Separated and Vortical Flow in Aircraft Wing Aerodynamics: Basic Principles and Unit Problems [1st ed.] 9783662613269, 9783662613283

Citation preview

Ernst Heinrich Hirschel Arthur Rizzi Christian Breitsamter Werner Staudacher

Basic Principles and Unit Problems

Separated and Vortical Flow in Aircraft Wing Aerodynamics

Ernst Heinrich Hirschel Arthur Rizzi Christian Breitsamter Werner Staudacher •

•

•

Separated and Vortical Flow in Aircraft Wing Aerodynamics Basic Principles and Unit Problems

123

Ernst Heinrich Hirschel Institute of Aerodynamics and Gasdynamics University Stuttgart Zorneding, Germany

Arthur Rizzi Department of Aeronautical and Vehicle Engineering Royal Institute of Technology Stockholm, Sweden

Christian Breitsamter Chair of Aerodynamics and Fluid Mechanics Technical University of Munich Munich, Germany

Werner Staudacher Institute of Flightsystems Bundeswehr University Munich Zorneding, Germany

ISBN 978-3-662-61326-9 ISBN 978-3-662-61328-3 https://doi.org/10.1007/978-3-662-61328-3

(eBook)

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Preface

This book is devoted to a presentation and discussion of the fluid mechanical aspects of separated and vortical flows in aircraft wing aerodynamics. The main focus is on large aspect-ratio wings and on small aspect-ratio delta-type wings. In general, we do not deal with aerodynamic design issues of these wings. Our aim is to foster an understanding of the basic properties of the wing vortex systems and their behavior. We further want to demonstrate the capabilities of numerical simulation methods for such flows and the interpretation of the results from a fluid-mechanical point of view. The economical and ecological pressures on, in particular, transport aircraft are progressively increasing. Military aircraft, on the other hand, faces large challenges with regard to agility and maneuverability, but also to economical operation. This all requires a refinement of the aerodynamic design and consequently an increasingly better handling of separation and vortical flow phenomena. We see, on the other hand, that since about two decades disciplinary and multi-disciplinary discrete numerical simulation methods play a progressively larger role in all aircraft design and development processes. Their potential is very large, but to realize it fully, a very good understanding of the flow phenomena, that govern the aerodynamic properties of an aircraft, is necessary. A tool is only as good as the person, who selects and wields it. Accordingly, we consider in the introduction the different mathematical models that underlie the aerodynamic computation methods from the linear panel method up to the most sophisticated Reynolds-averaged Navier–Stokes methods and the recently emerging scale-resolving methods. Particular methods are the discrete Euler methods, which as rather inexpensive methods take into account compressibility effects and without an explicit Kutta condition also permit to describe lifting-wing flow. The concept of the kinematically active and inactive local vorticity content of a shear layer is introduced. It gives important insight into a number of flow phenomena, but also, with the second break of symmetry—the first one is due to the Kutta condition—an explanation of essential phenomena, which are connected to lifting-wing flow fields. The prerequisite is an extended definition of separation, v

vi

Preface

which besides the classical—ordinary—separation introduces the concept of flow-off separation at sharp trailing edges of large aspect-ratio wings and sharp leading edges of small aspect-ratio delta-like wings. The concept of the kinematically active and inactive vorticity content, with a compatibility condition for the flow-off separation at sharp edges, permits to understand the properties of the evolving trailing vortex layer of large aspect-ratio wings. This vortex layer then leads to the pair of trailing vortices behind the aircraft. The concept, on the other hand, gives proof that discrete Euler methods at sharp delta or canard leading edges indeed lead to an exact simulation of the evolution of the primary lee-side vortex pair. The book treats three main topics. 1) Basic Principles are considered in introducing chapters. They concern relevant boundary-layer properties, vortex theory, the local vorticity content of shear layers, the matter of discrete Euler solutions for lifting wings, the Kutta condition in reality and an introduction to the topology of skin-friction and velocity fields. 2) Unit Problems concentrate on isolated flow phenomena, respectively configuration parts. In our case, these are the flow phenomena present at large aspect-ratio and small aspect-ratio delta-type wings. Capabilities of panel methods and discrete Euler methods are investigated. One Unit Problem is the flow past the wing of the Common Research Model. We demonstrate besides others that the tip-vortex system leads to a very small non-linear lift and that a bridge can be spanned over to the small aspect-ratio wing. Other Unit Problems concern the lee-side vortex system appearing on this type of wings with sharp and fully or partly round leading edges and also the vortical flow past a blunt-edged configuration at hypersonic speed. 3) Particular Flow Problems of large aspect-ratio and small aspect-ratio delta-type wings. In short sections, practical design problems are discussed. The treatment of separated and vortical flow past fuselages, although desirable, was not possible in the frame of this book. The authors of the book are from the aerospace field and were for many years— in teaching, research, as well as in industrial aircraft design—deeply involved in phenomenological, mathematical, computational and flight-vehicle shaping issues of separated and vortical flow. They wish to give the student and his teacher, the researcher, and in particular, also the practical aerospace engineer an in-depth knowledge about separated and vortical flow in aircraft wing aerodynamics. Zorneding, Germany Stockholm, Sweden Munich, Germany Zorneding, Germany September 2020

Ernst Heinrich Hirschel Arthur Rizzi Christian Breitsamter Werner Staudacher

Acknowledgements

The authors are much indebted to many colleagues, whose input was crucial for the book. First of all, we wish to thank S. Pfnür, whose master thesis work was the essential input to Chap. 8. Our doctoral students S. Crippa, J. Fischer, R. Hentschel, A. Hövelmann, B. Schulte-Werning and S. Riedelbauch generously made available large parts of their theses for Chap. 10. Substantial input was received from A. Büscher and A. Schütte, too. Many thanks to all of them. Directly and thankfully received was data, material, critical and constructive comments, input of all kind and advise from A. A. Allen, K. Becker, N. Bier, O. Brodersen, M. Drela, R. Friedrich, H. Fütterer, S. Görtz, W. Heinzerling, M. Herr, S. M. Hitzel, D. Hummel, R. Konrath, H.-P. Kreplin, J. M. Luckring, D. Niedermeier, R. Rudnik, and M. Tomac. C. Weiland made ad hoc computations and illustrations for the book. G. Simeonides and C. Weiland were reading all the chapters and gave critical and constructive comments. Our spouses deserve special thanks for their patience, which was tested over quite a long time. Lastly, special thanks go to Thomas Ditzinger, the Editorial Director of Interdisciplinary and Applied Sciences and Engineering from Springer, because for many years, he very effectively and helpfully supported the publication of the works of the first author of the present book. September 2020

Ernst Heinrich Hirschel Arthur Rizzi Christian Breitsamter Werner Staudacher

vii

Contents

1

2

3

. . . . . . . .

1 2 3 9 9 10 15 16

......

18

...... ......

24 25

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

29 30 34 36 37 37 39 41 42

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

45 45 46 47

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Content of the Book . . . . . . . . . . . . . . . . . . . . . . 1.2 The Application Background . . . . . . . . . . . . . . . . . . . 1.3 What Is Separation? . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Classical View . . . . . . . . . . . . . . . . . . . . 1.3.2 The View Taken in This Book . . . . . . . . . . . 1.3.3 The Definition of Separation in This Book . . . 1.4 A Wider View at Separation . . . . . . . . . . . . . . . . . . . 1.5 Flow Physical and Mathematical Models of Separated and Vortical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Basic Principles, Unit Problems and the Contents of the Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag . . . . . . . . . . . . . . . . . . . . . 2.1 Relevant Boundary-Layer Properties . . . . . . . . . . . . 2.2 Interaction Issues . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Special Issue: The Locality Principle . . . . . . . . . 2.4 Aspects of Drag . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Drag Components . . . . . . . . . . . . . . . . . . . 2.4.2 The Drag Divergence . . . . . . . . . . . . . . . . 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

Elements of Vortex Theory . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Models of Finite-Wing Theory of Flight 3.1.2 Benefits in Studying Vortex Dynamics . .

. . . .

. . . .

. . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

ix

x

Contents

3.2

The Concept of Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Designations and Definitions . . . . . . . . . . . . . . . . 3.3 Stokes’ Theorem and the Concept of Circulation . . . . . . . 3.3.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Origins of Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Kutta Condition . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Cambridge School—No-Slip Viscous Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The Göttingen School—Inviscid Vortex-Sheet Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Other Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Short Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Entropy and Total Enthalpy Gradients and Vorticity: Crocco’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Equations of Transport of Vorticity . . . . . . . . . . . . . . . . . 3.7 Helmholtz’s Vorticity Theorems . . . . . . . . . . . . . . . . . . . . 3.8 Kelvin’s Circulation Theorem . . . . . . . . . . . . . . . . . . . . . 3.9 Law of Biot-Savart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Vortex Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Structure of Trailing Vortices . . . . . . . . . . . . . . . . . . . . . . 3.11.1 SAAB 39 Gripen Wake Model . . . . . . . . . . . . . . 3.11.2 Trailing Vortex Instability . . . . . . . . . . . . . . . . . . 3.11.3 Crow Instability . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Vortex Layers and Vortices . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 Roll-Up of Shed Vortex Layers . . . . . . . . . . . . . . 3.12.2 Vortex Stretching . . . . . . . . . . . . . . . . . . . . . . . . 3.12.3 Vortex Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Vortex Breakdown, Vortex Re-configuration . . . . . . . . . . . 3.13.1 Fundamental Studies . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Computed Vortex Breakdown Over a Delta Wing 3.13.3 Vortex Re-connection . . . . . . . . . . . . . . . . . . . . . 3.14 Separation and Vortex Flow Control . . . . . . . . . . . . . . . . 3.15 Vortex Flows and Dynamic Structural Loads . . . . . . . . . . 3.16 Basic Quantities of Trailing-Vortex Flow Fields . . . . . . . . 3.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

. . . . . . . .

48 48 49 50 50 50 51 52

...

53

... ... ...

55 56 58

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

58 59 60 61 61 63 67 68 68 68 69 69 73 73 75 76 77 78 78 80 82 87 89

The Local Vorticity Content of a Shear Layer . . . . . . . . . . . . . . . 4.1 Definition and Derivation of the Local Vorticity-Content Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

94

Contents

xi

4.2

Kinematically Active and Inactive Vorticity Content: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Rankine Vortex . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Two-Dimensional Boundary Layer . . . . . . . . . . . 4.2.3 Near Wake of a Lifting Airfoil . . . . . . . . . . . . . . 4.2.4 Bound Vortex of a Lifting Airfoil . . . . . . . . . . . . 4.2.5 Three-Dimensional Boundary Layer . . . . . . . . . . . 4.2.6 Near Wake (Trailing Vortex Layer) of a Lifting Finite-Span Wing . . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Summary of the Results . . . . . . . . . . . . . . . . . . . 4.3 Lift and Induced Drag: Two Breaks of Symmetry . . . . . . . 4.3.1 First Symmetry Break: The Lifting Airfoil . . . . . . 4.3.2 Second Symmetry Break: The Lifting Finite-Span Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The Symmetry Breaks in the Reality of Aircraft Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Flow Pattern at the Trailing Edge of Large-Aspect Ratio Lifting Wings: A Compatibility Condition . . . . . . . . . . . . 4.5 Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

6

The Matter of Discrete Euler Solutions for Lifting Wings 5.1 Vorticity Creation in Euler Solutions of Lifting-Wing Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Vorticity and the Related Entropy Rise . . . . . . . . . . . 5.3 Critical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. 96 . 96 . 97 . 98 . 102 . 104

. . . .

. . . .

. . . .

105 107 108 108

. . . 110 . . . 112 . . . .

. . . .

. . . .

113 115 116 117

. . . . . . . 119 . . . . .

. . . . .

. . . . .

119 121 123 125 125

About the Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Decambering Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Boundary-Layer Decambering . . . . . . . . . . . . . . . . 6.1.2 Shock-Wave Decambering . . . . . . . . . . . . . . . . . . 6.2 Kutta Condition and Kutta Direction in Reality . . . . . . . . . . 6.3 Geometric Properties of Trailing and Leading Edges of Actual Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 When Can an Edge Be Considered as to Be Aerodynamically Sharp? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Implicit and Explicit Kutta Condition, Modeling and Grid Generation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

127 127 128 129 131

. . . . .

. . . . .

. . . . .

. . . . .

. . 132 . . 136 . . 140 . . 142 . . 144

xii

7

8

Contents

Topology of Skin-Friction and Velocity Fields . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Three-Dimensional Attachment and Separation . . 7.1.3 Detachment Points and Lines . . . . . . . . . . . . . . 7.1.4 Lighthill’s Separation Definition and Open-Type Separation and Attachment . . . . . . . . . . . . . . . . 7.2 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Flow-Field Continuation and Phase Portraits . . . 7.2.2 Off-Surface Flow-Field Portraits . . . . . . . . . . . . 7.2.3 Singular Points in Off-Surface Velocity Fields . . 7.3 Singular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Topological Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Surface Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Off-Surface Rules . . . . . . . . . . . . . . . . . . . . . . . 7.5 Structural Stability and Changes of Flow Fields . . . . . . . 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

147 148 148 148 150

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

151 152 152 156 161 162 165 165 165 166 170 177 177

Large Aspect-Ratio Wing Flow . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Five Flow Domains at and Behind the Lifting Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Influence of the Trailing Vortex Layer and Vortices on the Wing’s Performance . . . . . . . . . . . . . . . . . . 8.2 Panel Method (Model 4) Solutions—Proper and Improper Results in Flow Domain 0 . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Kolbe Wing . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Forward-Swept Wing . . . . . . . . . . . . . . . . . . . 8.3 Creation of Lift in an Euler Solution (Model 8) for a Lifting Large-Aspect Ratio Wing—Proof of Concept in Flow Domain 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 The Computation Case and Integral Results . . . . . . 8.3.2 Details of the Computed Flow Field of Domain 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 The Circulation and the Kinematically Active Vorticity Content . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2 . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Computation Method and Grid Properties . . . . . . . 8.4.3 Flow Domain 0: The Flow over the Wing, . . . . . . . 8.4.4 Excursion: The Wing-Tip Vortex System and Non-linear Lift . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 181 . . 181 . . 181 . . 185 . . 187 . . 188 . . 190

. . 193 . . 194 . . 194 . . 198 . . . .

. . . .

199 199 200 202

. . 212

Contents

xiii

8.4.5

Flow Domain 0 Contd.: Trailing-Edge Flow and the Compatibility Condition . . . . . . . . . . . 8.4.6 Flow Domain 1: The Trailing Vortex Layer in the Near Field . . . . . . . . . . . . . . . . . . . . . . 8.4.7 Flow Domain 2: The Trailing Vortices Appear . 8.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Particular Flow Problems of Large Aspect-Ratio Wings 9.1 Supercritical Airfoil—Shock-Wave/Boundary-Layer Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Flow Past a High-Lift System . . . . . . . . . . . . . . . . 9.3 The Wing in High-Lift Condition . . . . . . . . . . . . . . 9.4 The VHBR Engine and the Nacelle-Strake Vortex . 9.5 Wing-Tip Devices . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 The Wake-Vortex Hazard: The Problem and Means to Control It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 214 . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

219 223 229 230 231

. . . . . . . . 233 . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

233 239 244 248 253

. . . . . . . . 259 . . . . . . . . 265 . . . . . . . . 266

10 Small Aspect-Ratio Delta-Type Wing Flow . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Flight Vehicle Classes . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Notes on the History . . . . . . . . . . . . . . . . . . . . . . . 10.2 Non-linear Lift Exemplified with the Plain Delta Wing . . . . 10.2.1 Geometrical and Flow Parameters . . . . . . . . . . . . . 10.2.2 The Non-linear Lift as Aerodynamic Phenomenon . 10.2.3 The Matter of Secondary and Higher-Order Lee-Side Vortex Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Basic Influences of Sharp and Round Leading Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 Limits: Vortex Breakdown and Vortex Overlapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.6 Correlations of Lee-Side Flow Fields . . . . . . . . . . . 10.2.7 Flow-Physical Challenges . . . . . . . . . . . . . . . . . . . 10.2.8 Manipulation of Lee-Side Flow Fields, an Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Vortical Flow Past the Sharp-Edged VFE-1 Delta Wing—Different Models . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Creation of Lift in the Euler Solution (Model 8) for the Sharp-Edged VFE-1 Delta Wing—Proof of Concept . . . . . .

. . . . . . .

. . . . . . .

269 270 270 277 285 285 286

. . 289 . . 291 . . 293 . . 294 . . 297 . . 300 . . 300 . . 305

xiv

Contents

10.4.1 The Computation Case and Integral Results . . . . 10.4.2 Details of the Computed Flow Field . . . . . . . . . 10.4.3 The Circulation and the Kinematically Active Vorticity Content in the Euler Simulation . . . . . 10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing . 10.5.1 The Wing and the Subsonic Computation Case . 10.5.2 Two Pairs of Primary Vortices . . . . . . . . . . . . . 10.5.3 Vortex Breakdown . . . . . . . . . . . . . . . . . . . . . . 10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Laminar Hypersonic Flow Past a Round-Edged Delta Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 306 . . . . 308 . . . . .

. . . . .

. . . . .

. . . . .

312 315 315 317 323

. . . . 329 . . . .

. . . .

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Lift and Stability Problems Connected to Lee-Side Vortex Systems of Delta-Type Wings . . . . . . . . . . . . . . . . . . . . . . 11.2 Wing-Planform Shaping and Optimization . . . . . . . . . . . . . 11.2.1 Effects of the Wing Geometry . . . . . . . . . . . . . . . . 11.2.2 Wings of High-Speed Vehicles . . . . . . . . . . . . . . . 11.3 Wing Sections and Leading-Edge Flaps . . . . . . . . . . . . . . . 11.3.1 Wing Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Leading-Edge Flaps . . . . . . . . . . . . . . . . . . . . . . . 11.4 Fuselage Forebody Strakes . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Spanwise Blowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Blowing at a Pilot Model Without Strake and Forebody Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Blowing at the Pilot Model with Strake and Without Forebody Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Summarizing Remarks . . . . . . . . . . . . . . . . . . . . . 11.6 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Basic Configuration . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Modifications of the Wing Geometry . . . . . . . . . . . 11.6.3 The Final Configuration . . . . . . . . . . . . . . . . . . . . 11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

339 345 351 353

. . 359 . . . . . . . . .

. . . . . . . . .

360 364 364 366 368 369 371 373 376

. . 378 . . . . . . . .

. . . . . . . .

379 381 383 383 385 387 391 391

12 Solutions of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Contents

xv

Appendix A: Useful Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Appendix B: Constants, Atmosphere Data, Units, and Conversions . . . . 431 Appendix C: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Appendix D: Abbreviations, Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Permissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Chapter 1

Introduction

Vortex layers and vortices, although not occupying much space in the flow field around and behind an aircraft, can be seen as “the sinews and muscles of the fluid motion”, as D. Küchemann was putting it [1]. Separated and vortical flow is present at any aircraft. The largest part of the flow past a flight vehicle, however, can be considered as inviscid. Viscous effects are restricted to the boundary layer, which is a bound vortex layer, in contrast to the free or trailing vortex layer, which is the lifting wing’s wake. Despite of these vortex layers, the flow over a lifting wing, for instance, can well be described in the frame of inviscid theory, even only in the frame of potential flow theory. This all is the topic of the classical text books. The aerodynamic phenomena appearing in reality today are described with more or less fidelity by flow realizations in ground-simulation facilities or by advanced discrete numerical simulation methods (Computational Fluid Dynamics, CFD). This in particular also holds for the vortex layers and vortices which are present in any aircraft’s flow field. A book with a concise presentation of the basics of fluid mechanics, and in particular of vorticity fields, vortex dynamics, etc., with applications to airfoils and wings is that of M.J. Lighthill [2]. Newer books on vortex phenomena usually cover the whole range of phenomena down to laminar-turbulent transition and turbulence, which basically all are related to the phenomenon of vorticity, see, e.g., the books of H.J. Lugt (1996) [3] and of J.-Z. Wu, H.-Y. and M.-D. Zhou (2006) [4]. D.J. Peake and M. Tobak in 1980 published their NASA Technical Memorandum, later AGARDograph, on three-dimensional interactions and vortical flows [5]. A book specifically devoted to vortex flows in high angle-of-attack aircraft aerodynamics is that of J. Rom (1992) [6]. The topology of three-dimensional separated flow fields is treated by J. Délery (2013) [7]. Relevant conference proceedings are beside others those of AGARD symposia held in Göttingen, Germany in 1975 [8], © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_1

1

2

1 Introduction

and in Rotterdam, the Netherlands in 1983 [9], and that of an IUTAM Symposium in Novosibirsk, USSR in 1990 [10]. Monographs on boundary-layer and viscous flow in general are, for example, those of F.M. White (2005) [11], T. Cebeci and J. Cousteix (2005) [12], H. Schlichting and K. Gersten (2006) [13], and E.H. Hirschel, J. Cousteix and W. Kordulla (2014) [14]. Regarding aircraft aerodynamics in general we point to the classical works—still very rewarding to study—of H. Schlichting and E. Truckenbrodt (1959/1979) [15] and D. Küchemann (1978/2012) [16]. Recent publications are from J.D. Anderson, Jr. (2011) [17], M. Drela (2014) [18] and the relevant chapters in C.-C. Rossow, K. Wolf, P. Horst (eds.) (2014) [19]. To appear in 2020 is the book of A. Rizzi and J. Oppelstrup, who in particular present an approach to aerodynamics with discrete numerical methods—computational fluid dynamic—[20]. Since the 1990s large advancements were made in both discrete numerical simulation methods and experimental techniques. They led to new insights and also to new views at the topic of separated and vortical flow. About ten years ago economical and ecological pressures anew began to ask for more efficient transport aircraft. This not only holds for the cruise phase (fuel efficiency), but also for take-off, climb, approach and landing (reduction of noise and exhaust emission). In general also a better and more reliable determination of static and dynamic mechanical loads on the airframe and its components became desirable in order to reduce airframe mass and to increase the structural life time. Military aircraft basically underlie the same demands. Here, however, agility, stalled flight and post-stall flight lead to additional requirements. Unmanned fighter aircraft permit very high maneuverability which puts even stronger demands on the airframe design.

1.1 The Content of the Book For the aircraft designer and developer not only the numerical and experimental tools govern his or her success, but first of all the knowledge of and the insight into the aerodynamic phenomena present at the aircraft which is to be defined and developed. This holds the more, if in the flight-envelope opening process of the aircraft, problem diagnosis becomes necessary. Therefore the book concentrates on separated and vortical flow past aircraft configuration elements and total configurations. The background of application basically is given with large aspect-ratio wing aircraft on the one hand and small aspect-ratio, high leading-edge sweep, i.e., delta-type wing aircraft on the other hand. The former are characterized by a leading-edge sweep small enough, so that no lee-side vortices appear, the latter by a leading-edge sweep so high, that lee-side vortices will appear at higher angles of attack. The physical and mathematical basics and concepts of attached, separated and vortical flow are presented to the needed degree. The concept of kinematically active

1.1 The Content of the Book

3

and inactive vorticity content of a vortex layer is introduced. It permits on the one hand to connect the singularities of potential theory with the viscous reality of vortex sheets and vortices. On the other hand it allows to show that discrete numerical solutions of the potential equation (panel methods) and the Euler equations in principle are viable. Such methods today still have their place in conceptual and pre-design work. In particular the decades old question of how vortices appear in Euler solutions is answered. Basically the applicability of computational methods is considered. In much detail discussed are examples of flows past high and low aspect-ratio wings, and that, if given, also in view of the whole aircraft with its performance, flyability and controllability demands. In the following sections we first sketch the application background, then aspects of separation are considered, a wider view at it is given. The introduction closes with a classification of flow-physical and mathematical models of in particular separated and vortical flow and an overlook over the following chapters.

1.2 The Application Background The application background of the book are fixed-wing aircraft of all kind. However, we concentrate on the phenomena present at large transport aircraft and likewise at fighter aircraft, manned and unmanned. The reason for this is that much and detailed information and data are available for such aircraft from a host of former and present research programmes. Typical large transport aircraft are shown in Fig. 1.1. In cruise flight the angle of attack nominally is α = 0—as well as the flight-path angle γ—but the wing has a rigging angle of incidence, as well as an appropriate spanwise distribution of camber and twist of the airfoil sections, such that the demanded lift is obtained.

Fig. 1.1 Two typical transport aircraft. Left: Airbus A350 [21], right: Boeing 787 [22]

4

1 Introduction

Fig. 1.2 Schematic of the influence of configuration elements on the circulation distribution in span direction Γ (y), its change in y-direction dΓ (y)/dy and local discrete vortices [23]. Left side: clean wing, b0 indicates the distance between the fully developed trailing vortices, the tip vortex is not indicated. Right side: wing with deflected flap

While the aircraft is in cruising mode, the wing is clean, i.e., slats and flaps are not activated. A vortex sheet, the trailing vortex layer, separates from the trailing edge and rolls up downstream into the two trailing vortices, left side of Fig. 1.2. At the horizontal stabilizer, which as a rule exerts a downward directed trim force, a secondary counter-rotating vortex layer and trailing vortices emerge. These and the wing and stabilizer tip vortices are merged into the wing’s primary trailing vortex layer, respectively the trailing vortices. Tertiary vortices emerging, for instance, at the wing’s root, the flap tracks, the fuselage and at the nacelles of the turbo-jet engines, are also merged into the primary vortex layer/trailing vortices. If landing flaps, short flaps, and slats are activated, the situation changes drastically. The right side of Fig. 1.2 shows the effect for an isolated trailing-edge flap. Lift, hence circulation increases, but a flap vortex appears. Its strength is connected to the gradient dΓ (y)/dy at the spanwise end(s) of the flap. Important, however, is to note that flap deflection as a rule goes together with significantly high angles of attack. In order to get an idea what is to be expected, we look at a typical flight path of a transport aircraft with the major flight segments, Fig. 1.3. In this figure, 1 denotes the segment of climb-out from take-off to thrust reduction, 2 that above it. In between is the possible segment 1 , which we disregard. Segment 3 is the cruise segment, which ends with the approach 4 to the airport. 4 is a possible hold segment, when the aircraft has to wait for landing. We disregard that, too. 5, 6, and 7 are the segments immediately ahead of touch-down—where also the ground effect is an issue—8 is the special case of go-around, when the final approach of the aircraft is aborted. For medium-range and medium-size aircraft, like Boeing 737 and Airbus A320, we collect data in Table 1.1.

