nodes must be transformed into those of the Cartesian coordinate system. The conversion relationship between the two systems is as follows: ⎧ ⎨ x = r cosψ y = r sinψ ⎩ z=z

(8.1)

The wake coordinate system is a cylindrical coordinate system, which is introduced to represent the positions of the nodes of vortex elements, as shown in Fig. 8.1b. The positive direction of rw (the subscript w represents wake) points outward, and the positive direction of the age angle ζ is counterclockwise. Furthermore, ζ = 0 when the rw -axis coincides with the r-axis of the blade. When solving the entire flow field, the node coordinates must be converted into the Cartesian coordinate system, and the conversion relationship between the systems is as follows: ⎧ ⎪ ⎨ x = rw cos(ψ − ζ ) y = rw sin(ψ − ζ ) . ⎪ ⎩ z=z

(8.2)

When the wind turbine is in a non-axial inflow, the Cartesian coordinate system (x, y, z) of the wind is used to define the wake structure. The z-axis is defined as

8.2 Models of Vortices

135

Fig. 8.1 Definition of coordinate systems: a Cartesian and cylindrical coordinate systems of the rotor; b wake coordinate system

the direction of the inflow velocity, and the angle between the z-axis and rotation axis of the rotor is known as the yaw angle γ . The direction of the y-axis is defined as “plumb down.” Viewed upwind, the x-axis points horizontally to the right. When the x component of the inflow velocity is inpositive, the yaw angle γ is defined as positive. In the case of non-axial inflow, the Cartesian coordinate system of the rotor is written as (x', y', z'), as shown in Fig. 8.2. The parameters in the wind coordinate system (x, y, z) can be converted into the rotor coordinate system (x', y', z') as follows: ⎡ ⎤ ⎡ ⎤⎡ ⎤ x' cos γ 0 sin γ x ⎣ y' ⎦ = ⎣ 0 ⎦ ⎣ (8.3) 1 0 y ⎦. z' − sin γ 0 cos γ z

8.2 Models of Vortices 8.2.1 Models of Vortices Attached to Blades The aerodynamic models for a wind turbine blade include the lift-line and lift-surface models. In the lift-line model, the blade is represented by a vortex line (lift line) attached to the blade, and the strength of the vortex line varies spanwise. The aerodynamic performance of each section of the blade is assumed to be independent of each other; thus, the flow of each section is a two-dimensional (2D) flow. However, the 2D performances of different sections are allowed to differ. In other words, the flow is

136

8 Computational Models of Vortex Wake

Fig. 8.2 Definition of coordinate system for non-axial inflow conditions

regarded as 2D for individual sections of the blade but three-dimensional (3D) for the entire blade. Such a model is known as a quasi-2D model. The lift-line model is relatively simple and suitable for engineering applications; however, it cannot adequately consider the 3D effect of the blade tip. In the lift-surface method, the thickness of the blade is disregarded, and the blade is segmented into many quadrilateral surface elements. However, this method requires a significant amount of computation. In addition, it cannot reflect the viscous effect because it does not use the experimental data of airfoils. A simplified lift-surface model, known as the Weissinger–L (W– L) lift-surface model, is introduced in this chapter. The model associates the blade lift with the attached circulation using the Kuta–Zhukovsky theorem. The strength of the trailing vortex of a blade is determined by the circulation variation in the blade-attached vortex. In the W–L lift-surface model, the lift characteristics of the blade are simulated by an attached vortex line located at the line of the 1/4 chord of the blade, as shown in Fig. 8.3. The blade is partitioned into many segments along the span direction, and the circulation of each segment vortex is constant. Each blade element corresponds to a control point located at the 3/4 chord length of the blade element. A blade span can be segmented into several elements. Let the number of blade elements be NE . Considering that the spanwise variation in the blade circulation is more significant in the blade-tip area, the cosine distribution function can be used for

8.2 Models of Vortices

137

Fig. 8.3 Schematic diagram showing blade discretization

the segmentation; therefore, the segmented elements are denser at locations closer to the blade tip. If the radius of the rotor is R, and the radial position of the blade root is Rt , then the radial dimensionless positions of the boundary points of the blade elements can be determined as follows: { Rt i =1 ( ) ( ) 2 i−1 −1 , (8.4) r bp i = ( ) 1− −0.1 cos π NE 1 − R t i = 2, 3, . . . , N E + 1 Rt + 0.9 where R t = Rt /R. The radial dimensionless positions of the control points of the blade elements are determined as follows: [( ) ( ) ( ) ] r cp i = 21 r bp i + r bp i+1 i = 1, 2, . . . , N E . (8.5) If the circulation of the blade attached to the vortex line at a blade element is ┌b , then the lift force of the blade element can be determined based on the Kutta– Joukowski theorem: dL = ρW ┌b dr,

(8.6)

where W denotes the resultant velocity. According to blade element theory, the lift force can be expressed as dL =

1 ρW 2 cCl dr, 2

(8.7)

where c and Cl denote the chord length and lift coefficient of the blade element, respectively. Based on Eqs. (8.6) and (8.7), the attached vortex circulation of the ith blade element can be expressed as (┌b )i =

1 Wi Cl ci 2

(8.8)

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8 Computational Models of Vortex Wake

8.2.2 Models of Wake Vortices 1. Circulations of Wake Vortices According to the Helmholtz theorem, the circulation variation between adjacent blade elements must be separated from the blade in the form of trailing vortices at the boundary of the blade elements. The time-dependent variation in the circulation of the attached vortex is detached from the blade in the form of a detached vortex. The forms and discretization of the wake vortices are shown in Fig. 8.4. The strength of the trailing vortices is defined as the difference between the attached vortex circulations of adjacent blade elements; thus, the strength of the trailing vortex corresponding to the ith blade element boundary is

Fig. 8.4 Schematic diagram showing discretization of wake vortices

8.2 Models of Vortices

(┌t )i, j

139

⎧ i =1 ⎨ (┌b )1, j = (┌b )i, j − (┌b )i−1, j i = 2, 3, . . . , N E , ⎩ i = NE + 1 (┌b ) N E , j

(8.9)

where j = 1, 2, . . . , NT is the azimuth angle. The strength of the detached vortex is defined as the difference in the attached vortex circulation between the blade elements at adjacent azimuth angles. Therefore, the strength of the detached vortex at the jth azimuth angle is { (┌s )i, j =

(┌b )i,1 − (┌b )i,NT j = 1 , (┌b )i, j − (┌b )i, j−1 j = 2, 3, . . . , N T

(8.10)

where i = 1, 2, . . . , N E . 2. Wake Segmentation The wake of a wind turbine can be separated into two regions: near- and far-wake regions. The change in wake geometry is primarily performed in the near-wake region; meanwhile, the far-wake region represents the equilibrium state of the region far downstream. When segmenting the near- and far-wake regions, the physical reality and computational cost must be considered. If the segmented near-wake region is larger than the segmented far-wake region, then the calculation accuracy will be higher, but the number of vortex elements will increase significantly, thus resulting in a longer calculation time. In the PVW model, a near-wake region indicator Tnw is defined, which represents the time spent by the trailing vortex to propagate from the blade to the end of the near-wake region. Tnw is defined as a function of tip-speed ratio λ = ΩR/V0 , as follows: Tnw =

7π λ . 4Ω

(8.11)

The far-wake region extends downstream from the end of the near-wake region to infinity. Because the far-wake region represents the ultimate equilibrium state of the wake, it is regarded as a cylindrical axisymmetric flow field. According to one-dimensional momentum theory, the axial-induced velocity far downstream of the wake is twice the axial-induced velocity at the rotor, i.e., (vz )1 = 2vz .

(8.12)

To reflect the effect of the radial velocity gradient in the wake, the relationship above is not used in the PVW method. Instead, a factor F is introduced to describe the axial-induced velocity far downstream, i.e., (vz )1 = Fvz .

(8.13)

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8 Computational Models of Vortex Wake

When t ≥ Tnw , the flow field reflects the condition of the far wake, and the following are assumed: (1) The axial-induced velocity reaches Fvz and remains constant. (2) The radial-induced velocity decreases to zero and remains unchanged. In the FVW model, the region within three times the diameter of the rotor is usually defined as the near-wake region. The number of turns NC of a vortex in the near-wake region can be determined as follows: (

ΩD NC = int π V0

) + 1,

(8.14)

where D is the rotor diameter, and V0 is the wind speed. As shown in Eq. (8.14), NC is correlated with the tip-speed ratio λ.

8.3 PVW Model The axial velocity factor of the far wake in the PVW method can be expressed as follows: F = 1.1426 + 5.1906r − 8.9882r 2 + 4.0263r 3 .

(8.15)

The axial flow velocity in the entire wake can be described as follows: ⎧ 1 − a − 21 t ≤ 17 − 1)at ⎪ 5 (F ⎪ ⎨ 7 1 ( ) 1 − 2 (1 + F)a − 10 (F − 1)at 17 < t ≤ 47 Vz w = , 7 4 ⎪ 1 − 7+23F a − 30 (F − 1)at 1 1 − Fa

(8.16)

where a is the axial-induced velocity factor at the blade element boundaries. a=−

vz . V0

(8.17)

( ) Here, t and V z w are the dimensionless time and wake axial velocity, respectively. t= (

Vz

) w

t Tnw

=

(Vz )w . V0

(8.18) (8.19)

8.3 PVW Model

141

Based on Eq. (8.16), the near wake (t = 0 → 1) is partitioned into three subregions, and the axial flow velocity in each subregion is a linear function of time. The axial velocity in the far wake (t > 1) is equal to that at the end of the near wake. The axial displacement of the wake is obtained by integrating the axial velocity with time, as follows: ⎧ 2 7π ⎪ t ≤ 17 − 1)at (1 − a)t − 147π ⎪ 40 ( (F ⎪ ) ⎨ π4 2 1 a t − 49π < t ≤ 47 1 − 1+F (F − 1)a + 7π (F − 1)at 4 ( 2 7 ) 80 49π , (8.20) z w = 16 2 4 47π 7π 7+23F ⎪ − 1)a + 4 1 − 30 a t − 240 (F − 1)at 7 < t ≤ 1 ⎪ 240 (F ⎪ ⎩ 2π t >1 − 1)a + 7π − Fa)t 5 (F 4 (1 where zw =

zw . R

(8.21)

The radial-induced velocity of the wake is calculated as follows: { (vr )w =

[ ] vr 1 − t(2 − t) t ≤ 1 , t >1 0

(8.22)

where vr is the dimensionless radial-induced velocity at the blade element boundary, and (vr )w is the dimensionless radial-induced velocity in the wake. vr = (vr )w =

vr V0

(8.23)

(vr )w . V0

(8.24)

The radial displacement rw of the wake can be obtained by integrating Eq. (8.22). rw = rw = R

{

)] [ ( t ≤1 r + 47 πvr t 1 − t 1 − 3t r+

7 πvr 12

t >1

,

where r is the radial location of the dimensionless blade element boundary.

(8.25)

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8 Computational Models of Vortex Wake

During the generation of the initial wake structure, the radial-induced velocity at the blade cannot be obtained because a full wake has not been generated. In this case, the radial displacement of the wake cannot be obtained using Eq. (8.25) but can be expressed as follows: ⎧ r + 21 t ≤ 17 (r 1 − r )t ⎪ 5 ⎪ ⎨1 7 1 < t ≤ 47 (r + r 1 ) + 10 (r 1 − r )t 7 r w = 21 , 7 4 ⎪ 30 (7r + 23r 1 ) + 30 (r 1 − r )t 7 < t ≤ 1 ⎪ ⎩ t >1 r1

(8.26)

where r 1 is the radial position of a dimensionless wake element, which can be determined using the continuity equation, / r1 =

1−a · r. 1 − Fa

(8.27)

Notably, Eq. (8.26) is only applied for constructing the initial wake structure.

8.4 FVW Model 8.4.1 Governing Equations for Vortex Filaments For a potentially inviscid incompressible flow, the dynamic equation for vorticity can be reduced to the Helmholtz equation, as follows: D ω (⇀ ) ⇀ − ω ·∇ V = 0. Dt ⇀

(8.28)

The wake vortex behind the rotor is replaced by a vortex filament with a zero cross-sectional area. According to Helmholtz’s second vortex theorem, the vortex filament propagates with the flow; therefore, the governing equation of the vortex filament can be obtained as follows: ⇀

) ⇀( ⇀ d r (ψ, ζ ) = V r (ψ, ζ ), t , dt loc

(8.29)

− → where ψ denotes the azimuth angle of the blade, ζ the wake age angle, and V loc the local velocity. A complete differential expansion of the left side of the equation along the azimuth angle and age angle yields

8.5 Flow Field Computation

143 ⇀

⇀

⇀

d r (ψ, ζ ) ∂ r (ψ, ζ ) ∂ψ ∂ r (ψ, ζ ) ∂ζ = + . dt ∂ψ ∂t ∂ζ ∂t

(8.30)

The local velocity on the right side of Eq. (8.29) includes the inflow and induced velocities, as follows: ) ⇀ ⇀( ⇀ ) ⇀( ⇀ V r (ψ, ζ ), t = V + V r (ψ, ζ ), t ,

loc

∞

(8.31)

ind

− → → r is the position vector of the vortex filament control point, V ∞ the inflow where − − → velocity, and V ind the sum of the induced velocities of all vortex filaments at the control point. = dζ = Ω, the governing equation for the vortex filaments in Considering dψ dt dt partial differential form can be written as ] 1[→ ∂ r→(ψ, ζ ) ∂ r→(ψ, ζ ) + = V∞ + V→ind (→ r (ψ, ζ ), t) . ∂ψ ∂ζ Ω

(8.32)

8.4.2 Description of Initial Wake The initial wake may affect the stability and speed of the iterative solution. The initial wake can be a rigid or prescribed wake. In general, a cylindrical rigid wake with equal pitches is sufficient. Let the azimuth angle of the blade be ψ, the age angle of a node in the wake corresponding to the ith blade element be ζ , and the coordinates of the node in the Cartesian coordinate system be defined as follows: ⎧ ) ( ( ) ⎪ ⎨ x w (i, j, k) = (r bp )i cos( ψ j − ζk) y w (i, j, k) = r bp i sin ψ j − ζk . ⎪ ⎩ z (i, j, k) = Δt ζk V0 w Δζ R

(8.33)

8.5 Flow Field Computation 8.5.1 Wake Discretization As described in Sect. 8.2.1, the blade is discretized into N E elements with N E + 1 element boundaries. The first boundary corresponds to the position of the blade root, and the N E + 1 boundary corresponds to the position of the blade tip. Each blade element is replaced by an attached vortex segment located at the 1/4 chord length.

144

8 Computational Models of Vortex Wake

The positions of the control point and the boundary point of the ith blade element are denoted as (r cp )i and (r bp )i , respectively. To further discretize the flow field, one revolution of the rotor is segmented equally into N T time steps, and the azimuth angle of the jth time step is expressed as ψj =

2π ( j − 1), NT

(8.34)

where j = 1, 2, . . . , N T , and the azimuth angle of the kth blade at the jth time step is expressed as ψ j,k = ψ j +

2π (k − 1). B

(8.35)

Here, B is the number of rotor blades, and k = 1, 2, . . . , B. The wake contains N E + 1 helical vortex filaments. The vortex filament is formed by a series of straight-segment vortex elements, and each vortex element corresponds to a time step. When a vortex element propagates sufficiently far downstream, its induction on the blade is negligible. Therefore, the far wake is truncated after NC revolutions, and the wake after the truncation point is disregarded. Thus, each wake vortex filament contains (N T · NC ) vortex elements, and the position and length of each vortex element are determined by the nodes at its two ends. The time step can be determined as follows: 2π . ΩN T

(8.36)

Δt 8 = . Tnw 7λN T

(8.37)

Δt = Its dimensionless form is Δt =

Subsequently, the total time for a vortex element to propagate from the blade to the far end of the wake is expressed as T = N T NC Δt =

8NC . 7λ

(8.38)

The time corresponding to the jth vortex element on the nth revolution of the vortex filament is expressed as t j,n = [ j − 1 + (n − 1)N T ]Δt,

(8.39)

where n = 1, 2, . . . , NC . [( ) ( ) ( ) ] The Cartesian coordinates x bp i, j , y bp i, j , z bp i, j of the blade element boundary points can be expressed as

8.5 Flow Field Computation

145

⎧ ⎨ (x bp )i, j = (r bp )i cos ψ j (y ) = (r bp )i sin ψ j . ⎩ bp i, j (z bp )i, j = 0

(8.40)

After time t j,n , the coordinates of a vortex node in the wake are expressed as follows: ⎧ ⎨ x w (i, j, k, n) = (r w )i, j,k,n cos ψ j,k (8.41) y (i, j, k, n) = (r w )i, j,k,n sin ψ j,k , ⎩ w z w (i, j, k, n) = (z w )i, j,k,n where (z w )i, j,k,n and (r w )i, j,k,n are the axial and radial displacements of the node corresponding to the ith element of the kth blade, respectively. The last node at the end of one revolution of the vortex filament is the first node of the next revolution, and this relationship can be expressed as follows: x w (i, N T + 1, k, n) = x w (i, 1, k, n + 1) y w (i, N T + 1, k, n) = y w (i, 1, k, n + 1) z w (i, N T + 1, k, n) = z w (i, 1, k, n + 1).

(8.42)

8.5.2 Computation of Attached Vortex Circulation The Cartesian coordinates of the blade control point corresponding to the ith blade element in the jth phase are as follows: (

( ) = r cp i cos ψ j

(8.43)

(y cp )i, j = (r cp )i sin ψ j

(8.44)

(z cp )i, j = 0.

(8.45)

x cp

) i, j

The induced velocity at this control point is expressed as cp cp cp → (vcp )i, j = (vx )i, j →i + (v y )i, j →j + (vz )i, j k.

(8.46)

Its dimensionless form is (vcp )i, j cp cp cp → = (v x )i, j →i + (v y )i, j →j + (v z )i, j k. V0

(8.47)

146

8 Computational Models of Vortex Wake cp

Subsequently, the radial-induced velocity (vr )i, j and tangential-induced velocity cp (v ψ )i, j can be expressed as [

cp

(vr )i, j cp (v ψ )i, j

]

[ =

cp

(vr )i, j /V0 cp (vψ )i, j / V0

]

[

cos ψ j sin ψ j = − sin ψ j cos ψ j

][

] cp (v x )i, j cp . (v y )i, j

(8.48)

cp

cp

Meanwhile, the axial inflow velocity (Vz )i, j and tangential inflow velocity (Vψ )i, j can be expressed as ( (

Vψ

Vz

)cp i, j

cp

)cp

=

i, j

( =

(Vz )i, j

Vψ

cp

= 1 + (v z )i, j

V0 )cp i, j

V0

(8.49)

( ) ( )cp = λ r cp i − v ψ i, j .

(8.50)

cp

The resultant inflow velocity Wi, j is expressed as cp

cp W i, j

=

Wi, j V0

=

/[

cp

(vr )i, j

]2

+

[(

Vψ

)cp ]2 i, j

+

[(

Vz

)cp ]2 i, j

.

(8.51)

The strength of the attached vortex (┌b )i, j is expressed as ( ) (┌b )i, j 1 cp = ┌ b i, j = W ci (Cl )i, j , 4π V0 R 8π i, j

(8.52)

where ci = ci /R. Here, ci is the chord length of the ith blade element, and (Cl )i, j is the 2D lift coefficient of the ith blade element at the angle of attack αi, j , which is expressed as αi, j = φi, j − θi .

(8.53)

Here, θi is the pitch angle of the ith blade element, and φi, j is the inflow angle. ( φi, j = tan

−1

(

Vz

)cp

Vψ

i, j

)cp .

(8.54)

i, j

8.5.3 Computation of Rotor Aerodynamic Performance Using the angle of attack αi, j , blade element lift coefficient (Cl )i, j , and blade element drag coefficient (Cd )i, j , the normal force coefficient (Cn )i, j and tangential force

8.5 Flow Field Computation

147

coefficient (Ct )i, j of the blade element can be expressed as [

(Cn )i, j (Ct )i, j

]

[ =

cos φi, j sin φi, j sin φi, j − cos φi, j

][

(Cl )i, j (Cd )i, j

] (8.55)

and [

(Cn ')i, j (Ct ')i, j

]

[

cos αi, j sin αi, j = sin αi, j − cos αi, j

][

] (Cl )i, j . (Cd )i, j

(8.56)

In Eq. (8.55), the direction of (Cn )i, j is perpendicular to the rotor plane, and the direction of (Ct )i, j is parallel to the rotor plane. In Eq. (8.56), the direction of (Cn ')i, j is perpendicular to the chord of the blade element, and the direction of (Ct ')i, j is parallel to the chord. Let ⎧ N ⎨ l = j + BT (k − 1) (8.57) j = 1, 2, . . . , N T + 1 ⎩ k = 1, 2, . . . , B { l l ≤ NT m= . (8.58) l − NT l > NT Subsequently, 2

B NE [ ] ( ) ( ) [( ) ( )] 1 ∑∑ cp CQ j = W i,m (Ct )i, j ci r cp i r bp i+1 − r bp i π k=1 i=1

(8.59)

2

B NE [ [( ) ] ( )] 1 ∑∑ cp W i,m (Cn )i, j ci r bp i+1 − r bp i (C T ) j = π k=1 i=1

(8.60)

and ( ) (C P ) j = λ C Q j .