1.2 The Application Background

5

Fig. 1.3 Schematic of the flight path of a transport aircraft with different flight phases, after [24]

Table 1.1 Typical transport aircraft: flight-path segments and angle of attack α, flight-path angle γ, settings of flaps η f lap , and slats ηslat , the latter two downward deflected with positive sign. H and v are the characteristic altitude and speed. All numbers are approximate, in reality depending on a number of parameters No. Segment H u ∞ /M∞ α γ η f lap ηslat (m) (◦ ) (◦ ) (m/s)/(-) (◦ ) (◦ ) 1

2 3 4 5 6 7 8

Takeoff/climbout 1 Climb 2 Cruise Approach Final approach Flare Landing Go around

0. This happens if an adverse pressure gradient is given (the classical consideration), or surface-normal injection (blowing), or heating of the boundary layer [5]. If the three entities are simultaneously present, they can enlarge or reduce the magnitude of ∂ 2 u/∂ y 2 | y=0 . The wall temperature via the inverse of the viscosity reduces the magnitude, if the wall is hot, and enlarges it, if the wall is cold. The second derivative on the other hand becomes negative with a favorable pressure gradient, with suction, and with surface cooling. Here we take a broader view and say that the presence of a point of inflection of u(y) means a less full tangential velocity profile and hence a reduction of the tangential momentum flux, profile 3 in Fig. 2.2. This effect is the stronger the larger ∂ 2 u/∂ y 2 | y=0 is. If the second derivative is negative, the velocity profile is fuller, profiles 2 and 1 in Fig. 2.2. The momentum flux is the more enlarged, the more negative the second derivative at the wall is. (Profile 2 is the classical zero pressure gradient Blasius boundary-layer profile.) All this holds for both laminar and turbulent boundary layers, even if in Fig. 2.2 the tangential velocity profiles have been sketched only for the former. However, we have to note that we tacitly assumed hydraulically smooth surfaces, i.e., surfaces with no roughness, respectively roughness of sub-critical size. Supercritical roughness of a surface influences the stability and the transition behavior of laminar boundary layers. (Turbulence tripping, for instance, is applied in lowReynolds number wind-tunnel experiments, see, e.g., [7].) Turbulent boundary layers are strongly affected by super-critical roughness, showing a sizeable increase of the skin friction, and, where it applies, also of the surface

2.1 Relevant Boundary-Layer Properties

33

heat transfer. For attached turbulent boundary layers much knowledge is available [4, 8]. Regarding the separation behavior of boundary layers on surfaces with super-critical roughness, the picture is not unambiguous, see for instance the corresponding discussions in [9]. – (b) Regarding the density profile ρ(y), we also follow the discussion in [5]. In attached viscous flow past a flat or nearly flat surface the gradient of the static pressure normal to the wall is small and in the large Reynolds number limit of flat-plate flow goes to zero.3 This means that in the direction normal to the wall the pressure is constant—or nearly constant—and equal to that of the external inviscid flow4 p = pe .

(2.2)

Consequently, with the equation of state p = ρ R T , we have in the boundary layer ρ T = ρe Te = constant,

(2.3)

and hence it holds the proportionality ρ∝

1 . T

(2.4)

This indicates that a hot wall leads to a small density at and above the wall. At a cold wall the density is large. In this case the average tangential momentum flux < ρ u 2 > is larger than in the hot-wall case. The density profile ρ(y) hence must be regarded in Eq. (2.1), if a wall-normal temperature gradient exists in the boundary layer. Regarding the separation inclination of a boundary-layer, the magnitude of the Reynolds number is important, too. This can qualitatively be understood with the help of the empirical separation criterion of B. Thwaites [10]: 1 du e 2 δ = −0.09. ν dx 1

(2.5)

A deceleration of the external inviscid flow—negative du e /d x, equivalent to a pressure rise, positive dp/d x—leads to separation, once that criterion is fulfilled. Because the displacement thickness δ1 is inversely proportional to the Reynolds number, Appendix A.5.4, the Reynolds number counteracts the flow-deceleration effect. Increasing the Reynolds number hence reduces the separation tendency.

3 If locally the inverse of the boundary-layer thickness δ

is of the order of magnitude of the largest of the principal surface curvatures radii Ri , the pressure gradient is no more small due to the centrifugal forces induced by the surface curvature [5]. This is the best known boundary-layer higher-order effect. It can be treated with second-order boundary-layer theory. 4 External means at the outer edge of the boundary layer.

34

2 Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag

For turbulent flow the separation criterion of B.S. Stratford, [11], see also [12], actually shows the same effect, i.e., that an increasing Reynolds number counteracts the flow-deceleration effect and reduces the separation tendency. Finally we note that all the above also holds for the main-flow direction, i.e., streamwise, velocity profile of three-dimensional boundary layers, at least if the three-dimensionality is not too strong. Moreover, the three-dimensionality—with a given cross-flow pressure gradient—is directly influenced by the streamwise pressure gradient, suction, blowing, cooling and heating. A number of examples is discussed in [5], notably also a virtual boundary-layer fence due to distributed surface suction.

2.2 Interaction Issues When we consider separation, either flow-off or ordinary separation, we do that in boundary-layer terms. That means that we treat a two-domain problem: the boundary layer flow and the (external) inviscid flow. If the boundary layer is to serve as phenomenological model of attached viscous flow, and further, if a boundary-layer method is employed for the numerical simulation of this flow, one has to ask, under what conditions this is permissible. The boundary-layer concept considers the flow past a body as to be composed of the external inviscid flow and the boundary layer flow. The inviscid flow transmits its pressure field to the boundary layer. Two kinds of interaction between the two flow fields are usually distinguished: weak and strong interaction. Classical boundarylayer theory mainly treats two-dimensional flow at infinite or semi-infinite bodies, and if three-dimensional flow is considered, this also usually is not done for bodies of finite dimensions [4]. Our definition of separation in Sect. 1.3.3 brings in vortex layers and vortices which carry away from the body surface kinematically active and inactive vorticity. Hence we have to consider a third kind of interaction, that of the vortex layers and the vortices with the flow past the body. This kind of interaction we call global interaction. In this context we also have the issue of the locality principle, which we discuss in Sect. 2.3. The three kinds of interaction are characterized now, following the discussion in [5]: – Weak interaction: The presence of attached viscous flow virtually changes the contour of the body (displacement effect of the boundary layer). In two-dimensional flow the displacement δ1 thickness is given by   y=δ  ρu 1− d y. (2.6) δ1 = ρe u e y=0 The displacement of the inviscid flow in general is positive (possible exceptions are cold-wall cases), i.e., the displacement thickness δ1 > 0, which represents a

2.2 Interaction Issues

35

streamline, virtually enlarges the thickness of the body. In three-dimensional boundary layers δ1 is not found so easily. One has to solve a first-order differential equation over the body surface [5]. δ1 then represents a stream surface, which virtually enlarges the volume of the body. If the displacement effect is small—the Reynolds number is an important factor—, one speaks of weak interaction between the viscous flow—the boundary layer— and the inviscid flow past the body. The inviscid flow is only weakly, even negligibly, affected by the presence of the boundary layer. In boundary-layer computations the virtual thickening of the body can be taken into account—regarding the external inviscid flow—by adding the displacement thickness to the body contour, or, more elegantly, by introducing on the body surface an equivalent inviscid source distribution, e.g., the transpiration velocity [5]. This holds for both first-order and second-order boundary-layer computations. – Strong interaction: Separation, either flow-off separation at trailing edges of airfoils or wings (also at sharp leading edges of delta wings) or ordinary separation— the classical separation—, leads to a strong interaction between the original, but now separating boundary-layer flow and the external inviscid flow. This means that inviscid flow and viscous flow can no more be treated independently of each other. This holds locally and downstream of the separation location. A particular case of strong interaction is shock wave/boundary-layer interaction, which appears if supercritical flow past the wing is present with a pre-shock Mach number larger than M f oot ≈ 1.3−1.35 [13]. Then the boundary layer separates below the shock foot (in two dimensions: bubble separation). This is discussed in Sect. 9.1. Below that Mach number weak interaction is present without separation. The separation locations in two or in three dimensions usually can be determined to a sufficient approximation by boundary-layer computations. The separation process itself—strong interaction—cannot be described in the frame of classical boundary-layer theory, it cannot be treated by means of boundary-layer methods alone. The triple-deck theory, for instance, was devised for such flow situations. It led to a significant progress in the understanding and the description of separation and other strong flow interaction problems [5]. NS (Model 9) or RANS (Model 10) methods as one-domain computation methods—which in practice today are the methods of choice—treat flow separation inherently without any regard to the strong interaction which happens there, if one looks at these phenomenon from the side of boundary-layer theory. However, if the flow is turbulent, the turbulence modeling problem remains. – Global interaction: Attached viscous flow on realistic body shapes of finite extent separates from the body surface basically either by ordinary or by flow-off separation. Regardless of how and where this happens, kinematically active and inactive vorticity leaves the surface and then is present in the wake flow. At lifting large aspect-ratio wings, for instance, this wake is initially a trailing vortex layer, which soon rolls up to a pair of trailing vortices, the wing vortices. The kinematically active vorticity in this wake causes a global interaction, which results in the well understood downwash at the location of the wing and the induced drag of the lifting wing. The upstream and over-the-wing changes of the flow field,

36

2 Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag

however, usually are rather small. This is in contrast to wings with large leading-edge sweep, i.e., delta wings. For a certain combination of sweep angle and angle of attack a pair of lee-side vortices, usually accompanied by secondary and even higher-order vortex phenomena, is present, Sect. 10.2. At the leeward side of the wing, a global interaction happens, too. The flow field there is altered completely. At the windward side, however, the flow field generally is not changed much. Global interaction is described with one-domain methods. For large aspect-ratio wings these can even be inviscid (Model 4 and Model 8) methods but now in general they are NS/RANS (Model 9/10) methods. In particular for delta wings and for fuselages NS/RANS methods are to be employed, even if for pre-design purposes discrete modeled Euler methods (Model 8) still are in use. Examples of both are discussed in Chaps. 8 and 10. This discussion of the three kinds of interaction is supported by a large number of observations from experimental and theoretical/numerical flow field investigations. The observations point to the fact that to a certain extent a ‘locality principle’ exists. The concept of the locality principle was put forward in [14] and later independently in [15].

2.3 A Special Issue: The Locality Principle The locality principle means that a local change in body shape, or the separation of flow—with or without kinematically active and inactive vorticity in the wake— changes the flow only locally or downstream of that region, respectively the separation region. Upstream of that region, the changes generally are small, typically at the lifting wing the downwash at the location of the wing.5 This holds also for subsonic flows, although mathematically their characteristic propagation properties are of elliptic nature, such that always a global interaction occurs. In the one-domain NS/RANS methods (Model 9/10) all three interaction kinds are described correctly, problems may arise regarding turbulence modeling (RANS) in the very separation region, and if the wake flow is unsteady. Due to the mathematically parabolic character of the boundary-layer equations, the Model 2 level, however, implicitly assumes semi-infinite or infinite bodies. Therefore with Model 2 approaches one must ask, whether and how the interaction must be taken into account, either for phenomenological considerations or for simulation purposes. Of course on that level only the attached part of the flow can be described. The locality principle is observed to hold in general, except in cases, where massive separation occurs, or where a structurally unstable topology of the velocity field is present, Sect. 7.5. If the principle holds, it is possible to study attached viscous flow 5 This

contradicts van Dykes statement that a wake exerts a first-order influence even in the flow upstream [16]. Of course, the integral forces and moments, which the flow exerts on the body, are affected by separation.

2.3 A Special Issue: The Locality Principle

37

phenomena at bodies of finite length on the boundary-layer level without taking into account the wake flow. Then it is also permitted, as usually is done, to employ a rather coarse discretization of the wake domain behind a wing, see in this regard Sect. 5.3. However, if computations on the Model 2 level are performed, if possible the global interaction should be taken into account. In the case of large aspect-ratio lifting wings, for instance, this is done in most methods automatically with the computation of the inviscid flow. In linear (potential, Model 4) methods—today usually panel methods—an explicitly imposed Kutta condition at the trailing edge serves this purpose. In Euler methods (Model 8), it is the implicit Kutta condition, which is present at sharp trailing edges. In all these cases, kinematically active vorticity in one or the other form is present in the wing’s wake.

2.4 Aspects of Drag In this section we shortly discuss some aspects of the aerodynamic drag, which aircraft experience. At the center of the discussion is the need to overcome the so called drag divergence, which occurs when the vehicle enters the transonic flight regime.6 This need has led to some distinctive configurational aircraft features. These in turn govern separated and vortical flow at and behind the aircraft. Of course this strongly depends on the respective configuration.

2.4.1 Drag Components The drag of an aircraft is a particular topic. Different concepts exist to break it up and to analyze it. We look at the aerodynamic effects and list the most important components of the aerodynamic drag: – skin-friction drag (Dskin 6 When

f riction ),

in the late 1930s the potential of airbreathing jet propulsion emerged—recognized at that time predominantly in Germany [17]—the drag divergence was a matter of very high concern. In Germany up to 1945 frantic work at research organizations and in aircraft industry was conducted in order to shift in particular the drag divergence to as high as possible (sub-sonic) flight Mach numbers. Connected to the supercritical airfoil are the names K.H. Kawalki and B. Göthert, to the swept wing the names A. Busemann, H. Ludwieg, A. Betz, to the delta wing A. Lippisch, and to the area rule O. Frenzl. The area rule later was confirmed with the equivalence theorem by F. Keune and K. Oswatitsch (1953). An overview of the work is given in [17] and a very detailed account in [18]. When the war ended, the outcome of this work, which in Germany barely came to an application, found its way to the Allies. In [17] a short section “Transfer of the German Aeronautical Knowledge After 1945” is included, details regarding the American acquisition of data are given in [19]. Large research and development efforts in a short time then changed the geometrical appearance of all kind of aircraft. For the USA see in this respect the publication of J.D. Anderson, Jr., “The Airplane: A History of Its Technology” [20].

38

2 Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag

Fig. 2.3 Theoretical and experimental pressure distributions at the surface of a Joukowsky airfoil [21]

– – – –

viscous-effects induced pressure drag, usually called form drag (D f or m ), induced drag (Dinduced ), wave drag (Dwave ), interference drag (Dinter f ).

The first two components together constitute the viscous drag Dvisc = Dskin f riction + D f or m . In cruise it generally amounts up to about half of the whole aerodynamic drag of an aircraft. Regarding the drag divergence, the form drag is what needs to be considered in detail. The point of departure of our considerations is the drag of an airfoil at subcritical Mach number, hence without supersonic flow pockets and terminating shock waves. To understand it, consider Fig. 2.3. The theoretical pressure distribution was found with linear potential theory (Model 4). The airfoil has a trailing edge with a non-zero trailing-edge angle. The flow leaves the trailing edge in bi-sector direction. This means a corner flow on both sides and therefore a stagnation point each. That is clearly indicated with the pressure coefficient being c p = 1 at x/L = 1. The airfoil has lift but no drag. The agreement between the theoretically and the experimentally obtained pressure coefficient is very good up to x/l ≈ 0.8. That means that a weak interaction only is present between the boundary layer and the external inviscid flow. The disagreement at 0  x/l  0.3 is not explained. Laminar-turbulent transition over a separation bubble is the likely cause.

2.4 Aspects of Drag

39

At x/l  0.8 the theoretical and the experimental result strongly disagree. The boundary layers at the upper and the lower side in the reality of the experiment due to their displacement properties prevent the recompression to the stagnation point. At the trailing edge we now observe c p ≈ 0.15. The result is the form drag, which always is present also in flow-off separation. We too see that this kind of separation means that a strong interaction between the boundary layers and the external inviscid flow occurs. This result holds for the flow at the aft end of any body. At a round end of a fuselage of course ordinary separation occurs. Topologically it can be very complex compared to the flow-off separation. This also holds for the separation at the in reality slightly blunted trailing edge of a wing, Sect. 6.3.

2.4.2 The Drag Divergence The drag coefficient of a wing or fuselage is almost constant over the Mach number range from small to rather large subsonic free-stream Mach numbers. This is in contrast to the lift coefficient of an airfoil or wing, where a clear dependence on the Mach number is present. In linear potential theory (Model 4) that is modeled with the Prandtl–Glauert rule [3, 21]. Figure 2.4 over a large Mach-number span shows the almost constant drag coefficient C D of airfoils of different thickness. We observe, however, that above a certain ∗ the drag coefficient strongly begins to rise. The rise begins the Mach number M∞ earlier, the thicker the airfoil is. The phenomenon is called the transonic drag divergence. Accordingly we get a dip in the lift coefficient and also in the pitching moment coefficient (not shown). The cause of the drag divergence, which directly concerns the zero-lift drag D0 only, can be sketched as follows.7 (For details see Sect. 9.1.) Above the critical Mach number a pocket of supersonic flow appears over the airfoil’s surface. This pocket generally is terminated by a shock wave. (The shock-free airfoil is a special topic.) The resulting shock-wave/boundary-layer interaction—a strong-interaction phenomenon—causes an extra thickening of the down-stream boundary layer.8 The enlarged displacement thickness enlarges the form drag. This and the wave-drag increment due to the shock wave, the boundary-layer decambering and the associated shock-wave decambering combined, Sect. 6.1, cause the transonic drag divergence and the lift divergence.

7 The

zero-lift drag is the drag of the aircraft without the induced drag. shock wave terminates orthogonally to the airfoil’s surface, although at the boundary-layer edge particular phenomena can be present. For all pre-shock Mach numbers this leads to a reduction of the unit Reynolds number and hence to a thickening of the boundary layer.

8 The

40

2 Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag

Fig. 2.4 Drag coefficients C D of symmetric airfoils (NACA—00xx—1, 13 30) of different thickness ratios (6–18 per cent) at α = 0 as function of the free-stream Mach number M∞ [21]. Measurements of B. Göthert, DVL 1941 [22]

The drag-divergence Mach number Mdd is higher than the (lower9 ) critical Mach ∗ . This means that the effect in a sense is delayed. Different definitions number M∞ are in use of the drag-divergence Mach number, we just point to [23]. One means to shift the drag-divergence Mach number Mdd to Mach numbers as high as possible—and also to cap the drag—is evident when looking at Fig. 2.4. The thinner the airfoil, the lower is the excess speed past it (relative to the free-stream speed), and the higher is Mdd . This leads to the concept of the thin wing. (Usually this is a small aspect-ratio wing.) The thin-wing concept is limited to high-performance aircraft, the F-104 Starfighter being a prominent example. For transonic transport aircraft, which carry the engines at the wings, as well as the fuel in them, the thin wing is no option. The wing needs enough spar height in order to bear forces and moments. The now common solution is to back-sweep the wing. This in effect means that the free-stream “sees” a thinner wing. (The same effect is present at the forward-swept wing.) The wing section, the airfoil, moreover can be shaped in a way, that with a given thickness ratio the excess speed is reduced. This is the concept of the supercritical ∗ is distinguished from the upper critical M∞,l ∗ . At M ∗ supersonic flow begins to appear at the body, at M ∗ Mach number M∞,u ∞,u the flow past ∞,l the body is fully supersonic, at a blunt-nosed body except for the subsonic region at the nose.

9 In the literature often the lower critical Mach number

2.4 Aspects of Drag

41

airfoil/wing. Since long the combination of the swept wing and the supercritical airfoil is the rule. For supersonic flight the rule today is a wing with subsonic leading edge, i.e., the delta wing with its different shapes. Subsonic leading edge means that the apex half angle is smaller than the angle of the free-stream Mach cone μ∞ (sinμ∞ = 1/M∞ ). This amounts to a sweep angle ϕ0 < 90◦ − μ∞ . An aircraft is composed of subsystems. Geometrically these are the wing, the fuselage, the tail unit and the nacelles/engines. In supercritical flight interference effects between these subsystems are present, which go beyond the low-speed effects. To minimize these interference effects, the distribution of the cross-section area of the whole aircraft in the main longitudinal axis (free-stream) direction has to be as smooth as possible. This is the so called area rule. What have these shape particularities of transonic and supersonic aircraft to do with our topic, separated and vortical flow in aircraft aerodynamics? We give very short sketches only. The reader will find the details in the later chapters. – The wing sweep, either backward or forward, leads to distinctly different properties of the upper and lower boundary layers and the near-wake flow, the trailing vortex layer, behind the trailing edge. Even if earlier problems with flow separation, which led to the application of boundary-layer fences, have been overcome, the stall behavior is still a topic. – Regarding delta wings or canards of hybrid wings with their highly swept leading edges, the appearance of the lee-side vortices, Fig. 1.10b in Sect. 1.3.2, is a very prominent topic of this book. The provision of general flight quality and maneuverability demands a deep understanding of the lee-side flow, as well as high-fidelity aerodynamic design and verification tools. – The area rule finally demands appropriate engine-nacelle locations at the wing as well as special geometrical features of the fuselage. In particular at high angles of attack separation phenomena at the nacelles affect the wing flow and separation behavior.

2.5 Problems Problem 2.1 Assume an airfoil at a flight altitude of H = 10 km as to be a flat plate with a chord length of c = 5 m. Determine the wall shear stress at the locations x/c = 0.5 and 1.0 and also the displacement thicknesses δ1 at the two locations. At x/c = 1 strong interaction effects are neglected. Employ the approximate relations given in Appendix A.5 for the Mach numbers M∞ = 0.5 and 0.8 and at the wall the recovery temperature. Assume laminar flow throughout. The boundary-layer edge values can be approximated with the free-stream values. Problem 2.2 Repeat Problem 2.1 for turbulent flow.

42

2 Separation: Some Relevant Boundary-Layer Properties, Interaction Issues, and Drag

Problem 2.3 Repeat Problem 2.1 for a wall temperature twice as high as the recovery temperature. Problem 2.4 Repeat Problem 2.2 for a wall temperature twice as high as the recovery temperature. Problem 2.5 Give a short summary of the results of the Problems 2.1–2.4. Problem 2.6 The drag coefficient is not the same as the drag. Get from Fig. 2.4 for the section drag coefficient Cd for the Mach numbers M∞ = 0.2, 0.4 and 0.8. Assume Cd to be constant up to the critical Mach number. Assume flight at H = 10 km, a reference area Ar e f = 1 m2 , and compute the drag for the three Mach numbers. How does the drag behave? Problem 2.7 The 6 per cent thick airfoil at α = 2◦ at M = 0.3 has a section lift coefficient Cl = 0.2. How large is the section lift coefficient at the Mach numbers 0.2, 0.4, 0.8? Remember the Prandtl–Glauert rule. Problem 2.8 How large is the actual lift of the airfoil from Problem 2.7 at the three Mach numbers. Use the flight parameters from Problem 2.6. Problem 2.9 Give a short summary of the results of the Problems 2.6–2.9.

References 1. Shapiro, A.H.: Basic equations of fluid flow. In: Streeter, V.L. (ed.) Handbook of Fluid Dynamics. McGraw-Hill Book Company, New York, pp. 2-1–2-19 (1961) 2. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002) 3. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw-Hill Book Company, New York (2011) 4. Schlichting, H., Gersten, K.: Boundary Layer Theory, 8th edn. Springer, Berlin (2000) 5. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014) 6. Hirschel, E.H.: Basics of Aerothermodynamics, 2nd edn, revised. Springer, Cham (2015) 7. N.N.: Boundary-layer simulation and control in wind tunnels. AGARD-AR-224 (1988) 8. Cebeci, T., Cousteix, J.: Modeling and Computation of Boundary-Layer Flows, 2nd edn. Horizons Publishing, Long Beach and Springer, Berlin (2005) 9. Knopp, T., Eisfeld, B., Calvo, J.B.: A new extension for k-ω turbulence models to account for wall roughness. Int. J. Heat Fluid Flow 30, 54–65 (2009) 10. Thwaites, B.: Approximate calculation of the laminar boundary layer. Aeronaut. Q. 1(3), 245– 280 (1949) 11. Stratford, B.S.: The prediction of separation of the turbulent boundary layer. J. Fluid Mech. 5(1), 1–16 (1959) 12. Cebeci, T., Mosinskis, G.J., Smith, A.M.O.: Calculation of separation points in incompressible turbulent flow. J. Aircr. 9(9), 618–624 (1972) 13. Délery, J.: Transonic shock-wave boundary-layer interactions. In: Babinsky, H., Harvey, J.K. (eds.) Shock-Wave Boundary-Layer Interactions, pp. 5–86. Cambridge Universtity Press, Cambridge (2011)

References

43

14. Hirschel, E.H.: On the creation of vorticity and entropy in the solution of the Euler equations for lifting wings. MBB-LKE122-Aero-MT-716, Ottobrunn, Germany (1985) 15. Dallmann, U., Herberg, T., Gebing, H., Su, W.-H., Zhang, H.-Q.: Flow-field diagnostics: topological flow changes and spatio-temporal flow structure. AIAA Paper 95–0791, (1995) 16. Van Dyke, M.: Perturbation Methods in Fluid Mechanics. Academic, New York (1964) 17. Hirschel, E.H., Prem, H., Madelung, G. (eds.): Aeronautical Research in Germany–from Lilienthal until Today. Springer, Berlin (2004) 18. Meier, H.U. (ed.): German Development of the Swept Wing–1935–1945. Library of Flight. AIAA, Reston (2010) 19. Samuel, W.W.E.: American Raiders: The Race to Capture the Luftwaffe’s Secrets. University Press of Mississippi, Jackson (2004) 20. Anderson Jr., J.D.: The Airplane: A History of Its Technology. AIAA, Reston (2002) 21. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, vol. 1 and 2, Springer, Berlin (1959), also: Aerodynamics of the Aeroplane, 2nd edn (revised). McGraw Hill Higher Education, New York (1979) 22. Göthert, B.: Hochgeschwindigkeitsuntersuchungen an symmetrischen Profilen mit verschiedenen Dickenverhältnissen im DVL-Hochgeschwindigkeits-Windkanal (2.7 m ∅) und Vergleich mit Messungen in anderen Windkanälen. ZWB/FB 1506 (1941) 23. Vos, R., Farokhi, S.: Introduction to Transonic Aerodynamics. Springer Science+Business Media, Dordrecht (2015)

Chapter 3

Elements of Vortex Theory

This chapter is devoted to a brief introduction to elements of vortex theory, which are of relevance for the topic of the book. We shortly outline each item, give basic mathematical descriptions, illustrating sketches, and provide supporting references. General literature recommended for a deeper study of the topic as such are the monographs by, e.g., P.G. Saffmann (1992) [1], H.J. Lugt (1996) [2] and J.-Z. Wu, H.-Y. and M.-D. Zhou (2006) [3]. Regarding monographs on aerodynamics in view of potential-flow theory the reader will find ample material in, e.g., H. Schlichting and E. Truckenbrodt (1959/1979) [4], J.D. Anderson, Jr. (2011) [5], M. Drela (2014) [6] and in the Handbuch der Luftfahrzeugtechnik by C.-C. Rossow, K. Wolf and P. Horst (2014) [7].

3.1 Introduction It is interesting to note that the circulation theory, i.e., the vortex theory of the lifting airfoil, evolved with the work of M.W. Kutta and N. Joukowski in the years 1902 to 1912, see the sketch of its general development in Sect. 1.4. Their work regarded the lifting airfoil, and that at a time, when human heavier-than-air flight barely was at its beginning [8, 9]. For quite a time then all aerofoil data still had to be obtained from experimental work and fitted to wings with different aspect ratios, planforms, etc., by empirical formulae based on past experience with other airfoils and wings.