(8.61)

The thrust coefficient C T and torque coefficient C Q generated on the rotor are calculated as follows: CT =

NT / B T B ∑ = (C T ) j 1 N T j=1 ρV02 π R 2 2

(8.62)

CQ =

NT / B Q B ∑ (C Q ) j . = 1 N T j=1 ρV02 π R 3 2

(8.63)

148

8 Computational Models of Vortex Wake

Meanwhile, the power coefficient of the rotor is calculated as follows: CP =

P 1 ρV03 π R 2 2

= λC Q .

(8.64)

8.5.4 Computation of Induced Velocity A vortex element is defined[ by nodes at its two ends. The ]coordix w (i, j, k, n), y w (i, j,] k, n), z w (i, j, k, n) and [nates of the nodes are x w (i, j + 1, k, n), y w (i, j + 1, k, n), z w (i, j + 1, k, n) . Let [ ] r→A = [x − x w (i, j, k, n)]→i + y − y w (i, j, k, n) →j + [z − z w (i, j, k, n)]k→

(8.65)

[ ] r→B = [x − x w (i, j + 1, k, n)]→i + y − y w (i, j + 1, k, n) →j + [z − z w (i, j + 1, k, n)]k→

(8.66)

and

Therefore, r→AB = [x w (i, j + 1, k, n) − x w (i, j, k, n)]→i [ ] + y w (i, j + 1, k, n) − y w (i, j, k, n) →j → + [z w (i, j + 1, k, n) − z w (i, j, k, n)]k,

(8.67)

where →i, →j, and k→ are unit vectors in the x-, y-, and z-directions, respectively. According to the Biot–Savart law, the induced velocity v→i, j,k,n of a trailing vortex element with strength ┌t at point (x, y, z) can be calculated as v→i, j,k,n

) ( ┌t r→A × r→B r→A · r→AB r→B · r→AB , =− − |→ |→ r A| rB| 4π |→ r A × r→B |2

(8.68)

where the negative sign indicates that a positive trailing vortex induces a negative velocity. Let ) ( v→i, j,k,n r→A × r→B r→A · r→AB r→A · r→AB , (8.69) = −┌ t R − |→ |→ r A| rB| V0 |→ r A × r→B |2

8.5 Flow Field Computation

149

where ┌ t represents the dimensionless strength of the trailing vortex element of (i, j, k, n). The dimensionless strength is defined as ( ) ┌ t i, j

⎧( ) i =1 ⎪ ⎨ (┌ b )1, j ( ) (┌t )i, j ┌ b i, j − ┌ b i−1, j i = 1, . . . , N E . = = ⎪ 4πV0 R ⎩ (┌ ) i = NE + 1 b NE , j

(8.70)

Equation (8.69) can be re-expressed as (v x )i, j,k,n →i + (v y )i, j,k,n →j + (v z )i, j,k,n k→ = −┌ t R Ii,t j,k,n ( ) ( ) ( ) Ii,t j,k,n = Ixt i, j,k,n →i + I yt i, j,k,n →j + Izt i, j,k,n k→ ) ( r→A × r→B r→A · r→AB r→A · r→AB . = − |→ |→ r A| rB| |→ r A × r→B |2

(8.71)

(8.72)

In the above, t indicates ( ) the superscript ( ) ( ) the contribution of the trailing vortex. Meanwhile, Ixt i, j,k,n , I yt i, j,k,n , and Izt i, j,k,n are the impact factors of the induced velocity, which are defined as ⎧( ) t ⎪ ⎪ ⎪ Ix i, j,k,n = ⎨ ( t) I y i, j,k,n = ⎪ ⎪ ( t) ⎪ ⎩ I z i, j,k,n =

(

(→ r A ×→ r B )·→i r→A ·→ r AB rA| |→ r A ×→ r B |2 ( |→ (→ r A ×→ r B )· →j r→A ·→ r AB rA| |→ r A ×→ r B |2 ( |→ r A ×→ r B )·k→ r→A ·→ (→ r AB |→ rA| |→ r A ×→ r B |2

)

r→A ·→ r AB |→r B | ) − r→A|→r·→rBAB | ) r→A ·→ r AB − |→r B |

−

.

(8.73)

Based on Eqs. (8.57) and (8.58), the impact factors can be re-expressed as ⎧ ( ) ⎪ I tx (i, m, k, n) = Ixt i, j,k,n ⎪ ⎪ ( ) ⎪ ⎪ I ty (i, m, k, n) = I yt i, j,k,n ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎨ I tz (i, m, k, n) = Izt i, j,k,n . i = 1, 2, . . . , N E + 1 ⎪ ⎪ ⎪ ⎪ j = 1, 2, . . . , N T ⎪ ⎪ ⎪ ⎪ k = 1, 2, . . . , B ⎪ ⎪ ⎩ n = 1, 2, . . . , NC

(8.74)

The second subscript j in the original definition is changed to m to facilitate the subsequent expression of the induced velocity.

150

8 Computational Models of Vortex Wake

The intensity of the trailing vortex corresponding to the ith blade element boundary at the jth time step remains unchanged during the downstream movement. Therefore, the impact factors can be rewritten as follows: NC B ∑ ∑ ( t) ( t) Ix i, j = Ix i, j,k,n

(8.75)

k=1 n=1 NC B ∑ ∑ ( t) ( t) I y i, j = I y i, j,k,n

(8.76)

k=1 n=1 NC B ∑ ∑ ( t) ( t) Iz i, j = Iz i, j,k,n .

(8.77)

k=1 n=1

The velocity v→i, j,k,n induced by the wake vortices at point (x, y, z) can be calculated using the impact factors, as follows: v x (x, y, z) = −

N∑ NT E +1 ∑ i=1

v y (x, y, z) = −

v z (x, y, z) = −

( ) ( t) ┌ t i, j I y i, j

(8.79)

( ) ( t) ┌ t i, j Iz i, j

(8.80)

j=1

N∑ NT E +1 ∑ i=1

(8.78)

j=1

N∑ NT E +1 ∑ i=1

( ) ( t) ┌ t i, j Ix i, j

j=1

Subsequently, the dimensionless induced velocity at the (i, j ) blade element control point can be expressed as ⎧ (( ) ( ) ( ) ) cp ⎪ = v x cp , y , z cp (v ) ⎪ x x i, j ⎪ ⎨( ) (( )i, j ( cp )i, j ( )i, j ) cp v y i, j = v y x cp i, j , y cp i, j , z cp i, j ⎪ (( ) ( ) ( ) ) ⎪ ⎪ ⎩ (v z )cp = v z x cp , y cp , z cp i, j i, j i, j i, j

(8.81)

The induced velocity at the (i, j ) blade element boundary point can be similarly obtained, as follows: ⎧ (( ) ( ) ( ) ) bp ⎪ x bp , y , z bp = v (v ) ⎪ x x i, j ⎪ ⎨ (( )i, j ( bp )i, j ( )i, j ) ( )bp v y i, j = v y x bp i, j , y bp i, j , z bp i, j (8.82) ⎪ (( ) ( ) ( ) ) ⎪ ⎪ bp ⎩ (v z ) = v z x bp , y bp , z bp i, j i, j i, j i, j

References

151

The radial- and tangential-induced velocities at the blade element boundary point can be obtained using the following equation: [

bp

(vr ) ( )i,bpj v ψ i, j

]

[

cos ψ j sin ψ j = − sin ψ j cos ψ j

][

bp

(v x ) ( )i,bpj v y i, j

] (8.83)

Subsequently, the axial and tangential induction factors at the blade element boundary point are expressed as follows, respectively: ( ) bp abp i, j = −(v z )i, j

(8.84)

( )bp v ψ i, j ( ) abp ' i, j = − ( ) λ r bp i

(8.85)

References 1. Kocurek, D. (1987). Lifting surface performance analysis for horizontal axis wind turbines. NASA STI/Recon Technical Report N, p. 87. 2. Dumitrescu, H., & Cardos, V. (1998). Wind turbine aerodynamic performance by lifting line method. International Journal of Rotating Machinery, 4(3), 141–149. 3. Wang, T. G. (1999). Unsteady aerodynamic modelling of horizontal axis wind turbine performance. University of Glasgow. 4. Coton, F. N., & Wang, T. G. (1999). The prediction of horizontal axis wind turbine performance in yawed flow using an unsteady prescribed wake model. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 213(1), 33–43. 5. Wang, T. G., & Coton, F. N. (2001). A high resolution tower shadow model for downwind wind turbines. Journal of Wind Engineering and Industrial Aerodynamics, 89(10), 873–892. 6. Wang, T. G., & Coton, F. N. (1999). An unsteady aerodynamic model for HAWT performance including tower shadow effects. Wind Engineering, 23(5), 255–268. 7. Breton, S. P., Coton, F. N., & Moe, G. (2008). A study on rotational effects and different stall delay models using a prescribed wake vortex scheme and NREL phase VI experiment data. Wind Energy: An International Journal for Progress and Applications in Wind Power Conversion Technology, 11(5), 459–482. 8. Wang, T. G., & Coton, F. N. (2000). Prediction of the unsteady aerodynamic characteristics of horizontal axis wind turbines including three-dimensional effects. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 214(5), 385–400. 9. Rosen, A., Lavie, I., & Seginer, A. (1990). A general free-wake efficient analysis of horizontalaxis wind turbines. Wind Engineering, 14(6), 362–373. 10. Crouse, J. R. G., & Leishman, J. (1993). A new method for improved rotor free-wake convergence. In Proceedings of the 31st Aerospace Sciences Meeting, Keno. 11. Elgammi, M., & Sant, T. (2016). Combining unsteady blade pressure measurements and a freewake vortex model to investigate the cycle-to-cycle variations in wind turbine aerodynamic blade loads in yaw. Energies, 9(6), 460. 12. Gohard, J. C. (1978). Free wake analysis of a wind turbine aerodynamics. Massachusetts Institute of Technology, Department of Aeronautics and Astronautics.

152

8 Computational Models of Vortex Wake

13. Qiu, Y. X., Wang, X. D., Kang, S., et al. (2014). Predictions of unsteady HAWT aerodynamics in yawing and pitching using the free vortex method. Renewable Energy, 70, 93–106. 14. Sipcic, S. R., & Morino, L. (1986). Wake dynamics for incompressible and compressible flows. In L. Morino (Ed.), Computational methods in potential aerodynamics. Springer. 15. Wang, T. G., Wang, L., Zhong, W., et al. (2012). Large-scale wind turbine blade design and aerodynamic analysis. Chinese Science Bulletin, 57(5), 466–472. 16. Afjeh, A. A., & Keith, T. G. (1989). A simple computational method for performance prediction of tip-controlled horizontal axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics, 32(3), 231–245. 17. Afjeh, A. A., & Keith, T. G. (1986). A vortex lifting line method for the analysis of horizontal axis wind turbines. Journal of Solar Energy Engineering, 108(4), 303–309. 18. Miller, R. H. (1984) Application of fast free wake analysis techniques to rotors. Vertica. 19. Miller, R. H. (1985). Methods for rotor aerodynamic and dynamic analysis. Progress in Aerospace Sciences, 22(2), 113–160. 20. Simoes, F. J., & Graham, J. M. R. (1992). Application of a free vortex wake model to a horizontal axis wind turbine. Journal of Wind Engineering and Industrial Aerodynamics, 39(1–3), 129– 138. 21. Gupta, S. (2006) Development of a time-accurate viscous Lagrangian vortex wake model for wind turbine applications. University of Maryland. 22. Gupta, S., & Leishman, J. (2006). Performance predictions of NREL Phase VI combined experiment rotor using a free-vortex wake model. In Proceedings of the 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada. 23. Gupta, S., & Leishman, J. (2004). Stability of methods in the free-vortex wake analysis of wind turbines. In Proceedings of the 42nd AIAA aerospace sciences meeting and exhibit, Reno, Nevada. 24. Gupta, S., & Leishman, J. (2006). Validation of a free-vortex wake model for wind turbines in yawed flow. In Proceedings of the 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada.

Chapter 9

Solving Aerodynamic Performance of Wind Turbines

In the previous chapter, derivations of the prescribed-vortex wake (PVW) and freevortex wake (FVW) models are presented. This chapter introduces the solution methods and procedures of these two models under steady and unsteady conditions and provides the results calculated by considering the three-dimensional (3D) rotational effects or dynamic stall model. Additionally, the aerodynamic characteristics of wind turbines under steady or unsteady inflow conditions are briefly analyzed.

9.1 Solution of Steady PVW Model 9.1.1 Solution Process The specific procedures for calculating the aerodynamic characteristics of a horizontal-axis wind turbine under steady wind conditions using the PVW model are shown in Fig. 9.1. By performing a series of iterative processes, a convergent solution for the wake geometry and blade aerodynamic forces can be obtained. The complete computational procedures are described as follows: 1. Input Parameters Parameters to be input into the model are as follows: (1) (2) (3) (4) (5)

Number of blade elements, N E Number of time steps per revolution, N T Number of wake circles, N C Number of blades, B Tip-speed ratio, λ

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_9

153

154

9 Solving Aerodynamic Performance of Wind Turbines Input system parameters Calculate initial blade condition and trailing vorticity Calculate initial wake geometry Calculate induced velocities at blade element control points

Update trailing vorticity

Calculate blade loading pattern and rotor aerodynamic performance

Generate modified wake geometry

Converged?

N

Calculate induced velocities at blade element boundary points

Y Output calculated results

Fig. 9.1 Solving scheme of steady PVW method

(6) (7) (8) (9) (10)

Rotor radius, R Distance between the blade root and hub center, Rt Radial distribution of the chord length along the blade, ci Radial distribution of the twist angle along the blade, θ i Two-dimensional (2D) lift and drag coefficients of the wind turbine airfoils, C l and C d , respectively

Notably, the values of N T and B must be set such that N T /B values are integers. 2. Calculate Blade Initial Conditions The calculated results of the PVW model are generally not affected by the initial conditions unless backflow is involved in the initial conditions. However, suitable initial conditions can effectively reduce the computation time required to obtain a convergent solution. The initial value calculated using the PVW model can be derived using blade element method (BEM) theory. Wilson and Lissaman [1] proposed a simple computation program, i.e., the well-known PROP code, to obtain induction factors a and a' . The induction factors obtained using BEM theory are suitable for the initial inputs of the PVW model and can be iterated using linear differences. The iteration is performed only in the region that contains only one solution, and the specific process is described below. According to BEM theory, Eqs. (9.1) and (9.2) can be obtained as follows: ( ) f a a, a ' = σ Cn' (1 − a) − 8Fa sin2 φ = 0

(9.1)

( ) ( ) f t a, a ' = σ Ct' 1 + a ' − 8Fa ' sin φ cos φ = 0.

(9.2)

Therefore, the solutions of a = a∗ and a ' = a∗' must satisfy Eqs. (9.1) and (9.2), respectively. Denoting the axial induction factor a obtained via iterative procedure p as ap , the tangential induction factor satisfying Eq. (9.2) is a 'p , and the following formula can

9.1 Solution of Steady PVW Model

155

be derived: ⎧ ⎨ (a ' ) ⎩

p q+1

=

( ) ( ) a p ,(a 'p )q −(a 'p )q f t a p ,(a 'p )q−1 ( ) ( ) f t a p ,(a 'p )q − f t a p ,(a 'p )q−1

(a 'p )q−1 ft

(9.3)

q = 1, 2, . . .

( ( ) ) ( ) ( ) If a 'p q−1 and a 'p q are selected such that the signs of f t a p , a 'p q−1 and ( ( ) ) f t a p , a 'p q are opposite, then ( ( ) ) ( ( ) ) f t a p , a 'p q−1 · f t a p , a 'p q < 0

q = 1, 2, . . . .

(9.4)

( ) This formula holds throughout the iterative process, and a 'p q+1 can converge to a 'p . Based on Eq. (9.1), the induction factor a can be obtained iteratively using the following formula: {

a p+1 =

a p−1 f a (a p ,a 'p )−a p f a (a p−1 ,a 'p−1 ) a p−1 f a (a p ,a 'p )− f a (a p−1 ,a 'p−1 )

p = 1, 2, . . .

.

(9.5)

( ) ( ) If a p−1 and a p are selected such that the signs of f a a p−1 , a 'p−1 and f a a p , a 'p are opposite, then ( ) ( ) f a a p−1 , a 'p−1 · f a a p , a 'p < 0

p = 1, 2, . . . .

(9.6)

This formula holds throughout the iterative and ap+1 can converge to a* . ( process, ) ( ) ' Once the induction factors abp i, j and abp at the blade element boundary i, j

points are determined, the induction factors at the blade element control points can be approximated using the following formula: [( ) ] ( ) ( ) acp i, j = 21 abp i, j + abp i+1, j [( ) ] ( ) ( ) . ' ' ' acp + abp = 21 abp i, j

i, j

(9.7)

i+1, j

Subsequently, the formula for calculating the resultant velocity becomes cp W i, j

/ [ ( ) ]2 [ ( ) ]2 ( )2 ' 1 − acp i, j + λ2 r cp i 1 + acp = . i, j

(9.8)

( ) Next, the bound circulation ┌ b i, j and trailed vortex circulation can be calculated using Eqs. (8.52) and (8.9).

156

9 Solving Aerodynamic Performance of Wind Turbines

3. Calculate initial wake geometry Using the results obtained via BEM theory, the initial wake geometry of the rotor can be determined using the prescribed functions expressed in Eqs. (8.20) and (8.26), and the Cartesian coordinates of the wake vortex element can be calculated using Eq. (8.41). 4. Calculate induced velocities at blade element control points The induced velocities at the blade element control point are calculated using the Biot–Savart law expressed in Eq. (8.81). 5. Calculate blade loads and rotor aerodynamic performance The blade loads and rotor aerodynamic performance are calculated using the formulas in presented in Sects. 8.5.2 and 8.5.3, respectively. 6. Calculate induced velocities at blade element boundary points A new wake geometry is created, the Biot–Savart law is used to calculate the induced velocities at the blade element boundary point [Eq. (8.82)], and the axial-induced velocity is recalculated based on the new wake geometry [Eq. (8.84)] and tangentialinduced velocity [Eq. (8.83)]. 7. Update wake geometry Based on the newly calculated blade-induced velocities, the new wake geometry [Eq. (8.41)] at the current blade position can be updated using a prescribed function [Eqs. (8.20) and (8.25)]. 8. Update trailing vorticity Thus far, the trailed vortex circulation at the blade can be updated and calculated using Eq. (8.9), and the blade rotates to the next azimuthal position [Eqs. (8.40) and (8.43)–(8.45)]. The circulation calculation continues by returning to Step 4, and Steps 4–8 are repeated until the final convergent wake geometry and aerodynamic loads are obtained.

9.1.2 Computation Example Figure 9.2 shows the wake geometry of the NREL Phase VI blade [2] at a wind speed of 7 m/s. The solid points in the figure denote the blade-tip vortex position numerically calculated using the CFD method [3]. The blade-tip vortex position calculated using the PVW method is consistent with the CFD results in the axial development. Because some of the wind energy is absorbed after the wind passes through the rotor, the wake gradually expands outward behind the rotor, and the expansion radius of the prescribed-wake tip vortex is slightly smaller compared with the CFD results. Because the prescribed-vortex wake is limited by the description function, the wake geometry is not affected by distortion.

9.2 Solution of Steady FVW Model

157

Fig. 9.2 Wake geometry of NREL Phase VI wind turbine

9.2 Solution of Steady FVW Model 9.2.1 Relaxation Iterative Method The relaxation iterative method is a spatial iterative solution of the vortex line governing equations in the pseudo-time domain and necessitates periodic boundary conditions. The two remaining terms in the control equation [Eq. (8.32)] are discretized in the five-point central difference scheme. The differentiation of the time step can be expressed in the difference scheme as follows: ⇀

1 ∂r = ∂ψ 2Δψ

(

⇀

r

j+1,k+1

) − r + r −r . ⇀

⇀

⇀

j,k+1

j+1,k

j,k

(9.9)

The differentiation of space steps can be expressed in the difference scheme as follows: ⇀

1 ∂r = ∂ζ 2Δζ

(

⇀

r

j+1,k+1

) − r + r −r . ⇀

⇀

⇀

j+1,k

j,k+1

j,k

(9.10)

By substituting the two differential approximations above into Eq. (8.32), a discrete approximation control equation can be obtained, as follows: ⇀

r

j+1,k+1

) ) ( ( ⇀ Δψ − Δζ 2 ΔψΔζ ⇀ ⇀ ⇀ + = r + r − r V→0 + V . j,k j+1,k j,k+1 Δψ + Δζ Ω Δψ + Δζ ind

(9.11)

Let Δψ = Δζ ; therefore, the vortex line governing equation becomes ) ( ⇀ ⇀ ⇀n−1 r j+1,k+1 − r j,k = Δψ V 0 + V ind /Ω

⇀n

The periodic boundary condition is as follows:

(9.12)

158

9 Solving Aerodynamic Performance of Wind Turbines ⇀

⇀

r (ψ + 2π, ζ ) = r (ψ, ζ ).