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_3

45

46

3 Elements of Vortex Theory

Fig. 3.1 Various models of finite-span wing theory consisting of different vortex line arrangements. a lifting-line model b wing-sheet model, and c vortex-lattice model, after [2]

3.1.1 Models of Finite-Wing Theory of Flight The Laplace equation describes incompressible, irrotational and inviscid (potential) flow around the wing, but surprisingly no global force acts on it, because in a very basic sense no interesting motions can take place in incompressible homogeneous unbounded fluids without vorticity, Sect. 1.4. This is the Laplace equation, Model 3 of Table 1.3, and the d’Alembert paradox. With the independent work of F.W. Lanchester and L. Prandtl, which culminated in Prandtl’s lifting-line model, a theoretical model of the lifting wing was created.1 It consisted of a system of vortices that imparted to the surrounding air a motion similar to the actual flow, and that yielded a force equivalent to the lift known to be created. The vortex system consisted of four main parts: the bound vortex, the two trailing vortices and the starting vortex. Each of these may be treated separately but they are all component parts of one whole acting together, see Fig. 3.3. We use this theory to illustrate the connection of lift to vorticity, its creation and the dynamics that vortices undergo in the various Models 4, 6, 8 and 10 of lifting wing flow. Figure 3.1 displays three variants of Model 4 for finite-span wing theory consisting of different vortex line arrangements. The first is Prandtl’s lifting-line model that forms the basis for an analytical finite-wing theory, the second, also analytical, is a wing-sheet or lifting-surface model, and the third, a numerically discrete description of the wing sheet, is the vortex-lattice model. Since they do not account for vortices shed from side edges, the first two models in Fig. 3.1a, b are valid only for the flow sketched in Fig. 1.10a, while the vortex-lattice model, Fig. 3.1c is applicable for the flows in both Fig. 1.10a, b. As background to understanding how the Models 4 to 10 of Table 1.3 perform, the current chapter briefly summarizes some relevant fundamentals of vortex theory, discussing among others: 1. vortex sheets associated with lifting surfaces, 2. their creation and dynamics including roll-up into vortices associated with the flows shown in Fig. 1.10a, b, 1 In the literature often the term “Lanchester–Prandtl-Theory” is found, in Germany it reads “Prandtl–

Lanchester-Theory”, see also the remark on Sect. 1.4.

3.1 Introduction

47

3. stability and decay of vortices in the wake behind a wing, 4. vortex-flow technology applied in aircraft design.

3.1.2 Benefits in Studying Vortex Dynamics The first benefit is overcoming the d’Alembert paradox as Kutta and Joukowsky did. Model 3, the Laplace equation is linear. Hence vortex singularities can be superposed on a given solution to obtain desired features, as described further in this chapter. Secondly the motion of an incompressible fluid can be described either by the Navier–Stokes equations for the velocity and the pressure (primitive variables), or by the vorticity equation for the velocity and the vorticity. The two versions are equivalent. Historically it has been preferred to describe the flow field in terms of velocity and pressure, i.e., to solve the Navier–Stokes equations, because the velocity vector and the pressure are more tangible quantities than the vorticity vector. However, the alternative description using the vorticity concept has its advantages. In particular before numerical solutions of the compressible Navier–Stokes equations, Model 9, became so prevalent in the 1990s, it was thought that solving the vorticity transport equation offered a simpler approach. Admittedly, the concept of vorticity is less intuitive than that of the pressure, but we can argue that the grasping of a physical concept is to a large extent a matter of usage. Vorticity must be understood from its collective properties, and to convey such an understanding is the purpose of this chapter. For instance, vortex patterns can be described and analyzed better by vorticity, or in the limit by point vortices, since the vorticity field—also—is invariant with respect to Galilean and rotational transformations (except for a constant). Three-dimensional vorticity patterns reveal stretching and twisting processes, bringing about local intensification of vorticity. In general vortices serve as a paradigm for the development of dynamic flow patterns, since their theory is so well advanced. Perhaps Dietrich Küchemann’s well-known metaphor that ‘vortices are the sinews and muscles of fluid motion’ captures best the essence of what we mean by pattern recognition. Vortex theory has become a paradigm for pattern development even in general. A better understanding of vortical phenomena helps us to understand and judge the validity of our computed results, from Models 8 to 10 (Euler, RANS, URANS)—and of course also Model 11—that we present in later chapters. Here one must caution that our solutions are to the compressible equations, while the vortex dynamics discussed in this chapter derive from the incompressible equations. Keep in mind also that the most complete and elegant description of vorticity, however, exists for inviscid fluids, since vorticity is here attached to fluid elements and is not lost through diffusion. In such cases the Lagrangian description is the natural one and gives the best insight into the essence of vorticity, because it is conserved and can be traced back to its initial state.

48

3 Elements of Vortex Theory

This chapter therefore provides us with a bridge in our discussion from the fluidflow models outlined in Chap. 1 to the computed results of these models presented and analyzed as Unit Problems in later chapters.

3.2 The Concept of Vorticity 3.2.1 General Remarks The understanding of vortex theory requires the clarification of two central concepts, namely, what is a vortex and what is vorticity? Both are elusive concepts, but in opposite ways. A dictionary easily defines a vortex as the rotation of fluid elements around a common center, but these are just words without mathematical precision. In contrast to the vortex concept, the vorticity vector ω has an exact mathematical definition as the curl of the velocity vector ω = curl v, but its general physical meaning is difficult to grasp. The standard explanation is that vorticity is a measure for the angular velocity of a fluid element, and we see that a vortex requires vorticity for its existence. Metaphorically speaking vorticity is the building agent for vortices, but the reverse statement that a vorticity field represents a vortex, however, is not always true, since a straight shear layer, for example a boundary layer, has vorticity but is not a vortex. However, the vorticity field is a continuum, and to relate the vorticity to the angular velocity of a fluid element in this continuum, from which it can diffuse to neighboring elements, causes imaginative difficulties inherent in the meaning of a continuum. The way to proceed is to make the vortical flow field compatible with the vorticity by relating the vorticity concept (or any other mathematically defined quantity) to that of the vortex. Flow fields with three-dimensional vortices in general have complex structures, and they can be understood to a large extent by the movement, stretching, and interaction of vortex filaments (‘strings’). Although these processes are nonlinear and, hence, capable of abrupt, drastic changes within a flow field, they involve solenoidal-field properties along with measures like circulation that give us relations between vortices and vorticity, which then makes it easy to study mathematically the generation, behavior, and decay of vortices.2 In this book we are dealing with vortical patterns seen in computed solutions of Models 8, 9 and 10, which include the generation, behavior, and the decay of a single vortex as well as a system of vortices, vortex sheets, shear layers etc. of high and low kinematically active and/or inactive vorticity content, and we need to make some physical sense of what we observe. Vorticity, in short, is a crucially important feature of any shearing motion and provides a path towards further understanding. 2 Solenoidal

means source-free, or zero divergence in all points of a vector field.

3.2 The Concept of Vorticity

49

3.2.2 Designations and Definitions As stated above, a vorticity vector field, ω(x, y, z), is defined as the curl of the velocity vector ω = curl v = ∇ × v. (3.1) Vorticity appears in the decomposition of the velocity field as the rotation term in the fundamental theorem of kinematics, which states that the velocity of a fluid element v + dv in a continuum is composed of the rates of translation v, deformation D · d r and rotation 21 curlv × d r [10]. The vorticity represents a measure of the local rate of rotation of a fluid element. A vortex in a viscous fluid rotates like a solid body at the axis and thus requires vorticity. There is no vortex without vorticity. This seems to be a contradiction, when looking at a potential (point) vortex or a vortex sheet (slip line or surface). However, the concept of kinematically active and inactive vorticity content, Chap. 4, permits us to overcome this contradiction. A vorticity line is a line tangent to the vorticity vector—analogous to the streamline in a velocity vector field. If the whole field is irrotational except for an isolated vorticity line, this line is then called a vortex line. In three-dimensional flows a bundle of vortex lines is enclosed in a vortex tube, which is a surface comprising all the vortex lines passing through points of a particular loop as shown in Fig. 3.2. An isolated vorticity tube is a vortex tube, which incloses a vortex filament [11]. The Rankine vortex is an infinitely long straight vortex filament.

Fig. 3.2 A vortex tube is a surface comprising all the vortex lines passing through points of a particular loop [3]

50

3 Elements of Vortex Theory

3.3 Stokes’ Theorem and the Concept of Circulation 3.3.1 The Theorem From the definition ω = curl v it follows that the vorticity field is solenoidal— div ω = 0—since for any vector field u we have div curl u ≡ 0. The solenoidal property means that vorticity tubes (like stream tubes) cannot end inside the fluid. Next, for a general vortex tube, as sketched in Fig. 3.2, composed of the vortex lines passing through a particular loop C, the solenoidal property div ω = 0 implies that all along the vortex tube the product of its cross-sectional area with the magnitude of the vorticity remains constant. To see this, we apply the Divergence Theorem to this property over the volume V of the fluid, bounded by two different surfaces S1 and S2 , spanning the tube and by the part of the tube between them, and obtain 

 ∂V

ω · n dS =

V

div ω dV = 0

(3.2)

since div ω ≡ 0. The bounding surface ∂V of the volume comprises not only S1 and S2 , but also the part of the vortex tube between them, the latter making no contribution to the left-hand side of Eq. 3.2 since, by the definition of a vortex tube, the vorticity vector is tangential to it, thus giving ω · n = 0. It follows that  ω · n dS = constant, (3.3) ∂V

which gives the useful result that the inner product of S with the magnitude of the vorticity remains constant along the tube.

3.3.2 Circulation The concept of circulation was introduced by W. Thomson (Lord Kelvin) in 1869 and is defined in the following way. The quantity stated in Eq. 3.3 to take everywhere a constant value along a general vortex tube, is a characteristic property of that vortex tube, called its strength Γ , i.e., its circulation. It is the famous mathematical theorem due to Stokes, which gives a special interpretation of the left-hand side of Eq. 3.3. Stokes’s theorem applies to any surface S for which we are able to make a consistent, continuously varying selection of a direction for the unit normal vector n at each point of the surface and for which the boundary ∂S consists of one or more closed curves C, Fig. 3.2. Stokes’s theorem states that, for any vector field, and thus in particular for the velocity field v

3.3 Stokes’ Theorem and the Concept of Circulation



 S

curl v · n dS =

S

51

 ω · n dS =

v · d,

(3.4)

C

where  is the unit vector tangential to the closed curve C. Thus from Eq. 3.3 we conclude   ω · n dS = v · d = Γ, (3.5) S

C

that the circulation Γ is the strength of the vortex tube. The Kutta–Joukowski Theorem, underlying the foundation of the circulation theory of lift for the models in Fig. 3.1, states that an isolated two-dimensional wing section in an incompressible inviscid flow feels a lift force lu per unit width lu = ρv ∞ Γ,

(3.6)

see also Sect. 3.16.

3.4 Origins of Vorticity The basic foundation of the finite-span wing theory of flight is the creation of lift through flow circulation around the wing, that translates into a mathematical model, in the simplest case into the lifting-line model. In this model a ring vortex is formed, where one leg is placed inside the wing—the bound vortex—another leg—the starting vortex—is far downstream, connected on both ends by two trailing-vortex legs, illustrated in Fig. 3.3. The figure is the epitome of Prandtl’s lifting-line model, which permits to describe the flow downwash at the location of the wing and the induced drag. The figure demands a closer inspection. Globally the lifting-line model is a Model 4 (Table 1.3) theory for lifting wings with large aspect ratio. The perceived reality, Model 1, is reduced in the following way. The trailing vortex layer, Sects. 4.3 and 4.4, has undergone an immediate roll-up. The instantly present trailing vortices are

Fig. 3.3 The bound vortex, the two trailing vortices, and the starting vortex form a closed vortex ring, which is the simplest model of a lifting wing of finite span, after [2]

52

3 Elements of Vortex Theory

located at the wing tips at a chosen chord location and there they connect with the bound or lifting vortex. At the wing tips the tip vortices, or more in general the tipvortex systems, Sect. 8.4.3, are not present. Hence also not their relevance in view of the non-linear lift of wings with small aspect ratio, Sect. 8.4.4.3 The reader should be aware that often in the literature verbally and even conceptually the tip vortices are mixed up with the trailing vortices. For the purpose of discussing origins of vorticity, it suffices to explain the flow mechanism present in the lifting-line model that creates the bound vortex leading to circulation and lift. Let us look at various models for wings within the framework of potential-flow (Model 4) theory. The essence of two-dimensional wing theory contains the idea of the bound vortex or alternatively a circulation around the wing. In the mathematical formulation, the bound vortex is represented by a point, i.e., potential vortex located inside the wing section. In Sect. 4.2.4 we show that the boundary layers at the upper and the lower side of the airfoil contain the kinematically active vorticity connected to that vortex. In an unsteady two-dimensional flow, or in a steady or unsteady three-dimensional flow, also vortices outside of the wing must be considered. Because of Helmholtz’s law for persistence of circulation, the vortex lines must be closed by connecting the starting vortex with the trailing vortices.

3.4.1 The Kutta Condition In several papers published in the first decade of the 20th century, Kutta and Joukowsky independently proposed that the lift on an airfoil at incidence in a steady unseparated flow is given by potential-flow theory with the unique value of the circulation that removes the inverse-square-root velocity singularity at the trailing edge. This proposal—tantamount to saying that in the unsteady start-up phase the action of viscosity is such that, in the ultimate steady motion, viscosity can be explicitly ignored, but is implicitly incorporated in a single edge condition—is known as the Kutta–Joukowsky hypothesis, commonly called the ‘Kutta condition’.4 Although this suggests the ‘action of viscosity’ as the responsible mechanism, there is no fully universal convincing physical explanation for the production of the starting vortex and the generation of the circulation around the wing section. This issue has been debated for over a hundred years, as described in book-length by D. Bloor [12]. The most usual explanation, represented following Bloor as The Cambridge School, is based on the large viscous forces in the boundary layer, associated with the high velocities round the trailing edge invoked by the no-slip condition on the surface. 3 The simple lifting-line model anyway does hold only for wings with large aspect ratio, say Λ  3. 4 Regarding

the Kutta condition in the reality of aircraft see Chap. 6.

3.4 Origins of Vorticity

53

This line of reasoning infers that circulation cannot be generated—and no aerodynamic lift can be produced—in an inviscid fluid, because the first Helmholtz theorem, Sect. 3.7, says that it is not possible to endow a fluid particle with vorticity. Moreover, Kelvin’s circulation theorem, Sect. 3.8, shows that the circulation around a material circuit remains zero if initially zero. The question then arises whether vorticity can be created without violating these theorems, and without appeal to viscous or barotropic effects.5 Another line of reasoning, which we call The Göttingen School, however, argues that local flow acceleration is equally important and that this is sufficiently high to account for the failure of the flow to follow round the sharp trailing edge, without invoking viscosity. The following two sections present arguments on both sides of the debate whether the vortex-sheet creation is essentially viscous or inviscid. We note that in Chap. 4 we do not discuss the formation of the starting vortex and the development process of the circulation around the wing.

3.4.2 The Cambridge School—No-Slip Viscous Mechanism Fundamentally unsteady, Fig. 3.4 illustrates in more detail the formation process of the starting vortex [13]. Initially at rest the wing starts to accelerate, and in the first instant Fig. 3.4a presents a qualitative picture of the streamlines around any cross section of the wing. The Cambridge School argues that instead of irrotational flow around the trailing edge, as in Fig. 3.4b, the no-slip condition in the boundary layer generates vorticity causing the flow to break away in the starting vortex, Fig. 3.4f. • Lighthill’s Concept—Viscous Pressure Gradient M.J. Lighthill has developed a hypothesis for a surface-pressure gradient that determines the precise amount of vorticity that must be created in the boundary layer, consistent with the circulation around the wing [14]. He considered the two-dimensional unsteady flow over the airfoil and started from the Navier–Stokes equations in streamline coordinates (n, s), where s and n are the unit tangent and normal vectors of any streamline. At t = 0 the airfoil impulsively moves from rest. Since pressure waves travel with infinite speed in an incompressible fluid, an infinitely thin shear layer develops along the surface, which reduces the velocity to zero to satisfy the no-slip condition. The Navier–Stokes equations along the airfoil surface then instantly reduce to (s being the streamwise, n the surface-normal direction) ∂ω 1 ∂p =ν ≡ σ, ρ ∂s ∂n ∂ω 1 ∂p = −ν , ρ ∂n ∂s 5 The

barotropic fluid is a fluid, whose density is only a function of pressure.

(3.7) (3.8)

54

3 Elements of Vortex Theory

Fig. 3.4 Phenomenological sketch of streamlines of inviscid potential and rotational flow around a wing during its first instants of acceleration from rest [13]: a flow around sharp trailing edge S; b potential model, flow around S allowed; c compressible Euler equations, flow expands to vacuum in the limit; d separates with a tangential discontinuity; e rolls up into a vortex; and f is shed drawing the separation point B to the trailing-edge point S

referred to as Lighthill’s relations. σ is the vorticity flux in chord direction along the 2 = ν ∂∂nu2 , so the right hand side is the normal gradient wing’s surface. Note that ν ∂ω ∂n of the viscous shear force. The amount of vorticity that is diffused into the fluid next to the boundary is the flux of vorticity, σ, being equivalent to the vorticity source strength on the boundary. According to Lighthill, to fulfill the no-slip condition and the solenoidality of vorticity, the vorticity flux σ depends only on the pressure change along the wall. Not everyone, however, agrees with Lighthill’s conclusion. For example H.J. Lugt [2] disagrees with this creation of the vorticity flux by the pressure gradient, and argues instead that it are the shear stresses that drive the mechanism.

3.4 Origins of Vorticity

55

3.4.3 The Göttingen School—Inviscid Vortex-Sheet Mechanism Let us turn now to the Göttingen School and its arguments for the opposing concept— an inviscid mechanism. This line of reasoning begins with F. Klein in 1910, and was nicely described and summarized more recently by P.G. Saffmann [1]. • Klein’s Kaffee-Löffel Experiment In 1910 F. Klein addressed the origin-of vorticity problem with his Kaffee-Löffel (coffee spoon) experiment [15], further studied in 1950 by A. Betz [16]. In Klein’s experiment a two-dimensional plate (a line segment) is set in motion through a perfect incompressible fluid with velocity normal to the plate. Klein argues that if the plate is removed (by pulling out the coffee spoon) or dissolved (certainly possible in a gedankenexperiment), then a vortex sheet is left in the fluid. Non-zero circulation has been created around any of the fluid circuits and there is no contradiction with the Helmholtz-Kelvin laws, because no fluid particle has acquired vorticity, and no circulation has been generated about closed contours lying entirely within the fluid. The new contours with circulation were not closed beforehand; they intersected the body (spoon). This is an example in which vorticity has been created as a sheet— i.e. a singular distribution—by changing the topological properties of the flow. • Early Computational Experience with Model 8 In 1981 the second author of this book obtained one of the first numerical solutions of the Euler equations (Model 8) for transonic flow around the ONERA M6 wing, [13], remarkably without the implementation of a Kutta condition, which is essential in all existing computational methods solving the potential equations, Methods 4 and 5. In explaining why his method did not require a Kutta condition, Rizzi applied arguments, similar to those for the coffee spoon, to what happens when a wing begins to accelerate from rest. Consider again Fig. 3.4c at the instant when the wing is first set in motion. The air over the rear part of the wing section is required to change direction suddenly while still moving at high speed, the pressure around the trailing-edge point S will drop and draw towards it air from the upper surface, that then meets the airstream from the lower surface. If these two streams adjoin with differing velocities along their interface, they form a tangential discontinuity, i.e. a vortex sheet, that by self-induction rolls up into a vortex, Fig. 3.4d. The sense of the shed vorticity acts to impede the velocity around the trailing edge drawing B towards S in a self-correcting process, Fig. 3.4e, and the vortex is convected downstream by the surrounding stream, Fig. 3.4f. The shed vortex is equal in strength and opposite in sense to the circulation round the wing and is called the starting vortex. Subsequent computations with a highly refined grid confirmed that these results were adequately resolved and not grid dependent [17]. • Shear between Two Inviscid Flow Fields Boundary-layer computations of the first author of this book for finite-span wings led to observations of flow field

56

3 Elements of Vortex Theory

properties at the lower (pressure) and the upper (suction) side of lifting wings. These and the observations of the second author in his numerical solutions of the Euler equations for lifting wings led to the concept of the kinematically active and inactive vorticity content of two-dimensional and three-dimensional shear layers, Chap. 4. That chapter also presents a thorough and more general analysis of the origin of vorticity, which for finite-span wings essentially is seen to be due to the second break of symmetry, with the Kutta condition representing the first break of symmetry, Sect. 4.3. Examples in the form of Unit Problems are given in Chap. 8 for large aspectratio wings and in Chap. 10 for small aspect-ratio delta-type wings.

3.4.4 Other Origins In support of counter arguments to the mechanism of no-slip condition, other nonviscous mechanisms for creating circulation can be mentioned. • Inviscid Weis-Fogh Mechanism Insect flight usually is thought of as lowReynolds-number movement, hence with strong viscous effects, but surprisingly, the Klein mechanism may work also in an inviscid fluid. The work of T. Weis-Fogh, [18], on the hovering flight of certain wasps indicates that it is possible to generate circulation and lift in the complete absence of viscosity by the alternate separating and folding of two plates, see Fig. 3.5. Although the total circulation is zero for irrotational flow, according to Kelvin’s theorem, the two plates have, after separation from each other, the same amount of circulation but with opposite sign. Lighthill calls the process clap and fling as the hovering insect moves its wings in a horizontal plane, and computes the strength of the rolled-up vortex [19]. Here, in subscribing to this inviscid mechanism, Lighthill plants his feet in both sides of the controversy, Cambridge versus Göttingen schools of thought. • Across Shock Waves It must be mentioned, too, that vorticity is generated in compressible flow, when it passes through a curved shock wave in either the transonic, supersonic or hypersonic flow regime, producing an entropy-vorticity wake behind

Fig. 3.5 The Weis-Fogh mechanism of circulation generation producing lift, after [2]

3.4 Origins of Vorticity

57

shock

s = const. ω=0

s = const. ω=0

shock wake

viscous region Fig. 3.6 Airfoil at supercritical speed [6]: vorticity-free flow ahead of the shock wave, vorticity field behind the shock wave

it [20]. The connection between entropy and vorticity is the topic of Crocco’s law. This law is discussed in the following Sect. 3.5. Here we consider shortly two typical flow situations with shock waves, where vorticity is created. Figure 3.6 shows an airfoil at supercritical speed. Ahead of the shock wave, which terminates the supersonic flow pocket at the upper side of the airfoil, the entropy s is constant, also the vorticity ω. The flow is homentropic. Behind the shock wave, which is curved, the entropy is not constant, hence vorticity is present in the shock wake. The flow downstream of the shock wave, however, is isentropic, i.e., the entropy is constant along the streamlines, but differs from streamline to streamline. In the figure the boundary layers at the upper and the lower side of the airfoil, as well as the airfoil’s wake, are marked as viscous region. We argue in Sect. 4.2.3 that the shock wave’s curvature is such that the produced vorticity flux compensates the difference of the vorticity fluxes in the boundary layers. In this way the whole wake of the airfoil, the vorticity or entropy wake in Fig. 4.6, does not carry a kinematically active vorticity content, as is demanded for the steady flight situation. We note in addition that in the inviscid picture of the flow, the shock wave terminating the supersonic flow pocket on the upper side of the airfoil impinges orthogonally on the airfoil’s surface, Sect. 4.2.3. In the viscous reality a shock-wave/boundarylayer interaction is present, with potentially large consequences for the airfoil’s or wing’s performance, Sects. 2.4.2 and 9.1. The other flow situation is that at a blunt body at supersonic or hypersonic speed, Fig. 3.7. High supersonic or hypersonic flight vehicles as a rule have blunt noses in order to cope with the high thermal loads. Re-entry vehicles moreover enlarge the total drag with the blunt nose, since they essentially fly a braking mission [20]. The figure shows the bow-shock wave of a blunt body at angle of attack. At the windward side the shock wave is lying almost parallel to the body surface. The strongly curved bow shock leads to a distinctive entropy layer, hence a vorticity layer. The figure shows that the vorticity layer, in terms of the surface tangential velocity v(n)—n being locally the direction normal to the body surface—has different profiles at the windward and the leeward side. At the windward side the profile is wake-like, whereas it is slip-flow boundary-layer like at the leeward side. The different shapes

58

3 Elements of Vortex Theory

Fig. 3.7 Blunt body at angle of attack at high supersonic or hypersonic speed [20]: entropy layers at the lower and the upper side. Note that the boundary layers are not indicated

are due to the fact that the stagnation point S1 lies below the nose point of the body at the windward side. At zero angle of attack, the profile is slip-flow boundary-layer like everywhere. One effect of interest in this case is the so-called entropy-layer swallowing of the boundary layer, which leads to a decrease of the boundary-layer thickness, hence an increase of the thermal load, and moreover influences the laminar-turbulent transition process [20].

3.4.5 Short Conclusion In conclusion, the debate about whether the vorticity-origin mechanism is viscous or inviscid driven has become moot. Today the overwhelming majority of studies are computed with Methods 9 and 10, and also 11, because many other boundary-layer phenomena, including separation are involved and need to be considered. Method 8 is used only for certain specialized inviscid flow cases.

3.5 Entropy and Total Enthalpy Gradients and Vorticity: Crocco’s Theorem Combining the first law of thermodynamics with the momentum equations yields the Crocco theorem that connects vorticity, a kinematic flow property, to thermodynamic properties, the specific entropy s and the specific total enthalpy h o :

3.5 Entropy and Total Enthalpy Gradients and Vorticity: Crocco’s Theorem

T ∇s = ∇h o − v × ω,

59

(3.9)

valid for steady inviscid adiabatic flow. Crocco’s theorem shows that whenever an enthalpy or entropy gradient is present, the flow must be rotational. This explicitly confirms the equivalence between isentropy and irrotationality. The theorem is particularly important for transonic flow and supersonic/hypersonic blunt-body flow, where a shock wave either terminates a supersonic region, as indicated in Fig. 3.6, or envelops the body, Fig. 3.7. Of importance is that the entropy rise over a shock wave goes together with a drop of the total pressure, see, e.g., [20]. If locally the subscript ‘1’ stands for the flow ahead of the shock wave, and ‘2’ behind it, we obtain the total-pressure change over the shock wave to pt2 = pt1 e−(s2 −s1 )/R ,

(3.10)

with R being the gas constant. The total-pressure loss in the context of the topic of our book concerns for instance the shock-wave decambering effect of airfoils, respectively wings, Sect. 6.1.2, and also the prerequisites for the compatibility condition at the trailing edge of a lifting wing, Sect. 4.4. Also the performance of aerodynamic control surfaces is to be considered with regard to a possible total-pressure loss occurring ahead of such a surface.

3.6 Equations of Transport of Vorticity M. Drela, [6], derives the Helmholtz transport equation for the behavior of vorticity by formally taking the curl of the momentum equation, using vector identities, combining and rearranging terms: D Dt

      ω ω ∇ρ × ∇ p 1 ∇·τ = ·∇ v+ ∇ × . + ρ ρ ρ3 ρ ρ

(3.11)

The baroclinic source term ∇ρ × ∇p can cause vorticity to appear wherever there are density and pressure gradients present. However, in isentropic flow, where the viscous term is negligible, the isentropic f ( p, ρ) = 0 relation holds, the ρ and p gradients are parallel, and the baroclinic term vanishes. The simpler incompressible form of Eq. 3.11, resulting from ρ and μ being constant, is Dω = (ω · ∇) v + ν∇ 2 ω. (3.12) Dt The term (ω · ∇)v on the right-hand side represents vortex tilting and vortex stretching. Stretching causes a rotating fluid’s vorticity to intensify—think of a pirou-

60

3 Elements of Vortex Theory

etting ice-skater stretching his arms up—when stretched by the components of the velocity gradient matrix ∇ v, which are parallel to ω itself. However, if ω = 0 to begin with, then this term is disabled, since there is no initial vorticity to stretch or tilt. The Helmholtz vorticity equation 3.11 or 3.12 simplifies greatly for most aerodynamic problems. These typically have uniform flow and hence ω = 0 upstream, and their viscous stresses are negligible outside of viscous layers and outside of shock waves.