(9.13)

If the induced velocity of the wake node in the previous step is propagated directly by one step to obtain a new wake geometry, then the circulation and velocity field will change rapidly, which may result in numerical divergence. To solve this problem, periodic boundary conditions are introduced and the concept of “virtual period” is used to improve the iterative stability; i.e., the virtual period is regarded as a period of 2π , and the wake is propagated one period in the velocity field of the previous step to obtain the wake of the next step. The updated wake expression is as follows: ⇀'

⇀n−1

⇀'

⇀'

r j−NT+2,k−NT+2 = r j−NT+1,k−NT +1 + r j−NT+3,k−NT+3 = r j−NT+2,k−NT+2 + .. . ⇀n

' ⇀

r j+1,k+1 = r j,k +

Δψ Ω

Δψ Ω Δψ Ω

(

(

⇀

⇀

⇀

⇀

)n−1

V ∞ + V ind (

j−NT+1,k−NT+1 )n−1

V ∞ + V ind

⇀

j−NT+2,k−NT+2

⇀

,

(9.14)

)n−1

V ∞ + V ind j,k

where r ' is the intermediate transition amount, and NT is the number of steps in one cycle. At each step, the relaxation factor is used for the following correction: r→new r old r new j,k = (1 − ω)→ j,k + α→ j,k .

(9.15)

The computational cost of FVW model is proportional to the square of the number of nodes. If the number of vortex lines increases, then the computational cost increases significantly. To balance the convergence speed and computational stability, the adaptive relaxation factor method can be used. In the iterative process, a new relaxation factor is obtained using the positive and negative values of the residual derivative. When the derivative is negative, the wake geometry develops in the direction of convergence, and the relaxation factor is increased to accelerate convergence. When the derivative is positive, the wake geometry changes rapidly and diverges easily. Decreasing the relaxation factor increases the stability. At the beginning of the iteration, rapid convergence is more important, and at the end of the iteration, stability is the main focus. This is mathematically expressed as follows: α n = α n−1 + α n = α n−1 −

0.1+e(2−n)/ 4 , 4 1−e(2−n)/ 6 , 4

/ dRMS dn < 0 / dRMS dn > 0

(9.16)

During the entire iterative process, the condition 0.1 ≤ α ≤ 1 must be maintained.

9.2 Solution of Steady FVW Model

159

9.2.2 Solution Process The FVW model uses the relaxation iteration method to calculate the aerodynamic performance of a horizontal-axis wind turbine under steady wind conditions. The specific procedures are shown in Fig. 9.3. The computational process is described as follows: (1) Input the computational parameters, including the air density, inflow wind speed, number of blades, blade geometry, airfoil aerodynamic characteristics, number of blade elements, time step, and space step (generally the same step length). (2) Calculate the initial wake coordinates using Eq. (8.33). (3) Calculate the inflow characteristics of the blade using the BEM, and calculate the bound circulation, trailed vortex circulation, and shed vortex circulation using the aerodynamic models presented in Chap. 8. The shed vortex is zero when the axial inflow is constant. (4) Use the vortex model in Chap. 8 to calculate the induced velocities of each node and update the wake geometry using Eq. (9.14). (5) Calculate the induced velocities of the blade element control point based on the new wake geometry. Input system parameters Calculate initial wake coordinates Calculate blade inflow, bound circulation and trailing vortex circulation Calculate induced velocities of wake Update wake Calculate induced velocity at blade element control points Calculate residual error

Converged?

N

Y Calculate blade loads and aerodynamic performance Stop Fig. 9.3 Computation scheme of steady FVW and relaxation iteration methods

160

9 Solving Aerodynamic Performance of Wind Turbines

(6) Calculate the geometric residual, which is expressed by the mean square error of the coordinates of each node of the old and new wakes, as follows:

RMS

┌ |j k ( )2 max ∑ max |∑ new old | r − r j,k j,k | √ j=1 k=1 , = jmax kmax

(9.17)

where jmax = NT and kmax = NT · NC represent the discrete numbers of wake angles. When the geometric residual is < 1 × 10−4 , the wake geometry is regarded as convergent, and the aerodynamic loads of the blade and aerodynamic performance of the wind turbine are further calculated; otherwise, the calculation is continued by returning to Step 3.

9.2.3 Computation Example Figures 9.4 and 9.5 show the wake geometry of the NREL Phase VI [2] wind turbine calculated using the FVW method, and the inflow wind speeds are 7 and 10 m/s, respectively. The six dots denote the tip vortex positions calculated using the CFD method presented in [3], which are the positions of the tip vortices after the simulated wake extracted for one and a half circles. Generally, the positions of the tip vortices calculated using the FVW model are consistent with the CFD results in the axial development, and the predicted radial positions of the tip vortex core agree well with the CFD results. In the FVW method, the vortex line adheres to the governing equations. As the local velocity of the flow field propagates unrestrictedly, the wake geometry becomes distorted downstream of the wind rotor. A comparison between Figs. 9.4 and 9.5 shows that increasing the wind speed can reduce the tip-speed ratio of the rotor and result in a sparse wake behind the rotor. The FVW model can simulate the free rolling of the wake vortex. When the vortex line propagates downstream, the outer vortex line is induced to gradually propagate toward the blade tip, followed by the wake boundary and then Fig. 9.4 Wake geometry of NREL Phase VI blade at 7 m/s wind speed

9.3 Unsteady PVW Method

161

Fig. 9.5 Wake geometry of NREL Phase VI blade at 10 m/s wind speed

inward, thus forming a rolled-up concentrated vortex, as shown by the dotted circle in Fig. 9.5.

9.3 Unsteady PVW Method 9.3.1 Calculation of Inflow Wind Speed ( ) For the rotating system, using the cylindrical coordinate system, r, ψ, z ' allows one to define the local conditions of the blade flow. Axis r is defined along the blade radial direction, and ψ is defined as the azimuthal angle of the blade rotating clockwise along axis x ' . The boundary point and control point coordinates of the blade element in the wind-axis coordinate system are expressed as follows: ⎧ ⎪ (x bp )i, j = (r bp )i cos γ cos ψ j ⎪ ⎪ ⎪ ⎪ ⎨ (y bp )i, j = (r bp )i sin ψ j (z bp )i, j = (r bp )i sin γ cos ψ j ⎪ ⎪ ⎪ i = 1, 2, . . . , N E + 1 ⎪ ⎪ ⎩ j = 1, 2, . . . , N T

⎧ ⎪ (x cp )i, j = (r cp )i cos γ cos ψ j ⎪ ⎪ ⎪ ⎪ ⎨ (y cp )i, j = (r cp )i sin ψ j (z cp )i, j = (r cp )i sin γ cos ψ j , ⎪ ⎪ ⎪ i = 1, 2, . . . , N E ⎪ ⎪ ⎩ j = 1, 2, . . . , N T

(9.18)

where γ is the yaw angle − → The freestream velocity V ∞ can be decomposed into radial, tangential, and axial components, as shown in Fig. 9.6, and their expressions are shown in Eq. (9.19). V→∞ = V∞ sin γ cos ψ e→r − V∞ sin γ sin ψ e→ψ + V∞ cos γ e→z ' ,

(9.19)

where e→r , e→ψ , and e→z ' are unit vectors in the radial, tangential, and axial directions, respectively. In an actual wind field, wind turbines operate in the atmospheric boundary layer. The incoming flow is typically shear wind, and the change in wind speed with height can be expressed via a simple exponential law, as follows:

162

9 Solving Aerodynamic Performance of Wind Turbines

Fig. 9.6 Schematic illustration showing decomposition of freestream velocity

( V (h) = V (h 0 )

h h0

)η

,

(9.20)

where h is the height from the ground (considering roughness), h0 the reference height, and V (h0 ) the reference wind speed at height h0 . The exponent η is the shear index, which is determined by the terrain and is generally set as = 1/6 for open and flat grounds. For wind turbines, the hub height H is often used as the reference height, and the wind speed V 0 at this height is selected as the reference wind speed. Subsequently, the distribution of shear wind speed along the entire wind rotor plane can be expressed as

9.3 Unsteady PVW Method

163

V∞ =

) ( r sin ψ η V∞ = 1− . V0 H

(9.21)

9.3.2 Induced Velocities Under unsteady conditions, the blade flow changes with time t or the azimuthal angle ψ; ; consequently, the local angle of attack (AOA) of the blade is different at different azimuthal angles. To consider the time-varying AOA, a shed vortex is introduced into the wake, and the change in blade circulation over time is considered. Similar to the trailed vortex in the wake, the shed vortex is determined by its two endpoints; furthermore, their coordinates are expressed as [xw (i, j, k, n), yw (i, j, k, n), z w (i, j, k, n)] and respectively. [xw (i, j + 1, k, n), yw (i, j + 1, k, n), z w (i, j + 1, k, n)], Subsequently, the following are assumed: r→A = [x − xw (i, j, k, n)]→i + [y − yw (i, j, k, n)] →j − → + [z − z w (i, j, k, n)] k

(9.22)

− → r→B = [x − xw (i, j + 1, k, n)] i − → + [y − yw (i, j + 1, k, n)] j − → + [z − z w (i, j + 1, k, n)] k .

(9.23)

− → r→AB = [xw (i, j + 1, k, n) − xw (i, j, k, n)] i − → + [yw (i, j + 1, k, n) − yw (i, j, k, n)] j − → + [z w (i, j + 1, k, n) − z w (i, j, k, n)] k .

(9.24)

Therefore,

( ) ( ) ( ) The induced velocity factors Ixs i, j , I ys i, j , and Izs i, j generated by the shed vortex can be defined, and the induced velocities of the entire wake vortex system at the blade can be calculated as follows: vx = −

N∑ NT E +1 ∑ i=1

j=1

E ∑ T ∑ (┌s )i, j ( s ) (┌t )i, j ( t ) Ix i, j − Ix i, j 4π 4π i=1 j=1

N

N

(9.25)

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9 Solving Aerodynamic Performance of Wind Turbines

vy = −

N∑ NT E +1 ∑ i=1

vz = −

j=1

N∑ NT E +1 ∑ i=1

j=1

E ∑ T ∑ (┌s )i, j ( s ) (┌t )i, j ( t ) I y i, j − I y i, j , 4π 4π i=1 j=1

(9.26)

E ∑ T ∑ (┌t )i, j ( t ) (┌s )i, j ( s ) Iz i, j − Iz i, j 4π 4π i=1 j=1

(9.27)

N

N

N

N

where ┌s is the shed vorticity. The induced velocities at the blade element boundary in the wind-axis coordinate system can be used to determine the wake geometry. To calculate the aerodynamic loads of the blade, the induced velocities at the blade element control point in the wind-axis coordinate system are obtained using the method above, which are typically converted into radial-, tangential-, and axial-induced velocity components, as follows: ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ vr cos ψ sin ψ 0 cos γ 0 sin γ vx ⎣ vψ ⎦ = ⎣ − sin ψ cos ψ 0 ⎦⎣ 0 (9.28) 1 0 ⎦⎣ v y ⎦ vz ' vz 0 0 1 − sin γ 0 cos γ

9.3.3 Coupling of Dynamic Stall Models When the wind turbine operates under unsteady conditions, the airfoil motion can be periodic or aperiodic; thus, an unsteady dynamic stall may be presented [4, 5]. To simulate a dynamic stall, the dynamic stall model can be coupled using the PVW method. The method for coupling the Beddoes–Leishman and unsteady PVW models is presented below. The Beddoes–Leishman model requires the blade motion at a reduced pitch rate, the inflow velocity, and the instantaneous AOA as input conditions. These initial inputs are provided by the PVW model using 2D static data. Additionally, the PVW model generates the wake geometry behind the rotor as the basis for dynamic wake correction. The unsteady aerodynamic loads provided by the Beddoes–Leishman model are different from those obtained using the quasi-steady PVW model. This difference causes a redistribution of the bound vortex strength, which changes the strengths of the trailed and shed vortices in the wake. The change in the wake vorticity affects the induced velocities at the blade, which consequently affects the load distributions of the blade. Therefore, an iterative process must be introduced to obtain the appropriate blade-induced velocities and load distributions. In modeling the performance of an airfoil in an unsteady flow, the indicial technique implicitly includes the induced effect of the shed wake structure downstream of the airfoil. Meanwhile, the wake structure generated in the PVW method contains filaments of shed vorticity, whose induced effect on the blade flow field is calculated

9.3 Unsteady PVW Method

165

directly by applying the Biot–Savart law. Consequently, the coupling of the two schemes is hindered by this duplicative effect. This can be overcome by selectively excluding the shed wake terms from the PVW model and calculating the induced effect using the unsteady airfoil performance scheme of the Beddoes–Leishman model. Subsequently, the induced velocities at the (i, j)th blade element control point are estimated in the unsteady computation stage as follows: ⎡ cp

(v x )i, j

⎤

⎢ ⎥ ⎥ NE N T ⎢ N∑ ∑ ∑ ⎢ E +1 ( ) ( t ) ( ) ( s) ⎥ ⎢ ┌ t p,q Ix p,q + ┌ s p,q Ix p,q ⎥ =− ⎢ ⎥ ⎥ q=1 ⎢ p=1 ⎣ ⎦ p=1 p /= i ⎡ ⎤

⎢ ⎥ ⎥ NE N T ⎢ N∑ ∑ ∑ ⎢ E +1 ( ) ( t ) ( )cp ( ) ( s) ⎥ ⎢ ┌ t p,q I y p,q + ┌ s p,q I y p,q ⎥ v y i, j = − ⎢ ⎥ ⎢ ⎥ q=1 ⎣ p=1 ⎦ p=1 p /= i ⎡ ⎤ cp

(v z )i, j

⎢ ⎥ ⎥ NE N T ⎢ N∑ ∑ ∑ ⎢ E +1 ( ) ( t ) ( ) ( s) ⎥ ⎢ ┌ t p,q Iz p,q + ┌ s p,q Iz p,q ⎥ =− ⎢ ⎥. ⎢ ⎥ q=1 p=1 ⎣ ⎦ p=1 p /= i

(9.29)

(9.30)

(9.31)

9.3.4 Computation Example By coupling the PVW model with the Beddoes–Leishman model modified by Kirchhoff’s theory, the predictive accuracy can be improved significantly. Figure 9.7 shows a comparison of azimuthal load variations obtained using the PVW model and those obtained experimentally at different radial locations of the NREL Phase VI blade based on a wind speed of 15 m/s and a yaw angle of 10°. Notably, the correction model to account for the 3D rotational effects is not included. The results predicted using the modified model are more consistent to the experimental data.

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9 Solving Aerodynamic Performance of Wind Turbines

Fig. 9.7 Comparison between predicted aerodynamic loads and measured values at different radial locations (based on 5 m/s wind speed and 10° yaw angle)

9.4 Unsteady FVW Method 9.4.1 Time-Stepping Method The primary purpose of the time-stepping FVW method is to obtain the time-varying wake geometry and aerodynamic characteristics of wind turbines in an unsteady flow field [5]. This requires time-step differentiation, which demands a higher accuracy compared with space-step differentiation. To perform time-step differentiation, a five-point center-difference scheme is often used, which is expressed as follows: ⇀

1 ∂r = ∂ζ 2Δζ

(

⇀

⇀

⇀

⇀

i, j

i, j−1

i−1, j

i−1, j−1

r − r + r −

r

) .

(9.32)

To obtain the numerical solution for the time-step differential equations, the estimation–correction scheme can be used, and the derivations involved are presented below. Equation 8.32 can be written as ] ∂ r→(ψ, ζ ) 1[→ ∂ r→(ψ, ζ ) =− + V0 + V→ind (→ r (ψ, ζ )) . ∂ψ ∂ζ Ω

(9.33)

The first term on the right-hand side can be expressed using the five-point centerdifference scheme [Eq. (9.32)]. To simplify the derivation, the formula above is written as an ordinary differential equation in the following general form:

9.4 Unsteady FVW Method

167

dy = f (x, y). dx

(9.34)

Subsequently, the linear multistep method with equal step sizes, which is expressed as xn = x0 + nh, where n is the step size, is adopted. Let f n = (xn , yn ); therefore, the general form of the linear multistep method can be expressed as k ∑

α j yn+ j = h

j=0

k ∑

βi f n+ j ,

(9.35)

j=0

where α j and β j ( j = 0, 1, …, k) are constants, αk /= 0, and the values of α 0 and β 0 are not zero. Assuming α k = 1 and that the coefficient of yn+k is 1, Eq. (9.35) becomes yn+k = −

k−1 ∑

α j yn+ j + h

j=0

k ∑

βi f n+ j .

(9.36)

j=0

If βk = 0, the linear multistep method is an explicit method; if βk /= 0, the linear multistep method is an implicit method. When the implicit method is used to solve the problem, the approximate value at each node must be obtained iteratively, and the amount of computation is relatively significant. In actual computations, implicit methods can be used to improve results obtained via explicit methods, and they are known as prediction–correction methods. Explicit and implicit linear multistep methods can be constructed separately using the undetermined coefficient method. Let y(x) be the smooth solution of the differential equation. The local truncation error of Eq. (9.36) at x n+k is Tn+k = y(xn+k ) + α0 y(xn ) + α1 y(xn+1 ) + · · · + αk−1 y(xn+k−1 ) [ ] − h β0 y ' (xn ) + β1 y ' (xn+1 ) + · · · + βk−1 y ' (xn+k−1 ) + βk y ' (xn+k ) . (9.37) Using Taylor expansion, we obtain ) ( ( j h)2 '' y (xn ) + · · · y xn+ j = y(xn + j h) = y(xn ) + j hy ' (xn ) + 2!

(9.38)

( ) ( j h)2 ''' y (xn ) + · · · . y ' xn+ j = y ' (xn + j h) = y ' (xn ) + j hy '' (xn ) + 2!

(9.39)

Substituting these two equations into the local truncation error expressed in Eq. (9.37) yields Tn+k = c0 y(xn ) + c1 hy ' (xn ) + · · · + cl h l y (l) (xn ) + · · · ,

(9.40)

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9 Solving Aerodynamic Performance of Wind Turbines

where ⎧ ⎪ ⎪ c0 = α0 + α1 + · · · + αk , αk = 1 ⎪ ⎪ ⎪ c1 = α1(+ 2α2 + · · · + kαk − (β)0 + β1 + · · · + βk ) ⎪ ⎪ ⎪ ⎪ c2 = 2!1 (α1 + 22 α2 + · · · + k 2 αk ) − (β(0 + 2β1 + · · · + kβk ) ) ⎪ ⎪ ⎨ c3 = 3!1 α1 + 23 α2 + · · · + k 3 αk − 2!1 β0 + 22 β1 + · · · + k 2 βk . (9.41) ⎪ .. ⎪ ⎪ . ⎪ ( ) ( ) ⎪ ⎪ 1 1 l l l−1 l−1 ⎪ α − β + 2 α + · · · + k α + 2 β + · · · + k β = c l 1 2 k 0 1 k ⎪ l! (l−1)! ⎪ ⎪ ⎪ ⎩ .. . The undetermined coefficient method can be used flexibly to construct a highorder linear multistep method. For a fixed number of steps k, the order of the linear multistep method can be maximized by selecting the values of α i and β i . When k = 1, the linear multistep method is transformed into a linear single-step method. Using different values of α 0 , β 0 , and β 1 , the explicit Euler method, implicit Euler method, and trapezoidal formula can be obtained. Although the single-step method is simple, its numerical stability is inadequate. However, the multistep difference algorithm offers good stability and fast convergence. Next, we introduce the derivation of a three-step difference method, i.e., substituting k = 3 in Eq. (9.36). Therefore, we obtain yn+3 = −α0 yn − α1 yn+1 − α2 yn+2 + h(β0 f n + β1 f n+1 + β2 f n+2 + β3 f n+3 ).

(9.42)

For the three-step method, if the accuracy is lower than the third order, then the efficiency is relatively low; however, forcibly improving the computational accuracy will affect the stability. Thus, by selecting appropriate values of α i and β I , the threestep difference algorithm exhibits third-order accuracy. The requirement to achieve third-order accuracy is c0 = c1 = c2 = c3 = 0. Therefore, we have ⎧ α0 + α1 + α2 + 1 = 0 ⎪ ⎪ ⎨ α1 + 2α2 + 3 − (β0 + β1 + β2 + β3 ) = 0 ⎪ + 4α2 + 9 − 2(β1 + 2β2 + 3β3 ) = 0 α ⎪ ⎩ 1 α1 + 8α2 + 27 − 3(β1 + 4β2 + 9β3 ) = 0

(9.43)

This system of equations contains seven unknowns and four equations. Let β 0 = β 1 = β 3 = 0. Therefore, we obtain α 0 = 1/2, α 1 = − 3, α 2 = 3/2, and β 2 = 3. Subsequently, Eq. (9.42) becomes

9.4 Unsteady FVW Method

169

1 yn+3 = − (3yn+2 − 6yn+1 + yn ) + 3h f n+2 , 2

(9.44)

which expresses an explicit linear three-step method with a local truncation error of Tn+3 =

( ) 1 4 (4) h y (xn ) + O h 6 . 2

(9.45)

Clearly, this explicit linear three-step method exhibits third-order accuracy. Let β 0 = β 1 = β 2 = 0; therefore, we obtain α 0 = − 2/11, α 1 = 9/11, α 2 = − 18/11, and β 3 = 6/11. Subsequently, Eq. (9.42) becomes yn+3 =

1 6 (18yn+2 − 9yn+1 + 2yn ) + h f n+3 , 11 11

(9.46)

which expresses an implicit linear three-step method with a local truncation error of ( ) 3 Tn+3 = − h 4 y (4) (xn ) + O h 6 . 2

(9.47)

This implicit linear three-step method also exhibits third-order accuracy. Using the third-order explicit linear three-step method for prediction and the thirdorder implicit linear three-step method for correction, a three-step third-order prediction–correction scheme can be achieved. This difference format can be abbreviated as “D3PC,” which represents the following: (0) P(Prediction) : yn+3 = − (21 (3yn+2 − 6y ) n+1 + yn ) + 3h f n+2 (0) (0) E(Estimation) : f n+3 = f xn+3 , yn+3

1 C(Correction) : yn+3 = 11 (18yn+2 − 9yn+1 + 2yn ) + E(Estimation) : f n+3 = f (xn+3 , yn+3 ).

(0) 6 h f n+3 11

(9.48)

Next, we apply the D3PC scheme to the difference of the time-step differential equation in the wake governing equation, and then substitute the result into Eq. (9.32). By setting Δψ = Δζ , the discretization scheme of the governing equations is expressed as follows: Estimated steps: ) 1( −9˜ri−1, j + 12˜ri−2, j − 2˜ri−3, j + 3˜ri, j−1 + 3˜ri−1, j−1 7 [ )] 1 ( n−1 6 Δψ n−1 n−1 n−1 V0 + V + Vind(i−1, j) + Vind(i, j−1) + Vind(i−1, j−1) + 7 Ω 4 ind(i, j) (9.49)

r˜i, j =

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9 Solving Aerodynamic Performance of Wind Turbines

Corrected steps: ri, j =

) 1( 15ri−1, j − 9ri−2, j + 2ri−3, j + 3ri, j−1 + 3ri−1, j−1 14 3 Δψ 1 n−1 n−1 n−1 n−1 + [V0 + (Vind(i, j) + Vind(i−1, j ) + Vind(i, j−1) + Vind(i−1, j−1) ) 7 Ω 8 1 + (V˜ind(i, j) + V˜ind(i−1, j) + V˜ind(i, j−1) + V˜ind(i−1, j−1) )], (9.50) 8

~ind represents the induced velocity at the nodes of the new wake calculated where V using the estimated step. First, the new wake geometry is estimated using Eq. (9.49). Subsequently, the induced velocity of the new wake at each node is calculated, and Eq. (9.50) is used to correct it. Because the D3PC format is a three-step method, the classical Runge–Kutta method can be used to calculate the values of r 1,j , r 2,j , and r 3,j , as follows: ri, j = ri−1, j +

) Δψ ( 6V0 + Vind(i−1, j) + 2Vind(i+1, j ) + 2Vind(i, j−1) + Vind(i, j+1) 6Ω (9.51)

9.4.2 Computation Steps The computation procedures for the time-stepping method are as follows: (1) Input the computational parameters, including the air density, inflow wind speed, number of blades, blade geometry, airfoil aerodynamic characteristics, number of blade elements, time step, and space step (typically the same step length). (2) Calculate the initial wake coordinates using Eq. (8.33). (3) Calculate the inflow characteristics of the blade using BEM theory and calculate the bound, trailed vortex, and shed vortex circulations using blade aerodynamic models. The shed vortex is zero when the axial inflow is constant. (4) Calculate the induced velocities of each node of the wake using vortex models. (5) Step along the azimuthal angle and use the estimated value calculated using Eq. (9.49) to estimate the new wake geometry; subsequently, calculate the induced velocities of the blade element control points. (6) Calculate the blade inflow characteristics as well as the bound, trailed vortex, and shed vortex circulations under the effect of the estimated-step wake geometry; subsequently, calculate the induced velocities of each node at this time step. (7) Use Eq. (9.50) to correct the wake geometry at the estimated step. (8) When the azimuthal angle is not an integer multiple of 2π, return to Step (3) to continue stepping. When the azimuthal angle is an integer multiple of 2π, calculate the geometric residual between the wake geometry at this moment and

9.4 Unsteady FVW Method

171

that at the previous cycle. If the geometric residual is less than the set value (e.g., 1 × 10–4 , which implies that the wake geometry is convergent), then calculate the aerodynamic loads and aerodynamic performance of the blade; otherwise, return to Step (3).

9.4.3 Computation Example Figure 9.8 shows the trajectory of the tip vortex at different times behind the rotor under extreme wind-direction changes [6, 7]. The analysis corresponding to each label in the figure is as follows: (a) At this moment, the inflow wind speed is in the axial direction, and the wake is an axisymmetric regular helix.

Fig. 9.8 Tip vortex trajectory at different times under extreme wind-direction changes

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9 Solving Aerodynamic Performance of Wind Turbines

(b) At this moment, the wind-direction angle is approximately 30°. The wake geometry near the wind rotor within one revolution is deflected, and the downstream wake maintains the axial speed and propagates rearward. However, the downstream wake geometry is changed slightly owing to the induction of a skewed wake. (c) At this moment, although the wind-direction angle is maintained at 30° for 9 s, the wake geometry has not been reorganized; therefore, the performance of the wind turbine is not yet stable. (d) At this moment, the wake is reorganized, and the aerodynamic performance of the wind turbine reaches a new equilibrium state.

References 1. Wilson, R. E., & Lissaman, P. (1974). Applied aerodynamics of wind power machines. Renewable Energy, 16, 13–70. 2. Hand, M. M., Simms, D. A., Fingersh, L. J., et al. (2001) Unsteady aerodynamics experiment phase VI: Wind tunnel test configurations and available data campaigns. National Renewable Energy Lab., Golden, CO (US). 3. Sezer-Uzol, N., & Long, L. (2006). 3-D time-accurate CFD simulations of wind turbine rotor flow fields. 4. Xu, B., Wang, T., Yuan, Y., et al. (2018). A simplified free vortex wake model of wind turbines for axial steady conditions. Applied Sciences., 8(6), 866. 5. Xu, B. F., Yuan, Y., & Wang, T. G. (2014). Development and application of a dynamic stall model for rotating wind turbine blades. IOP Publishing. 6. Xu, B., Yuan, Y., & Wang, T. (2016). Unsteady wake simulation of wind turbines using the free vortex wake model. World Scientific. 7. Xu, B. F., Wang, T. G., Yuan, Y., et al. (2015). Unsteady aerodynamic analysis for offshore floating wind turbines under different wind conditions. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373(2035), 20140080.

Part IV

Computational Fluid Dynamics Method

Owing to the development of wind turbines toward the super large-scale, aerodynamic problems are becoming increasingly complex and have exceeded the application scope of the blade element momentum and vortex wake methods. For example, the turbulent characteristics of flow in the vicinity of large wind turbines are prominent, and the interaction between turbulence in the atmospheric boundary layer and the above-mentioned flow is significant. To investigate these complex flow problems, computational fluid dynamics (CFD) has been increasingly applied in recent years. CFD uses numerical methods to solve the governing equations of fluid mechanics and can obtain almost all information pertaining to the entire flow field of wind turbines. It offers significant advantages over other numerical methods in terms of the analysis of flow characteristics. This part includes the fundamentals of CFD, the application of the Reynolds-averaged method, as well as the large-eddy and detached simulation methods used for the numerical simulation of wind turbines.

Chapter 10

Fundamentals of Computational Fluid Dynamics

The motion of a fluid adheres to three basic laws: the laws of mass, momentum, and energy conservation. These laws are used in relevant constitutive models and equations of state to establish partial differential or integral equations to describe the motion of the fluid; these equations are known as the governing equations of fluid motion. Owing to the development of fluid mechanics, the mathematical solution for fluid motion with different properties and flow states is becoming increasingly mature. The discrete solution of the governing equations of fluid motion is the primary problem in CFD. This chapter introduces the fundamentals of CFD, such as control equation discretization, numerical schemes, turbulence simulation, preprocessing, and post-processing.

10.1 Brief Introduction to CFD Fluid motion is one of the most complex forms of natural motion. The highly nonlinear governing equations and complex geometry of the flow region render it difficult to obtain analytical solutions for most flow problems in science and engineering. Owing to the rapid development of high-performance computers since the 1950s, an independent discipline has been developed that uses numerical methods combined with computer technology to solve the governing equations of fluid mechanics and simulate the mechanical problems of fluid flow, i.e., computational fluid mechanics (CFD). CFD is an interdisciplinary discipline that integrates mathematics, fluid mechanics, computer technology, and scientific visualization. The basic idea of the CFD method is to replace a continuous physical quantity field (e.g., the velocity field, pressure field, and temperature field) with a variable value set of a finite number of discrete points in the computational domain, followed by using numerical methods to

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_10

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10 Fundamentals of Computational Fluid Dynamics

discretize the governing equations. Subsequently, the corresponding algebraic equations relating the discrete points are established and, finally, the equations are solved to obtain the approximate values of the physical field. The development of CFD has been promoted by its increasing industrial demand, in particular the demand from the aerospace industry. The development history of CFD can be classified into three stages: The first stage was from 1965 to 1974. The main research objective of this stage was to solve basic theoretical problems in CFD, such as establishing model equations (turbulence, heat transfer, radiation, gas–particle interaction, chemical reaction, and combustion), numerical methods (difference schemes and methods for solving algebraic equations), mesh division, programming, and implementation. At this stage, the numerical results were compared with the results of classical fluid mechanics experiments and exact solutions to determine the reliability, accuracy, and effect of the numerical prediction method. Meanwhile, to solve flow problems involving complex geometries in engineering, researchers began to investigate the problem of grid transformation, and a specific technology named “grid generation technology” was formed gradually. The second stage was the industrial application stage (1975–1984). Owing to continuous improvements in numerical prediction, principles, and methods, the primary focus of this stage was to evaluate the feasibility, reliability, and industrialization of CFD in solving practical engineering problems. Researchers began to develop CFD technology for solving various flow-based engineering problems such as multiphase flows, non-Newtonian flows, chemical reaction flows, and combustion. However, CFD could only be applied by professional research teams. The CFD codes were not universal; thus, the users were typically the developers of the codes. In 1977, Spalding et al. developed a public program named GENMIX. However, they later realized that protecting their intellectual property rights would be difficult if the source code was published. Thus, a company called Concentration, Heat, and Momentum Limited (CHAM) was founded in 1981 to officially release the packaged software PHONNICS (short for Parabolic Hyperbolic or Elliptic Numerical Integration Code Series) to the market, which serves as a precedent for CFD commercial software. The third stage was the rapid development stage (1984–present). The application of CFD in engineering has yielded meaningful results, and CFD has been fully recognized by the academic community. Numerous CFD commercial and opensource software have been developed and utilized successfully (e.g., FLUENT, CFX, STAR-CCM+ , NUMECA, TAU, SU2, and OpenFOAM). Additionally, the advancement of computer graphics has enabled the development of software for performing pre- and post-processing in CFD simulations (e.g., GRAPHER, GRAPHER TOOL, ICEM-CFD, POINTWISE, etc.). The CFD method is widely applicable and exhibits high adaptability. It can solve numerous complex problems such as computational domain with complex geometries and boundary conditions, problems with multiple unknowns, and nonlinear flow problems. In addition, the CFD method is relatively inexpensive compared with physical experiments and offers high flexibility and controllability. It can acquire data

10.2 Mathematical Description of Incompressible Viscous Flow

177

comprehensively, which facilitates a comprehensive analysis of problems. It has been widely used in the aerospace, transportation, shipbuilding, meteorology, ocean, water conservancy, hydraulics, and petrochemical industries and will continue to contribute significantly to academia and industry.

10.2 Mathematical Description of Incompressible Viscous Flow Generally, the linear velocity of the rotational motion of the blade tip of a wind turbine does not exceed 100 m/s, and the corresponding Mach number is less than 0.3. The aerodynamic noise produced by wind turbines increases significantly with the blade-tip speed. Considering the effect of aeroacoustic noise, the blade-tip velocity of on-shore wind turbines is typically limited to below 65 m/s [1]. Therefore, in investigations pertaining to the wind turbine flow field, the compressibility of air is negligible, and the incompressible Navier–Stokes (N–S) equation is used as the governing equation. For a finite control volume Ω, without considering the effects of external heating and penetrating forces, the N–S equation in the full integral form can be written as ∂ ∂t

∮ Ω

→ dΩ + W

∮ ∂Ω

(− → − → →) − FC − FV d S =0,

(10.1)

− → − → where ∂Ω is the boundary of the control volume Ω; W is the conserved variable; FC − → and FV are the convective and viscous fluxes, respectively. They can be expressed as follows: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ρV 0 ρ ⎢ ρuV + n p ⎥ ⎢n τ + n τ + n τ ⎥ ⎢ ρu ⎥ x ⎥ y xy z xz⎥ ⎢ ⎢ x xx ⎥ → =⎢ W ⎢ ⎥, F→C = ⎢ ⎥, ⎥ F→V = ⎢ ⎣ ρvV + n y p ⎦ ⎣ n x τ yx + n y τ yy + n z τ yz ⎦ ⎣ ρv ⎦ ρw

ρwV + n z p

n x τzx + n y τzy + n z τzz (10.2)

where ρ is the density, p is the pressure, n x , n y , and n z are three components of the unit external normal vector n→ on the surface of the control volume, u, v, and w are three components of the velocity vector. The contravariant velocity V is expressed as V = u→ · n→ = n x u + n y v + n z w.

(10.3)

τi j is the shear stress tensor, which includes the effects of laminar and turbulent viscosities, and the specific expressions for each item are as follows:

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10 Fundamentals of Computational Fluid Dynamics

) ( ) ( + 2μ ∂∂ux τx y = τ yx = μ ∂u + ∂v τx x = 23 μ ∂∂ux + ∂v + ∂w ∂ y ∂ z ∂x ) ( ) ( ∂u∂ y ∂w ∂w ∂v + 2μ τ yy = 23 μ ∂∂ux + ∂v , + = τ = μ + τ x z zx ∂ y ∂z ∂ y ∂z ∂ x ), ( ) ( ∂w ∂v τ yz = τzy = μ ∂z + ∂ y + 2μ ∂w τzz = 23 μ ∂∂ux + ∂v + ∂w ∂y ∂z ∂z

(10.4)

where μ is the viscosity coefficient, which is the sum of the laminar viscosity coefficient μ L and turbulent viscosity coefficient μT , namely, μ = μ L + μT .

(10.5)

When investigating incompressible fluids, the density of air is considered constant.

10.3 Turbulence Modeling Fluid motion can be classified into three types: laminar, turbulent, and transitional. Generally, these three flow states are distinguished by the Reynolds number (Re). When Re is lower than a certain critical value, the flow tends to be laminar. When Re is higher than this critical value, the flow tends to be turbulent. The phenomenon in which fluid motion evolves from laminar to turbulent is known as a “transition,” which is extremely unstable. Turbulence is a highly complex, three-dimensional (3D) unsteady flow. In addition to lateral flow, a reverse movement occurs in the direction opposite to the overall movement. The heat and mass transfer rates of turbulent flows are orders of magnitude higher than those of laminar flow. Similarly, the frictional resistance of turbulent flows is significantly higher than that of laminar flows. Turbulence is characterized by a highly dissipative, multiscale, and swirling irregular flow. In terms of physical structure, turbulent flow is a multiscale flow composed of eddies of different superimposed scales, and the scales and rotation directions of the eddies are random and irregular. The maximum size of large eddies is determined by the boundary conditions of the flow. The large eddies are primarily affected by inertia and result in low-frequency flows. Small eddies are primarily determined by viscous forces and are results in high-frequency pulsations. In turbulent flows, eddies at various scales disintegrate, thus resulting in eddies smaller than the original scale. Turbulent motion refers to the continuous generation, development, and dissipation of eddies. In a fully developed turbulent flow, the size of eddies in the fluid exhibits varies continuously within a certain range. Large eddies continuously absorb energy from the mainstream, and because of the interaction among various eddy structures, energy is gradually transferred to the small eddies. Owing to viscosity, the small eddies gradually dissipate and disappear, and the mechanical energy of turbulence is dissipated and finally converted into fluid thermal energy. Simultaneously, new eddies are generated owing to the effects of the boundary, viscosity, and velocity gradient.

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Turbulent flow is ubiquitous in nature, such as in oceans and the atmosphere. It is also the most typically investigated flow in engineering problems. The airflow in the vicinity of a wind turbine is typically turbulent. Currently, turbulence simulation methods can be classified into the following three categories: direct numerical simulation (DNS), large-eddy simulation (LES), and Reynolds-averaged Navier–Stokes simulation (RANS).

10.3.1 DNS The DNS method does not require a turbulence model, and the 3D transient N– S equation is solved directly to obtain the instantaneous flow field of turbulence. DNS solves the turbulent pulsations of all eddy scales to obtain all flow information in space and time. Therefore, the DNS method requires extremely high spatial and temporal resolutions. In specific simulations, extremely small time steps are required to distinguish the detailed spatial structure and temporal characteristics of turbulent flow. DNS demands extremely high computing requirements; thus, it is currently only suitable for investigating simple flows with low Re, such as plate boundary layer and circular tube flows.

10.3.2 LES The LES method was first proposed in the 1970s, and its main concept is to calculate the structures of large and small eddies separately. The large eddies are simulated by directly solving the transient N–S equation, whereas the effect of the small eddies is modeled based on the subgrid-scale stress. The large-eddy structure is significantly affected by the boundary conditions and exhibits high anisotropy. Because a universal turbulence model that describes the structure of large eddies does not exist, a direct solution is typically applied. In contrast, small eddies are affected less significantly by boundary conditions and can be regarded as isotropic. Therefore, sub-grid scale models can be used to simulate small eddies. In the numerical process of LES, the operation for separating large and small eddies is known as filtering. The grid scale is typically selected as the filter scale; therefore, the grid scale must be of the same order of magnitude as the inertia subregion scale. Compared with DNS, LES only solves information whose scale exceeds that of the inertial subregion. Although the computer resources required for LES are demanding, they are less demanding than those required for DNS. Hence, LES have been used increasingly in recent years.

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10.3.3 RANS Method In the RANS method, the physical quantities in the governing equations are averaged via time averaging, space averaging, or ensemble averaging. A quantity can be decomposed into the sum of the average and pulsation quantities, as presented in the following equation: u i = u i + u i' ,

p = p + p' ,

(10.6)

where the superscripts “−” and “' ” represent the average and pulsation quantities, respectively. Using the time average as an example, the average velocity of the flow can be written as ∮ 1 t+T u i = lim u i dt (10.7) T →∞ T t Subsequently, the incompressible RANS equation in differential form can be obtained, as follows: ) ∂u i ∂u i 1 ∂p ∂ ( + vj =− + τ i j + τijR ∂x j ρ ∂ xi ∂x j ∂t

(10.8)

In Eq. (10.8), τ i j is the laminar viscous stress, and τiRj is the Reynolds stress tensor, which are expressed as ) ( ∂u j ∂u i τ i j = 2μS i j = μ + ∂x j ∂ xi ) ( τiRj = −u i' u 'j = u i u j − u i u j

(10.9) (10.10)

The form of the RANS equation is similar to that of the full-form N–S equation. The only difference is that the RANS equation contains an additional term, i.e., the Reynolds stress tensor. Based on the theoretical fundamentals of turbulence, experimental data, and DNS results, researchers have proposed various closures for the RANS equation. The RANS method simulates eddies at all scales using various models. RANS turbulence models can be classified into two categories. One is the secondorder Reynolds stress model with transport equations for solving the Reynolds stress and dissipation ε. The other is based on Boussinesq’s eddy-viscosity hypothesis, which expresses the turbulent stress as a function of turbulent viscosity. Turbulence models in this category are known as eddy-viscosity models. 1. Reynolds stress models

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181

The Reynolds stress tensor is directly computed using Reynolds stress models, which rely on Reynolds stress transport equations and can account for complex interactions in turbulence. A typical Reynolds stress model contains 15 equations and involves a high amount of computation. Owing to the rapid development of modern computer technology, Reynolds stress models have been used increasingly in engineering applications. 2. Eddy-viscosity models Boussinesq compared the turbulent pulsation of fluid elements with the thermal motion of molecules and proposed the following eddy-viscous hypothesis [2]: 2 −u i' u 'j = 2vt Si j − kδi j , 3

(10.11)

where νt is the eddy-viscosity coefficient, Si j the strain tensor of the mean flow, k the turbulent kinetic energy, and δi j the Kronecker delta function. Additional differential equations must be introduced to ensure that the equation is closed. Based on the number of differential equations, the following models exist: zero-equation models; one-equation models, such as the Spalart–Allmaras model; and two-equation models, such as the k − ε and k − ω models. The accuracy of the turbulence model should not be assumed to be directly related to the number of equations. An appropriate turbulence model should be selected based on a specific flow. Currently, a universal turbulence model does not exist, and various turbulence models are still being improved. Compared with the LES method, the RANS method can only provide the averaged parameters of turbulence, such as the averaged velocity and pressure. However, the calculation accuracy and efficiency of the RANS method render it the most widely used method for investigating turbulence in engineering.