3.7 Helmholtz’s Vorticity Theorems The vorticity theorems of H. von Helmholtz (1858) concern the motion of fluid at and in the vicinity of vortex filaments [1–3]. The theorems hold for inviscid flow and flow, where the influence of viscosity can be ignored. In real flows all vorticity phenomena are affected by the dissipative effects of viscosity. • First Helmholtz Vorticity Theorem The First Helmholtz Vorticity Theorem states: The integrated vorticity flux over a cross surface of a vorticity tube, or the circulation of the tube, is constant, independent of the shape and location of the cross surface or its boundary over which the integrals are estimated. In other words the vortex strength is constant along the vortex tube (or filament), and therefore, Helmholtz’s first theorem simply reflects the same solenoidal nature of the vorticity field as the velocity field for an incompressible fluid. Since the vortex strength is constant along the vortex filament, its strength cannot suddenly go to zero. Thus, a vortex cannot end in the fluid. It can only end on a boundary or extend to infinity. Of course in a real, viscous fluid, the vorticity is diffused through the action of viscosity and the width of the vortex filament can become large, until it is hardly recognized as a vortex filament. For example a tornado has one end on the ground, but at the other end, the vortex diffuses over a large area with distributed vorticity. A ring vortex, which forms a closed path, is another example. • Second Helmholtz Vorticity Theorem The Second Helmholtz Vorticity Theorem states: If and only if the flow is circulation-preserving, then a material vorticity tube moves with the fluid. Singularities such as vortices in the flow move along with the local flow velocity. An example here is the way vortices in the trailing vortex layer (wake) of a wing interact and curve around each other forming a non-planar wake as they are convected. • Third Helmholtz Vorticity Theorem The Third Helmholtz Vorticity Theorem states: If and only if the flow is circulation-preserving, a material vorticity tube has constant strength.

3.7 Helmholtz’s Vorticity Theorems

61

As we shall see in the next section, this is quite similar to what Kelvin derived independently eleven years later in his theorem.

3.8 Kelvin’s Circulation Theorem Previously we have seen how the circulation Γ is related kinematically to the vorticity through Stokes’ theorem. In 1869 Kelvin derived his circulation theorem on the persistence of circulation showing how it is dynamically governed: If and only if the acceleration is curl-free, the circulation along any material loop is time invariant:  D D Γ = v · d = 0, (3.13) Dt Dt C where the closed curve C consists always of the same particles of fluid moving with the flow velocity. This condition defines a special class of flows of significant interest, known as circulation-preserving flows. It is similar to Helmholtz third law which, historically, had been obtained otherwise and earlier. • Movement of Vortex Lines One especially valuable deduction from Kelvin’s theorem concerns the movement of vortex lines. This is Helmholtz’s third theorem, an exact consequence of the Euler equations, which states that vortex lines move with the fluid.

3.9 Law of Biot-Savart The definition of vorticity ω = curlv provides us with the vorticity field through the differentiation of the velocity field. What does the reverse function look like, if the vorticity field is given, and the velocity field is sought? The answer requires an integration process, and Lugt provides the answer [2], p. 90.

62

3 Elements of Vortex Theory

Fig. 3.8 Sketch for the variables in the vorticity-induction equation, after [2]

The dynamic equation of vorticity transport, Eq. 3.12, describes the convection of the vorticity with the velocity induced by the distribution of vorticity itself. A distribution of vorticity ω(r  ) induces the velocity v(r) by the relation 1 v(r) = − 4π

 V

ω(r  ) ×

s dV, s3

(3.14)

where the variables are indicated in Fig. 3.8. This is the vorticity-induction equation. The vorticity field may occupy only a part or the whole of the space over which the velocity field is nonzero. For instance, in a potential vortex, the vorticity field is confined to the singular point with infinite ω. A confined vorticity field, which exponentially decays away from its boundary, is called a compact field. If the vorticity is concentrated to a single line filament L of circulation Γ , its strength is constant by Kelvin’s theorem, and Eq. 3.14 reduces to the Biot-Savart law v(r) = −

Γ 4π

 L

dr  × s , s3

(3.15)

where the vector dr  is tangential to the vortex filament. Note that the integral itself is purely geometric, and can be evaluated without knowing the filament circulation Γ a priori. • Self-Induced Motion of Thin Vortex Filaments In general, the prediction of the free motion of vorticity is a difficult unsolved problem governed by unsteady nonlinear equations. The solutions that have been obtained involve one or more of the following simplifications: steady or quasi-steady flow, small perturbations on a straight filament, small vortex-core size, and simple vorticity distributions and/or flow configurations. It is well known that if self-induced motion of the line filament is calculated by evaluating the velocity from Eq. 3.15 on the filament itself, the result will be logarithmically infinite, if the filament is curved, and zero, if it is straight. Thus, self-induced motion occurs only for curved filaments, but to obtain the correct value for the velocity, further considerations of the finite size of the vortex core as well as the vorticity distribution are required.

3.9 Law of Biot-Savart

63

• Vortex-Core Cutoff Method The Biot-Savart law is a valid description of the velocity field induced by a thin vortex filament, if proper account is taken of the flow within and near the vortex core. One of the important interpretations of this is—under the restrictions mentioned previously—that the concept of a cutoff distance is asymptotically valid. Very often, the motion of curved vortex filaments has been calculated by integrating the BiotSavart law, but excluding a small segment of the filament of length  on either side of the point on the filament itself to avoid the logarithmic singularity. If  is chosen appropriately, the correct self-induced motion will be predicted by the cutoff method. Another important result of this work is the concept of effective core size. A vortex of core size a with a particular distribution of vorticity and axial velocity is kinematically—i.e. for motions well away from it—equivalent to a vortex of core size a  with constant vorticity. This requires that the size of the vortex core be small in comparison with the vortex-core radius of curvature and with the scale of axial variation along the filament. The next section makes use of the concept of effective core size.

3.10 Vortex Models Figure 3.9 characterizes the structure of trailing vortices in a cross-section of a wing’s wake. Relevant are two radii, which are defined on the basis of their velocity, respectively vorticity distributions, see, e.g., [1].

Fig. 3.9 Trailing vortices: definition of the viscous core radius rc and the vorticity or outer core radius rv [21]. s is the span-wise load factor

64

3 Elements of Vortex Theory

1. Viscous or inner core radius rc . The viscous core radius rc defines the inner vortex core. The circumferential velocity vθ increases linearly from 0 at r = 0 up to vθmax at r = rc , so the inner core rotates like a rigid body. 2. Vorticity or outer core radius rv . The distribution of the circumferential velocity of a vortex in reality is not that of the Rankine vortex, see below and also Sect. 4.2.1. For r > rv the flow can be considered free from viscosity and vorticity effects, therefore the radial distribution of circumferential velocity is that of the potential vortex. rv is typically several times larger that rc . The region between rc and rv contains the transition from the viscous core of the vortex to the potential vortex. It contains a large part of its circulation, and viscosity effects are small. The following discussion indicates some of the requirements that determine these radii as well as other parameters, and some of the commonly used vortex models are then presented. • Finite-Core Vortex As mentioned above, in theoretical studies and numerical computations the singularity of a point or potential vortex is often replaced by a core of finite vorticity called a vortex blob or a vortex patch. A vortex with such a non-singular center is called a finite-core vortex. The Rankine vortex with a core of finite uniform vorticity is an example. Other vorticity distributions of the core are possible, which do not necessarily satisfy the Euler equation but which may simulate viscous flow properties or other core behavior. • Wake Trailing Vortices: Model Requirements The direct way to determine the structure of trailing vortices in the wake behind the wing is to calculate accurately the roll-up process of the trailing vortex layer as it convects downstream. This however is an extremely demanding computation, and other approximate methods are sought without attempting to follow the roll-up. One such process assumes a final configuration for the flow and requires conservation of some chosen flow quantities between the initial and final configurations, both modeled as two-dimensional flows. For example, if the trailing vortex layer from each half of the wing rolls up into a single vortex, Stokes’ theorem gives the magnitude of the circulation. Then conservation of momentum requires that the impulse of the vortex layer equals the impulse of the two trailing vortices, thus locating each vortex at the center of gravity of the shed vorticity in its half of the vortex layer. Consider a finite core vortex with radius rc . As rc → 0, while Γ0 = const., the kinetic energy increases without bound, therefore implying a wing with infinite induced drag. This is not an adequate flow model; the vorticity in trailing vortices is finite and distributed in some manner. The following are some examples fulfilling conditions for adequate models. For further details the reader should consult S.E. Widnall [11] and the publications [21, 22] of the third author of the book. • Rankine Vortex The Rankine vortex is an infinitely long straight vortex filament whose interior rotates like a rigid body, thus with constant vorticity. Most early discussions of the far-field structure of trailing vortices were based on the vortexwake model of J.R. Spreiter and A.H. Sacks [23], in which after roll-up the wake

3.10 Vortex Models

65

consists of two Rankine vortices: the vorticity being uniform in two circular vortex cores and zero outside. The circumferential velocity vθ is Γ0 r , 2πrc rc Γ0 r > rc : vθ = . 2πr r  rc : vθ =

(3.16)

This vortex has a clear definition of core size rc : all of the vorticity is in the core, the peak tangential velocity is at the edge of the core, and the total circulation is proportional to the product of peak velocity and core size. The core radius rc is determined by requiring kinetic energy to be conserved in the roll-up process. For a wing with elliptic circulation distribution the core radius rc is easily determined. • Lamb–Oseen Vortex The Lamb–Oseen vortex analytically satisfies the unsteady Navier–Stokes equations. It has been used extensively for initializing large-eddy simulations and in the design of LIDAR (light detection and ranging) matched filters. The only non-zero velocity component, the tangential velocity vθ as a function of the radial distance r from the vortex center, is

vθ (r ) =

r 2 Γ0 [1 − e−β( rc (t) ) ], 2πr

(3.17)

 2 + 4βν(t − t0 ) is the vortex core radius, which is defined as the where rc (t) = rc,0 radius, where the tangential velocity is maximum. Γ0 is the vortex circulation, and r is the distance from the vortex center, ν the kinematic viscosity, t the time, β = 1.256 usually. Note that the Lamb–Oseen vortex has a core that grows with time, so it is never steady. It may be employed however by making use of the two-dimensional unsteady analogy to steady three-dimensional wake flow. • Burgers’ Vortex Burgers’ vortex is an exact incompressible Navier–Stokes flow. It demonstrates the existence of extrema of vorticity in a steady three-dimensional flow: κ0 ar 2 [1 − e− 2ν ], r vr (r ) = −a r, vz (z) = 2 a z,

vθ (r ) =

(3.18)

Γ , hence the parameter a is determined by the circulation and the where κ0 = νa = 2π kinematic viscosity ν. The tangential velocity, vθ , has a finite value in the vortex core, which reduces to the potential vortex as ν → 0. The extremum in three-dimensional flow is due to stretching of the vorticity lines.

66

3 Elements of Vortex Theory

The constant a can have only positive values and is a measure of the radial influx. The meridional flow, vr , vz is a potential flow that is meaningful only for ν > 0. In other words, it is an example of a laminar potential flow. If however ν → 0, the tangential velocity represents a potential flow for an inviscid fluid. The meridional flow, vr , vz , may be considered the lower part of a circulatory motion, called a cell. Notice that the vorticity in Burgers’ vortex is in the z-direction. As in two-dimensional flows, all vorticity lines are parallel to each other. • Batchelor Vortex The Batchelor vortex was obtained by G. Batchelor (1964) as an approximate solution to the Navier–Stokes equations under a boundary-layer type approximation. It has been used extensively as a typical mathematical model of vortices, for example in vortex stability studies and also as models for trailing or jet-like vortices. The Batchelor vortex can be represented in the cylindrical coordinates (z, r, θ) as q 2 [1 − e−r ], r vr (r ) = 0, 2 vz (z) = a + e−r .

vθ (r ) =

(3.19)

Non-dimensionalized velocities and lengths, and time scaled by the ratio of length to velocity scales give q 4t 2 [1 − e−r /(1+ Re ) ], r vr (r ) = 0, 1 4t 2 vz (z, t) = a + e−r /(1+ Re ) , 4t 1 + Re vθ (r, t) =

(3.20)

where Re is the Reynolds number and q is the swirl strength as the ratio of the maximum tangential velocity to the core velocity. The parameter a here designates the free stream velocity. It has been noted that the translation and inversion of the axial velocity vz do not affect the instability of the Batchelor vortex, so that one can set a = 0. Batchelor found this vortex suitable to describe a trailing vortex far downstream of an aircraft. We note again that this is an unsteady vortex with growing core. • Burnham-Hallock Vortex The Burnham-Hallock vortex is the most widely used model for wake vortex applications, which include processing of LIDAR observations, initialization of large-eddy simulations and modeling of aircraft response to wake encounters. The tangential velocity field is given by vθ (r ) =

r2 Γ0 , 2πr r 2 + rc 2

(3.21)

3.10 Vortex Models

67

where rc is the vortex core radius. Also numerical vortex-lattice models often use this formulation for the calculation of the influence coefficients, e.g., to avoid infinities, where trailing vortices could intersect bound vortices.

3.11 Structure of Trailing Vortices Flight through trailing wing vortex wakes is by no means a new experience having occurred since WW I, when pilots encountered vortex wakes trailing from maneuvering aircraft, or sometimes even flew into their own wakes.6 Figure 3.10 illustrates the flight environment of an aircraft with large aspect-ratio wing, see also Figs. 8.1 and 9.25. The control of the so-called wake-vortex hazard is the topic of Sect. 9.6, where the latter figure is discussed in some detail. For several reasons, the subject is of great importance today. With increasing air traffic around airports and in flight corridors, there are safety aspects, ranging from dynamic flight behavior to concern about structural loads or time separation for takeoffs and landings. Modern fighters with relaxed longitudinal static stability are also extensively relying on active flight control systems. This trend is seen also with modern civil aircraft. All this together has given new dimensions to the problem. Flight control systems process information from sensors like accelerometers, angle-of-attack sensors and pressure gauges, and the system responds on this infor-

Fig. 3.10 Wake-vortex hazard encountered by aircraft flying behind another aircraft [21]

6 Regarding

the development and the structure of the wake of a large aspect-ratio wing see the discussion of the corresponding Unit Problem in Sect. 8.4.

68

3 Elements of Vortex Theory

mation. The pilot also is acting, interacting and reacting in this scenario. How this all will take place in wake encounters, and what response there will be, is not easy to foresee, so simulation models are required. The SAAB Gripen fighter is a good example of this.

3.11.1 SAAB 39 Gripen Wake Model In 1999 a SAAB 39 Gripen aircraft was lost due to vortex-wake encounter while practicing air combat below 3,000 ft. Thus the SAAB company launched a project to develop a model for dynamic analysis and real-time simulation of flight through wake vortices. The aerodynamic model consists of both the wake flow model behind the generating aircraft as well as the computational model for estimating incremental forces and moments induced by the wake on the incoming aircraft. Desktop as well as real-time models for flight simulation of vortex-wake encounters were developed. Simulations were demonstrated with a six degrees of freedom aircraft model. The model has been used for development of flight control systems, control laws, flight safety investigations as well as for demonstrations of wake encounters. The reader is referred to Sedin et al. for details [24].

3.11.2 Trailing Vortex Instability Recent concern over the hazard presented by the trailing vortices produced by large aircraft has stimulated the study of flows with concentrated vorticity in free motion. Since these vortices can be strong and persistent enough to pose a safety hazard to other aircraft, it is clearly desirable to be able to predict the structure, position, and persistence of such vortices as well as to understand the mechanisms by which vortex wakes are dissipated, see also Sect. 8.4. Under most conditions, trailing vortices undergo a natural sinusoidal instability, the Crow instability, that eventually causes them to touch and break into a series of crude vortex rings. This process destroys the initial wake structure more rapidly than viscous or turbulent decay of the individual filaments. An example of practical interest is the case of two counter-rotating vortices. Means to control the wake-vortex hazard are discussed in Sect. 9.6.

3.11.3 Crow Instability The Crow model considers the decay of a vortex pair in an inviscid linearly stratified atmosphere. The model treats the generation of vorticity both in the wake due to

3.11 Structure of Trailing Vortices

69

nonuniform density and at the interface between the oval of fluid carried by the pair and the surrounding fluid. The first quantitative analysis of the three-dimensional instability of a vortex pair in an ideal homogeneous fluid was given by S.C. Crow [25]. The results of his analysis are in good agreement with observations of general features of aircraft wake instability and measurements of amplification rates. This instability is due to the mutual inductance of the sinusoidally perturbed pair. A single sinusoidally perturbed vortex filament will rotate with angular velocity Ω about its own axis. Instability occurs when the velocity field induced at the filament by the presence and perturbation of the other filament annuls the self-induced rotation. The perturbations then diverge on two planes tipped symmetrically at an angle of about 45◦ with the horizontal. Both anti-symmetric and symmetric modes are predicted by theory, but only the symmetric modes have been observed in flight. General features of the modes are discussed in [11], see also [22]. The self-induced rotation Ω depends on the details of the vorticity distribution in the core.

3.12 Vortex Layers and Vortices We look first at the roll-up process of vortex layers. These are the trailing vortex layers of lifting large aspect-ratio wings, which roll up into the pair of trailing vortices, but also the so-called feeding layers, which lead to the lee-side vortices of small aspect-ratio delta-like wings. We only give an overview over some items, the reader interested in the theoretical details of roll-up of vortex layers should consult, e.g., [1–3]. Once the trailing or the lee-side vortices are established, vortex stretching and vortex pairing are phenomena, which can occur. These are shortly considered, too.

3.12.1 Roll-Up of Shed Vortex Layers Given the difficulty to state a precise definition of a vortex due to its elusive nature, we focus here—where we are most secure in our understanding—on the development of the trailing vortex layer, which is present behind a finite-span lifting wing. We discuss the behavior of aircraft wakes following the generation of this initial vortex layer: the roll-up of the sheet to form regions of concentrated vorticity, the trailing vortices, and their motion. Figure 3.11 presents the overall problem; the left side shows the spiraling of the sheet in a plan-wise cut through the sheet, the right side shows a numerical model for the spiral discussed below. Under most conditions the shed vortex sheet will roll up, and the wake structure will be established within a few wing spans. A step-by-step calculation of the roll-up process then is the way to predict the initial wake structure.

70

3 Elements of Vortex Theory

Fig. 3.11 Overall vortex-sheet roll-up. Left: roll-up of a wing’s trailing vortex layer, after [2]. Right: a wake or free-vortex sheet in a computational method (Model 4 of Table 1.3) [26]

For lightly loaded wings at high Reynolds numbers, roll-up takes place gradually enough to justify a local two-dimensional model of the wake flow, and yet rapidly enough to use an inviscid model of the flow. Unfortunately, even this simplification results in a problem that has no known solutions. Moreover in a remarkable paper from 1951, appreciating the high kinematicallyactive vorticity content in leading-edge vortex sheets, J.R. Spreiter and A.H. Sacks, [23], estimated that these sheets roll up 18 times more rapidly, in terms of chord lengths, behind a low-aspect-ratio triangular wing than behind a large aspect-ratio rectangular wing, see also Sect. 8.4.4. The problem for the low-aspect-ratio wing therefore is even more difficult. The complete three-dimensional determination of the shape of the trailing vortex layer throughout the rolling-up process presents a problem of extreme difficulty, and amenable only to computational solutions using Model 8 and higher models. • Initial Computational Models for Sheet Roll-Up During the early 1980s two computational approaches to the roll-up problem were feasible: either Model 4 or Model 8 methods. Use of panel representations of high-order make the vortex-sheet model amenable to calculation with Model 4 methods, since it reduces or eliminates singular behavior of the velocity field at the edges of panels, so that unrealistic disturbances do not arise from the close approach of two turns of a spiral or of a sheet to a wall. The representation usually adopted comprises a line vortex of varying strength, and a nearly planar feeding vortex element that maintains continuity of circulation between the free edge of the finite, outer part of the sheet and the line vortex along the axis. In this formulation the leading-edge vortex consists of a tightly wound spiraling vortex sheet of infinite extent. In order to simulate the flow field outside of this vortex, the vortex sheet is cut as it has reached some angular extent, while the remaining inner region is modeled by an isolated line vortex connected to the sheet by a socalled feeding sheet. This leads to the free-vortex sheet potential flow model depicted

3.12 Vortex Layers and Vortices

71

Fig. 3.12 Vortex sheet tracked with two methods for the flow past a flat 70◦ swept delta wing, M∞ = 0, α = 20◦ . Upper part: The Model 4 (VORSEP) result [26]. Lower part: The captured sheet in Model 8 [27]. Vorticity magnitude contours were computed with a medium mesh with 80 × 24 × 40 cells

on the right in Fig. 3.11 showing how the wing (or wake) and the free vortex sheet are discretized by panel elements. Both the line vortex and the feeding element experience forces, but the position of the line vortex is chosen so that the total transverse force on each lengthwise element of the combination is zero. With the core representation described above, a complete discretization of the vortex sheet is possible—in the framework of the Prandtl–Glauert approximation—in many different ways. The potential flow problem is solved by employing a surface doublet distribution μ(x) on both Sw —representing the wing and the trailing vortex layer—and Sv — representing the feeding vortex layers shed from the swept leading edges. The velocity field defined by the surface doublet distribution is an exact solution of the incompressible flow equations and satisfies the far-field boundary condition. Further details of this Model 4 computational method, VORSEP, are found in [27]. The second method computes the numerical solution to the incompressible Euler equations, Method 8, and captures the roll-up process in the discrete solution. H.W.M. Hoeijmakers and the second author of this book carried out an important comparison of results computed with these two methods for the flow past a flat 70◦ swept delta wing, M∞ = 0, α = 20◦ [26, 27]. The comparison in Fig. 3.12 shows a three-dimensional view of the computed tracked vortex sheet in VORSEP as well as the vorticity magnitude contours in three planes of the medium resolution grid for the captured vortex in the Euler solution. The computed geometry of the tracked vortex sheet, Fig. 3.12, is very nearly conical, up to 70 per cent of the root chord.

72

3 Elements of Vortex Theory

Figure 3.13 superimposes the shape of the tracked vortex sheet onto the vorticity contour in the spanwise cross-flow plane x/c = 0.6 in both the coarse and fine-grid Euler solutions [28]. The vortex-sheet shapes are cross-sections with planes x/c = constant, whereas the vorticity contours are projections on these planes of the surface of the constant mesh coordinate, which intersects the wing at the corresponding stream-wise station. Due to the artificial viscosity implied in the numerical method, the vortex sheet is spread over a number of cells in the medium grid solution, and as expected, substantially less in the fine-grid solution. This means that in general the vortical flow region as computed by the Euler code on the medium grid occupies a larger region than enclosed by the vortex sheet tracked by VORSEP. But as the truncation error as well as the numerical viscosity decreases due to decreasing grid size, the agreement is much better for the finer grid. It follows from Fig. 3.13 that in this respect the two solutions agree quite well. The strength and position of the vortex features of both computation models are in good agreement. Figure 3.13 is central to our discussion, because it correlates two world views, namely the view from Model 4, where the natural element is that hard-to-define concept of a vortex, with the view from Model 8, where the natural element is the mathematically-precise concept of vorticity over a compact field, and shows that, in this case, they are equivalent. The linking of these two concepts is very reassuring for all our discussions in this book, see in this regard also Chap. 5. We note concluding that present-day discrete numerical computation methods, Model 8 and higher, need to have a sufficiently fine discretization—grid independence—of the domain where the roll-up happens, see, e.g., Sect. 8.4.2 and Fig. 8.50, where beginning at x ∗ = 9 the original grid resolution is shown to be insufficient.

Fig. 3.13 Spanwise section x/c = 0.6: comparison of captured vorticity contours, solid line, [28], and tracked vortex sheet in the panel method VORSEP solution, dashed line. Left part: medium mesh with 80 × 24 × 40 cells. Right part: fine mesh with 160 × 48 × 80 cells

3.12 Vortex Layers and Vortices

73

3.12.2 Vortex Stretching Vortex Stretching was mentioned above in Sect. 3.6. The term (ω · ∇)v on the righthand side of Eq. 3.12 can be interpreted physically as the stretching and twisting of vortex lines. The appearance of this nonlinear term in the vorticity-transport equation is unique in the sense that it does not occur in the other forms of the Navier–Stokes equations. Consider an example. If in three-dimensional flow only the ωz component is nonzero, that is, ω = ωz k, the vorticity-transport equation Eq. 3.12 simplifies to ∂ω Dωz = ωz . Dt ∂z

(3.22)

The vortex line is stretched when ω changes with z. Stretching a vortex line makes the fluid spin faster.

3.12.3 Vortex Pairing A single vortex filament does not move in an infinite fluid, since the velocity at the singularity is cutoff to zero. Two or more vortices, however, move due to their mutual interaction. The problem of describing such interaction will be discussed briefly here. In an infinite, inviscid fluid, individual vortex pairs and rings move with constant speed. In reality, friction causes the velocity to decay. We call two vortices of opposite rotation in a two-dimensional flow a vortex pair. The strengths of the two vortices need not be the same, and the core can be either singular, as in the case of the point or potential vortex, or a blob. Examples of practical interest are, for the case of two counter-rotating vortices, the trailing vortices behind the wings of large aspect-ratio airplanes. Another example is the combination of leading-edge and trailing vortices on delta wings, which we look at more closely below. • Vortex Pairing Behind a Delta Wing In his doctoral thesis, Y. Le Moigne, a doctoral student of the second author of this book, studied vortex interactions above and behind a delta wing at an angle of attack of α = 20◦ in low subsonic flow at M∞ = 0.2 [29]. He solved the Euler equations—Model 8—using an automatic mesh refinement procedure with a vortex sensor based on the eigenvalue analysis of the tensor of the velocity gradients to improve the mesh resolution in the vortical regions. Figure 3.14 shows the usual strong leading-edge vortex pair and a second vortex pair, the trailing vortices, formed by the roll-up of the trailing vortex layer shed from the trailing edge. The vortices are visualized with streamlines. The trailing vortices are seen to interact with the leading-edge vortices and to roll around them, a particular case of vortex pairing.

74

3 Elements of Vortex Theory

Fig. 3.14 Streamline visualization of the full-span solution using the vortex-adapted refined grid [29]

The interesting point to notice is that the streamlines in the trailing vortices spiral in the opposite direction to the one in the leading-edge vortices, corroborating D. Hummel’s observation, [30], which is sketched in Fig. 3.15. The flow field at the upper surface of the wing is restructured by the lee-side vortices in such a way that finally at the trailing edge the general flow direction is symmetrically outward, as sketched for the closed lee-side flow field in Fig. 7.12. The effect of course is present also with an open lee-side flow field, Fig. 7.11, but there it is restricted to the immediate vicinity of the outer edges, see also the figures in Sect. 10.4.2. At the lower side of the wing the flow field more or less is not affected by the lee-side phenomena. All this is in contrast to the flow-field patterns always observed on lifting large aspect-ratio wings. In Sect. 4.3.2 it is shown in a paradigmatic way that the flow at the lower side is symmetrically outward and inward at the upper side. The resulting

Fig. 3.15 Schematic of the interaction of the lee-side vortex pair with the trailing vortex pair originating from the trailing vortex layer behind the wing’s trailing edge, after [28]

3.12 Vortex Layers and Vortices

75

trailing vortex layer, hence the trailing vortices have the same sense of rotation as the tip vortices. In Fig. 3.15 the outward sense of rotation in the trailing vortex sheet is indicated as being opposite to the lee-side vortices. The whole picture then results in doublebranched inner vortex cores (marked as ‘trailing edge vortex’) and the usual singlebranched outer vortex cores (marked as ‘leading edge vortex’). Coming back to Fig. 3.14 one can see in addition that just behind the trailing edge the trailing vortex is linked to the leading-edge vortex, forming somewhat like a vortex pair. Further downstream it seems to be further separated from the leadingedge vortex. These results are an example of the interactions between two vortex pairs.

3.13 Vortex Breakdown, Vortex Re-configuration A spectacular phenomenon of vortex flows, termed vortex breakdown or burst, was first visualized by H. Werlé in a water tunnel for the leading-edge vortex over a delta wing [31], and was later confirmed by D.H. Peckham and S.A. Atkinson, [32], and also B.J. Elle [33]. The vortex breakdown appears, when at some point along its axis, a well-ordered vortex suddenly becomes chaotic, as Fig. 3.16 indicates schematically. Initially high-momentum jet-type vortex-core flow is present. An adverse axial pressure gradient over the wing causes the breakdown, which can be characterized as the development of a stagnation point on the vortex axis. It follows a region of reversed axial flow encapsulated by a greatly swollen stream surface. A lowmomentum wake-type vortex-core flow ensues. Experiments reveal essentially two basic types of vortex breakdown: the axisymmetric bubble-type and the non-axisymmetric helical (spiral) type. Most experiments were carried out with delta wings but also in pipes.

Fig. 3.16 Schematic illustration of vortex breakdown over a delta wing [34]

76

3 Elements of Vortex Theory

Characteristic features of vortex breakdown over delta wings are: • The bubble type is characterized by a stagnation point on the vortex axis followed by an oval-shaped recirculation zone. The bubble is single-celled with the recirculation flow moving upstream along the vortex axis. The bubble is usually two or three (upstream) core diameters in length. Downstream of the bubble the vortex is turbulent and diffuses rapidly with distance. • In the spiral type of breakdown a rapid deceleration of the core flow takes place. Immediately downstream, the vortex core filament abruptly kinks and starts to spiral around the axis of the structure, forming a corkscrew-like distortion of the vortex core. The spiral structure can persist for one or two turns before breaking up into large-scale turbulence. For leading-edge vortices the sense of the spiral winding is opposite to the direction of rotation of the upstream vortex, however, the rotation of the winding is in the same direction as the rotation of the upstream vortex. • The flow upstream of breakdown is steady, whereas numerous researchers have observed unsteady flow downstream of the breakdown point. In Sect. 10.5.3 we discuss vortex breakdown in the frame of Unit Problems. Finally we note that vortex re-configuration happens, if the vortex structure after breakdown enters an area with favorable pressure gradient. This was observed in pipe flow. Whether it can occur with delta wings, is not known.

3.13.1 Fundamental Studies E. Krause has carried out a number of fundamental studies for incompressible flow and established some basic conclusions about vortex breakdown [35–37]. In 1990 a time-dependent numerical solution of the incompressible Navier–Stokes equations described the process of vortex breakdown of isolated slender vortices [35]. It showed the transition from bubble-type to spiral-type breakdown, when sideboundary conditions compatible with stable breakdown were implemented. In another incompressible study of a slender columnar vortex with its axis parallel to the oncoming axial flow, Krause shows that in inviscid flow the azimuthal velocity component near the axis vanishes with the axial velocity component [37]. This close coupling between the axial and the azimuthal flow motion, referred to as the onset of vortex breakdown for vanishing axial flow, is described by the balance of the convective acceleration and vortex stretching terms in the vorticity transport equation, formulated by von Helmholtz in 1858, and represents a fundamental result for inviscid incompressible flow. It can be formally stated as: In axially symmetric, incompressible, inviscid flow of a columnar vortex with its axis parallel to the main flow, the angular velocity near the axis vanishes, when the axial velocity component vanishes and a stagnation point is formed. However, in laminar flow, with viscous forces acting near the stagnation point, the angular velocity does not necessarily vanish with the axial velocity going to zero.

3.13 Vortex Breakdown, Vortex Re-configuration

77

Nevertheless this result links the occurrence of breakdown with the presence of a stagnation point, and this can be checked in computations of more complex setting.

3.13.2 Computed Vortex Breakdown Over a Delta Wing In his doctoral dissertation S. Görtz, a doctoral student of the second author of this book, carried out a comprehensive and exhaustive study of vortex breakdown over delta wings using Models 9 and 10 [38]. The detailed discussion of lee-side vortex flow past delta wings, including the vortex-breakdown phenomenon, is given in our Chap. 10. Presented here is one case that supports Krause’s claim of a stagnation point occurring even in the practical situation of compressible high Reynolds-number turbulent flow over a delta wing. Figure 3.17 shows a three-dimensional view of breakdown for a turbulent computation of Model 10 type at 35◦ angle of attack using the one-equation turbulence model of Spalart and Allmaras. The left-hand side of the picture presents an iso-surface of total pressure colored in magnitude of axial velocity. The feeding shear layer can be seen to emanate from the sharp, highly swept leading edge. The primary vortex itself emanates from the apex of the wing. It curves slightly inboard when intercepting the flat upper surface but curves back

Fig. 3.17 Vortex breakdown: computed total pressure iso-surface and vortex core filament [38]. M∞ = 0.16, Re = 1.97 × 106 , α = 35◦ , 1-eq. turbulence model

78

3 Elements of Vortex Theory

further downstream. The fluid entrained in the core accelerates gradually along the core until the magnitude of the axial velocity reaches a value of about M = 0.4, i.e. 2.5 times the freestream velocity. Notice the following expansion of the core and the preceding deceleration of the axial fluid. Although it is difficult to identify complete stagnation here, the color contours definitely show that the axial flow component is strongly reduced, becoming even negative (i.e. flow upstream) inside the spiral core, in line with the Krause result [37]. On the right hand side of the wing a recirculation zone is seen downstream of the breakdown point. The spiraling nature of the core behind the recirculation zone suggests that a bubble type of breakdown with a “spiraling tail” has been predicted.

3.13.3 Vortex Re-connection Reconnection encompasses merging and splitting of vortex filaments, but since a precise and rigorous definition of a vortex eludes us, a commonly agreed upon and clear definition of vortex reconnection is lacking, as well as a consensus on relevant physical mechanisms. Nevertheless, reconnection has been observed in laboratory experiments and in the sky in condensation trails of airplanes after the Crow instability occurred. The two trailing vortices touch, split, and rearrange themselves into a row of irregularly shaped vortex rings, see, e.g., Fig. 8.1. The study of vortex reconnection is of particular importance in understanding the evolution of vortical structures in turbulence.

3.14 Separation and Vortex Flow Control In modern aircraft technology control of boundary-layer separation and of vortex flow is an important engineering tool. An aircraft today is unthinkable without such control. The benefits of separation and vortex control include drag reduction, lift increase, stabilization and performance. Control can be a direct design means or a “repair” solution, after test flights have shown a need for correction. We list and comment on a few significant control means without going into details. • Vortex generators: control of boundary-layer separation Vortex generators are passive devices, which are embedded in a boundary layer in order to prevent separation. They can differ in size and shape and can be denoted by different names, however their mechanism is usually the same—to generate longitudinal vortices in order to reduce or even avoid separation. The effect of the longitudinal vortices is the transport of high momentum flow from the outer domain of the boundary layer into wall-near flow portions. In this way

3.14 Separation and Vortex Flow Control

79

the momentum of the wall-near boundary-layer flow is enhanced and its proneness to separation is reduced. Recently sub-boundary-layer vortex generators (SBLVGs) found much attention. They are fully submerged in the boundary layer. In Sect. 9.2 we give a short account of them. For a more recent and thorough account of the modeling and use of vortex generators in the design and performance of aircraft, the interested reader is referred for instance to A. Jiràsek [39], a doctoral student of the second author of this book. General discussions of means to influence boundary-layer flow in particular in three dimensions, can be found in [40]. • Geometrical shaping of configuration elements A passive means to avoid the separation at the wing-root/fuselage intersection is the wing-root fairing. Today it can be found on almost all airplanes. Without this fairing separation in the form of a horse-shoe vortex occurs, which causes a drag increment and leads to buffet at high angles of attack. We give a short discussion of this issue in Sect. 8.4.3. Another topic is the wing planform shaping and optimization of small aspect-ratio delta-like wings. We sketch this topic in Sect. 11.2. A particular problem of such slender wings is the pitch-up behavior at higher angles of attack. We present a design example in Sect. 11.6. • Suction and Blowing Suction through a surface can be used to control, for instance, laminar-turbulent transition—hybrid laminar flow control (HLFC)—or to suppress flow separation over an aerodynamic control surface. In that case just ahead of the hinge line boundary-layer flow is removed by suction and attached flow is ensured over the flap. Another application is at supersonic and hypersonic engine ramp inlets, where at the ramp corners shock/boundary-layer separation is controlled by suction. In [40] results of an old study are shown regarding suction as a virtual boundary-layer fence. Blowing is used at a number of separation problems. We do not go into details. We note, however, a particular means to achieve separation-free flow over highlift systems, the “tangential blowing” at the slat and the Fowler flap. The effect is that the flow through a gap, driven by the pressure difference between the lower (pressure) and the upper (suction) side of the wing, prevents boundary-layer separation at the upper side of the wing behind the slat and at the Fowler flap, Sect. 9.2. Regarding small aspect-ratio delta-like wings spanwise blowing was an important topic in the 1970s. In Sect. 11.5 we give an account of some work in this regard. Noteworthy also is the topic of jets in cross flow, which regards vehicle flight control issues. • Strakes Strakes are devices to generate vortices. These are used, for instance, to generate side forces on forebodies and to stabilize delta-type aircraft at high and even post-stall angles of attack in the lateral/directional motion, see Sect. 11.4. A now prominent example is the nacelle strake, which behind the nacelles of high bypass-ratio engines prevents separation when the wing is in high-lift condition. In Sect. 9.4 the topic of the nacelle-strake vortex is discussed in some detail. Figure 3.18 shows a combination of nacelle-strakes and vortex generators, which

80

3 Elements of Vortex Theory

Fig. 3.18 Boeing 707 during high-lift test: flow control with nacelle strakes and vortex generators [41]. Note the visualized nacelle-strake vortices

re-energizes the flow over the upper surface of the wing behind the nacelles, the two together preventing flow separation there. Nacelle strakes were possibly for the first time used on the DC-10 as a solution to avoid a significant loss in maximum lift coefficient, when flaps and landing slat extensions are deployed [39]. They can lower stall speed in approach configuration and reduce the required take-off and landing runway length, a reduction which for example in the case of the DC-10 aircraft was about 6 per cent.

3.15 Vortex Flows and Dynamic Structural Loads Turbulent boundary layers, separation regions, vortex sheets and vortices have rugged edges and large sub-structures, which are unsteady and can cause buffeting, structural fatigue, aileron buzz (vibration of control surfaces), etc. All of these phenomena warrant a brief general problem discussion of dynamic structural loads. At delta wings at high angle of attack, the phenomenon of leading-edge vortex breakdown over the wing planform is of specific interest. The transition from stable to unstable core flow, evident by the rapid change in the axial velocity profiles from jet- to wake-type with increasing angle of attack, leads to extremely high turbulence intensities at the breakdown position and to increased turbulence levels further downstream. Hence, the buffet excitation level increases strongly above a certain angle of attack, and wing and fin normal-force spectra may exhibit narrow-frequency band peaked distributions.

3.15 Vortex Flows and Dynamic Structural Loads

81

Figure 3.19 schematically indicates the problem of vertical tail buffet for the F/A18. The unsteady aerodynamic loads excite the vertical tail structure or even the wing structure in their natural frequencies, resulting in increased fatigue loads, reduced service life and raised maintenance costs. For example, the fin buffeting problem plagues twin-fin configurations like the F-15 and F/A-18, but single-fin aircraft are also affected. Therefore, comprehensive research programs have been undertaken aimed at understanding the buffeting loads and reducing the structural response. The related vortex-flow features are carefully analyzed using wind tunnel tests on small- and full-scale models supplemented by flight tests and detailed numerical flow simulations. In addition, design and analysis methods have been developed to describe the fin-buffet environment and to predict buffet loads for use in aircraft design. The buffet loads do not only decrease the fatigue life of the airframe, but may, in turn, limit the angle of attack envelope of the aircraft. To counter buffeting problems, several methods have been suggested. They deal with alterations of the structural properties like stiffness and damping, and aerodynamic modifications for passive or active control of vortex trajectories to avoid a direct impact of the burst vortex flow. The structural dynamic loads can be reduced to increase the service life and to enhance the maneuverability by allowing to extend the angle of attack envelope.

Fig. 3.19 Schematic illustration of the vertical tail buffet problem on the F/A-18 aircraft [34]. LEX: leading-edge extension

82

3 Elements of Vortex Theory

The interaction with a fin involves the following phenomena: • the time-averaged vortex breakdown location depending on the adverse pressure gradient set by the recompression at the wing trailing edge and/or by the blockage of the fin, • the helical mode instability of the breakdown flow, • quasi-periodic oscillations of the breakdown location, distortion of structural dynamic properties. For a thorough account of the aero-structural modeling of the F/A-18 the reader should turn to [42, 43]. In this regard also the work reported in [44] should be mentioned, and in a wider context the publications [45–47]. In closing this section we note that the buffeting problem also exists for any other aircraft and not only in relation to the vertical fin. When the undercarriage is extended, wheel-well doors and the wheel well (as large cavity) are exposed to the air flow. On fighter aircraft, like bombers, the bays for internal weapon carriage pose the same problem during weapon release. In all cases buffeting means dynamic structural loads, which are of high concern.

3.16 Basic Quantities of Trailing-Vortex Flow Fields In order to get basic quantities and relations of trailing-vortex flow fields potential flow theory (Model 4 of Table 1.3) is useful, even if in principle it is restricted to the low subsonic flight domain. For general applied aerodynamic purposes see, e. g., [4–7]. The relation between the spanwise lift L(y) and circulation Γ (y) distributions is given by the Kutta–Joukowsky theorem: d L = ρ∞ u ∞ Γ (y)dy. The lift coefficient is CL =

L , q∞ A

(3.23)

(3.24)

where A is the reference area, usually the wing’s surface projected into the x-y plane, and ρ∞ 2 u . q∞ = (3.25) 2 ∞ the dynamic pressure.7 The wing loading Ws is defined as

7 Regarding

the reference area we note that Airbus uses the ‘Airbus Gross’ definition and Boeing the ‘Wimpress’ definition (after its inventor at Boeing), www.lissys.demon.co.uk/pug/c03.html.

3.16 Basic Quantities of Trailing-Vortex Flow Fields

83

Fig. 3.20 Local chord element of a finite-span wing [4]

Ws =

L Mg = [N /m 2 ]. A A

(3.26)

The dimensionless circulation is, with b being the wing span σ(y) =

Γ (y) . u ∞ b/2

(3.27)

Consider now the local chord element in Fig. 3.20. The local normal force coefficient C z is found by the integration of the Δ-pressure coefficient along the chord section8 : C z (y) =

1 c(y)

 Δc p dy,

(3.28)

c(y)

with Δc P = c pu − c pl being the pressure-coefficient difference between the upper and the lower chord side and c(y) the local chord length. For small angles of attack we have Cl ≈ C z , and find the lift distribution in span direction to be dL = Cl c(y) q∞ . (3.29) dy Generally the local lift coefficient Cl can be written with Eq. (3.23) as Cl (y) =

2Γ (y) dL 1 = . c(y) q∞ dy u ∞ c(y)

(3.30)

If at a spanwise location y the local lift coefficient Cl is given, the local value of the circulation reads 1 (3.31) Γ (y) = Cl (y)c(y)u ∞ . 2 The dimensionless circulation σ(y) then is

8 Local

force and moment coefficients are denoted with lower-case letter indices.

84

3 Elements of Vortex Theory

Fig. 3.21 Schematic of finite-span lifting wing [4]. Upper part: Circulation distribution. Lower part: trailing vortex layer behind the wing

σ(y) =

Cl (y)c(y) . b

(3.32)

For a lifting wing the circulation distribution in spanwise direction, Γ (y), is sketched in the upper part of Fig. 3.21. The lower part shows in planform view the vortex layer with the local strength being dΓ = (dΓ /dy)dy  , see also the compatibility condition in Sect. 4.4. The lift L of the wing then is expressed as  b/2  b/2 Cl c(y) dy = ρ∞ u ∞ Γ (y) dy. (3.33) L = q∞ −b/2

−b/2

Consider Fig. 3.22. It schematically shows the roll-up process of the trailing vortex layer which leaves the lifting wing. At the end of the extended near field, x/b ≈ 10, Sect. 8.1, the fully established trailing vortices are present. The bound or root circulation at the wing in Fig. 3.22, Γ0 , can be written in terms of the lift coefficient C L of the wing, the spanwise load factor s, and the aspect ratio Λ = b2 /A, as L C L u∞b Γ0 = = , (3.34) 2sΛ ρ∞ u ∞ b0 with the load factor s: 1 s= b



b/2

−b/2

Γ (y) b0 dy = . Γ0 b

(3.35)

3.16 Basic Quantities of Trailing-Vortex Flow Fields

85

Fig. 3.22 Idealized circulation distribution and roll-up process of the trailing vortex layer, resulting in the trailing-vortex pair [21]

The quantity b0 denotes the lateral distance of the barycenters of the two trailing vortices. If the circulation distribution is elliptic  Γ (y) = Γ0 1 − (2y/b)2 ,

(3.36)

the spanwise load factor s is (the superscript ∗ referring to the elliptical distribution) s = s∗ =

π b0 = . b 4

(3.37)

The result is that the lateral distance b0 of the rolled-up vortices is smaller than the wing span b. Only for wings with a very large aspect ratio we get b0 /b → 1 [4]. With an elliptic circulation distribution the root circulation becomes Γ0∗ =

2C L u ∞ b . πΛ

(3.38)

Γ . Γ0

(3.39)

The normalized circulations G is G=

86

3 Elements of Vortex Theory

These relations are helpful to compare wake-vortex properties due to a given circulation distribution with those of an elliptical distribution. The induced downward velocity w0 on the vortex centers (law of Biot-Savart, Sect. 3.9) is w0 =

Γ0 CL u∞ = . 2πb0 4πΛs 2

(3.40)

For the elliptic circulation distribution we obtain w0∗ =

4C L u ∞ . π3 Λ

(3.41)

With the induced velocity w0 and the lateral distance b0 a timescale t0 can be defined: t0 =

Λb b0 2π(sb)2 = = 4πs 3 . w0 Γ0 CL u∞

(3.42)

The time scale t0 defines the time interval at which the vortex pair moves downward by the distance equal to the vortex spacing b0 . For the elliptic circulation distribution it reads t0∗ =

π4 Λ b . 16 C L u ∞

(3.43)

The time scale can be seen as a measure of the trailing vortex age. It depends strongly on the load factor s: t0 ∝ s 3 . The dimensionless timescale related to a downstream location x = u ∞ t, respectively dimensionless x ∗ = x/b = u ∞ t/b, is τ=

1 CL x t . = = x∗ 3 t0 u ∞ t0 s 4πΛ

(3.44)

For the elliptic circulation distribution this reads τ∗ =

16C L t = x∗ 4 . t0∗ π Λ

(3.45)

The dimensionless time τ permits an analysis of the trailing vortex pair at the location x ∗ = x/b, taking into account the load factor s, the lift coefficient C L and the aspect ratio Λ. For the elliptical circulation distribution as aerodynamic reference distribution, C L and Λ are the only relevant configuration parameters. In the design of large aspect-ratio wings generally the elliptical circulation distribution is the target distribution, because it means minimum induced drag. Assuming such a distribution, we obtain for the lift L ∗ and for the induced drag Di∗ the relations

3.16 Basic Quantities of Trailing-Vortex Flow Fields

L∗ = and

87

π ρ∞ b u ∞ Γ0∗ 4

(3.46)

π ρ∞ Γ0∗ 2 , 8

(3.47)

L∗ 2 , πq∞ b2

(3.48)

Di∗ =

respectively Di∗ =

q∞ being the dynamic pressure of the freestream. The coefficient of the induced drag at elliptical circulation distribution reads C D∗ i =

C L∗ 2 , πΛ

(3.49)

C L∗ =

L∗ . q∞ A

(3.50)

with the lift coefficient C L being

In design aerodynamics the drag coefficient usually is composed of the zero-lift drag coefficient C D0 and the induced drag coefficient C Di . The latter then reads C Di =

C L∗ 2 , πeΛ

(3.51)

with e being span-efficiency factor, if the isolated wing is the topic, and e being the Oswald efficiency factor, if the whole aircraft is considered. In the literature the matter appears to be presented somewhat varying [5–7]. In any case the Oswald efficiency factor usually is given with 0.7  e  0.85 for the subsonic/transonic aircraft with large aspect-ratio wings, which we consider here, whereas the span-efficiency factor is given with 0.9  e  1. These relations permit quick estimates of aerodynamic properties of large aspectratio wings. We note that in cruise at optimal lift-to-drag ratio the induced drag is less than half of the total drag.

3.17 Problems Problem 3.1 Consider Fig. 3.3. The lifting-line model is the simplest model of a lifting wing in the frame of circulation theory. (a) What flow model is behind it. (b) Is the circulation Γ0 constant and the same on all four legs? (c) What vorticity theorems does the lifting-line model obey? (d) How is the situation in the two-dimensional

88

3 Elements of Vortex Theory

Fig. 3.23 Sketch of the airfoil

case of an airfoil? (e) How is the situation in the quasi-three-dimensional case of the infinite swept wing? Problem 3.2 Consider steady level flight of a large transport aircraft with a mass of m = 230,000 kg, a wing span of b = 60 m and an aspect ratio of Λ = 9 flying at the Mach number of M∞ = 0.82 at an altitude of H = 10 km. Consider further an elliptical circulation distribution. By how many meters moves the wake vortex system approximately downward within a flight path length of x = 50 km? Problem 3.3 Consider now for this aircraft configuration an approach flight with u ∞ = 80 m/s at a lift coefficient of C L = 1.4, assuming again an elliptical circulation distribution. Which maximum circumferential velocity is attributed to the trailing vortices applying on the one hand the Lamb–Oseen vortex model and on the other hand the Burnham-Hallock vortex model? How can the difference be judged? The viscous core radius rc is estimated to about 3 percent of the wing span b. Problem 3.4 Calculate the lateral distance b0 of the rolled-up wake vortex system for a parabolic circulation distribution with respect to the wing span b. How may this affect the wake vortex decay scenario? The spanwise parabolic circulation distribution is given by Γ (y) = Γ0 (1 − (2y/b)2 ). Problem 3.5 Simulate the incompressible flow around a 10 per cent thick biconvex circular arc airfoil at zero incidence with three constant strength source distributions of equal length on the cord line. What is the pressure coefficient in the most upstream collocation point? Problem 3.6 Consider a flat plate placed in an air stream at a small angle of attack, α, as sketched (Fig. 3.23). Model the flow with a discrete vortex at x/c = 0.25. What is the strength Γ of the vortex? Problem 3.7 Place two plane airfoils in a tandem arrangement separated by a distance . Use the lumped vortex method, Appendix A.2.2, to determine the lift on each airfoil and the variation with the spacing .

3.17 Problems

89

Problem 3.8 Consider an aircraft flying at constant speed. The flight Mach number is M∞ = 0.8, the flight altitude H = 10 km. The wing span is b = 60 m. The mass of the aircraft is 200,000 kg. Assume an elliptic circulation distribution, an Oswald efficiency factor e = 0.7 and compute the coefficients of lift and induced drag as well as the lift-to-drag ratio, assuming that the induced drag is about one third of the total drag.