10.4 Methods of Numerical Discretization The basic idea of CFD is to discretize the computational domain, replace the space– time continuous physical quantity field with a set of variables on a finite number of discrete points, and establish a discrete set of algebraic equations for unknown variables. The flow field information in the computational domain is obtained by solving discrete algebraic equations. During discretization, the properties of the continuous flow field are transformed into those of the grid elements, and changes in the continuous flow field are described by changes in the physical quantities inside the grid elements, boundaries, and cell nodes. If the flow field is finely meshed and the grid cells are sufficiently small, then the computational results will be reasonably accurate.

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Three different discrete methods are generally used in CFD simulations: the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM).

10.4.1 FDM The FDM is a classical numerical method for solving differential equations. It involves performing a Taylor series expansion to discretize the derivatives of flow variables. Suppose we are required to compute the first derivative of a scalar function u(x) at a point located at x 0 . Therefore, we can write u(x0 + Δx) as a Taylor series in x, as follows: u(x0 + Δx) = u(x0 ) + Δx

| | Δx 2 ∂ 2 u || ∂u || + + ... ∂ x |x0 2 ∂ x 2 |x0

(10.12)

The first derivative can be approximated as | ∂u || u(x0 + Δx) − u(x0 ) + O(Δx). = ∂ x |x0 Δx

(10.13)

The approximation above is a first-order approximation. The same procedure can be applied to derive more accurate finite difference formulae and obtain approximations of higher-order derivatives. Compared with the FEM and FVM, the FDM is simpler and achieves high-order approximations for spatial discretization more easily. However, the FDM cannot be directly applied to body-fitted (curvilinear) coordinates. The governing equations for the FDM must first be transformed into the Cartesian coordinate system; i.e., they must be transformed from the physical to the computational space. The FDM is relatively mature and suitable for hyperbolic and parabolic problems; however, it is not as suitable as the FEM and FVM for investigating flows in the vicinity of bodies with complex geometries.

10.4.2 FEM The FEM was originally employed for structural analysis. Its application in structural and elastic mechanics is mature. However, it was applied extensively in solving N–S equations beginning only in the late 1990s. In the FEM, the computational domain is partitioned into a set of unstructured finite elements. These elements are typically triangles or quadrilaterals for solving two-dimensional (2D) cases, and tetrahedra or hexahedra for solving 3D cases. The

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183

most distinguishing feature of the FEM is that the equations are multiplied by a weight function before they are integrated over the entire domain. For example, the solution can be approximated by a linear shape function within each element to guarantee the continuity of the solution across the element boundaries. In practical applications, the FEM has been used to develop geometric models for different unit body shapes corresponding to triangular, quadrilateral, and polygonal meshes. Depending on the different weight functions and interpolation functions used, the FEM can be classified into the configuration, moment, least-square, and Galerkin methods. Meanwhile, based on the interpolation accuracy, the interpolation functions can be classified into linear and high-order interpolation functions. Compared with the FDM, the FEM can accommodate arbitrary geometries. In the FEM, the grids can be easily refined, and each element can be partitioned easily. In particular, it allows practical engineering problems involving complex boundary regions to be solved conveniently. The FEM features a rigorous mathematical foundation, particularly for elliptic and parabolic problems. Its approximation for each element is continuously analytic, which distinguishes it from the FDM.

10.4.3 FVM The FVM directly utilizes conservation laws; therefore, it is based on the integral form of conservation equations. The computational domain is partitioned into a finite number of non-overlapping contiguous control volumes (see Fig. 10.1), and conservation equations are applied to each control volume. This method ensures that the conservation laws are satisfied for each control volume in the entire computational domain. In contrast to the FDM, the FVM not only considers the numerical values on grid nodes, but also specifies the variations in the flow variables within the control volume, which is similar to the FEM. Several methods can be used to define the shape and position of a control volume with respect to the grid, e.g., the cell-center and cell-vertex schemes. In the cell-center scheme, the variable values are stored at the centroids of the grid cells, and each control volume is identical to each grid cell. In the cell-vertex scheme, variable values are stored at the nodes of the grid cells, Fig. 10.1 Control volumes for FVM

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and a control volume can be defined as the union of all cells sharing one node or a volume centered on the node. The accuracy of spatial discretization depends on the scheme by which the fluxes are evaluated. Using a scalar φ as an example, the integral of a control volume can be transformed into a closed surface integral using Gauss’ theorem, as follows: ∮

∮ Ω

∇ · φdΩ =

∂Ω

→ φd S.

(10.14)

In the equation above, ∂Ω represents the surface of the control volume Ω, and − → S is the surface area vector of the control volume. Using the control volume P in Fig. 10.1 as an example, the scalar value at a certain location or time can be expressed as follows: ( ) ∂φ t φ(x) = φ P + (x − x P ) · (∇φ) P , φ(t + Δt) = φ + Δt , (10.15) ∂t t where x denotes the position vector. The surface integral for the conservation equations can be approximated by summing the fluxes crossing the individual faces of the control volume. ∮ φ(x)dΩ = φ P Ω P (10.16) ΩP

∮

∮ ΩP

∇ · φdΩ =

∂Ω P

d S→ · φ =

∑ (∮

d S→ · φ f

f

) =

∑

S→ f · φ f ,

(10.17)

f

− → where φ f is the value of φ at face f , and S f represents the surface normal vector. Therefore, the calculation of the convection term can be transformed into solving the face values of the control volume, as follows: ∮ ∑ ∑ ∑ ∇ · (→ v φ)dΩ = v φ) f = ( S→ f · v→ f )φ f = Fφ f , (10.18) S→ f ·(→ ΩP

f

f

f

where F represents flux, which is expressed as F = S→ f · v→ f . A few interpolation schemes can be used to obtain the value of φ f . Typically, the schemes used for incompressible flows include center, upwind, and hybrid schemes. The physical meaning of the FVM is clear, as is each of its discrete term. This is an advantage of the FVM over the FDM and FEM. In addition, the control equations for the FVM are in the integral form, which reflects the conservation characteristics of each variable in the computational domain. The FVM performs well in solving flow and heat transfer and has been employed in most engineering software for CFD simulations.

10.5 Algorithms for Velocity–Pressure Coupling

185

10.5 Algorithms for Velocity–Pressure Coupling A segregated solver is typically employed in the FVM to solve the N–S equations for an incompressible flow. The pressure is assumed to be known when solving the velocity field, and vice versa. However, the velocity and pressure are coupled with each other, and such a pressure–velocity coupling must be addressed when solving the equations. The solution of the pressure (or velocity) should be corrected based on the most recent results of the velocity (or pressure) field.

10.5.1 SIMPLE Algorithm When solving the governing equations, two first-order derivative terms must be considered: the nonlinear convection term and the pressure gradient term in the momentum equation, which are particularly important to the solution of incompressible flows. The nonlinear convection term can be solved using a different approach. The pressure gradient term involves the coupling of velocity and pressure; therefore, the computed pressure must be corrected in each step until the solution converges. This correction can be realized using the SIMPLE algorithm. The SIMPLE algorithm was proposed by Patanakar and Spalding [3] in 1972. In this algorithm, the velocity field is approximated by solving the momentum equation with the pressure gradient from the previous iteration or via an initial estimation. Subsequently, the pressure equation is formulated and solved to obtain a new pressure field. Next, the velocity is corrected using the most recent pressure to satisfy the continuity equation. A flowchart of the SIMPLE algorithm is shown in Fig. 10.2. The primary steps of the solution are as follows: (1) Boundary conditions are established, and initial assumptions for the velocity and pressure field are introduced. (2) The coefficients in the momentum equation are calculated based on the velocity and pressure, and the momentum equation is solved to compute the intermediate velocity u im . . (3) The uncorrected mass fluxes at the faces are computed using u im . . (4) Subsequently, the pressure correction equation is solved to obtain the values of the pressure correction p ' . . (5) The pressure and velocity are corrected to u imm and p m , respectively, and the boundary values are updated. (6) This iterative process is repeated until the obtained velocity field converges. In most codes used for the SIMPLE algorithm, the velocity components are stored at the interfaces of adjacent control volumes, whereas other solved variables are stored at the centers of the control volumes. Several sets of coefficients must be computed and stored during the discretization of the momentum equation, which is not conducive to achieving the solution. Hence, Rhie and Chow [4] proposed a pressure-weighted interpolation method, which enables the velocity values at the faces of control volumes to satisfy the local momentum equation more effectively. In their method, the velocity comprises two aspects: one is linear interpolation using a central difference scheme, and the other is correction to the interpolation using

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Fig. 10.2 Flowchart of SIMPLE algorithm

convection, diffusion, and pressure gradient terms. The obtained velocity is used to compute the coefficients in the pressure correction equation, which significantly reduces the number of iterations required and improves the convergence rate. In addition to the standard SIMPLE algorithm, modified versions, such as SIMPLEC and SIMPLER, have been developed, which focus on reducing the computation time by improving convergence. The SIMPLEC algorithm considers convective diffusion in velocity correction, which accelerates the convergence of the velocity

10.6 Mesh Generation and Post-processing

187

field. In the SIMPLER algorithm, the discrete continuity equation is used to establish a discrete equation of pressure such that the pressure can be obtained directly without correction. This reduces the number of iterations for the SIMPLER algorithm, although it requires more time per iteration than the SIMPLE algorithm. Consequently, the SIMPLER algorithm is overall faster than the SIMPLE algorithm.

10.5.2 PISO Algorithm Pressure implicit with splitting of operators (PISO), which was proposed by Issa in 1986, is another widely used algorithm for velocity–pressure coupling. It is a pressure-based implicit segregated algorithm that is particularly suitable for transient computations. The primary difference between the PISO and SIMPLE algorithms is that the SIMPLE algorithm is a two-step algorithm, i.e., one step for prediction and another step for correction, whereas the PISO algorithm contains one prediction step and two correction steps, which allow the velocity to satisfy both the momentum and continuous equations more effectively. The PISO algorithm can accelerate the convergence of iteration since it involves three steps. A flowchart of the PISO algorithm is shown in Fig. 10.3. The computational accuracy of the PISO algorithm depends on the time step used. If the time step is extremely large, then convergence cannot be achieved within a limited number of correction steps, which consequently results in divergence. Generally, the Courant number is less than 1. In LESs with high-precision requirements, a smaller Courant number is recommended to ensure that the flow field fluctuation is fully resolved. In the later development of the PISO algorithm, techniques such as pressure under-relaxation and SIMPLE subiterations were introduced, and they were developed primarily for large-step transient computations. This type of PISO algorithm can enhance the computational convergence for iterations involving larger Courant numbers.

10.6 Mesh Generation and Post-processing 10.6.1 Mesh Generation A high-quality grid is key to the computational accuracy of flow simulations. Mesh types that are typically used include structured, unstructured, and Cartesian grids. Structured and unstructured grids are the body-fitted grid type. The data structure of the structured grid is simple, requires small storage space, and is easy to index. It typically comprises grid cells with large aspect ratios and good orthogonality, which are suitable for characterizing the boundary-layer flow. However, in structured grids,

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Fig. 10.3 Flowchart of PISO algorithm

generating a suitable mesh topology for flow simulations using complex geometries is difficult. Meanwhile, the unstructured grid allows meshes to generate easily, can accommodate complex shapes, and is conducive to possible subsequent adaptive processing. However, compared with structured grids, unstructured grids require more computational time and larger computer memory, and cannot simulate the boundary-layer flow as effectively as structured grids. The Cartesian grid exhibits the simplest structure and allows grid generation to be automated easily. It is widely used in the adaptive mesh refinement for high-precision numerical simulations. However, because the Cartesian grid is often not conformal to wall surfaces, it requires additional treatment in the vicinity of the wall surfaces, e.g., the immersed boundary, cutting element, and hybrid Cartesian grid methods. These methods are extremely effective for inviscid flows or viscous flows at low Reynolds numbers; however, they pose several difficulties when used for solving engineering problems with high Re.

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189

10.6.2 Post-processing Results of CFD simulations must be displayed graphically in the form of graphs comprising curves, contours, vectors, streamlines, etc. This process is known as the post-processing of data. The most widely used CFD post-processing software are Origin, Tecplot, and ParaView. Origin is a professional function graphical software developed by OriginLab. It can not only satisfy the drawing requirements of general users, but also fulfill the demands of advanced users for data analysis and function fitting. Origin offers mathematical functions such as statistics, signal processing, curve fitting, and peak analysis and can allow various data formats, including ASCII, Excel, and pClamp, to be imported. Tecplot is an effective scientific drawing software program launched by Amtec. Its application fields include aerospace engineering, automotive, and other industries, as well as scientific research fields such as fluid mechanics and heat transfer. Tecplot has a dedicated data interface that can be directly read as *.cas and *.dat files for the CFD software FLUENT. ParaView is an open-source cross-platform visualization program. It supports parallelism and can be executed on a mainframe computer with a distributed memory. ParaView features a dedicated OpenFOAM interface. OpenFOAM is an objectoriented CFD open-source code written in C++ and is executed under Linux. It is increasingly used in wind turbine flow simulations, particularly wake simulations. Support for the OpenFOAM data format is one of the advantages of ParaView.

10.7 Applications of CFD in Wind Turbine Aerodynamics The application of the CFD method to wind turbines is based primarily on its application to helicopters. Previously, researchers solved steady and unsteady 2D axisymmetric incompressible Euler and incompressible RANS equations using the finite difference method. In these simulations, the wind turbine rotor is simplified to an actuating disk. By the mid-1980s, numerical methods for solving the viscous flows of rotating blades had been gradually developed. To conserve computing resources, the flow field was segmented into three regions: the viscous boundary-layer region close to the blade wall, wake inviscid region, and outer inviscid region. The boundarylayer equations are solved using a turbulence model in the boundary-layer region, whereas the Euler equations are solved in the inviscid regions. Results show that rotation delayed the onset of blade flow separation and thus significantly increased the maximum lift coefficient. Hence, the CFD method is advantageous for improving our understanding of the physics associated with rotating flow fields. Since the 1990s, numerous numerical simulations based on the FVM have been performed to investigate the wind turbine flow field [5–7], the most popular of which is to solve the RANS equations using a body-resolved mesh. In the early 1990s, the

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simulation of wind turbine flows was limited to axial inflows. It was later extended to the solution of unsteady states, such as yaw, and significant achievements have been realized in aerodynamic performance prediction and flow field mechanism research. Additionally, CFD simulations have been performed to investigate the 3D rotational effect and tip loss [8–11]. Scholars have used a quasi-3D N–S equation solution method to investigate the 3D rotational effect and discovered that the latter significantly affected the separation flow on blades but barely affected the attached flow. Subsequently, 3D rotational effect models were developed based on CFD simulation results. CFD has contributed significantly to wind turbine aerodynamic research in other aspects, such as the aerodynamic properties of airfoils and blades, the flow in the vicinity of wind turbine nacelles, blade flow control, and the characteristics of a ducted wind turbine rotor. In addition to wind turbine aerodynamic performance prediction, CFD is vital to wind turbine wake research [12–14]. The combination of actuator theory and the RANS method has been widely used to investigate the unsteady aerodynamics of rotors and towers, wind turbine wake interactions, and flows in wind farms. To characterize turbulent flows more comprehensively, LES was used for turbulence modeling instead of the RANS method. Many researchers have combined LES and actuator theory to investigate the wake in wind turbines. The topics investigated include the development and stability of the wake of a single wind turbine, the effect of turbulence intensity on wake interactions, and the aerodynamics of wind turbine arrays in atmospheric turbulence.

References 1. Burton, T., Sharpe, D., Jenkins, N., et al. (2011). Wind Energy Handbook. Wiley Publishing. 2. Hinze, J. O. (1975). Turbulence. McGraw-Hill. 3. Patankar, S. V., & Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer, 15(10), 1787–1806. 4. Rhie, C., & Chow, W. L. (1983). Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA journal, 21(11), 1525–1532. 5. Troldborg, N., Zahle, F., Rethore, P. E., et al. (2015). Comparison of wind turbine wake properties in non-sheared inflow predicted by different computational fluid dynamics rotor models. Wind Energy, 18(7), 1239–1250. 6. Troldborg, N., Zahle, F., Réthoré, P. –E., et al. (2012). Comparison of the wake of different types of wind turbine CFD models. In Proceedings of the 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. American Institute of Aeronautics and Astronautics. 7. Mittal, A., Sreenivas, K., Taylor, L. K., et al. (2016). Blade-resolved simulations of a model wind turbine: Effect of temporal convergence. Wind Energy, 19(10), 1761–1783. 8. Du, Z., & Selig, M. S. (1998). A 3D stall-delay model for horizontal axis wind turbine performance prediction. In Proceedings of the ASME Wind Energy Symposium. Reno, United States, F 1/12–1/15. 9. Glauert, H. (1927). A general theory of the autogyro. Journal of the Royal Aeronautical Society, 31(198), 483–508.

References

191

10. Pitt, D. M., & Peters, D. A. (1980). Theoretical prediction of dynamic-inflow derivatives. In Proceedings of the 6th European Rotorcraft and Powered Lift Aircraft Forum. Bristol, England, F 16/9–19/9/. 11. Coleman, R. P., Feingold, A. M., & Stempin, C. W. (1945). Evaluation of the induced-velocity field of an idealized helicoptor rotor. National Advisory Committee for Aeronautics, NACAWR-L-126. 12. Fleming, P., Gebraad, P. M. O., Lee, S., et al. (2015). Simulation comparison of wake mitigation control strategies for a two-turbine case. Wind Energy, 18(12), 2135–2143. 13. Sørensen, N. N., Bechmann, A., Réthoré, P.-E., et al. (2014). Near wake Reynolds-averaged Navier-Stokes predictions of the wake behind the MEXICO rotor in axial and yawed flow conditions. Wind Energy, 17(1), 75–86. 14. Wu, Y. T., & Porte-Agel, F. (2011). Large-eddy simulation of wind-turbine wakes: evaluation of turbine parametrisations. Boundary Layer Meteorol, 138(3), 345–366.

Chapter 11

Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

Predicting the aerodynamic performance of wind turbines is key in the design of wind turbine rotors. The prediction accuracy directly affects the design outcome. In this chapter, numerical issues arising from the application of the Reynolds-averaged Navier–Stokes (RANS) method to wind turbine flow simulations are introduced.

11.1 Governing Equations and Discretization 11.1.1 Governing Equations In the computational fluid dynamics (CFD) simulation of a wind turbine, the rotor blades exhibit rotational motion relative to the ground coordinate system, which must be described in the simulation. If the ground system is used as a reference system for solving, the rotational motion of the blades must be described directly, which requires grid motion. By contrast, if we introduce a reference frame that rotates with the blades, the rotor flow field under axial flow conditions can be transformed into a quasi-steady problem. The latter approach is typically referred to as the multiple reference frame (MRF) model. In the MRF model, the rotation axis (and speed) of the rotating reference frame coincides with the rotation axis (and speed) of the wind turbine rotor. Under a rotating reference frame, the forms of the governing equations remain the same. The rotating motion is reflected in the velocity definition, and velocity can be defined in two ways.

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_11

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

In the first approach, the velocity vector is defined in the rotating reference frame and the expression for the contravariant velocity V remains unchanged. The centrifugal and Coriolis forces can be expressed by adding a source term to the right-hand side of the incompressible Navier–Stokes (N–S) equation, as in Eq. (11.1). ∂ ∂t

∮ Ω

→ dΩ + W

∮

(

∂Ω

∮ ) → F→c − F→v dS = QdΩ

(11.1)

Ω

The source term Q is defined as →= Q

[

] 0 , −ρ[2ω → × u→ + ω → × (ω → × r→)]

(11.2)

where ω → is the angular velocity of the reference frame, and r→ denotes the position vector. The second approach is to define u→ as the absolute velocity, where the source term Q is defined as →= Q

[

] 0 . −ρ(ω → × u→)

(11.3)

Therefore, the contravariant velocity V in Eq. (10.3) is expressed as V = (→ u−ω → × r→) · n→.

(11.4)

Considering the low absolute rotation speed of air in most areas, except near the wind turbine rotor, the latter method is presented in this chapter.

11.1.2 Spatial Discretization For a scalar ϕ, the incompressible conservation transport equation of the finite volume method (FVM) can be expressed in the following integral form: ∮ ρ

Ω

∂ϕ dΩ + ρ ∂t

∮

ϕ u→ · d S→ =

∮

┌ϕ ∇ϕ · d S→ +

∮ Ω

Q ϕ dΩ.