References 1. Saffmann, P.G.: Vortex Dynamics. Cambridge University Press (1992) 2. Lugt, H.J.: Introduction to Vortex Theory. Vortex Flow Press, Potomac, Maryland, USA (1996) 3. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin Heidelberg New York (2006) 4. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, Vol. 1 and 2, SpringerVerlag, Berlin/Göttingen/Heidelberg, 1959, also: Aerodynamics of the Aeroplane, 2nd edition (revised). McGraw Hill Higher Education, New York (1979) 5. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011) 6. Drela, M.: Flight Vehicle Aerodynamics. The MIT Press, Cambridge, MA (2014) 7. Rossow, C.-C., Wolf, K., Horst, P. (eds.): Handbuch der Luftfahrzeugtechnik. Carl Hanser Verlag, München, Germany (2014) 8. Anderson Jr., J.D.: The Airplane: A History of Its Technology. AIAA, Reston Va (2002) 9. Hirschel, E.H., Prem, H., Madelung, G. (eds.): Aeronautical Research in Germany–from Lilienthal until Today. Springer-Verlag, Berlin Heidelberg New York (2004) 10. Serrin, J.B.: Mathematical Principles of Classical Fluid Mechanics. In S. Flügge (ed.): Handbuch der Physik, Band VIII/1, Strömungsmechanik 1, Springer-Verlag (1959) 11. Widnall, S.E.: The Structure and Dynamics of Vortex Filaments. Ann. Rev. Fluid Mech. Palo Alto, CA 7, 141–165 (1975) 12. Bloor, D.: The Enigma of the Aerofoil-Rival Theories in Aerodynamics, 1909–1930. The University of Chicago Press, Chicago and London (2011) 13. Rizzi, A.: Damped Euler-Equation Method to Compute Transonic Flow Around Wing-Body Configurations. AIAA J. 20(10), 1321–1328 (1982) 14. Lighthill, J.: Introduction Boundary-Layer Theory. In: Rosenhead, L. (ed.) Laminar Boundary Layers, pp. 46–113. Clarendon Press, Oxford (1963) 15. Klein, F.: Über die Bildung von Wirbeln in reibungslosen Flüssigkeiten. Zeitschrift für Math. und Physik 59, 259–62 (1910) 16. Betz, A.: Wie ensteht ein Wirbel in einer wenig zähen Flüssigkeiten? Die Naturwissenschaften 9, 193–96 (1950) 17. Rizzi, A.: Three-Dimensional Solutions to the Euler Equations with one Million Grid Points. AIAA Journal 23, 1986–1987 (1985) 18. Weis-Fogh, T.: Quick Estimates of Flight Fitness in Hovering Animals, Including Novel Mechanisms for Lift Production. Journal Exp. Biol 59, 169–230 (1973) 19. Lighthill, J.: On the Weis-Fogh Mechanism of Lift Generation. Journal Fluid Mech. 60, 1–17 (1973) 20. Hirschel, E.H.: Basics of Aerothermodynamics. 2nd, revised edition. Springer, Cham Heidelberg New York (2015) 21. Breitsamter, C.: Nachlaufwirbelsysteme großer Transportflugzeuge - Experimentelle Charakterisierung und Beeinflussung (Wake-Vortex Systems of Large Transport Aircraft— Experimental Characterization and Manipulation). Inaugural Thesis, Technische Universität München, 2007, utzverlag, München, Germany (2007)

90

3 Elements of Vortex Theory

22. Breitsamter, C.: Wake Vortex Characteristics of Transport Aircraft. Progress in Aerospace Sciences 47(1), 89–134 (2011) 23. Spreiter, J.R., Sacks, A.H.: The Rolling Up of the Trailing Vortex and its Effect on the Downwash Behind Wings. Journal Aero. Sci. 18, 21–32 (1951) 24. Sedin, Y.C.J., Grasjo, I., Kullberg, E., Larsson, R.: A Model for Simulation of Flight Passages through Trailing Tip Vortices. Paper ICAS 9(3), 2002–7 (2002) 25. Crow, S.C.: Stability Theory for a Pair of Trailing Vortices. AIAA Journal 8(12), 2172–2179 (1970) 26. Hoeijmakers, H.W.M.: Computational Aerodynamics of Ordered Vortex Flows. Doctoral Thesis, TU Delft, Rept. TR diss 1729, Delft, The Netherlands (1989) 27. Hoeijmakers, H.W.M., Rizzi, A.: Vortex-Fitted Potential and Vortex-Captured Euler Solution for Leading-Edge Vortex Flow. AIAA Journal 23, 1983–1985 (1985) 28. Rizzi, A.: Multi-cell Vortices Computed in Large-Scale Difference Solution to the Incompressible Euler Equations. Journal Comp. Phys. 77, 207–220 (1988) 29. Le Moigne, Y.: Adaptive Mesh Refinement and Simulations of Unsteady Delta-Wing Aerodynamics. Doctoral Thesis, KTH Royal Institute of Technology, Rep. TRITA-AVE 2004:17, Stockholm, Sweden (2004) 30. Hummel, D.: On the Vortex Formation over a Slender Wing at Large Angles of Incidence. In: High Angle of Attack Aerodynamics, Conference Proceedings AGARD CP-247, 15-1–15-17 (1978) 31. Werlé, H.: Quelques résultats expérimentaux sur les ailes en flèches, aux faibles vitesses, obtenus en tunnel hydrodynamique. La Recherche Aéronautique 41, (1954) 32. Peckham, D.H., Atkinson, S.A.: Preliminary Results of Low Speed Wind Tunnel Tests on a Gothic Wing of Aspect Ratio 1.0. Report CP-508, Aeronautical Research Council (1957) 33. Elle, B.J.: An Investigation at Low Speed of the Flow Near the Apex of Thin Delta Wings with Sharp Leading Edges. Reports and Memoranda 3176, Aeronautical Research Council (1958) 34. Breitsamter, C.: Unsteady Flow Phenomena Associated with Leading-Edge Vortices. Progress in Aerospace Sciences 44(1), 48–65 (2008) 35. Krause, E.: The Solution to the Problem of Vortex Breakdown. Lecture Notes in Physics Vol 371, Springer-Verlag, Berlin, 35–50 (1990) 36. Krause, E.: On an Analogy to the Area-Velocity Relation of Gasdynamics in Slender Vortices. Acta Mech. 201, 23–30 (2008) 37. Krause, E.: Stagnant Vortex Flow. Acta Mech. 209, 345–351 (2010) 38. Görtz, S.: Realistic Simulations of Delta Wing Aerodynamics Using Novel CFD Methods. Doctoral Thesis, KTH Royal Institute of Technology, Rep TRITA-AVE 2005:01, Stockholm, Sweden (2005) 39. Jiràsek, A.: Vortex Generator Modeling and its Application to Optimal Control of Airflow in Inlet. Doctoral Thesis, KTH Royal Institute of Technology, Rep TRITA-AVE 2006:66, Stockholm, Sweden (2006) 40. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin Heidelberg (2014) 41. Campbell, J.F., Chambers, J.R.: Patterns in the Sky–Natural Visualizations in Aircraft Flow Fields. NASA SP-514 (1994) 42. Guillaume, M., Gehri, A., Stephani, P., Vos, J.B., Mandanis, G.: F/A-18 Vertical Tail Buffeting Calculation Using Unsteady Fluid Structure Interaction. Aero J., Vol 115, No. 1166 (2011) 43. Vos, J.B., Charbonnier, D., Ludwig, T., Merazzi, S., Gehri, A., Stephani, P.: Recent Developments on Fluid Structure Interaction Using the Navier-Stokes Multi Block (NSMB) CFD Solver. AIAA-Paper 2017–4458, (2017) 44. Breitsamter, C.: Turbulente Strömungsstrukturen an Flugzeugkonfigurationen mit Vorderkantenwirbeln. (Turbulent Flow Structures at Aircraft Configurations with Leading-Edges Vortices). Doctoral Thesis, Technische Universität München, 1996, utzverlag, München, Germany (1997) 45. Mabey, D.G.: Some Aspects of Dynamic Loads Due to Flow Separation. AGARD-R-750 (1988)

References

91

46. Luber, W., Becker, J., Sensburg, O.: The Impact of Dynamic Loads on the Design of Military Aircraft. AGARD-R-815, 8-1–8-27 (1996) 47. Breitsamter, C., Schmid, A.: Airbrake Induced Fin Buffet Loads on Figther Aircraft. Journal of Aircraft 45(5), 1619–1630 (2008)

Chapter 4

The Local Vorticity Content of a Shear Layer

In the mid 1980s computer speed and storage had developed to a degree that Euler methods (Model 8 in Table 1.3) became a viable tool for aerodynamic design work. At that time very much discussed was the fact that at delta wings with sharp leading edges lee-side vortices resulted in the relevant angle of attack and Mach number regime. The question was, where is the apparent vorticity coming from, and accordingly the associated entropy rise. Oddly enough this was not an issue for Euler solutions—like for modeled potential-flow (Model 4) solutions—for lifting large aspect-ratio wings. In these cases at that time Euler methods produced reasonable results regarding lift, pitching moment and induced drag. The involved trailing vortex layers and vortices, however, remained unobserved. Back then the first two authors of this book argued that the vorticity over delta wings was produced in the same way as over large aspect-ratio wings. Accordingly at the Military Aircraft Division of Messerschmitt–Bölkow–Blohm (MBB) the concept of the local vorticity content of a shear layer was developed and tested. With this concept it could be shown how the problem may be understood. In the present chapter the concept of the local vorticity content of a shear layer is defined and explained. The concept permits to connect in a simple and descriptive way viscous phenomena of reality in the form of Model 2 in Table 1.3 to singularities of the classical potential theory, Model 4, which analogously holds for Euler solutions, too, Chap. 5.1 This means that in general viscous phenomena can be explained in terms of potential theory and vice versa. In particular the circulation theory of lifting airfoils and finite-span wings can be connected to viscous phenomena, which carry either kinematically active or inactive vorticity. To any vortex-flow singularity a unique amount of vorticity can be assigned. This mainly is a question how the limit Rer e f → ∞ is obtained. Behind this are the prerequisites of Model 2: high Reynolds numbers, and no strong or global interactions. 1 See

also the introductory discussion in Sect. 1.4. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_4

93

94

4 The Local Vorticity Content of a Shear Layer

The presentation and discussion of the local vorticity content of a shear layer in this chapter follows the publications [1–6]. The chapter has four sections. In the first two the concept of the local vorticity content of shear layers is introduced. The next section gives an interpretation of lift and induced drag as breaks of symmetry, followed in the last section by a discussion of the flow pattern at the trailing edge of large aspect-ratio lifting wings. A compatibility condition is introduced for this purpose. In Chap. 5 the matter of discrete numerical solutions of the Euler equations (Model 8) for lifting-wing flows is treated. Discussed in particular are the appearance of vorticity and the entropy rise in such solutions. Applications regarding the different aspects of the local vorticity content regarding large aspect-ratio wings then are given in Chap. 8 and regarding small aspect-ratio delta-type wings in Chap. 10.

4.1 Definition and Derivation of the Local Vorticity-Content Vector Figure 4.1 shows a generalized shear layer, which may be either a laminar or a turbulent one. The curvilinear orthogonal x, y, z-coordinate system has been placed with its origin P0 onto the skeletal surface of the shear layer. The x, y-coordinates lie in this surface. The coordinate z is rectilinear and normal to the surface. The velocity components are u, v, w. The x  , y  , z  ,-system is the reference coordinate system [7]. The external inviscid velocity vector V eu is located at the upper edge of the shear layer at z = δu , and the vector V el at the lower edge at z = δl . (We keep the vector notation of the original publications.) Note that generally the upper and the lower edge of the shear layer are not stream surfaces [7]. The following considerations assume a negligible curvature of the shear layer, so that for convenience the considerations can be made in Cartesian coordinates. For an exact approach see, e.g., [5]. We assume high Reynolds-number flow and consequently a very thin shear layer, i.e., a small extent in the direction normal to the surface, the z-direction, and a small velocity w in that direction. Hence we introduce the boundary-layer stretching [7], z˜ = z



Rer e f , w˜ = w



Rer e f .

(4.1)

z˜ and w˜ are dimensional entities. The reference Reynolds number is Rer e f =

u r e f ρr e f L r e f . μr e f

(4.2)

Like in boundary-layer theory now all velocities and lengths are nondimensionalized with vr e f and L r e f , respectively, and the density with ρr e f and the viscosity with μr e f .

4.1 Definition and Derivation of the Local Vorticity-Content Vector

95

Fig. 4.1 Element of a generalized shear layer in orthogonal curvilinear coordinates [3]. The coordinate x generally denotes the main flow direction

The vorticity vector in Cartesian coordinates reads  ω = r ot V = [ωx ; ω y ; ωz ] =

 ∂w ∂v ∂u ∂w ∂v ∂u − ; − ; − . ∂y ∂z ∂z ∂x ∂x ∂y

(4.3)

Applying the stretching leads to  ω=

Rer0.5 ef

1 ∂ w˜ ∂v ∂u 1 ∂ w˜ 1 − ; − ; 0.5 Rer e f ∂ y ∂ z˜ ∂ z˜ Rer e f ∂x Rer e f



∂v ∂u − ∂x ∂y

 .

(4.4)

We introduce now the local vorticity content of a shear layer, the vorticity-content vector Ω, as—at a given location x—the integral of the vorticity across the shear layer in z-direction [3]: Ω = [Ωx ; Ω y ; Ωz ] =

z˜ =δ˜ u z˜ =δ˜l

ω d z˜ .

(4.5)

With Eq. (4.4) we finally obtain for the high Reynolds-number limit Rer e f → ∞ (without the stretching indications at the lower and the upper bound) Ω| Rer e f →∞ = [−v; u; 0]δδlu .

(4.6)

96

4 The Local Vorticity Content of a Shear Layer

Note that the shapes of the functions u(˜z ) and v(˜z ) do not play a role. However, the functions must be continuous and they must continuously blend into the external inviscid flow u e (δl ), ve (δl ) and u e (δu ), ve (δu ).

4.2 Kinematically Active and Inactive Vorticity Content: Examples The vorticity content of a shear layer in terms of the components of Ω can be nonzero or zero. In the first case we call the vorticity content kinematically active, in the second case kinematically inactive. Kinematically active vorticity globally influences the flow field (vortex dynamics), whereas kinematically inactive vorticity locally cancels out. We illustrate these two kinds of vorticity content with several examples. We consider boundary layers and airfoil wakes. First the Rankine vortex is discussed, then the two-dimensional boundary layer and the lifting airfoil together with its near wake both in subcritical and supercritical steady motion. The discussion of the three-dimensional boundary layer and the trailing vortex layer of the lifting finite-span wing follows. A summary closes the section.

4.2.1 Rankine Vortex We begin with the seemingly trivial application of the vorticity content concept to the Rankine vortex, Fig. 4.2. Actually it is the application of Stokes’ theorem, Sect. 3.3. The vortex is idealized in the sense of Model 2, Table 1.3. The vorticity content vector—now without the high Reynolds-number limit and in cylindrical coordinates—reads

r0



2 0

0

2π

ω r dϕ dr = 2 0

r0

0

2π

1 1 d(r v) r dϕ dr = Γ = 2πr0 v0 . 2 r dr

(4.7)

The vorticity content hence is equal to the circulation Γ . That of course is the line integral around the vortex core: Γ = 2πr0 v0 . The result means that with a given circulation Γ , i.e., a given amount of vorticity in the vortex core, the core can have any diameter, because r0 v0 = constant. To a given potential flow singularity with circulation Γ , Model 4, thus belongs a finite vorticity content. Although the diameter of the singularity is zero and v0 → ∞, the above vorticity content can be assigned to it. In the reality, Model 1, however, the core can have any diameter because due to diffusion the diameter may grow, Sect. 3.10, and under certain circumstances the vortex even may break down, Sect. 3.13.

4.2 Kinematically Active and Inactive Vorticity Content: Examples

97

Fig. 4.2 Schematic of the Rankine vortex and its radial velocity (v) and circulation (Γ ) distribution [3, 4]

We can suppose already that a connection exists to the bound vortex of a lifting airfoil, as it is found in the frame of potential theory. Before we treat this case in Sect. 4.2.4, we look at the vorticity content of the boundary layer, Sect. 4.2.2, and of the airfoil wake, Sect. 4.2.3.

4.2.2 Two-Dimensional Boundary Layer In Fig. 4.3 the profile u(z) of a two-dimensional boundary layer is shown. The external inviscid streamline is oriented along the x-coordinate of the element of the generalized shear layer shown in Fig. 4.1. The skeletal surface of that element (z = 0) now constitutes the solid body surface with the no-slip condition u (z = 0) = 0. The upper bound δu is the boundary-layer thickness δ, where u (z = δ) = u e .2 The vorticity content vector reads Ω| Rer e f →∞ = [0; u e ; 0] .

(4.8)

The result is that the boundary layer has a finite local vorticity content. The local vorticity vector points normal to the plane of the two-dimensional boundary layer in the y-direction.

2 Regarding

the definition of the boundary-layer thickness see Appendix A.5.4.

98

4 The Local Vorticity Content of a Shear Layer

Fig. 4.3 Schematic of the tangential velocity profile u(z) of a two-dimensional boundary layer [3, 4]

4.2.3 Near Wake of a Lifting Airfoil Figure 4.4 shows two idealized airfoil near wakes.3 Case (a) stands for an airfoil in steady subcritical motion, and case (b) for an airfoil in steady, but supercritical motion, with a supersonic flow pocket and a terminating shock wave at the upper (suction) side. At the trailing edge in both cases the static pressure at the lower and the upper side of the wake is the same, regardless whether the airfoil is lifting or not. This also holds for both wakes in the limit Rer e f → ∞, even if they are slightly curved. In case (a) the equal pressures lead to the same velocity of the external inviscid flow at the lower and the upper side of the wake. The vorticity content hence is Ω| Rer e f →∞ = [0; 0; 0] .

Fig. 4.4 Schematic of two-dimensional wakes [3, 4]: a wake of an airfoil in steady subcritical motion, b wake of an airfoil in steady supercritical motion

3A

discussion of the flow at the trailing edges of real airfoils and wings is given in Chap. 6.

(4.9)

4.2 Kinematically Active and Inactive Vorticity Content: Examples

99

Fig. 4.5 The wake of Fig. 4.4b in the limit Rer e f → ∞ as the two-dimensional vortex sheet (slip line) of potential theory [3, 4]

Wake (a) therefore is considered to be kinematically inactive, i.e., the vorticity leaving the trailing edge cancels out. This holds for both the lifting and the non-lifting airfoil. Only during lift changes, due for an angle of attack change or for a speed change, does the wake carry kinematically active vorticity. For the lifting airfoil this is in accordance with potential theory, Model 4. (The non-lifting case is trivial.) Once the steady motion of the airfoil has been established, the bound vortex of the airfoil and the opposite starting vortex at infinity behind the airfoil do no more change the strength of their circulations. The wake by no means must be symmetric, it is only the two edge values of the external inviscid velocity which are symmetric. In case (b) the total pressure loss due to the terminating shock wave at the upper side of the airfoil leads to different external inviscid velocities at the lower and the upper side of the wake, as the static pressure has to be the same. In this case a finite local vorticity content is present:

 Ω| Rer e f →∞ = 0; u eu − u el ; 0 .

(4.10)

In the high Reynolds-number limit thus the wake collapses into the twodimensional vortex sheet (slip line) of potential theory, Fig. 4.5. The kinematically active vorticity content Ω y is hidden in this layer. This case needs a closer inspection, because also in the steady supercritical lifting case no kinematically active vorticity is to leave the airfoil via its wake. This must be seen in the following way. Consider Fig. 4.6. A supersonic flow pocket is present at the suction side of the airfoil. The terminating shock wave leads to a total pressure loss. Hence we find at the trailing edge of the airfoil a smaller external inviscid velocity at the upper side than at the lower side of the wake. The result is a wake flow situation like that of case (b) in Fig. 4.4.

100

4 The Local Vorticity Content of a Shear Layer

Fig. 4.6 Schematic of the flow at the trailing edge of a lifting airfoil in steady supercritical motion [5] Table 4.1 Parameters of the CAST 7 airfoil case [8] M∞ Re L ,∞ L (m) T∞ (K) 0.7

4·106

1

300

x/L|trans.

α (◦ )

0.07

2

If we only consider this wake, we would conclude that kinematically active vorticity leaves the airfoil’s trailing edge, which, however, is not permitted, because we consider the lift as being unchanging. In this case the effect of the shock wave must be taken into account. Actually not only the boundary-layer wake is to be considered, but also the shock wake. The two together constitute a vorticity or entropy wake, which indeed does carry only kinematically inactive vorticity, Fig. 4.6. Hence we postulate that the vorticity created by the shock wave must cancel the kinematically active vorticity of the boundary layers, which leaves the trailing edge4 : Ω|s =

z=δs

 r ot v dz = 0; −(u eu − u el ); 0 .

(4.11)

z=δu

For kinematic reasons thus for lifting wings at transonic (and supersonic) speeds, wake considerations must include the shock wake. Obviously the bounds of the boundary-layer wake are adjusted to constitute the much thicker vorticity or entropy wake behind the airfoil or wing. The mechanism which brings about the necessary shock form and its change of strength to achieve the cancelation, is not known. In the doctoral thesis on self-adaptive grid refinement by J. Fischer, a student of the first author of this book, besides others the viscous flow past a supercritical airfoil was studied [8]. The considered airfoil was the so-called CAST 7 airfoil. A large number of experimental data is available for this airfoil [9]. The parameters chosen in the thesis are given in Table 4.1. Figure 4.7 shows the airfoil with the generated iso-Mach lines (viscous solution). Despite the presence of the boundary layer, the terminating shock wave at 4 The

amount of vorticity which must be canceled is small compared to the amount of vorticity of each of the involved boundary layers, usually only a few per cent.

4.2 Kinematically Active and Inactive Vorticity Content: Examples

101

Fig. 4.7 Cast 7 airfoil, viscous case [8]: Mach number isolines, Mmin = 0, Mmax = 1.3, M = 0.02. The broken line is the sonic line M = 1

the upper side is seen to impinge orthogonally on the airfoil’s surface, as inviscid theory demands [10]. Away from the surface the shock wave is slightly curved, the pre-shock Mach number diminishes away from the surface.5 The shock wave finally tapers off and at its upper end it blends into the sonic line. We discuss now the vorticity wake at the trailing edge of the CAST 7 airfoil, x = L, see Fig. 4.6. Because the velocity difference between the lower and upper external inviscid flows at the trailing edge is small, in [8] only the vorticity wake of the inviscid case was considered. Nevertheless, the result illustrates well the situation. We look at the y-component of the vorticity-content vector Ω. The simplification of the vorticity content integral, Eq. (4.6), is not adequate in this case, because we do not consider a thin boundary layer. Instead the full y-component of the vorticity must be taken into account:  x=L ,z=0.55L x=L ,z=0.55L  ∂u ∂w − dz. (4.12) ω y dz = Ωy = ∂z ∂x x=L ,z=−0.55L x=L ,z=−0.55L The lower and the upper bound of the vorticity integral were placed at z = −0.55 L, and z = 0.55 L. Figure 4.8 shows the evolution of the integral Ω y . We look only at the solution on the fully refined grid, represented by the broken line. Between z/L = −0.55 and the trailing edge (z/L = 0) the value of the integral is zero. At the lower

5 These

properties are important, because behind a straight shock wave with constant pre-shock Mach number, a total pressure loss is present, but no vorticity. See in this regard Crocco’s theorem, Sect. 3.5.

102

4 The Local Vorticity Content of a Shear Layer

Fig. 4.8 Cast 7 airfoil [8]: development of the integral Eq. (4.12) at the airfoil’s trailing edge, x=L

side of the airfoil no shock wave is present, hence no vorticity. At z/L = 0 a steep descent to Ω y ≈ −7.5 m/s occurs.6 For z/L > 0 the integral rises and finally reaches Ω y ≈ 0. Although the solution exhibits wiggles and Ω y does not exactly reach zero, this result can be seen to prove that the vorticity wake is kinematically inactive. A new investigation with today’s available algorithms and computer power appears to be desirable. In summary we state that the embedded wake profile, Fig. 4.4b, does not change the conclusion. The vorticity content of the boundary-layer wake Ω y = u eu − u el is compensated by the vorticity content of the wake of the shock wave. The whole wake, i.e., the vorticity or entropy wake—indicated in Fig. 4.6—in steady supercritical flow is kinematically inactive, as is demanded.

4.2.4 Bound Vortex of a Lifting Airfoil Consider the flow situation over a lifting airfoil in steady subcritical motion, Fig. 4.9. The flow is two-dimensional throughout. As we have seen above, a two-dimensional boundary layer has a finite kinematically active vorticity content, which locally is—in the high Reynolds-number limit—Ω| Rer e f →∞ = [0; u e ; 0]. The wake downstream of the trailing edge, on the other hand, does carry only kinematically inactive vorticity, as was discussed in the preceding sub-section. Hence it is no more considered. We determine the kinematically active vorticity content of the two boundary layers over the length of the airfoil L in the high Reynolds-number limit by taking the double integral over the boundary-layer thicknesses (z-direction) and along the 6 Note

that a discrete (Model 8) Euler solution close to a solid surface always exhibits a thin total-pressure loss layer, casually called Euler boundary layer. That is an artifact due to the finite discretization of the computation domain and the flow variables used.

4.2 Kinematically Active and Inactive Vorticity Content: Examples

103

Fig. 4.9 Schematic of the boundary layers and the wake at an lifting airfoil in steady subcritical motion [3]

boundary/shear-layer edges (s-direction). The integration is made in the sense of Model 2, not in the sense of Model 1 or 9, because the trailing-edge interaction—and its upstream influence—between the boundary layers and the external inviscid flow is not taken into account:

s=L s=0



z=δl ,δu z=0

u(z, s) ds dz| Rer e f →∞ =

s=L

u e ds = Γ.

(4.13)

s=0

The conclusion is that the double integral of the vorticity in the boundary layers for Rer e f → ∞ reduces to the line integral of the external inviscid velocity of the boundary layers which gives the circulation Γ of the bound vortex of the lifting airfoil. This is an application of Stokes’ theorem via the concept of the local vorticity content of a boundary layer. The result was given first in 1986 by M.J. Lighthill [11], as we mentioned it in Sect. 1.4. The interpretation is that the two boundary layers in terms of their vorticity content over the length of the airfoil constitute the rotational core of the bound vortex. Actually it is the excess of the vorticity content of the upper over that of the lower boundary layer. This is also evident in terms of the line integral. At the upper side of the airfoil the external inviscid velocity is larger than at the lower side. The excess accounts for the actual amount of circulation. For a better understanding it may help to consider the case of a symmetric airfoil at zero angle of attack and hence with zero lift. Then the vorticity content at the lower side of the airfoil has the same amount as that at the upper side, but with an opposite sign. It cancels out. The same holds for the external inviscid velocity. Consequently the circulation and hence the lift are zero.

104

4 The Local Vorticity Content of a Shear Layer

Fig. 4.10 Schematic of the profile of a three-dimensional boundary layer in external inviscid stream-line oriented coordinates [7]

4.2.5 Three-Dimensional Boundary Layer In Fig. 4.10 the profile v(z) of a three-dimensional boundary layer is shown.7 The profile is given in the external inviscid streamline-oriented (t, n) coordinate system [7]. (The t-coordinate locally lies tangential to the external inviscid streamline, the ncoordinate normal to it. This coordinate system is kind of a natural coordinate system for three-dimensional boundary layers. Other systems, of course, can be employed. The result below will be same.) The external inviscid streamline (t-direction) is oriented along the x-coordinate of the element of the generalized shear layer shown in Fig. 4.1. The velocity profile v(z) can be decomposed into the main-flow profile (streamwise) vt (z), which resembles the profile of a two-dimensional boundary layer, and into the cross-flow profile vn (z), which can have quite different shapes [7]. In any case vn is zero at the body surface (z = 0) and at the outer edge of the boundary layer (z = δ). The skeletal surface of the element of the generalized shear layer in Fig. 4.1 (z = 0) again constitutes the solid body surface with the no-slip condition vt (z = 0) = 0, vn (z = 0) = 0. The upper bound δu is the boundary-layer thickness δ, where vt (z = δ) = |V e | = vte (main-flow profile), vn (z = δ) = 0 (cross-flow profile). The vorticity content vector in the high Reynolds-number limit reads

 Ω| Rer e f →∞ = 0; |V e | = vte ; 0 .

(4.14)

The result is that the three-dimensional boundary layer has a finite local vorticity content which is that of the main-flow profile. The cross-flow profile’s vorticity 7 Note that the stream surface of the boundary layer does not lie in a plane, as in the two-dimensional

case, Fig. 4.3. Even more complex forms of the stream surface are possible [7].

4.2 Kinematically Active and Inactive Vorticity Content: Examples

105

content is zero. The local vorticity-content vector Ω points in the n-direction, i.e., the direction normal to the t-z-plane.

4.2.6 Near Wake (Trailing Vortex Layer) of a Lifting Finite-Span Wing Consider the flow past the canonical lifting wing in Fig. 1.10a. Intuition tells us— because of the finite span of the wing—that a pressure relaxation takes place between the high pressure on the lower side and the low pressure on the upper side. This relaxation leads at the lower side to a general flow deflection toward the wing’s tip and at the upper side away from it toward the wing’s root, see Sect. 4.3.2. The result is that the inviscid wing upper and lower flow fields are shearing. We look at the flow as it locally leaves the trailing edge of a back-swept finitespan lifting wing in steady subcritical motion, Fig. 4.11. (The shown s-t-coordinate system is the local wake coordinate system. It may not lie in the datum plane of the wing. In particular near the wing tip this must be regarded.) Because we discuss on the basis of Model 2, we disregard strong interaction phenomena and a possible finite, if small, thickness of the trailing edge (Sect. 6.3). The flow over the lower side generally has a direction toward the wing tip, that over the upper side away from the wing tip. We assume that the velocity vectors in the figure lie in the skeletal plane of the trailing vortex layer, which leaves the trailing edge. Because we have assumed subcritical motion, the total pressure, like the static pressure, is considered to be the same on the upper and the lower side of the trailing edge. The magnitudes of the external inviscid velocity vectors at the upper side (V eu ) and at the lower side (V el ) of the trailing edge are the same: |V eu | = |V el |. We define the angle between the two external inviscid velocity vectors V eu and V el as the (local) trailing-edge flow shear angle ψe . Locally the magnitude of the

Fig. 4.11 Local wake (trailing vortex layer) coordinate system (view from above onto the upper surface of the wing near the trailing edge) [7]: idealized situation of the flow at the trailing edge of a back-swept finite-span lifting wing in steady subcritical motion. The coordinate y points in span direction, the coordinate x in free-stream or chord direction

106

4 The Local Vorticity Content of a Shear Layer

Fig. 4.12 Schematic of the decomposed near-wake structure of a backward-swept finite-span lifting wing [3, 4]. Left: local wake coordinate system; right: wake in idealized reality

shear angle is a measure of the strength of the trailing vortex layer leaving the trailing edge, Sect. 4.4. The angle ε between the s-direction and the chord direction x, i.e., the (local) vortex-line angle, is small but not necessarily zero. Its sign is governed by the local sweep of the trailing edge [12]: positive (toward the wing tip) for back-swept trailing edges, negative (toward the wing root) for forward swept trailing edges. The magnitude of ε at given lift depends on the thickness of the wing.8 We note that in potential-flow theories of lifting wings, for instance lifting-surface theories, the vortex-line angle ε is assumed to be zero, see, e.g., [13]. The structure of the wake between the sheared external inviscid flows at the lower and the upper side directly at the trailing edge is complex. We show it schematically in a decomposed manner in Fig. 4.12. The two vectors of the external inviscid flow V eu and V el are decomposed in such a way that they have the components u eu = u el in s-direction (bi-sector direction) and veu = −vel in t-direction (normal to the bi-sector direction). The angles ψeu and ψel are found from tan ψeu =

veu ve , tan ψel = l , u eu u el

(4.15)

where the u e are the components of the two velocity vectors in s-direction and the ve those in n-direction. Note that ψeu = −ψel and, Fig. 4.11, ψe = ψeu + ψel . 8 See

the examples in Chap. 8.

4.2 Kinematically Active and Inactive Vorticity Content: Examples

107

Fig. 4.13 The wake of Fig. 4.12 right in the limit Rer e f → ∞ as vortex sheet of potential theory [3, 4]

The velocity component u(z) in s-direction resembles the (two-dimensional) wake of a lifting or non-lifting airfoil in steady subcritical motion. Locally we can relate to this 2-D wake the viscous drag, i.e., the skin-friction drag plus the form drag. The velocity component v(z) in t-direction, on the other hand, resembles a vortex with the axis in s-direction. To this we can relate the induced drag. This is a topic of Sect. 4.3.2. The vorticity content vector in the high Reynolds-number limit reads

 

Ω| Rer e f →∞ = Ωs ; Ωt ; Ωz = −(veu − vel ); 0; 0 ,

(4.16)

and with veu = −vel because of the bisector orientation



 Ω| Rer e f →∞ = 2vel ; 0; 0 = −2veu ; 0; 0 .

(4.17)

This means that the 2-D wake-like profile u(z) indeed has a zero vorticity content, whereas the vortex-like profile v(z) has a finite vorticity content. Figure 4.13 shows the decomposed wake, Fig. 4.12 right, in the high Reynoldsnumber limit collapsed into the vortex sheet of potential theory. The kinematically active vorticity content Ωs is hidden in this layer. The scoordinate as a streamline obviously represents a vortex line [11], because there with u eu |t=0,z=0 = u el |t=0,z=0 being considered as the flow vector V e |t=0,z=0 Ω| Rer e f →∞ × V e |t=0,z=0 = 0.

(4.18)

4.2.7 Summary of the Results We summarize the results: 1. In the frame of Model 2 the concept of the local vorticity content of a shear layer yields in the limit Rer e f → ∞ that this content is equal to the external inviscid

108

2. 3.

4. 5.

6.

7.

4 The Local Vorticity Content of a Shear Layer

velocity in the case of boundary layers, or to the difference of these velocities in the case of wakes. The thickness of the shear layer goes to zero for Rer e f → ∞. The shapes of the velocity profiles in the shear layer do not play a role, provided the profiles are continuous up to the second derivative. The concept of the local vorticity content permits to distinguish between kinematically active and inactive vorticity. The former influences the surrounding flow field (vortex dynamics), the influence of the latter locally cancels out. Wakes of airfoils—non-lifting or lifting—in steady motion carry no kinematically active vorticity. In the case of an airfoil or a finite-span wing in steady supercritical motion the concept of the local vorticity content leads to the conclusion that the vorticity wake behind the—by necessity variable in strength—shock wave must be taken into account regarding the wake properties. Wakes, i.e., trailing vortex layers, of lifting finite-span wings in steady motion locally can be decomposed into a kinematically active and a kinematically inactive part. The inactive part along the main-flow direction can be considered to locally carry the viscous drag, and the active part in the cross-flow direction the induced drag. The concept of the local vorticity content permits to connect a singularity of potential theory with a finite amount of vorticity, which is hidden in the singularity. This holds for vortices, boundary layers and wakes, which in the high Reynoldsnumber limit become vortex singularities and vortex sheets, respectively.

4.3 Lift and Induced Drag: Two Breaks of Symmetry Preliminary remark: symmetry in this section has two meanings. First it is meant in the sense that the surface pressure distribution is such that no net force acts on the airfoil or wing, and secondly it is also meant in the classical geometrical sense.

4.3.1 First Symmetry Break: The Lifting Airfoil In 1752 d’Alembert did prove that no net force is exerted on a body, which steadily moves through an incompressible and inviscid fluid. This situation is described by potential flow theory (Model 3, Table 1.3). An airfoil in such a potential flow schematically is shown in Fig. 4.14. The stagnation-point streamline impinges vertically on the airfoil’s surface. At the stagnation point a half-saddle point (S1 ) is present, as well as at the rearward stagnation point (S2 ), where the flow leaves the surface, again normal to the surface, see Chap. 7. The flow turns around the sharp trailing edge. In potential flow this means an infinitely large speed, which, however, is permitted in such a flow model.

4.3 Lift and Induced Drag: Two Breaks of Symmetry

109

Fig. 4.14 Schematic of steady incompressible and inviscid flow past an airfoil. No net force is exerted on the airfoil Fig. 4.15 Schematic of steady incompressible and inviscid flow past an airfoil with a Kutta condition. A lift force is exerted on the airfoil

The flow far behind the airfoil has the same direction as that ahead of the airfoil, i.e., no momentum flux is deflected downwards. This indicates that no airfoil lift is present. That no drag is present, is not indicated. That would have required to show that no wake exists. We consider the flow situation in Fig. 4.14 as being symmetrical. A break of that symmetry happens, if a Kutta condition is inserted at the airfoil’s trailing edge, Fig. 4.15. In 1902 Kutta introduced this feature into potential lift theory. The Kutta condition reflects what is observed in reality: the flow leaves smoothly the sharp trailing edge of an airfoil, flow-off separation. The basic topology of the surface flow field is as in Fig. 4.14. Now, however, the rearward stagnation point (S2 ) lies at the trailing edge. If the trailing edge has a finite opening angle and the flow leaves the surface in bi-sector direction, the velocity at S2 —in the frame of potential theory—indeed is zero. That a lift force is present, is indicated by the downward deflection of the flow behind the airfoil. However, no drag is present. (That there is no wake is not indicated.) Circulation theory, Model 4 in Table 1.3, models the flow past an airfoil with a Kutta condition by, for instance, combining the undisturbed free-stream with a potential vortex, the bound vortex, see, e.g., [13]. The bound vortex has a circulation Γ with a strength such that the induced velocity shifts the rearward stagnation point S2 from its location on the upper side of the airfoil, Fig. 4.14, to the trailing edge, Fig. 4.15. In this way flow-off separation at the trailing edge is enforced. The resulting lift force per unit span l is l = ρ∞ u ∞ Γ.

110

4 The Local Vorticity Content of a Shear Layer

4.3.2 Second Symmetry Break: The Lifting Finite-Span Wing Consider the surface stream-line pattern of the inviscid incompressible flow past the wing-like 3:1:0.125 ellipsoid shown in Fig. 4.16. The angle of attack is α = 15◦ .9 We look from above at the right-hand side half-span of the ellipsoid, 0  y  b/2. The forward stagnation point is located on the lower side (the pressure side) below—in this view—the apex (x = 0, y = 0). Also located on the lower side is the attachment line. From this line the streamlines diverge on the lower side (pressure side) in almost chord direction and around the leading edge on the upper side (suction side) also in almost chord direction. At the trailing edge we see—on the upper side of the ellipsoid—the convergence of the streamlines into the detachment line, Sect. 7.1.3. The streamlines on the upper side converge directly into that line, those from the lower side also, around the trailing edge. The detachment line enters the rear stagnation point, which lies—in this view—on the upper side above the antapex (x = c (y = 0), y = 0). We observe in the view from above that overall a shear is present in the streamline pattern—the wing upper and lower side flow-fields shear—, which at the trailing edge manifests itself in the shear angel ψe between the streamlines over the lower and the upper side of the ellipsoid. This shear angle is zero in the symmetry plane (y = 0) and increases toward the wing’s tip.

Fig. 4.16 Surface streamlines of the inviscid incompressible flow past a wing-like 3:1:0.125 ellipsoid at angle of attack α = 15◦ (Model 3) [5]. The z-coordinate lies normal to the x, y-coordinates

9 The figure is based on a computation case in [14]. The flow field was computed with exact potential

theory (Model 3 of Table 1.3) [15, 16].

4.3 Lift and Induced Drag: Two Breaks of Symmetry

111

The origin of the shear can be imagined in the following way. Because we deal with a linear potential flow problem, we find the streamline pattern in Fig. 4.16 by superimposing the solution for the free-stream component v∞,x = cos α v∞ with that for the component v∞,z = sin α v∞ , see also Sect. 7.4.3. For zero angle of attack no shear at all is present. With lift at a given span location y > 0 the shear increases with increasing angle of attack, attains a maximum, and for 90◦ angle of attack again it becomes zero. We now come back to Model 1. Results from experimental and theoretical/numerical flow-field investigations on several configurations suggest the existence of a locality principle, Sect. 2.3. In our context it says that a local change of a body shape and the resulting flow separation affects the flow only at that location and downstream of it. Of course the flow is changed also upstream—in subsonic flow fields due to their elliptical properties—but these changes are small in general. However, although being small, they can be non-negligible, if the ensuing flow wake carries kinematically active vorticity, which leads, for instance, to the downwash at the location of the lifting wing and to the induced drag. If the trailing edge of our wing-like ellipsoid is sufficiently thin and the flow is sufficiently real, the latter undoubtedly will break away from the surface at the trailing edge in the general form of flow-off separation.10 Because of the locality principle the flow upstream of the trailing edge will retain the general shear between the lower and upper (inviscid) streamlines in terms of the (local) trailing-edge flow shear angle ψe (= ψe,u + ψe,l ), Sect. 4.2.6. That shear—i.e. prior to the convergence to the potential-flow detachment line in Fig. 4.16—then extends downstream into the wing’s wake—the trailing vortex layer—leading to a flow geometry like that sketched in Fig. 4.17. This is the effect of the second break of symmetry. In Fig. 4.17 also indicated is the local wake coordinate system, Sect. 4.2.6, Fig. 4.11. The indicated (local) vortex-line angle  is negative, because the trailing edge at that location is swept forward. The skeleton surface of the trailing vortex layer is assumed to lie initially in the wing plane, not in the Kutta direction, Sect. 6.2. We summarize our observations. They have been obtained by discussions of a wing-like ellipsoid, however, they hold for every finite-span steadily moving lifting wing: 1. The inviscid streamlines (Model 3) on the surface of the wing-like ellipsoid at angle of attack without a Kutta condition are (odd) symmetrical with regard to the x ≈ 0.5 c, y, z - plane. This also holds for realistic wing shapes, then however the symmetry is not a strictly geometric one. No net force acts on the configuration. 2. If the flow is a realistic one (Model 1), it does not turn around the trailing edge but breaks away from it (flow-off separation). This is in accord with the Kutta condition, which represents the first break of symmetry, leading to a lift force. At the same time also the (odd) symmetry of the inviscid streamlines with regard to the x ≈ 0.5 c, y, z - plane is broken: second break of symmetry. 10 Trailing

edge properties as well as the Kutta condition in reality are treated in Chap. 6.

112

4 The Local Vorticity Content of a Shear Layer

Fig. 4.17 Surface streamlines of the inviscid incompressible flow past a wing-like 3:1:0.125 ellipsoid at angle of attack α = 15◦ : expected real inviscid flow pattern at and downstream of the trailing edge (Model 1) [5]. The “vortex layer” is the wing’s trailing vortex layer. The z-coordinate lies normal to the x, y-coordinates

3. The second break of symmetry, however, retains an important property of the inviscid surface streamline pattern of Model 1, viz. the shear of the inviscid streamlines, the wing upper and lower flow-fields shear. The shear in terms of the (local) trailing-edge flow shear angle ψe in Model 1 is approximately that of Model 3 ahead of the rear convergence to the detachment line shown in Fig. 4.16. 4. The shear due to the second break of symmetry introduces into the wake—due to the flow-off separation—a kinematically active vorticity content, Sect. 4.2.6. The result is the induced kinematically active wake, the trailing vortex layer, and hence the downwash at the location of the wing and the induced drag. The induced drag is a generic property of the inviscid (!) flow past the lifting wing, not of the separating three-dimensional boundary layers at the upper and the lower side of the wing, which constitute the wake flow. Hence it can be found with any properly formulated inviscid flow model, in particular with the methods of potential flow theory (Model 4) and with discrete numerical Euler methods (Model 8) as long as the Kutta condition implicitly or explicitly is present.

4.3.3 The Symmetry Breaks in the Reality of Aircraft Wings Regarding the situation in the reality of lifting wings we make two observations: 1. The Kutta condition is very complex, Chap. 6. Present are (a) the in spanwise direction varying trailing-edge flow shear angle ψe , (b) the vortex-line angle ,

4.3 Lift and Induced Drag: Two Breaks of Symmetry

113

which may be constant only over a portion of the trailing edge, (c) decambering effects, (d) a finite thickness of the trailing edge, and (e) at delta-type wings swept round leading edges. 2. The shear of the inviscid flow fields at the upper and lower side of the lifting wing is not so obvious as in the case of our flat ellipsoid. This in particular holds for small aspect-ratio delta-type wings. There the lee-side vortex system, which is present at higher angels of attack, Chap. 10, completely changes the picture. At the lifting large aspect-ratio wing, on the other hand, we get the tip-vortex system, Sect. 8.4.4, which has its own influence, which becomes larger with smaller aspect ratio.

4.4 Flow Pattern at the Trailing Edge of Large-Aspect Ratio Lifting Wings: A Compatibility Condition We study the connection of the local kinematically active vorticity content, which leaves the trailing edge, with the spanwise circulation distribution of the lifting wing. This connection is described by a compatibility condition. Our objective is to link aspects of Model 4 to aspects of Model 2 of Table 1.3. In Sect. 4.2.6 we locally decomposed the flow at the trailing edge into a 2-D wakelike profile u(z) and a vortex-like profile v(z), Fig. 4.12. We found with Eq. (4.17) 



 Ω| Rer e f →∞ = Ωs ; Ωt ; Ωz = 2vel ; 0; 0 = −2veu ; 0; 0 that the vortex-like profile has a finite—kinematically active—vorticity content, whereas the 2-D wake-like profile has zero vorticity content—hence is kinematically inactive. Figure 4.13 shows the decomposed wake of Fig. 4.12 right, in the high Reynoldsnumber limit collapsed into the vortex sheet of potential theory. The kinematically active vorticity content Ωs is hidden in this layer. We ask now how the kinematically active vorticity content Ωs locally is connected to the span-wise circulation distribution Γ (y) of the wing. The overall situation over a lifting wing is sketched in Fig. 4.18. Here Γ0 is the so-called root circulation. Prandtl in his lifting-line wing model located the trailing vortices, fully developed, at the wing tips [18], see the discussion of this matter in Sect. 3.4. In reality, however, a vortex sheet leaves the trailing edge, with in span-wise direction increasing strength— the trailing vortex layer in Fig. 4.18. This sheet rolls up to the two discrete trailing vortices. The (initial) horizontal distance b0 of the axes of the fully developed trailing vortices is smaller than the wing span b. For a wing with elliptical circulation distribution along the wing span, the spanwise load factor, Sect. 3.16, reads

114

4 The Local Vorticity Content of a Shear Layer

Fig. 4.18 Schematic of a lifting wing: depiction of span-wise circulation distribution Γ (y), trailing vortex layer (wake) behind the wing, and the two counterrotating trailing vortices with circulation Γ0 [17]

s=

π b0 = . b 4

(4.19)

The ratio b0 /b → 1 is only reached for a wing with very large aspect ratio. The location, where the roll-up process of the trailing vortex layer toward the pair of trailing vortices is completed, can not be defined exactly. The process is an asymptotic one, depending on the magnitude of the lift, the span loading and the wing’s aspect ratio. In the case of large aspect-ratio swept or unswept wings the approximate location can be a few half-span distances downstream of the wing, [17], in the case of low-aspect ratio wings the location can be much closer to the trailing edge at one chord length or less behind the wing [19], see in this regard also Sect. 8.4.4. Above we have noted that locally in the trailing vortex layer the s-coordinate, Fig. 4.13, is both a streamline and a vortex line. The kinematically active vorticity leaving the trailing edge, the trailing vorticity, therefore can be considered as streamwise vorticity [11].11 Hence, locally the connection of the kinematically active vorticity Ωs and the circulation Γ (y) obviously is via the span-wise gradient of the latter. If the vortex-

11 Lighthill

calls the trailing vorticity also residual vorticity.

4.4 Flow Pattern at the Trailing Edge …

115

line angle  is small, Ωs at a location y must be equal to the change of the circulation in the wing’s spanwise direction: dΓ = −Ωs = −(veu + vel ) = 2 |V eu | sin ψeu , dy

(4.20)

which in [2] was introduced as a compatibility condition. In principle this is a discussion of the lifting wing’s discontinuity surface of potential theory, Model 4, from the view of viscous modeled flow, Model 2. However, it is not necessary to assume  = 0, as in most of the potential-flow wing theories. Equation (4.20) shows that with increasing dΓ /dy either |V eu | must increase or the trailing-edge shear angle ψeu . As we have seen above, the second break of symmetry essentially and approximately leaves the increase of ψeu in spanwise direction unchanged. Hence it basically is the change of ψeu , which leads to the fulfillment of the compatibility condition. This is another aspect of the locality principle, Sect. 2.3. Regarding the compatibility condition, Eq. (4.20), we shortly look at the situation at the lifting infinite swept wing. The infinite swept wing and the locally infinite swept wing are simple and convenient quasi-two-dimensional local approximations of the geometry of swept wings [7]. Because in the infinite swept wing approximation all flow parameters and in particular the circulation Γ do not change in spanwise direction, the trailing-edge shear angle ψeu must be zero. This means that the flow field close to and at the wing’s trailing edge is different from that of the approximated lifting finite-span wing. No kinematically active vorticity leaves the trailing edge. Nevertheless, the infinite swept wing is useful in, for instance, investigations of boundary-layer stability and laminarturbulent transition at and close to the leading edge.

4.5 Final Remark The concept of the kinematically active vorticity permits to connect fundamental singularities of potential theory (Model 4) with the viscous and vortical flow phenomena found in the perceived reality (Model 1). Partly we have shown that with very simple examples. The elliptical wing shape of Sect. 4.3.2, for instance, is far away from that of present-days large aspect-ratio wings. Nevertheless, the concept permits to demonstrate important flow properties. These are present on realistic wing shapes, too, as we show in Chap. 8. The concept moreover permits to assess the capabilities of computation methods of several model levels. For flows over small aspect-ratio delta-type wings with sharp leading edges, for instance, it permits to prove that discrete numerical solutions of the Euler equations, Model 8, indeed lead to an in principle correct presentation of the primary lee-side vortex pair, Chap. 10. The introduction of the concept of flow-off separation at sharp trailing or leading edges on the other hand is the precondition for some of these findings, even if in

116

4 The Local Vorticity Content of a Shear Layer

reality the edges may not be sharp in the exact sense of the word, Chap. 6. The Kutta condition in the reality of lifting wings, too, is a much more intricate concept as it usually is perceived. It is important to be aware of these facts when treating flow problems of lifting wings.

4.6 Problems Problem 4.1 At the beginning of Sect. 4.1 it is assumed that the considered shear layer is very thin. Check this for a boundary-layer flow. Assume a flat wing surface with a chord length of L = 5 m and flight at H = 10 km altitude with the Mach number M∞ = 0.8. Assume at the wing’s surface the recovery temperature. How thick is the boundary layer for (a) laminar and (b) turbulent flow at x/L = 0.5 and at x/L = 1, there with the strong interaction being disregarded. Is the assumption justified? Problem 4.2 Derive the vorticity-content integral of the Rankine vortex, Sect. 4.2.1. The z- component of vorticity in cylindrical coordinates reads: ωz =

1 d(r vθ ) . r dr

Problem 4.3 The upper side of an airfoil often is called the suction side, and the lower side the pressure side. What actually is the situation? Is there suction in the sense of the word at the upper side, how comes the lift into being? Problem 4.4 Prove that the s-direction in Fig. 4.12 is a vortex line. Problem 4.5 Circulation theory, basically belonging to potential theory (Model 4), finds the lift of an airfoil with the help of a circulation Γ . How can this be interpreted physically when looking at Figs. 4.14 and 4.15 in Sect. 4.3. Remember that potential theory permits to superimpose different flow fields/solutions. Problem 4.6 In Fig. 4.18 the spanwise circulation distribution is shown with the circulation Γ0 at y = 0 and a diminishing Γ (y) to the left and the right side. Why is Γ (y = 0) smaller than Γ0 ? This is reflected mathematically in a condition at the trailing edge. What is its name? Problem 4.7 Consider Fig. 4.18. How must the trailing vortex layer be sketched in the frame of the lifting-line theory, steady movement of the wing assumed? What kind of vorticity content does it carry? Kinematically active or inactive or both of them? Where does it come from? Problem 4.8 In Fig. 4.18, lower part, at the symmetry line y = 0 the vortex layer is indicated with a finite extent. How can this be interpreted?

References

117

References 1. Hirschel, E.H.: Considerations of the vorticity field on wings. In: Haase, W. (ed.) Recent Contributions to Fluid Mechanics, pp. 129–137. Springer, Berlin (1982) 2. Hirschel, E.H., Fornasier, L.: Flowfield and vorticity distribution near wing trailing edges. AIAA-Paper, 1984–0421 (1984) 3. Hirschel, E.H.: On the creation of vorticity and entropy in the solution of the Euler equations for lifting wings. MBB-LKE122-AERO-MT-716, Ottobrunn, Germany (1985) 4. Hirschel, E.H., Rizzi, A.: The mechanisms of vorticity creation in Euler solutions for lifting wings. In: Elsenaar, A., Eriksson, G. (eds.) International Vortex-Flow Experiment on Euler Code Validation. FFA, Bromma (1987) 5. Eberle, A., Rizzi, A., Hirschel, E.H.: Numerical Solutions of the Euler Equations for Steady Flow Problems. Notes on Numerical Fluid Mechanics, Vol. 34. Vieweg, Braunschweig Wiesbaden (1992) 6. Hirschel, E.H.: Vortex flows: some general properties, and modelling. Configurational and manipulation aspects. AIAA-Paper, 96–2514 (1996) 7. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014) 8. Fischer, J.: Selbstadaptive, lokale Netzverfeinerung für die numerische Simulation kompressibler, reibungsbehafteter Strömungen (Self-Adaptive Local Grid Refinement for the Numerical Simulation of Compressible Viscous Flows). Doctoral Thesis, University Stuttgart, Germany (1993) 9. Stanewsky, E., Puffert, W., Müller, R., Batemann, T.E.B.: Supercritical Airfoil CAST 7— Surface Pressure, Wake and Boundary Layer Measurements. AGARD AR-138, A3-1–A3-35 (1979) 10. Zierep, J.: Der senkrechte Verdichtungsstoß am gekrömmten Profil. ZAMP, vol. IXb, pp. 764– 776 (1958) 11. Lighthill, J.: An Informal Introduction to Theoretical Fluid Mechanics. Clarendon Press, Oxford (1986) 12. Mangler, K.W., Smith, J.H.B.: Behaviour of the vortex sheet at the trailing edge of a lifting wing. Aeronaut. J. R. Aeronaut. Soc. 74, 906–908 (1970) 13. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011) 14. Schwamborn, D.: Laminare Grenzschichten in der Nähe der Anlegelinie an Flügeln und flügelänlichen Körpern mit Anstellung (Laminar Boundary Layers in the Vicinity of the Attachment Line at Wings and Wing-Like Bodies at Angle of Attack). Doctoral thesis, RWTH Aachen, Germany, 1981, also DFVLR-FB 81-31 (1981) 15. Zahm, A.F.: Flow and force equations for a body revolving in a fluid. NACA Rep. No. 323 (1930) 16. Maruhn, K.: Druckverteilung an elliptischen Rümpfen und in ihrem Außenraum. Deutsche Luftfahrtforchung, Jahrbuch 1941, SI, pp. 135–147 (1941) 17. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, vol. 1 and 2, Springer, Berlin/Göttingen/Heidelberg, 1959, also: Aerodynamics of the Aeroplane, 2nd edn (revised). McGraw Hill Higher Education, New York (1979) 18. Prandtl, L.: Tragflügeltheorie, I. und II. Mitteilung. Nachrichten der Kgl. Ges. Wiss. Göttingen, Math.-Phys. Klasse, 451–477 (1918) und 107–137 (1919) 19. Spreiter, J.R., Sacks, A.H.: The rolling up of the trailing vortex sheet and its effect on the downwash behind wings. J. Aeronaut. Sci. 18, 21–32 (1951)

Chapter 5

The Matter of Discrete Euler Solutions for Lifting Wings

With the development in particular of fighter aircraft with delta-type wings the phenomenon of lee-side vortices became a topic in the field of aerodynamics. (Configurational, fluid-mechanical and operational topics related to aircraft with such wings are treated in Chaps. 10 and 11.) For the description of the lee-side vortices analytical and semi-empirical methods, as well as panel methods (Model 4 in Table 1.3) were developed, which made necessary the geometrical modeling of the vortex sheets (feeding layers) emerging from the (sharp) leading edges of the delta wing [1]. These methods at the beginning of the 1980s gave way to discrete numerical solutions of the Euler equations (Model 8 of Table 1.3). Now since long of course methods based on Model 10 and higher govern the field. Nevertheless it is still appropriate to look at the Euler methods in view of the general problem of lifting-wing flow simulations. Here we study the topics, which emerged when the first Euler solutions came into use, viz. the creation of vorticity and the entropy rise in the flow field. These topics are treated in the following three sections using the concept of kinematically active vorticity. Examples are given then in Sect. 8.3 for a large aspect-ratio wing and in Sect. 10.4 for a delta wing.

5.1 Vorticity Creation in Euler Solutions of Lifting-Wing Flow The discrete modeled Euler solution, Model 8, for lifting-wing flow must include the wake (trailing vortex layer) of reality (Model 1) in one or the other form. We have discussed in Sect. 4.2.6—with the help of Model 2—how this wake for high Reynolds numbers shrinks into the discontinuity layer—or trailing vortex layer—of potential wing theory (Model 4). © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_5

119

120

5 The Matter of Discrete Euler Solutions for Lifting Wings

Fig. 5.1 View of the three-dimensional wake of a lifting finite-span wing in steady sub-critical flow [4]. Left: the wake in the limit Rer e f → ∞ as discontinuity layer of potential theory, see Fig. 4.13. Right: the discontinuity layer widened up by numerical diffusive transport of vorticity: the Euler wake

Hidden in this discontinuity layer is the kinematically active vorticity of reality. Analog to potential wing theory this layer should appear also in an ideal Euler solution (Model 7). That would be a weak solution of the Euler equations (Model 8) as was shown in [2]. In a discrete modeled conservatively formulated Euler solution the discontinuity layer is smeared out over some grid points. That is in analogy to the captured shock wave in supersonic flow [3]. For subcritical flow we show this situation schematically in Fig. 5.1.1 At the left side we see the discontinuity layer as discussed in Sect. 4.2.6. Hidden in this layer is the kinematically active vorticity content Ωs . At the right side we see the by the diffusive transport widened discontinuity layer resulting from the discrete Euler solution. The diffusive transport in the strict sense is a “false” diffusive transport. Nevertheless, the “Euler wake” in principle has the right properties. The u(z)-profile now is uniform, because in an inviscid flow model no skin-friction drag and no viscosity induced pressure or form (profile) drag is present, hence no kinematically inactive vorticity. Actually the u(z)-profile in Fig. 5.1 should be shown as being non-uniform, because of the properties of the Euler solution upstream of the wake on the wing’s surface.2 There we have on both the upper and the lower side what we loosely call 1 The following discussion also holds for supercritical flow. There the whole vorticity wake including

the entropy-layer wake must be considered, Sect. 4.2.3. in this regard also the discussion in [5].