(11.5)

In Eq. (11.5), ρ denotes the air density, u→ the velocity vector, S→ the surface area vector, ┌ϕ the diffusion coefficient of ϕ, Q ϕ the source of ϕ, and Ω the control volume. Equation (11.5) can be rewritten in discretized form as follows: ρ

N N faces faces ∑ ∑ ∂ϕ dΩ + ρ u→ f ϕ f · S→ f = ┌ϕ ∇ϕ f · S→ f + Q ϕ Ω. ∂t f f

(11.6)

11.2 Turbulence Models

195

In Eq. (11.6), Nfaces is the face number of the control volume, ϕ f the value of ϕ at face f , and S→ f the area vector of face f . ϕ f can be obtained by interpolating from the value at the cell center using the second-order upwind scheme. ⇀

ϕ f = ϕup + ∇ϕup · r ,

(11.7)

where ϕup and ∇ϕup denote the value of ϕ and its gradient, respectively, in the ⇀

upstream element; r is the displacement vector from the center of the upstream element to the center of the surface. The gradient of ϕ at the cell center c0 is calculated as (∇ϕ)c0 =

Nfaces ϕc0 + ϕc1 → 1 ∑ Sf , Ω f 2

(11.8)

where ϕc1 is the value at the center c1 of the right element (c0 is the left element) of the face f .

11.1.3 Temporal Discretization Considering scalar ϕ as an example, the rate of change of its value in the control volume over time can be written as ∂ϕ = F(ϕ), ∂t

(11.9)

where F(ϕ) is the result of spatial discretization. If an explicit time integration scheme is used, the value of F(ϕ) is the same as that of the current time step. By contrast, if an implicit time integration scheme is used, the value of F(ϕ) is that of the next time step. ϕ n+1 = ϕ n + Δt F(ϕ n+1 )

(11.10)

Because implicit time integration uses the spatial discretization results of the next time step, it must be solved iteratively at each time step.

11.2 Turbulence Models The most frequently used turbulence models are the one-equation Spalart–Allmaras model and two-equation k − ε and k − ω models.

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

11.2.1 One-Equation Model The one-equation model is also known as the single-equation eddy-viscosity transport model. The most commonly used one-equation model is the Spalart–Allmaras (S– A) model. The S–A model was proposed by Spalart based on practical engineering experience. In this model, the primary task is to solve the transport equation for the turbulent kinematic viscosity ν˜ . ] ( ) } ) μL ∂ ν˜ 2 ∂ ν˜ + Cb2 + ν˜ ρ ∂x j ∂x j ( )2 ν˜ + Cb1 S˜ ν˜ − Cω1 f ω d

∂ ν˜ 1 ∂ (ν˜ u i ) = + ∂t ∂ xi σ

{

∂ ∂x j

[(

(11.11)

The turbulent viscosity coefficient can be calculated as follows: ⎫ μt = ρ ν˜ f v1 ⎪ ⎬ χ3 f v1 = χ 3 +C 3 v1 ⎪ ⎭ χ = ν˜

(11.12)

μ L /ρ

The parameters in Eqs. (11.11) and (11.12) are expressed as follows: S˜ = S +

ν˜ f v2 , κ 2d 2

(11.13)

χ , 1 + χ f v1 ) ( √ ∂u j 1 ∂u i , − S = 2Ωi j Ωi j , Ωi j = 2 ∂x j ∂ xi f v2 = 1 −

( fω = g

6 1 + Cω3 6 g 6 + Cω3

)1/6 , g = r + Cω2 (r 6 − r ), r =

The constants used in the model are as follows: Cb1 = 0.1355, Cb2 = 0.622, σ = 2/3, Cv1 = 7.1, b2 ) Cω1 = Cκb12 + (1+C , Cω2 = 0.3, Cω3 = 2.0, κ = 0.4187. σ

(11.14) (11.15) ν˜ . ˜Sκ 2 d 2

(11.16)

11.2 Turbulence Models

197

11.2.2 Two-Equation Model 1. k − ε model The k − ε turbulence model is one of the most widely used two-equation eddyviscosity models. It is based on the solution of equations for the turbulent kinetic energy k and turbulent dissipation rate ε. The eddy-viscosity coefficient can be obtained as μT = ρCμ

k2 . ε

(11.17)

The transport equations for turbulent kinetic energy k and turbulent dissipation rate ε are expressed as follows: [( ) ] μT ∂k μ+ + μT S 2 − ρε, σk ∂ x j [( ) ] ∂ ∂ μT ∂ε ∂ (ρεu i ) = μ+ (ρε) + ∂t ∂ xi ∂x j σε ∂ x j

∂ ∂ ∂ (ρku i ) = (ρk) + ∂t ∂ xi ∂x j

ε ε2 + C1ε μT S 2 − C2ε ρ . k k

(11.18)

(11.19)

Here, the strain rate tensor is defined as } √ S = ( 2Si j Si j ) . ∂u Si j = 21 ∂ xij + ∂∂ux ij

(11.20)

The constants used in the model are as follows: C1ε = 1.44, C2ε = 1.92, Cμ = 0.09, σk =1.0, σε =1.3. The k − ε model is based on the equilibrium between turbulent kinetic energy generation and dissipation and is suitable for simulations with high Reynolds numbers (Re). 2. k − ω model The k − ω model solves the transport equations of turbulent kinetic energy k and its dissipation rate ω. The turbulent kinetic energy dissipation rate and eddy-viscosity coefficient are expressed as ω=

ε , k

k μT = ρCμ . ω

(11.22) (11.22)

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

The k − ω model exhibits better numerical stability than the k − ε model in the viscous sublayer, and no decay function is required at the wall boundary. Studies have shown that when the adverse pressure gradient is within acceptable limits, the results are consistent with the experimental data. Menter [1] proposed a mixed two-equation turbulence model, i.e., the shear stress transport (SST) k − ω model, which combines the advantages of the k − ω and k − ε models by using mixing functions in the boundary layer. The corresponding transport equations of turbulent kinetic energy k and its dissipation rate ω are as follows: [ ] ∂(ρk) ∂(ρu i k) ∂ ∂k (μ + σk μt ) , + = Pk − Dk + ∂t ∂ xi ∂ xi ∂ xi [ ] ∂(ρω) ∂(ρu i ω) ∂ ∂ω 2 2 (μ + σω μt ) + = αρ S − βρω + ∂t ∂ xi ∂ xi ∂ xi 1 ∂k ∂ω + 2(1 − F1 )ρσω2 , ω ∂ xi ∂ xi

(11.23)

(11.24)

where (

∂u i Pk = min μt ∂x j

(

) ) ∂u j ∂u i ∗ , 10β ρkω , + ∂x j ∂ xi

Dk = β ∗ ρkω,

(11.25) (11.26)

σk =

1 , F1 /σk1 + (1 − F1 )/σk2

(11.27)

σω =

1 , F1 /σω1 + (1 − F1 )/σω2

(11.28)

α = F1 α1 + (1 − F1 )α2 ,

(11.29)

β = F1 β1 + (1 − F1 )β2 .

(11.30)

Here, S is the modulus of the mean strain rate tensor and is defined as } √ S = ( 2Si j Si j ) . ∂u Si j = 21 ∂ xij + ∂∂ux ij

(11.31)

11.2 Turbulence Models

199

The mixing function F1 is defined as follows: ⎫ ⎪ ⎪ ⎪ ⎪ ) ]⎬ [ ( √ 500μ L 4ρσω2 k k 4 arg1 = min max β ∗ ωy , ρωy 2 , C Dkω y 2 ⎪. ) ⎪ ( ⎪ ⎪ C Dkω = max 2ρσω2 1 ∂k ∂ω , 10−10 ⎭ F1 = tanh(arg41 )

(11.32)

ω ∂ xi ∂ xi

In Eq. (11.32), y represents the distance from the nearest wall. The turbulent viscosity coefficient is defined as follows: μT =

a1 ρk . max(a1 ω, S F2 )

(11.33)

The second mixing function F2 is defined as } 2 F2 = tanh(arg ( √ 2) ) . L arg2 = max β2∗ ωyk , 500μ ρωy 2

(11.34)

The values of the constants used in the model are as follows: β ∗ = 0.09, a1 = 0.31, α1 = 5/9, α2 = 0.44, β1 = 3/40, β2 = 0.0828, σk1 = 0.85, σk2 = 1, σω1 = 0.5, σω2 = 0.856.

11.2.3 Selection of Turbulence Models The accuracy of the turbulence model does not necessarily increase with the number of equations. An appropriate turbulence model should be selected based on the actual flow conditions. As a one-equation turbulence model, the S–A model only solves the transport equation of the turbulent viscosity without considering the length scale of the local shear layer thickness. Therefore, it is not suitable for flows with significant variations in the turbulent length scale. Studies have shown that the S–A model performs well in simulations of attached flows but fails to accurately predict flow separation. For the k−ε model, the relaxation effect of Reynolds stress along the flow direction cannot be reflected under the assumption that the Reynolds stress is proportional to the local average shear. Because the model is isotropic, it cannot reflect the anisotropic property of Reynolds stress. Additionally, the model demands a high mesh resolution in the near-wall region and requires appropriate wall functions. To overcome these limitations, a revised nonlinear k−ε model has been proposed, in which the functional expression of Reynolds stress is replaced by an algebraic expression involving a second-order Taylor series expansion. The improved model not only includes the

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

anisotropy of the eddy-viscosity coefficient but also considers the effect of the average vorticity. The k − ω model is widely used for wall-resolved simulations. The SST k − ω model offers the advantages of both the k − ω and k − ε models. In particular, the model uses the k − ω and k − ε models to solve the flow in the near-wall and freeflow regions, respectively. The adjacent region is solved by constructing a weighted mixing function between the two models. This model couples a low Re model with a high Re model, which can not only accurately predict the flow in the boundary layer but also achieve a satisfactory computation result for the free flow. The SST k − ω model is widely used in the numerical simulation of wind turbine flows.

11.3 Transition Prediction A transition from laminar to turbulent flow occurs in the blade boundary layer. Recently, the transition effect on the aerodynamic characteristics of wind turbine airfoils and blades has been investigated. The results indicate that the transition process is not negligible when simulating blade aerodynamic performance. The two transition prediction models are described next.

11.3.1 Michel Transition Model The Michel model is a simple yet widely used classical transition prediction model. The occurrence of the transition is determined by the momentum thickness of the local boundary layer. To describe the flow at the transient point, Re based on the local momentum thickness Reθ and that based on the reference length between the local position and stagnation point Rex is introduced. The relationship between the two variables at the transition point is as follows: Reθ = 1.718Re0.435 x

(11.35)

When Reθ < 1.718Re0.435 , the flow is considered laminar, and the eddy viscosity x is set to zero. Otherwise, the flow is considered turbulent, and the eddy viscosity is calculated using a turbulence model. To avoid an overly abrupt transition process, the eddy-viscosity coefficient is multiplied by the following transition factor in a region after the transition: ⎤ ⎡ ∮x dx ⎦ (11.36) γtr = 1 − exp⎣−G(x − xtr ) ue xtr

in which the variables are defined as follows:

11.3 Transition Prediction

201

[ G=

]

u 3e −1.34 R , 213(log Rxtr − 4.7323) ν 2 xtr (u x ) e Rxtr = . ν tr 3

(11.37) (11.38)

Here, u e is the velocity of the local flow, and the subscript “tr” represents the parameter at the transition point.

11.3.2 γ–Reθ Transition Model The transition model developed by Menter et al. [2] has been widely used in RANS simulations. The transition model contains two transport equations—one for the intermittent factor γ and the other for the transition momentum-thickness Reynolds ˜ qt . A parameter known as the strain rate Re is introduced as follows: number Re Rev =

ρy 2 S, μ

(11.39)

where ρ is the density, μ the laminar viscosity coefficient, y the distance from the nearest wall, and S the strain rate modulus. The maximum value of Rev is proportional to the Reynolds number Reθ , which is based on the local momentum thickness θ and is expressed as Reθ =

Rev,max . 2.193

(11.40)

Based on Eq. (11.40), the non-local parameter Reθ can be calculated using the local parameter Rev . The transport equation for the intermittent factor γ is ) ( [( ) ] μt ∂γ ∂(ργ ) ∂ ρU j γ ∂ μ+ . + = Pγ 1 − E γ 1 + Pγ 2 − E γ 2 + ∂t ∂x j ∂x j σγ ∂ x j (11.41) The parameters in the equation are defined as follows: Pγ = Flength ca1 ρ S(γ Fonset )0.5 (1 − γ ),

(11.42)

Er = ca2 ρΩγ Fturb (ce2 γ − 1),

(11.43)

Fonset1 =

Rev , 2.193Reθc

(11.44)

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations (

)

4 R ρy 2 S ρk − T , RT = , Fturb = e 4 , Rev = μ μω [ ] 4 Fonset2 = min max(Fonset1 , Fonset1 ), 2 ,

[ Fonset3 = max 1 −

(

RT 2.5

)3

(11.45) (11.46)

] ,0 ,

(11.47)

Fonset = max(Fonset2 − Fonset3 , 0),

(11.48)

( ) √ 1 ∂u j ∂u i , S = 2Si j Si j , Si j = + 2 ∂ xi ∂x j ) ( √ ∂u j 1 ∂u i , − Ω = 2Ωi j Ωi j , Ωi j = 2 ∂x j ∂ xi

(11.49) (11.50)

where Flength and Reqc are functions of Reqc and are defined as (

) ( ) ˜ qt + 12000 ˜ qt + 100 Re 7Re Reqc = β + (1 − β) , 25 10 ⎧ ⎫ ⎡ ( )1.2 ⎤ ⎨ ⎬ ˜ qt Re ⎦ + 0.1, 30 , Flength = min 150 exp⎣− ⎩ ⎭ 120 ⎡(

˜ qt − 100 Re β = tanh⎣ 400

)4 ⎤ ⎦.

(11.51)

(11.52)

(11.53)

The transport equation for the transition momentum-thickness Reynolds number ˜ qt is Re ) ( ˜ qt ∂ ρ Re ∂t

+

( ) ˜ qt ∂ ρU j Re ∂x j

[ ] ˜ qt ∂ Re ∂ = Pθ t + σθ t (μ + μT ) . ∂x j ∂x j

(11.54)

The parameters in Eq. (11.54) are defined as follows: ρ ~θ t )(1.0 − Fθt ), Pθ t = cθ t (Reθ t − Re t t=

500μ , ρU 2

(11.55) (11.56)

11.3 Transition Prediction

{

203

[

Fθt = min max Fwake e θBL =

−( δy )

4

( , 1.0 −

γ − 1/ce2 1.0 − 1/ce2

} )2 ] , 1.0 ,

(11.57)

~θ t μ Re 15 50Ωy ρωy 2 , δBL = θBL , δ = δBL , Reω = , ρU 2 U μ Fwake = e

( −

Reω 1×105

(11.58)

)2

,

(11.59)

where U is the magnitude of the local velocity, and Reqt is the Re of the transition momentum-thickness outside the boundary layer, which can be expressed empirically as follows: Reqt = 803.73(T u + 0.6067)−1.027 F(λθ , K ), √ T u = 100

(11.60)

2k/3 ρθ 2 dU μ dU , λθ = ,K = . U μ ds ρU 2 ds

(11.61)

Here, s is the flow-direction coordinate, and θ is the momentum thickness. When λθ ≤ 0, F(λθ , K ) = 1 − (−10.32λθ − 89.47λ2θ − 89.47λ3θ )e

−T u 3.0

.

(11.62)

When λθ > 0, F(λθ , K ) =1 + (0.0962 K˜ − 0.148 K˜ 2 − 0.0141 K˜ 3 )(1 − e + 0.556(1 − e−23.9λθ )e

−T u 1.5

,

K˜ = 106 K .

−T u 3.0

) (11.63) (11.64)

To enhance the stability of the numerical simulation, the following restrictions are imposed on λθ , K, and Reθ t : −0.1 ≤ λθ ≤ 0.1

(11.65)

−3 × 10−6 ≤ K ≤ 3 × 10−6

(11.66)

Reθ t ≥ 20.

(11.67)

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

The constants used in the transport equation for the intermittent factor γ and ˜ θ t are as follows: transition momentum-thickness Reynolds number Re ca1 = 1.0, ca2 = 0.03, ce2 = 50, σ f = 1.0, σθt = 2.0. In addition, separation-induced transitions are considered, as follows: {

γsep

[ ( = min s1 max 0,

Rev 3.235Reθ c

Freattach = e

)

( )4 R − 20T

] } − 1 Freattach , 2 Fθt ,

(11.68)

, s1 = 2,

(11.69)

γeff = max(γ , γsep ).

(11.70)

The effective intermittent factor γeff , solved using the transition model, is coupled to the k-equation of the SST turbulence model. The turbulent kinetic energy production Pk and dissipation Dk , and the mixing function F 1 are modified as follows: [ ] ∂(ρk) ∂(ρu i k) ∂ ∂k ˜ ˜ (μ + σk μt ) , + = Pk − Dk + ∂t ∂ xi ∂ xi ∂ xi

(11.71)

P˜k = γeff Pk , D˜ k = min[max(γeff , 0.1), 1.0]Dk ,

(11.72)

√ ρy k 8 , F3 = e−(R y /120) , F1 = max(F1orig , F3 ). Ry = μ

(11.73)

11.4 Initial and Boundary Conditions Appropriate initial and boundary conditions are required for the numerical simulations. Prior to the iteration process, the velocity, pressure, and density fields of the initial state of the flow field must be specified. The initial value of each variable of the flow field at time t = 0 is set as follows: V (x, 0) = V (x), p(x, 0) = p(x), ρ(x, 0) = const.

(11.74)

In addition to the initial conditions, suitable boundary conditions are required. In wind turbine simulations, boundary conditions are typically imposed for the inlet, outlet, rotational periodic boundaries, and walls.

11.4 Initial and Boundary Conditions

205

11.4.1 Inlet and Outlet Boundary Conditions In the numerical simulation of a wind turbine associated with a two-dimensional (2D) airfoil or three-dimensional flow field, the windward surface is set as the velocity inlet boundary, and the values of the velocity and turbulence parameters are specified. The mass flow and momentum at the inlet boundary are calculated based on the velocity. For example, the mass flow into the computational domain per unit time is calculated as ∮ ⇀ ⇀ m˙ = ρ v ·d A. (11.75) In numerical simulations of airfoils or rotors, the leeward surface is typically set as the pressure outlet boundary, and the static pressure value on the boundary is specified.

11.4.2 Rotational Periodic Boundary Condition Under axial flow conditions, the wind turbine flow field exhibits periodicity. For a rotor with N blades, the flow field can be discretized equally into N zones, and the flow in each zone is identical. Therefore, the rotational periodic boundary condition can be employed to reduce the computational domain to 1/N of the entire flow field. The periodic boundary condition can be used to interpolate the velocity, pressure, and flux parameters of two identically shaped boundaries. The cell nodes on both sides of the periodic boundary group may correspond one-to-one or may not completely correspond. If a one-to-one correspondence exists, data transfer from one periodic boundary to the corresponding periodic boundary is simple, and precision is preserved. Otherwise, data transfer is realized via interpolation, which is accompanied by precision loss. The one-to-one correspondence of the periodic boundary condition is discussed in this section. The variables in the dummy “mirror cell” neighboring a periodic boundary are provided by the corresponding periodic boundary. Scalar parameters, such as pressure, can be directly assigned, but the velocity vector must be rotated first. If the rotation axis is set to the z-axis, the rotation transformation matrix is ⎡

⎤ cos θ − sin θ 0 ⇀ v 2 = ⎣ sin θ cos θ 0 ⎦v 1 0 0 1 ⇀

⇀

⇀

(11.76)

where v 2 is the velocity vector of the “mirror cell,” / v 1 the velocity vector of the corresponding periodic boundary, and θ = 2π N the angle between periodic boundaries.

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

11.4.3 Wall Boundary Condition The no-slip wall boundary condition is applied to the wall surfaces of the airfoils or blades. The fluid velocity v→ satisfies the following condition at the walls: ⇀

⇀

v = v grid ,

(11.77)

⇀

where v grid is the velocity of the wall mesh motion. When mesh motion is not ⇀ involved in the calculation, v grid equals zero.

11.5 Mesh for Simulation 11.5.1 2D Mesh for Airfoils When examining the aerodynamic characteristics of 2D airfoils via CFD simulations, meshing strategies involving H-, O-, and C-shaped topologies are typically used. The H-shaped topology shown in Fig. 11.1a is particularly suitable for airfoils with sharp leading and trailing edges. For an airfoil with a curved leading edge, the H-shaped topology results in singularities, which can be alleviated by partitioning the mesh topology in the direction perpendicular to the leading edge. However, this results in a high-aspect-ratio mesh, which limits the local time-step size to an extremely small value and consequently reduces the computational efficiency. The O-shaped topology, as shown in Fig. 11.1b, exhibits a relatively uniform grid distribution in the vicinity of the airfoil, which is beneficial for solving the flow field when separation exists or when the airfoil undergoes a pitching motion. However, for sharp trailing-edge airfoils, this topology causes high skewness, which deteriorates the robustness of the CFD solver and solution accuracy. Another meshing strategy is to use the C-shaped mesh topology, as shown in Fig. 11.1c, which conforms to the curvature of the leading edge. The C-shaped topology is more widely used for near-wall meshing, although it results in a high aspect ratio of the grid cells at the airfoil trailing edge. In addition, this type of grid is suitable for capturing airfoil wakes. Specific flow conditions must be considered when determining the size of the computational domain. If the walls of the wind tunnel experimental section are considered, the size of the computational domain is determined by the size of the wind tunnel experimental section. If free far-field boundaries are used, the inlet boundaries of the computational domain should be set at least 10 times the chord length away from the leading edge of the airfoil, the outlet boundaries should be at least 15 times the chord length from the airfoil trailing edge, and the lateral boundaries should be 10 times the chord length away from the airfoil surface. If flow separation exists, 200 of more mesh points should be specified on the airfoil up- or down-surface. To

11.5 Mesh for Simulation

207

Fig. 11.1 Different meshing topologies

capture the flow in the boundary layer accurately, one must refine the meshes near the airfoil surfaces. In particular, 40 mesh layers or more must be specified in the boundary layer, and the height of the first mesh layer adjacent to the walls should satisfy the condition y + < 1, in which y + is defined as y+ =

ρu τ y . μ

(11.78)

Here, y denotes the mesh height of the first mesh layer. The fraction velocity u τ is related to the friction stress τw on the wall and is defined as / uτ =

τw . ρ

(11.79)

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

11.5.2 Mesh for Wind Turbine Rotor In the numerical simulation of a wind turbine rotor, the computational domain is— segmented into two zones—a rotational zone containing the rotor and a stationary zone. Figure 11.2 shows the computational domain of a two-blade wind rotor under axial inflow conditions. Owing to the use of periodic boundaries, the computational domain is only one-half of the entire flow field. The distance between the outer boundary of the computational domain and the rotor is set as 20R (where R is the radius of the rotor), the distance from the inlet boundary to the rotor is set as 10R, and the distance from the downstream outlet boundary to the rotor is set as 20R. To capture the flow in the vicinity of the blade accurately, one must use more than 200 grid points along the blade chord and 50 grid points or more along the blade span. In addition, at least 40 mesh layers within the boundary layer are required, and the height of the first mesh layer should satisfy the condition y + < 1. Figure 11.3 shows the mesh in the vicinity of the wind turbine blade. The rotation of the wind turbine rotor can be modeled via two approaches, namely the multiple reference frame (MRF) model and sliding mesh. As discussed earlier, the MRF model is an approximate steady-state model that is typically used for steady simulations. Because the MRF model is employed, no

Fig. 11.2 Computational domain for two-blade rotor

Fig. 11.3 Mesh in the vicinity of wind turbine blade

11.6 Simulation Examples

209

mesh motion is required, and the rotor appears to be frozen. Therefore, a method based on the MRF model is known as the frozen-rotor method. The sliding mesh is a dynamic mesh technique, in which all mesh cells in the rotational zone rotate as rigid bodies. Because no additional assumptions are imposed on the rotational zone, the sliding mesh technique can inherently account for unsteady flow crossing the rotational and stationary zones and is typically used for transient simulations.

11.6 Simulation Examples As examples, numerical simulations of an S809 airfoil and the NREL Phase VI rotor are introduced. The S809 airfoil has a thickness of 21% and is used to construct the NREL Phase VI rotor blade. Wind tunnel experiments were performed at the Delft University of Technology, Ohio State University, and Colorado State University [3–7] to investigate the aerodynamics of the airfoil under both static and dynamic conditions; consequently, comprehensive measurement data were obtained. The prediction of the laminar–turbulent transition and the correction of the turbulence model parameter are described in the following sections. More accurate numerical results of wind turbine aerodynamic performance can be obtained by comprehensively investigating these two aspects.

11.6.1 Fully Turbulent Simulation Versus Transitional Simulation The aerodynamic performance of the S809 airfoil was simulated by solving RANS equations. The lift and drag coefficients obtained using the fully turbulent SST k − ω model and γ − Reθ transition model are presented in Fig. 11.4. Compared with the fully turbulent model, the transition model yields a relatively lower error for the lift and drag coefficients relative to the experimental data. At small angles of attack, the lift-coefficient error decreases from approximately 10% to within 1%, and the drag coefficient error decreases from 100 to 10%. At larger angles of attack, flow separation occurs at the trailing edge of the airfoil, and the lift and drag coefficients of the transitional simulations gradually approach those achieved by the fully turbulent simulations.

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11 Reynolds-Averaged Navier–Stokes Method for Wind Turbine Simulations

Fig. 11.4 Lift-coefficient curve (left) and lift–drag polar curve (right) of S809 airfoil

11.6.2 Parameter Correction for SST Turbulence Model All turbulence models based on turbulence transport equations must include closure constants. The constants used in the turbulent models are primarily derived from experiments. The empirical nature of these parameters cannot guarantee their universality in any practical situation. For specific flow conditions, better numerical simulation results can be obtained by adjusting the closure parameters. Based on the SST model as an example, the original closure constants are β ∗ = 0.09, a1 = 0.31, α1 = 5/9, α2 = 0.44, β1 = 3/40, β2 = 0.0828, σk1 = 0.85, σk2 = 1, σω1 = 0.5, σω2 = 0.856. The closure parameter β ∗ is selected as the key parameter to be corrected. The value of this parameter affects the numerical results of turbulent kinetic energy k and dissipation rate ω. Figure 11.5 shows the computational lift-coefficient curve of the S809 airfoil at Re = 1×106 . In the light stall stage (where the angle of attack range is 8°–20°, which corresponds to the occurrence of trailing-edge separation on the airfoil), when the value of β ∗ value is corrected to 0.11 from 0.09, the computational lift coefficients agree with the experimental data. By applying the corrected β ∗ value to the simulation of the NREL Phase VI rotor, a significant improvement is observed, as shown in Fig. 11.6. The results indicate that using the corrected parameter significantly improves the accuracy of the computational rotor torque, particularly when the wind speed is between 9 and 13 m/s. The corrected simulation accurately captures the mechanism underlying blade stall.

References

211

Fig. 11.5 Lift-coefficient curves computed using different β ∗ values

Fig. 11.6 Rotor torque simulated using different β ∗ values

References 1. Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8), 1598–1605. 2. Menter, F., Langtry, R., & Völker, S. (2006). Transition modelling for general purpose CFD codes. Flow, Turbulence and Combustion, 77(1–4), 277–303. 3. Ramsay, R., Hoffman, M., Gregorek, G. (1995). Effects of grit roughness and pitch oscillations on the S809 airfoil. National Renewable Energy Lab., Golden, CO 4. Somers, D. M. (1997). Design and experimental results for the S809 airfoil. National Renewable Energy Lab., Golden, CO 5. Sheng, W., Galbraith, R., Coton, F., et al. (2006). The collected data for tests on an S809 airfoil, volume I: pressure data from static, ramp and triangular wave tests. (Glasgow: University of Glasgow). 6. Sheng, W., Galbraith, R., Coton, F., et al. (2006). The collected data for tests on an S809 airfoil, volume II: pressure data from static and oscillatory tests. (Glasgow: University of Glasgow). 7. Sheng, W., Galbraith, R., Coton, F., et al. (2006). The collected data for tests on the sand stripped S809 airfoil, volume III: pressure data from static, ramp and oscillatory tests. (Glasgow: University of Glasgow).

Chapter 12

Large- and Detached-Eddy Simulation Methods for Wind Turbine Simulations

Wind turbines operate in the atmospheric boundary layer near the ground. The velocity and turbulence profiles in the atmospheric boundary layer are affected significantly by surface roughness, geography, topography, climate, and the mutual interference between wind turbine wakes in wind farms. The inflow conditions of wind turbines are extremely complex and include highly unsteady turbulent flows, which necessitate higher-order turbulence modeling techniques. Owing to the inadequacy of the Reynolds-averaged Navier–Stokes (RANS) method in simulating complex turbulence flow fields, large-eddy simulation (LES) and detached-eddy simulation (DES) methods have garnered significant attention. Because of its superiority in turbulence modeling, LES has become an effective method for investigating the complex and unsteady flows of wind turbines, particularly for wind turbine wakes. However, for body-resolved simulations performed via LES, the calculations involved require numerous grid cells and thus incur a high computational cost for engineering research. Hence, the actuator line method (ALM) has been proposed, in which the actual blades are equivalently represented by body forces equal to the forces exerted by air to the blades. This method significantly reduces both the difficulty in generating grids in the complex flow field of wind turbines and the number of mesh cells in the vicinity of the rotor, thus allowing a much finer mesh to be generated in the wake region. The combination of the ALM and LES offers significant prospects in wind turbine wake research, wind farm efficiency evaluation, and wind farm layout optimization.

© Science Press 2023 T. Wang et al., Wind Turbine Aerodynamic Performance Calculation, https://doi.org/10.1007/978-981-99-3509-3_12

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

When subjected to complex inflow conditions, the angle of attack of wind turbine blades changes continuously owing to transient changes in the wind speed and direction. Consequently, flow separation and dynamic stall occur on the blades. The RANS method is not sufficiently accurate for performing turbulence simulation corresponding to separated flow, whereas the computational cost of LESs for bladeresolved simulations is high. By contrast, the DES method combines the advantages of the RANS and LES methods. In the DES method, the flow in the boundary-layer region is solved using the RANS method, and LES is employed in other regions to capture large-scale eddies. In this chapter, the ALM and two turbulence solution methods, i.e., LES and DES, are introduced comprehensively.

12.1 ALM for Wind Turbine Simulations 12.1.1 Actuator Line Model Wind turbines extract energy from the wind, and the energy reduction in the wind can be reflected by the change in its momentum. In numerical simulations, the effect of wind on a wind turbine can be reflected by adding a momentum source term to the Navier–Stokes (N–S) equation. In the ALM, each wind turbine blade is substituted by an actuator line, and the body force representing the effect of air on the blade is applied to the mesh cells in the vicinity of the actuator line. Using the two-blade wind turbine in Fig. 12.1a as an example, the rotor blades are simplified into two actuator lines. Subsequently, each actuator line is segmented into several elements via control points. The forces of each element are determined based on the local flow information.

(a) Schematic illustration of actuator line model (b) Aerodynamic force on a blade cross-section

Fig. 12.1 Theoretical model for actuator line method

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215

The blade cross-section at a radial position r is shown in Fig. 12.1b. The local relative velocity Vrel can be calculated as follows: Vrel =

/

Vn2 + (Ωr − Vθ )2 ,

(12.1)

where Vn and Vθ represent the normal velocity perpendicular to the rotor plane and the tangential velocity parallel to the rotor plane, respectively. The local angle of attack α can be obtained as follows: ) ( Vn (12.2) α = φ − θ, φ = arc tan wr − Vθ Once the local angle of attack is determined, the lift and drag coefficients of the local airfoil can be obtained by interpolating the tabulated aerodynamic force data of the two-dimensional (2D) airfoil. Subsequently, the force on the blade element per unit length can be calculated as follows: 1 − → → → 2 f 2D = ρVrel e L + CD− e D ), c(C L − 2

(12.3)

→ → where − e L and − e D are the unit vectors in the lift and drag directions, respectively. To − → avoid the discontinuity problem in the computation, the force f 2D can be converted − → into a three-dimensional (3D) force distribution f 3D,ε using the 3D Gaussian distribution function. Subsequently, the distributed force is added to the N–S equation as the body force in the momentum source term. f→3D,ε = f→2D · ηε [ ( ) ] d 2 ηε (d) = ε−2 π −3/2 exp − ε

(12.4)

(12.5)

In the equation above, ε is the distribution factor associated with the mesh size and blade geometry. In most investigations, the value of ε is typically set to greater than 2Δ (where Δ is the local grid scale).

12.1.2 Nacelle and Tower Model The presence of a nacelle and tower results in additional disturbance, which would affect wake development. To model the corresponding body forces of the nacelle and the tower, a technique similar to the ALM can be used. The nacelle can be simplified to a cylinder with round caps, and its drag coefficient can be derived from the aerodynamic profile of the cylinder. The body force

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

representing the nacelle can be solved using the following equation: Fnacelle =

1 ρV 2 c D,nacelle Anacelle , 2 mag

(12.6)

where Anacelle represents the cross-sectional area of the nacelle. The wind turbine tower can be regarded similarly as the blade, except that the airfoil profile is replaced by the data of the cylinder. Considering the unsteady characteristics of the flow in the vicinity of the cylinder, the aerodynamic force of the tower should include the periodic oscillating lift force. The unsteady characteristics can be characterized by the Strouhal number St = f Dcyl /U∞ , where f represents the oscillation period, and Dcyl the local section diameter. The lift and drag coefficients of the tower can be calculated using Eqs. (12.7) and (12.8). The drag coefficients of the nacelle and tower generally range between 0.8 and 1.2. Cl,tower = Asin(2π f t)

(12.7)

Cd,tower = const

(12.8)

Notably, the nacelle and tower models of the body force method cannot satisfy the condition of no penetration at wall surfaces. To achieve a body force distribution that is closely similar to the solid geometry, the body forces must be distributed evenly within the nacelle and tower geometry, and beyond the geometric entity, a Gaussian distribution is involved. The distribution function can be easily described in the cylindrical coordinate system as follows, in which z represents the height direction: ⎧ ) ( z−z 0 2 − ⎨ Ce εz if r ≤ R )2 )2 ( ( η(r, θ, z) = (12.9) ⎩ − z−zεz 0 − r −R Ce e εr if r > R The parameter C is calculated as follows: ( )−1 C = εz R 2 π 3/2 + εz εr2 π 3/2 + εz εr Rπ 2 .

(12.10)

12.2 Large-Eddy Simulation LES is an intermediate turbulent flow simulation between the direct numerical simulation (DNS) and the RANS simulation. The primary principle of LES is to separate large eddies in regions with energy and the small eddies in the dissipative region of turbulent flow. Based on Kolmogorov’s [1] theory of self-similarity, large eddies in a

12.2 Large-Eddy Simulation

217

flow depend on the geometry, whereas small eddies are more universal. This feature allows one to explicitly solve for large eddies and implicitly account for small eddies using a subgrid stress (SGS) model. A filter operation is employed to decompose turbulent instantaneous motions into large-scale motions and small-scale fluctuations. The filtering operation is typically performed in Fourier wavenumber or physical space via low-pass filtering. Large-scale motion parameters are denoted by a “−” superscript, whereas smallscale motion parameters are denoted by a “' ” superscript. For example, the velocity v can be decomposed into large-scale motion velocity v and small-scale motion velocity v ' . The filtered large-scale motion satisfies the N–S equation, whereas the small-scale motion is solved using the subgrid-scale model. The governing equation used in LESs for wind turbine flow is the 3D filtered incompressible N–S equation, which describes the evolution of the large-scale motion in space and time. ∂u i =0 ∂ xi

(12.11)

∂τiSj ) ∂ ( ∂u i 1 ∂p + + ν∇ 2 u i − ui u j = − ∂t ∂x j ρ ∂ xi ∂x j

(12.12)

In Eq. (12.12), τiSj represents the SGS tensor and it is expressed as follows: τiSj = u i u j − u i u j

(12.13)

The SGS tensor represents the momentum transport between filtered small-scale fluctuations and the large-scale turbulence, and can be decomposed into three terms, namely τiSj = L i j + Ci j + τiSR j

(12.14)

The expressions and meanings of each term are as follows: • L i j = u i u j − u i u j is the Leonard stress, which represents the interaction between large-scale motions; it is the only term among the three terms that can be directly solved. • Ci j = u i u 'j + u i' u j is the cross-stress term, which represents the interaction between large- and small-scale eddies. ' ' • τiSR j = u i u j is the Reynolds stress tensor term, which reflects the interaction between small-scale eddies. Since the Leonard stress and cross-stress terms cannot satisfy the Galilean invariance, the decomposition shown above is only as a theoretical decomposition with physical significance; i.e., it is not employed in simulations. The specific method for computing the SGS will be described later.

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

12.2.1 Filtering Method The filter scale for LESs should be limited to the inertial subregion. In LESs, the dissipative effect of small eddies within the filter scale is modeled. In practical simulations, the filter scale typically coincides with the mesh cell scale; consequently, the required filter width depends on the grid resolution. Figure 12.2 shows the specific forms of filtering operations in the physical and Fourier spectral space. The filtering operation is typically implemented mathematically via integration. Let the filtering size be Δ. At space position (→ r0 , t), a physical parameter v for large-scale eddies can be expressed as follows: ∫ v(→ r0 , t) =

v(→ r , t)G(→ r0 , r→, Δ)d→ r,

(12.15)

D

where G(→ r0 , r→, Δ) is a filter function. The most typically used filter functions include the top-hat filter, Fourier cut-off filter, and Gaussian filter functions, which are described as follows: (1) Top-hat filter / { / 3 1 Δ |(x0 )i − xi | ≤ Δi / 2 G= |(x0 )i − xi | > Δi 2 0 (2) Fourier cut-off filter

Fig. 12.2 Schematic diagram showing filtering operations in physical and fourier space

(12.16)

12.2 Large-Eddy Simulation

219

G=

3 sin ∏ i=1

(

π Δi

[(x0 )i − xi ]

) (12.17)

π [(x0 )i − xi ]

(3) Gaussian filter ( G=

6 π Δ2

)3/2

(

6|→ r0 − r→|22 exp − Δ2

) (12.18)

In Eq. (12.16), Δi represents the filter scale in the i-direction and is calculated as Δ = (Δ1 Δ2 Δ3 )1/ 3 .

12.2.2 SGS Models The SGS model, which is derived based on Boussinesq’s eddy-viscosity assumption, is vital to LESs. Here, τiSj can be expressed as a combination of the deviatoric stress and normal stress, i.e., ) ( 1 S 1 S S S δi j , (12.19) τi j = τi j − τkk δi j + τkk 3 3 S S where τiSj − 13 τkk δi j is the deviatoric stress term, 13 τkk δi j the isotropic normal stress function. The deviatoric stress is proportional to the term, and δi j the Kronecker ( ) ∂u

strain rate tensor S i j = 21 ∂∂ux ij + ∂ xij of the large-scale eddies, and the deviatoric stress can be correlated to the subgrid kinetic energy kSGS as follows: 2 1 S δi j = −2νSGS S i j + νSGS Sii δii τiSj − τkk 3 3

(12.20)

1 S 2 1 S 2 τkk δi j = ( τkk )δi j = kSGS δi j . 3 3 2 3

(12.21)

Subsequently, we obtain kSGS =

1 S τ , 2 kk

(12.22)

and the SGS can be expressed as 2 2 τiSj = −2νSGS S i j + νSGS Sii δii + kSGS δi j . 3 3

(12.23)

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

The core of the SGS model is the appropriate closure of νSGS and kSGS . In most SGS models, νSGS is described as a turbulent viscosity coefficient. The most typically used SGS models include the standard Smagorinsky, dynamic Smagorinsky, scale similarity, dynamic k-equation, mixture, and wall-adapting local eddy-viscosity (WALE) models. In wind turbine wake simulations, the standard Smagorinsky and dynamic Smagorinsky models are the most widely used SGS models. Detailed descriptions of the two SGS models are presented as follows: 1. Smagorinsky Model The Smagorinsky model was proposed by Smagorinsky [2] in 1963 based on local equilibrium and eddy-viscosity assumptions. The eddy-viscosity coefficient is expressed as ( )2 | | νSGS = Cs Δ | S |,

(12.24)

| | ( )1/2 is the magnitude of the strain rate tensor, and Cs is the where | S | = 2S i j S i j Smagorinsky model constant. Researchers have attempted to determine a suitable value for Cs . In 1966, Lilly derived the constant Cs = 0.18 from the Kolmogorov spectrum. In 1970, Deardorff [3, 4] used the Smagorinsky model to simulate turbulent flow in a 3D channel and discovered that the value of Cs proposed by Lilly would result in an extremely high viscosity of the subgrid Reynolds stress. When Cs was set as 0.1, the simulated results agreed well with the experimental results. Deardorff attributed this difference to the presence of primary shear, which was not considered in Lilly’s analysis. Furthermore, Deardorff discovered that a Cs value of 0.21 was more suitable for simulating inhomogeneous flows without the presence of shear. Meanwhile, in applying LESs to isotropic turbulence, Kwak et al. [5], Shaanan et al. [6], and Ferziger [7, 8] matched the simulated energy decay rate with experimental data and concluded that the reasonable range for Cs was 0.19–0.24. To account for the reduced growth of small scales near walls, the value of the eddy viscosity νSGS must be reduced. Moin and Kim [9] added the van Driest function to the Smagorinsky model for near-wall modifications. The modified model is expressed as follows: [ / + ]2 | | + νSGS = Cs Δ(1 − e y A ) | S |,

(12.25)

where y + represents the closest distance to the wall, and A+ is a semiempirical constant. 2. Dynamic Smagorinsky Model The dynamic Smagorinsky model was proposed by Germano et al. [10] in 1991. It relies on a basic subgrid model, such as the Smagorinsky model. Based on the standard Smagorinsky model, the eddy-viscosity constant Cs in the dynamic Smagorinsky model is replaced with a function Cd (→ r , t) that changes with time and

12.2 Large-Eddy Simulation

221

space. The eddy-viscosity coefficient in the dynamic Smagorinsky model is expressed as νT = Cd (→ r , t)Δ2 |S|.

(12.26)

The dynamic eddy-viscosity coefficient Cd (→ r , t) is calculated based on the energy content of the smallest turbulence scale. The flow field is filtered for the second time, ˆ is typically set as Δ ˆ = 2Δ, which is known as “test filtering.” Its filtering scale Δ which is larger than the size of local mesh cells. The test filtering approach yields the test filtered stress term τiST j , as follows: Ʌ

ɅɅ

τiST j = ui u j − ui u j

(12.27)

S The test filter stress τiST j and SGS τi j are correlated as follows: Ʌ

Ʌ

ɅɅ

S L i j = τiST j − τˆi j = u i u j − u i u j ,

(12.28)

Ʌ

where L i j is the Leonard stress term after test filtering. It represents the Reynolds stress of the vortex structure, whose scale is between the filter scale Δ and test filter ˆ scale Δ. δi j L kk = −2Cd Mi j 3 | | [ 2 | | ]∧ 2| | | | Mi j = Δ |S |S i j − Δ |S |S i j . Ʌ

Li j −

Ʌ

Ʌ

Ʌ

Ʌ

Ʌ

(12.29)

Ʌ

(12.30)

The symbol “[]∧ ” indicates that the parameters in parentheses are subject to test filtering. The full expression for Cd (→ r , t) can be derived using the least-squares method. (→ ) 1 L i j Mi j r ,t = − . Cd − 2 Mmn Mmn

(12.31)

The dynamic Smagorinsky model overcomes some disadvantages of the standard Smagorinsky model. It does not require the model coefficients to be corrected nearwall surfaces. However, the model coefficient can be negative in some regions where energy backscatter is not excluded. To maintain numerical stability, an averaging r , t) in homogeneous directions. However, one might approach is performed on Cd (→ not be able to determine the homogeneity direction in complicated three-dimensional flows via wind turbine simulations. ⟩ ⟨ (− ) 1 L i j Mi j → Cd r , t = − , (12.32) 2 ⟨Mmn Mmn ⟩

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

where the braces < > represent the average approach. The scale-dependent dynamic model is superior to the standard dynamic model for eddy-viscosity prediction, although it incurs a higher computational cost owing to an additional filtering operation.

12.2.3 Turbulent Inflow Generation for LES The turbulent inflow condition is important in LESs. Variations in the mean velocity, velocity fluctuations, and pressure must be described at the inlet boundary. The flow variables should vary randomly in time and space and be compatible with the fluid governing equations. The flow associated with a wind turbine generally features a high Reynolds number (Re) and a high turbulence level. The instantaneous velocity of the inflow should satisfy certain spectral characteristics with strong spatial and temporal correlations. The generation of turbulent inflow has become an important topic in LES research. The corresponding methods can be classified into two types: precursor and synthetic methods. 1. Precursor Method The precursor method was proposed by Lund et al. [11] in 1998. The basic idea of this method is to create a geometrically similar computational domain and execute a presimulation to obtain the turbulence parameters for the inflow of the main simulation. A schematic illustration of the precursor method is shown in Fig. 12.3. In the presimulation, the instantaneous velocity, the pressure, and other variables in a plane perpendicular to the flow direction (P plane in the figure) are stored in a time series to generate a corresponding database. As the main simulation is implemented, all the instantaneous variables in the P plane are mapped to the inlet plane of the main simulation in a time series. The precursor method is advantageous in that the variables transferred to the inlet plane are derived from an actual turbulence simulation that satisfies both the

Fig. 12.3 Schematic illustration of precursor method

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223

N–S equation and energy spectrum characteristics. However, the precursor method presents some limitations. For example, the main simulation requires a sufficiently long time-series input to achieve accurate turbulence statistics, which implies that a significant amount of time is required during the presimulation to generate the database, and thus the necessity for large storage. 2. Synthesis Method In the synthesis method, turbulent variables are artificially synthesized with specified characteristics and then imposed on the inlet boundary of the computational domain. In contrast to the precursor method, the synthesis method does not require the generation of a turbulence database prior to simulation. This is because a synthetic turbulent wind has been specified to satisfy the continuity equation, and it is suitable for parallel computations. The synthesis mentioned above can be achieved via spectral synthesis, POD reconstruction, or the vortex method. For the vortex method, as an example, the basic principle is to add a random 2D vortex field to the mean velocity field at the inlet boundary. The vortex method applies the following 2D Lagrangian vortex equation: ∂ω + (→ v · ∇)ω = ν∇ 2 ω, ∂t

(12.33)

where ν is the kinematic viscosity coefficient. The equation is solved at the inlet boundary. The vorticity at point x = (x, y) on the inlet boundary is obtained by summing up the inductions of all vortex points. Suppose N vortex points exist on the inlet boundary; therefore, the vorticity at x is can be obtained as follows: ωi (→ x , t) =

N ∑

┌i (t)η(|→ x − x→i |, t),

(12.34)

i=1

where ┌i represents a function related to the local turbulent kinetic energy, and η the spatial vortex distribution. They are expressed as /

π Sk(→ xi ) 3N (2ln(3) − 3ln(2)) ( ) |→ |→ x |2 x |2 1 − 2σ − 2σ 2 2 2c − 1 2c , η(→ x) = 2π σ 2 ┌i (→ xi ) ≈ 4

(12.35) (12.36)

where σ can be a constant or calculated based on the turbulence-length scale as 3/4

σ=

Cμ k 3/2 ε

(12.37)

The tangential velocity at the inlet boundary can be solved based on the Biot– Savart law, as follows:

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

u(→ x) =

( ) N |x→i −→x |2 |x→i −→x |2 xi − x→) × n→ z 1 ∑ (→ 2σ 2 ┌i 1 − e e 2σ 2 . 2 2π i=1 |→ xi − x→|

(12.38)

Figure 12.4 shows the implementation steps for generating a turbulent inlet boundary using the vortex method, and Fig. 12.5 presents the corresponding results.

Fig. 12.4 Schematic illustration of turbulent inflow generated via vortex method

12.3 Detached-Eddy Simulation

225

Fig. 12.5 Results obtained from turbulent inflow generated using via vortex method

12.3 Detached-Eddy Simulation The treatment of the subgrid model at wall boundaries is a key issue in LESs. For complex turbulent flows with high Re, the scale of turbulence close to the walls is extremely small. To accurately capture the turbulence near the walls, the mesh scale requirements of DNSs are to be adopted. As an alternative, a hybrid method named DES has been developed, which was first proposed by Spalart [12] based on the Spalart–Allmaras (S–A) turbulence model in 1997. The DES method combines the advantages of the RANS and LES methods. Its primary idea is to use the RANS equation to simulate turbulent flow in the boundarylayer and near-wall region, and to apply LESs in the other regions.

12.3.1 S–A DES Model In the S–A turbulence model, the turbulent equilibrium is expressed as Cb1 Ων = Cω1 f ω

( ν )2 d

,

(12.39)

and the eddy viscosity is proportional to Ωd 2 , i.e., ν ∝ Ωd 2 ,

(12.40)

where d represents the distance from the wall surface. In the Smagorinsky subgrid model, the relationship among the subgrid turbulent viscosity, local rotation rate Ω, and grid scale Δ is as follows: νSGS ∝ ΩΔ2 , Δ = max(Δx, Δy, Δz)

(12.41)

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

Based on Eqs. (12.39–12.41), if d in Eq. (12.40) is replaced by Δ, then the S–A model behaves as the Smagorinsky model in LESs. Therefore, the DES length scale can be expressed as d˜ = min(d, CDES Δ), CDES = 0.65

(12.42)

When d > Δ, the DES model behaves as the Smagorinsky LES.

12.3.2 SST–DES Model Strelets [13] developed a hybrid SST–DES model. The length scale lk−ω of the SST model and the length scale l˜ of the DES model are defined as follows: lk−ω = k 1/ 2

/( ) β ∗ω

(12.43)

l˜ = min(lk−ω , CDES Δ),

(12.44)

where Δ = max(Δx, Δy, Δz) represents the maximum grid step size of the mesh cells, and CDES is a constant typically set as 0.61. The SST model can be converted to the SST–DES by modifying the dissipation term in the k-equation. The original dissipation term in the SST model is k DRANS = β ∗ kω = k 3/ 2

/ lk−ω

(12.45)

˜ which results in In the SST–DES model, lk−ω is replaced by l, k DDES = β ∗ kω · FDES = k 3/ 2

(

FDES

lk−ω ,1 = max CDES Δ

/

l˜

(12.46)

) (12.47)

Here, C DES is expressed as k−ω CDES = (1 − F1 )Ck−ε DES + F1 CDES ,

(12.48)

where F 1 is a mixing function defined in the SST model; meanwhile, Ck−ε DES = 0.61 and Ck−ω = 0.78 are often used in practical simulations. DES Menter et al. [14] further modified the DES model above to the following: k DDES = β ∗ kω · FDES−CFX

(12.49)

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227

(

FDES−CFX

) lk−ω = max (1 − FSST ), 1 , FSST = 0, F1 , F2 , CDES Δ

(12.50)

where F 1 and F 2 are the mixing functions defined in the SST model. When F SST = 0, Eq. (12.50) becomes Eq. (12.47). When F SST = F 2 , mesh-induced separation in the boundary layer can be avoided effectively. Spalart further proposed a delayed detached-eddy simulation model, in which a length-scale mixing function is introduced, as follows: lDDES = dw − f d max(0, dw − CDES Δ),

(12.51)

where d w denotes the distance from the nearest wall. The parameter C DES is calculated using Eq. (12.48), and the mixing function f d is computed using f d = 1 − tanh(8rd3 ) rd =

νt + ν √ . max[ Ui j Ui j , 10−10 ]κ 2 dw2

(12.52) (12.53)

Here, κ is the Karman constant, whose value is 0.41. The function f d is used to determine whether the concerning mesh cell is in the boundary layer; f d = 0 implies that it is in the boundary layer, whereas f d = 1 implies that it is away from the boundary layer. The DES method has been used increasingly in practical engineering flow simulations owing to its superiority in solving problems involving high Re and turbulence in separated flows.

12.4 Simulation Examples 12.4.1 Unsteady Performance Simulations Using DES DESs were conducted for the MEXICO wind turbine model of the EU MexNext project [15]. The unsteady aerodynamic characteristics of the wind turbine in the yaw state were investigated, and the DES results were compared with the results obtained using the RANS method. The rotor geometry was included in the simulations and was defined using a bodyfitted multiblock structured mesh. The rotational motion of the rotor was realized via sliding mesh technology. The computational domain was segmented into two zones: a cylindrical zone that rotates with a rotor therein, and a zone that remains stationary. The mesh around the rotor blades was generated using the O-type topology, and the local mesh at the vicinity of the blades was refined. All mesh cells were hexahedral. Figure 12.6 shows the computational domain and the mesh on the rotor surfaces.

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

Fig. 12.6 Computational domain and mesh

(a) schematic illustration of computational domain; (b) details of blade and hub surface mesh; (c) details of blade-tip surface mesh. In the simulations, the rotor blades operated at a uniform pitch angle of − 2.3° and rotated at a speed of 425.1 rpm. The inflow wind speed was 15 m/s, and the yaw angle between the wind direction and rotor axis was 30°. The third-order MUSCL scheme was applied to solve the convective term of the governing equation, and the second-order Crank–Nicolson scheme was adopted for time marching. The inflow velocity was specified at the inlet boundary, and an equilibrium pressure was assumed at the outlet boundary. The cyclicAMI boundary condition was imposed at the sliding interfaces between the rotational and stationary zones. The time step of the numerical marching was set as Δt = 0.0004 s, which exactly corresponds to the time required for the rotor to rotate by 1°. In other words, the rotor requires 360 time steps to perform one revolution. Menter’s SST–DES model was used to solve the turbulence in the DES. Additionally, simulations based on the RANS method using the SST k–ω turbulence model were performed, and the results were compared with those obtained from the DES. Figure 12.7 shows the variation in the normal force of the blade cross-sections during one revolution, and Fig. 12.8 shows the tangential force. The results at five cross-sections, i.e., 25%R, 35%R, 60%R, 82%R, and 92%R, are shown. Data from the simulation based on the RANS method and from the experiment [15] are provided in the figures for comparison. Based on the figures, compared with the simulation using the RANS method, the DES significantly improved the prediction accuracy of the unsteady aerodynamic forces of the wind turbine. This is because the DES can more accurately simulate the unsteady separation flow on the blades and the turbulence in the wake.

12.4 Simulation Examples

229

Fig. 12.7 Normal forces varying with azimuth angle at five cross-sections under yaw angle of 30°, wind speed of 15 m/s, and tip-speed ratio of 6.68

Fig. 12.8 Tangential forces varying with azimuth angle at five cross-sections under yaw angle of 30°, wind speed of 15 m/s, and tip-speed ratio of 6.68

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12 Large- and Detached-Eddy Simulation Methods for Wind Turbine …

12.4.2 Wind Turbine Wake Simulated Using LES Dozens of wind turbines are arranged in large wind farms, and the wake of the upstream wind turbines significantly reduces the output power and wind energy extraction efficiency of the downstream wind turbines. In a wind turbine wake, the flow velocity is significantly lower than the wind speed, and the turbulence intensity is high. The low-velocity results in power output loss in the downstream wind turbines. The high turbulence intensity increases the fatigue load of downstream wind turbines, which reduces the operating life. Therefore, studies pertaining to wind turbine wakes and wake interactions are vital to the design of wind farm layouts. The ALM-LES method was employed to numerically investigate wind turbine wakes. Simulations of the wake characteristics of a single wind turbine and those of wake interactions between two wind turbines are presented and discussed in this chapter. The simulation object is the wind turbine models of a wake experiment performed at the Norwegian University of Science and Technology. 1. Simulation Settings The wind turbines investigated, i.e., an upstream wind turbine denoted as T 1 and a downstream wind turbine denoted as T 2, were three-blade horizontal-axis wind turbines. The diameters of T 1 and T 2 were D1 = 0.994 m and D2 = 0.894 m, respectively. The blades of both wind turbines were identical, the cross-profiles of which were formed from an NREL S826 airfoil with 14% thickness. The tip-speed ratio of the rotors was designed as 6 at the wind speed of 10 m/s, which corresponds to the rotation speed of approximately 1200 rpm. The size of the computational domain was set to 11.15 m × 2.7 m × 1.8 m, which is identical to the size of the wind tunnel section where the wind turbines were tested. A uniform incoming flow with a velocity of 10 m/s and a turbulent intensity of 0.23% was applied at the inlet boundary. Figure 12.9 shows a schematic illustration of the computational domain used for the wake simulation of a single wind turbine T 1. The simulations were performed using a Cartesian mesh that filled the computational domain with 2.2 × 107 cells. An image of the mesh viewed from the side is shown in Fig. 12.10. To simulate the wind turbine wake interactions, in-line and staggered layouts were considered for the upstream and downstream wind turbines, as shown in Fig. 12.11. The x-direction is the incoming flow direction, z-direction the vertical upward direction, and y-direction the lateral direction. The upstream wind turbine T 1 was located 2D2 downstream of the inlet boundary, and the distance between the T 2 and T 1 was 3D2 . A lateral offset of 0.4 m, which is approximately one-half the diameter of the wind turbines, was specified for the staggered layout case. A Cartesian mesh comprising 2.5 × 107 cells was used. The wind turbine geometries were not directly simulated; instead, they were defined using the ALM. Each actuator line represents one blade with 30 spanwise control points. The lift and drag coefficients of the blade cross-profiles were specified based on the wind tunnel experimental data of an S826 airfoil with Re = 105 .

12.4 Simulation Examples

231

Fig. 12.9 Schematic illustration of computational domain used for single wind turbine

Fig. 12.10 Schematic illustration of Cartesian mesh used

The aerodynamic effects of the wind turbine towers were modeled using virtual equivalent cylinders generated via the action of corresponding body forces in the air. The third-order MUSCL scheme was used to discretize the convective term, and the second-order Crank–Nicolson scheme was used for time marching. Meanwhile, the PISO algorithm was used for pressure correction, and the dynamic Smagorinsky SGS model was used for turbulence modeling. The time step was set as Δt = 10–4 s. 2. Simulation Results (1) Simulation of single wind turbine Figure 12.12 shows the simulation results of the power and thrust coefficients of the T 1 wind turbine, along with the experimental data for comparison. The simulation results agreed well with the experimental data, which demonstrates the high accuracy of the numerical method employed. Figure 12.13 shows the time-averaged profiles of the dimensionless xdirection velocity at 1D, 3D, and 5D downstream of the wind turbine, which similarly demonstrate the high accuracy of the simulation. Figure 12.14 shows the wake vortices at various tip-speed ratios. As shown, the simulation results clearly depict the structure and evolution of the vortices in the turbulence wake.

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Fig. 12.11 Layouts for upstream and downstream wind turbines

Fig. 12.12 Power and thrust coefficients of single wind turbine

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Fig. 12.13 Velocity profiles at X/D = 1, 3, and 5 downstream of wind turbine (λ = 6)

Fig. 12.14 Vortices in the wake of single wind turbine

(2) Simulation of in-line wind turbines Figure 12.15 shows the power and thrust coefficients of upstream turbine T 1 and in-line downstream turbine T 2. At various tip-speed ratios, the computational coefficients of both wind turbines were consistent with the experimental data. Owing to the velocity deficit in the wake of the upstream wind turbine, the maximum power coefficient of the downstream wind turbine was only approximately 1/4 that of the upstream wind turbine. Figure 12.16 shows the wake vortices of the two wind turbines, as well as their wake interactions. Four SGS models were employed in the simulation for comparison, i.e., the Smagorinsky, dynamic k-equation, dynamic mixed Smagorinsky, and dynamic Smagorinsky models. All the simulation results clearly depicted the structure of the wake vortices of the upstream wind turbine, including the blade tip/root vortices and the shedding vortices generated. The downstream wind turbine rotor was completely enshrouded by the wake of the upstream wind turbine, and the turbulent characteristics of its wake were more significant than that of the upstream wind turbine.

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Power coefficient

Thrust coefficient

Fig. 12.15 Power and thrust coefficients of in-line wind turbines

Fig. 12.16 Vorticity isosurfaces simulated using various subgrid stress models

(3) Simulation of staggered wind turbines Figure 12.17 shows the power and thrust coefficients of the two wind turbines in a staggered layout. The coefficients of downstream wind turbine T 2 were significantly higher than those based on the in-line layout. The maximum power coefficient of T 2 was approximately 62% that of upstream wind turbine T 1. The thrust coefficients of the two wind turbines were similar.

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Fig. 12.17 Power and thrust coefficients of the two wind turbines in staggered layout

Figure 12.19 shows the wake vorticity in a horizontal plane of the computational domain containing the two staggered wind turbines, and Fig. 12.18 shows the vorticity isosurfaces of their wakes. As shown, the present simulation accurately captured the vortex structures and turbulence details of the wakes.

Fig. 12.18 Vorticity in horizontal plane for staggered wind turbines

Fig. 12.19 Vorticity isosurfaces of wakes for staggered wind turbines

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References 1. Kolmogorov, A. N. (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In Proceedings of the Dokl Akad Nauk SSSR. 2. Smagorinsky, J. (1963). General circulation experiments with the primitive equations: I. The basic experiment. Monthly Weather Review 91(3), 99–164. 3. Deardorff, J. W. (1970). A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. Journal of Fluid Mechanics, 41(2), 453–480. 4. Deardorff, J. W. (1970). Preliminary results from numerical integrations of the unstable planetary boundary layer. Journal of the Atmospheric Sciences, 27(8), 1209–1211. 5. Kwak, D., Reynolds, W., & Ferziger, J. (1975). Large eddy simulation of strained turbulence: Technical Report TF-5. Stanford University, California. 6. Shaanan, S., Ferziger, J., Reynolds, W. (1975). Numerical simulation of turbulence in the presence of shear 7. Ferziger, J., Mehta, U., Reynolds, W. (1977). Large eddy simulation of homogeneous isotropic turbulence. In Proceedings of the Symposium on Turbulent Shear Flows. 8. Ferziger, J. H. (1977). Large eddy numerical simulations of turbulent flows. AIAA Journal, 15(9), 1261–1267. 9. Moin, P., & Kim, J. (1982). Numerical investigation of turbulent channel flow. Journal of Fluid Mechanics, 118, (341–377). 10. Germano, M., Piomelli, U., Moin, P., et al. (1991). A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A: Fluid Dynamics, 3(7), 1760–1765. 11. Lund, T. S., Wu, X., & Squires, K. D. (1998). Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations. Journal of Computational Physics, 140(2), 233–258. 12. Spalart, P. R. (1997). Comments on the feasibility of LES for wings, and on a hybrid RANS/ LES approach. In Proceedings of the Proceedings of first AFOSR International Conference on DNS/LES. Greyden Press. 13. Strelets, M. (2001). Detached eddy simulation of massively separated flows. In 39th Aerospace Sciences Meeting and Exhibit 14. Menter, F. R., Kuntz, M., & Langtry, R. (2003). Ten years of industrial experience with the sst turbulence model. Turbulence, Heat and Mass Transfer, 4(1), 625–632. 15. Schepers, J., Boorsma, K., Bon, A., et al. (2011). Results from mexnext: Analysis of detailed aerodynamic measurements on a 4.5 m diameter rotor placed in the large German Dutch Wind Tunnel DNW. In Energy Research Center of the Netherlands. Petten, Netherlands, Technical Report No. ECN-M-11-034