2 See

5.1 Vorticity Creation in Euler Solutions of Lifting-Wing Flow

121

an Euler boundary layer. Usually that is seen as a total-pressure loss of the flow along the body surface, and also of a total-temperature loss [4]. In regions of flow deceleration this can be rather pronounced.3 However, as long as at the trailing edge the external inviscid velocities u eu and u el are the same, the vorticity content of the u(z)-profile is kinematically inactive, which means that the u(z)-profile can have any form. The v(z)-profile is the vortex-like profile shown in Fig. 4.12, right. The precise form of this profile also does not matter. What matters is that the kinematically active vorticity content, which was hidden in the discontinuity layer, has reappeared, Fig. 5.1, right. This vorticity content “carries” the induced drag, and its appearance is compatible with a discrete inviscid model of finite-span lifting-wing flow (Model 8). Provided a Kutta condition is present, the Euler wake appearing in discrete numerical solutions of the Euler equations for lifting-wing flow—past either large or small (delta) aspect-ratio wings—thus is a necessary and sufficient property of such solutions. The examples presented in Sect. 8.3 (large aspect-ratio wing) and in Sect. 10.4 (delta wing with sharp leading edges) give proof of the applicability of Model 8 methods, see also Sect. 3.12.1. The appearing vorticity content in principle has the right amount. To it belongs an entropy rise, or equivalently, a total pressure loss. This is treated in the following section.

5.2 Vorticity and the Related Entropy Rise The vorticity present in the Euler wake is accompanied by an entropy rise, respectively a total pressure loss. We investigate that now with the help of Crocco’s theorem, Sect. 3.5. This law for steady inviscid iso-enthalpic flow reads v × r ot v = v × ω = −T grad s,

(5.1)

where v is the velocity vector, ω the vorticity vector, T the temperature and s the entropy. We introduce ω in the stretched form, Eq. (4.4), and for Rer e f → ∞ (note that w → 0 in this case) and find4   ∂s 1 ∂ u 2 + v2 = −T . (5.2) 2 ∂z ∂z

3 The

total-pressure loss depends on the chosen variables, the discretization scheme, and the gridfineness. Depending on the simulation problem at hand, one has to be aware of that. 4 z is the stretched coordinate, however, we have omitted the tilde.

122

5 The Matter of Discrete Euler Solutions for Lifting Wings

Assuming a perfect gas, we relate the temperature T to the total temperature Tt and the velocities u and v: T = Tt −

u 2 + v2 . 2c p

(5.3)

We return to the element of a generalized shear layer, Fig. 4.1, integrate Eq. (5.2), beginning at the lower edge z = δl and obtain s(z) − s(z = δl ) = c p ln

−2 c p Tt + u 2 (z) + v 2 (z) . −2 c p Tt + u 2el + ve2l

(5.4)

At the lower edge the external inviscid velocity can be written as u 2el + ve2l = |V el |2 .

(5.5)

The Mach number Mel of the external inviscid flow then reads: Mel =

|V el | (γ R Tel )0.5

.

(5.6)

We introduce these two expressions into Eq. (5.4) and finally arrive at s(z) − s(z = δl ) Δs(z) = = cp cp   u 2 (z) + v 2 (z) γ−1 2 Mel (1 − )+1 . = ln 2 |V el |2

(5.7)

The equivalent total pressure loss Δpt (z) is, see, e.g. [3] Δpt (z) = ptel − pt (z) = ptel (1 − e

γ − γ−1

Δs(z) cp

).

(5.8)

Thus the entropy rise and the equivalent total pressure loss are functions of the squares of the velocities u and v in the wake and the edge velocity |V el |. For the thermodynamically singular case of zero Mach number—incompressible flow—no entropy rise happens. A total pressure loss, however, occurs, but it cannot be prescribed in the frame of our consideration. We look now at four wake cases, the third one being the Euler wake: 1. Two-dimensional wake, Fig. 4.4a: Eq. (5.7) simply reduces to   γ−1 2 u 2 (z) Δs(z) Mel (1 − 2 ) + 1 , = ln cp 2 u el

(5.9)

5.2 Vorticity and the Related Entropy Rise

123

Although the wake is kinematically inactive, we observe, as expected, an entropy rise throughout the wake and hence a total pressure loss. Regarding the wake in Fig. 4.4b, which is present in steady supercritical motion, we have to remember that in that case the entropy wake includes the wake of the shock wave, Fig. 4.6. 2. Three-dimensional lifting-wing wake, Fig. 4.12 right: In the decomposed view both the kinematically active wake v(z) and the kinematically inactive wake u(z) are contributing to the entropy rise and hence to the total pressure loss. Equation (5.7) explicitly contains these wake parts. 3. Euler wake, Fig. 5.1 right: In this case we have assumed u(z) = u el = const. and Eq. (5.7) becomes   u 2el + v 2 (z) γ−1 2 Δs(z) Mel (1 − | Rer e f →∞ = ln )+1 . (5.10) cp 2 |V el |2 We see that with u(z) = u el = const. the entropy rise is reduced compared to that of the real wake, but because v(z) < vel , it does not vanish. We note that here the remark in [5], see also above, Sect. 5.1, regarding the uniformity or non-uniformity of u(z) has a point. Although the u-component vorticity content still remains to be zero, the entropy production is affected. 4. Euler wake of the infinite swept wing: On Sect. 4.4 we have noted that no kinematically active vorticity leaves the trailing edge of the infinite swept wing. The trailing-edge shear angle ψe is zero. This means, Fig. 5.1, that vel and veu as well as v(z) are zero. Because we assume for the Euler wake u(z) = u el = const., Eq. (5.7) becomes   u 2el γ−1 2 Δs(z) Mel (1 − 2 ) + 1 = 0. | Rer e f →∞ = ln (5.11) cp 2 u el The result is that in this case no entropy rise happens, except for the upper and lower Euler boundary layer.

5.3 Critical Evaluation Euler codes, i.e., Model 8 methods, Table 1.3, now are for a long time in use, originally even for data-set generation, currently at most only in the earlier phases of flightvehicle shape definition. Nevertheless, a look at their capabilities and shortcomings is appropriate. We basically follow the discussion, which is given in [4]. In Sect. 5.1 of this chapter we have argued that, in analogy to potential theory (Model 4), an Euler solution for a finite-span lifting wing exhibits a discontinuity surface downstream of the trailing edge, a computational realization of the trailing vortex layer. At a wing with highly swept leading edges—the main geometrical property of delta wings—above a critical combination of the “normal angle of attack

124

5 The Matter of Discrete Euler Solutions for Lifting Wings

α N ” and the “normal leading-edge Mach number M N ” also a discontinuity surface is leaving each leading edge, Sect. 10.1. These discontinuity surfaces are the “feeding layers” of the lee-side vortices. We note in passing that over both wing types also secondary and even tertiary vortex phenomena can be present, Model 1. In addition we note that also at the trailing edge of the delta wing a trailing vortex layer is present, similar to the situation of large aspect-ratio wing, Fig. 1.10b. The said discontinuity surface is the vorticity wake, Model 1, in reality, but now in the limit Rer e f → ∞. True discontinuity surfaces—vortex sheets, as in potential theory—treated in Euler methods would be exact, but would hamper very much the solution methods for general wings and aircraft. The reason is that they would have to be “fitted”, i.e., presented by coordinate surfaces, like shock fitting. Therefore the appearance of Euler wakes of finite thickness, which are “captured” by the solution— similar to shock capturing—is acceptable. Indeed, to capture Euler wakes is the only practicable approach, and that in view of the fact that the kinematically active vorticity content appearing in them in principle is described exactly. A study of numerical Euler solutions, however, shows at least two problem areas regarding the phenomenon “Euler wake”: 1. In the ideal case the thickness of the Euler wake (Model 8) should be that of the real wake (Model 1). It is evident that, if the Euler wake is smeared out over, say, four cells, it depends on the size of the cell, i.e., the local fineness of the discretization of the computation domain, how thick the wake is. In the case of an isolated wing, the thickness of the Euler wake does not matter, as long as the wake transports the right vorticity content. If, however, vortex sheets or vortices interact with each other or with parts of the configuration—typical, for instance, for the lee-side vortices of delta wings—the solution will be erroneous, if the thickness of the domain of the vortex sheets or the vortices is inadequately represented. 2. Due to the kind of discretization of the computation domain behind a wing, a part of the kinematically active vorticity may be lost and hence also a certain amount of the circulation in the wake. This in general holds for all discrete numerical solutions of such flow cases, see, e.g., the CRM wing wake case in Sect. 8.4. However, in both panel (Model 4) and in RANS (Model 10) and higher model solutions usually a rather coarse discretization is applied in the flow domain behind the wing. Despite this, at the wing forces and moments as well as flow properties can be computed to a sufficient degree of accuracy. This also holds for Euler methods, of course only within the inherent limitations of the Model 8 approach. This concerns an isolated large aspect-ratio wing. But if the trailing vortex layers or vortices interact with each other or with the empennage, the solution will have deficits, if the discretization is inadequate. This problem of course is present also with low aspect-ratio wings, in particular with delta wings. If lee-side vortex

5.3 Critical Evaluation

125

phenomena are present, a suitable lee-side discretization or an automatic mesh adaptation is of utmost importance. The first problem area has to do with the diffusive properties of Model 8 Euler solutions, as to a degree also has the second one. In addition it must be ensured that the far-field boundary conditions, which in most of the methods in use are characteristic boundary conditions, do not influence the conservation of circulation in the far wake, see, e.g., [4]. Even for lifting airfoils in some codes the circulation is found to vanish near the outer boundary of the computation domain. All this is a topic also for Model 10 (RANS/URANS) and for Model 11 (Scale-resolving) simulations.

5.4 Problems Problem 5.1 Why can u(z) in Fig. 5.1 have any shape as long as u(δu ) = u(δl ). Problem 5.2 Why are discrete numerical solutions of the Euler equations (Model 8) of interest in design work? What are the differences in terms of capabilities compared to those of Model 4 and Model 10/11 methods? Problem 5.3 What are the differences in terms of costs? Look at the whole processes of the different methods. Problem 5.4 If u(z) is not constant, what wake case regarding the entropy rise holds? Why does the function u(z) matter? Problem 5.5 Why is a proper grid resolution necessary in discrete numerical solutions? Which flow models in Table 4.1 are concerned.

References 1. Rom, J.: High Angle of Attack Aerodynamics. Springer, Heidelberg (1992) 2. Powell, K.G., Murman, E.M., Perez, E.S., Baron, J.R.: Total pressure loss in vortical solutions of the conical Euler equations. AIAA J. 25, 360–368 (1987) 3. Hirschel, E.H.: Basics of Aerothermodynamics, 2nd edn (revised). Springer, Cham (2015) 4. Eberle, A., Rizzi, A., Hirschel, E.H.: Numerical Solutions of the Euler Equations for Steady Flow Problems. Notes on Numerical Fluid Mechanics, vol. 34. Vieweg, Braunschweig Wiesbaden (1992) 5. Hoeijmakers, H.W.M.: Modelling and numerical simulation of vortex flow in aerodynamics. AGARD-CP-494, 1-1–1-16 (1991)

Chapter 6

About the Kutta Condition

The Kutta condition in all physical and mathematical flow models defined in Sect. 1.5 is generally understood as appearing at acute edges of aerodynamic surfaces. That can be the “zero thickness” trailing edge of a large aspect-ratio wing or the “sharp” leading edge of a delta wing. Usually it is assumed that the flow leaves the trailing edge in bisector direction. The reality, however, demands a closer look, because the Kutta direction—we call it this way—is not necessarily the same as the bisector direction. The closer look is necessary for an understanding of the related flow phenomena in general. And then it holds for the application of the physical and mathematical models of Sect. 1.5 to the real shapes of trailing (and leading) edges and hence also for the grid generation for discrete numerical methods. Accordingly we treat two different topics in the following sections: decambering phenomena and the Kutta condition and the matter of the Kutta condition and direction in reality.

6.1 Decambering Phenomena In the subcritical case (subsonic flight regime) inviscid flow-off separation from an acute edge happens in the bi-sector direction. However, at the trailing edge of a lifting wing in reality, the actual flow-off direction—the Kutta direction—is slightly deflected upward. This leads to a reduction of the lift. This effect is called decambering. There are two basic mechanisms: boundary-layer decambering and shock-wave decambering, the latter appearing in the supercritical case (transonic and supersonic flight regime). We discuss and illustrate these kinds of decambering at airfoils, i.e., in two dimensions. The results hold in three dimensions, too, and on wings and aerodynamic trim, stabilization and control surfaces of all kinds, as well as lift enhancement surfaces. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_6

127

128

6 About the Kutta Condition

6.1.1 Boundary-Layer Decambering Boundary-layer decambering is due to the influence of the boundary-layer displacement properties [1].1 The boundary-layer decambering effect at an airfoil in subcritical motion manifests itself in a slight reduction of the lift slope dC L /dα, which (inviscid) thin airfoil theory obtains as dC L /dα = 2π [3]. The effect decreases with increasing Reynolds number. To understand the effect, we consider the situation at a symmetric airfoil. We look first at the situation, which we find at zero angle of attack. There the boundarylayer displacement thicknesses—small, but non-negligible—at the upper and the lower side of the airfoil are the same at the trailing edge of the airfoil.2 The flow-off direction is that of the bisector direction of the trailing edge, which is the same as the freestream direction, decambering is not present. (However, with a non-symmetric airfoil, boundary-layer decambering already is an issue at the nominal non-zero lift situation.) At angle of attack, we observe that the displacement thickness at the trailing edge is larger on the upper side, the suction side, than on the lower side, the pressure side. Hence the flow-off direction shifts somewhat toward the upper side—decambering happens. The reasons for the different displacement thicknesses are the following: (a) With positive angle of attack, the forward stagnation point moves away from the nose point to the lower side of the airfoil. This means an increase of the boundarylayer running length over the upper side and hence a thicker boundary layer—and larger displacement thickness—at the upper side of the trailing edge. (b) The boundary layer over the suction side—because of the larger external inviscid velocity there—is higher loaded with an adverse pressure gradient than that over the pressure side. This means an additional increase of the displacement thickness of the upper-side boundary layer at the trailing edge compared to that of the lower-side boundary layer. (c) Depending on the actual surface pressure and hence the external inviscid velocity distributions, the locations of the laminar-turbulent transition on the upper and the lower side are different. If, and that generally is the case, the transition location on the upper side is more forward than that on the lower side, this results in a longer running length of the turbulent boundary layer over the upper side, and hence a further increment of the displacement thickness at the upper side of the trailing edge.

1 The

concept of boundary-layer decambering seems to have been developed by M. J. Lighthill [2]. separation, like ordinary separation, locally is characterized by strong interaction of the external inviscid and the boundary-layer flow, Sect. 2.2. Nevertheless we can speak here simply of the boundary layers at the trailing edge.

2 Flow-off

6.1 Decambering Phenomena

129

Regarding boundary-layer decambering we summarize: 1. Boundary-layer decambering occurs in subcritical and supercritical cases, in the latter case together with shock-wave decambering (see next sub-section). It is due to the displacement effect of the boundary layers in the vicinity of the trailing edge. 2. The magnitude of the displacement effect depends on the shape of the airfoil and the angle of attack, with that on the surface pressure distribution and then on the location of laminar-turbulent transition. The primary flow parameter is the Reynolds number. The higher the Reynolds number, the smaller is the displacement thickness and hence the boundary-layer decambering. 3. This all holds for both airfoils and wings as well as for stabilization and trim surfaces, see above.

6.1.2 Shock-Wave Decambering Shock-wave decambering is an effect, which appears—in addition to boundarylayer decambering—in the supercritical case, i.e., in the transonic flight regime. To understand shock-wave decambering, we have to look at the flow situation at the trailing edge of a lifting airfoil. Consider as example the case illustrated in Fig. 4.6 on Sect. 4.2.3. There on the suction side of the airfoil a supersonic flow pocket with a terminating shock wave is present. We treat the inviscid case, however in a very simplified way. For all steady motion cases, which we consider here, at the upper and at the lower side of the trailing edge the (static) surface pressures p are equal, if we disregard possible near-wake curvature effects. In the subcritical case also the total pressures pt are the same on both sides of the trailing edge, also the velocities V u and V l , and hence also the absolute values of the momentum-flux vectors |Q|. To define the latter we consider the flow in a small domain n at and parallel to the surface. If a x is the unit vector of a surface-oriented curvilinear grid at the surface in downstream direction, [1], we approximately obtain the momentum-flux vector as Q = ρ|V 2 |n a x . In the subcritical case the absolute values of the momentum-flux vectors are the same at the upper (u) and the lower (l) side of the trailing edge: |Q u | = |Q l |. This situation is present in all subcritical cases, but also in the supercritical case of a symmetric airfoil at zero angle of attack (non-lifting case). In the supercritical case shock waves are present, (a) either only one at the suction side of the airfoil as shown in Fig. 4.6, or, (b) at both the upper and the lower side, that on the suction side having a larger strength.

130

6 About the Kutta Condition

Fig. 6.1 Inviscid flow at the trailing edge of a lifting supercritical airfoil with an embedded shock wave at the suction side: deflection of the flow (Kutta direction) out of the bi-sector (shock-wave decambering) [4]

In case (a) the ensuing total pressure loss leads to a reduction of the suction side velocity and hence of the momentum flux |Q u | at the trailing edge (this reasoning also holds for case (b)): |Q u | < |Q l |. (The magnitude of the total pressure loss at the surface can be found [4], if the pre-shock Mach number has been determined, because the terminating shock wave impinges vertically on the surface, inviscid flow [5].) The consequence in both cases is the upward deflection of the flow out of the bi-sector of the trailing edge, the shock-wave decambering, as illustrated in Fig. 6.1. Shock-wave decambering came into view in the late 1970s, early 1980s, when discrete numerical methods based on the compressible full potential equation, Model 5 of Table 1.3, were developed. These methods, employing the conservative formulation of the potential equation, were able to describe embedded shock waves, although with problems regarding their strength and location. Hence, the predictions of the aerodynamic properties of airfoils were erroneous. The problem with the full potential equation methods is that, although the shock waves are captured, the in reality occurring total pressure loss does not arise. (Potential methods are not able to describe a total pressure loss.) Hence the flow leaves at the trailing edge in bi-sector direction, i.e., the shock-wave decambering is not described. Although schemes were developed to overcome this problem, see, e.g., [6–8], full potential equation methods finally gave way to discrete numerical Euler methods (Model 8) and Navier–Stokes/RANS methods (Model 9/10), which both inherently describe shock-wave decambering. Regarding shock-wave decambering we summarize: 1. Shock-wave decambering occurs in supercritical cases. The effect is the larger, the larger the pre-shock Mach number is. Transonic airfoils may have supersonic flow pockets and hence terminating shock waves on both sides. Then the effect becomes weakened and in extreme cases may even be canceled.

6.1 Decambering Phenomena

131

2. The terminating shock wave has another effect. Below a pre-shock Mach number at the shock foot of M f oot ≈ 1.3−1.35 the boundary layer does not separate, a weak shock-wave/boundary-layer interaction is present, Sect. 9.1. In the inviscid picture the shock terminates at right angle to the surface, see above. For all supersonic Mach numbers the unit Reynolds number decreases over a normal shock wave [9]. The boundary layer behind the shock hence gets a thickness increment compared to that in front of the shock.3 This causes an extra thickening of the downstream boundary layer. At M f oot  1.3–1.35 a strong shock/boundary-layer interaction with a separation bubble happens. The in both cases also enlarged displacement thickness enlarges the boundary-layer decambering. (At even higher Mach numbers we get shock stall, i.e., a complete separation behind the shock foot and in consequence transonic buffet, Sect. 9.1. Then our considerations become invalid.) For an example of the combined effect of boundary-layer and shock-wave decambering see below Fig. 6.8 in Sect. 6.5.

6.2 Kutta Condition and Kutta Direction in Reality The Kutta condition and the Kutta direction in the perceived reality (Model 1 of Table 1.3) are different from that considered in Model 4 and 7. There the inviscid flow leaves smoothly in the bi-sector direction of the trailing edge, Fig. 4.15. We first consider the two-dimensional case. Regarding the external inviscid flows over the upper and the lower side of the trailing edge, we have in viscous modeled flow, Model 2—and also in Navier–Stokes flow and derivatives flow (Model 9/10)— in a sense two Kutta directions. They are, moreover, not in bisector direction due to the decambering effects. Regarding the actual Kutta direction, it helps to define it as the direction of the near-wake center line, Fig. 4.6 in Sect. 4.2.3. In the three-dimensional case we also have two external inviscid Kutta directions due to the decambering effects. Moreover, in the lifting case of the finite-span wing they are sheared by the trailing-edge flow shear angle ψe , which is a function of the span-wise location. Between them the viscous flow has a complex pattern, Fig. 4.12 in Sect. 4.2.6. Also here we define as Kutta direction the direction of the near-wake center line, which in that figure is the s-coordinate. This line is a vortex line, which is deflected from the x-coordinate (chord direction) by the positive or negative vortexline angle . In the following section we will see that in practice on operational aircraft the situation is even more complex. Trailing edges in reality have a finite thickness.

3 All

boundary-layer thicknesses depend on the inverse of some power of the Reynolds number, Appendix A.5.4.

132

6 About the Kutta Condition

6.3 Geometric Properties of Trailing and Leading Edges of Actual Wings Trailing edges usually are considered as having acute angles or being sharp edges. In reality they are not sharp in the sense of the word.4 Manufacturing demands lead to a certain bluntness, but also practical demands do so. Leading edges of subsonic and transonic transport aircraft are well rounded. This also holds if aircraft have swept wings in order to shift the transonic drag divergence to Mach numbers as high as possible. This shift can also be achieved with the unswept “thin wing”. Then the leading edge has to be sharp. Highly swept leading edges are found on supersonic aircraft. The leading edges of their wings or strakes usually have a rather small bluntness. However, often they are considered as being sharp. We look at several aspects of geometric properties of trailing edges and leading edges: (a) what wing trailing-edge properties are found on existing aircraft, (b) how are the aerodynamic properties of an airfoil or wing affected by a given trailing-edge thickness, (c) when can an edge be considered as to be “aerodynamically” sharp in the sense of Küchemann (Sect. 6.4). The literature is vast. However, we do not intend to give a systematic survey. For airfoil and wing layout see, e.g., [11, 12]. We note that NACA in 1945 published airfoil shapes with finite trailing-edge thickness [13]. They have, for instance, ratios ‘trailing-edge thickness’ to ‘local chord length’ h/c = 0.126 per cent (NACA 0006 Basic Thickness Form), or h/c = 0.252 per cent (NACA 0012 Basic Thickness Form). A matter is also the question of finite-angle versus cusp trailing edges of supercritical airfoils, the latter also for practical reasons with finite thickness. The reader is referred in this regard to [11]. In the following considerations we assume the typical slender airfoil or chord section of a wing. Of course the airfoil shape plays a role, in particular also the trailing-edge angle.5 Detailed airfoil and wing design parameters cannot be treated in the frame of this book. (a) Trailing and leading edge properties on existing aircraft. Wing trailing-edge thicknesses found on subsonic transport aircraft range from h ≈ 0.5 to 1 cm.6 The ratio h/c lies between 0.1 and 0.5 per cent.7 On transonic transport aircraft (moderate leading-edge sweep) one finds h ≈ 1–1.5 cm, with local

4 Although

D. Küchemann with the term “aerodynamically sharp edge”, [10], to some degree circumvented the topic of geometrical properties of trailing edges, we believe that a closer look is justified. 5 A large trailing-edge angle θ may even lead to a negative dC /dα at small angles of attack [14]. L 6 The authors are grateful to the Deutsches Museum in München, where it was possible to investigate the wings of a large number of aircraft. Thanks go also to colleagues who made data available. 7 The actual shapes of blunt trailing edges vary from sharply cut off to well rounded. Although the detailed shapes locally may play a role, we do not take them into account.

6.3 Geometric Properties of Trailing and Leading Edges of Actual Wings

133

ratios h/c ≈ 0.2–0.5 per cent. It is intended in the future to come down to h ≈ 0.5 cm in order to reduce the drag penalties, which are associated with finite trailing-edge thicknesses. We also note the data of the Common Research Model (CRM), see Sect. 8.4, which is a test configuration of the international AIAA CFD Drag Prediction Workshop. At the location of the Yehudi break of the trailing edge, the thickness is h = 1.31 cm and the ratio h/c ≈ 0.15 per cent. On the CRM wind-tunnel model near the wing tip the ratio is h/c ≈ 0.48 per cent. This thickness is needed in order to place a gauge there. The situation is similar on modern fighter aircraft with highly swept leading edges. Trailing-edge thicknesses vary between h ≈ 0.5 and 2 cm. The thinnest trailing edge was found at a MiG 21 fighter with h ≈ 0.15 cm. However, the delta wing shapes of all aircraft lead to small local values h/c. At the root region of the wings h/c ≈ 0.1 per cent is found, at the tip region h/c ≈ 0.3–0.5 per cent. With wings possessing highly swept leading edges also the leading-edge radius r L E is of interest. That usually varies from the root region to the tip region. In the root region we find r L E ≈ 2–3 cm, and in the tip region r L E ≈ 0.7–1.5 cm. The ratio r L E /c accordingly ranges from 0.5 to 0.1 per cent. The drag divergence problem, which arises in the transonic flight regime, has led to the supercritical airfoil design in combination with leading-edge sweep. An alternative is to use the already mentioned “thin wing”. The leading edge of the trapezoidal wing of the F-104 Starfighter—the only example of a thin wing for which it was possible to obtain data—has a radius r L E < 0.1 cm. The ratio ‘maximum wing thickness’ to ‘chord length’ of the wing is d/c = 3.36 per cent. The trailing-edge thickness is h ≈ 0.25 cm. With chord lengths of c ≈ 3.2 m (root) and c ≈ 1.45 m (tip) we have trailing-edge values h/c ≈ 0.1–0.2 per cent. A special situation arises with hypersonic flight vehicles. There thermal loads and issues of structural integrity may demand quite blunt leading and trailing edges. A notable example is the trailing edge of the wedge-like vertical stabilizer of the X-15, which, however, was needed in order to obtain at hypersonic speeds the necessary directional stability.8 (b) Trailing-edge thickness and aerodynamic performance of airfoils. An early systematic investigation of the influence of a blunt trailing edge of an airfoil was reported in 1927 [15]. At the Aerodynamische Versuchsanstalt (AVA) Göttingen wind tunnel experiments were made with the asymmetrical airfoil No. 508 and with the symmetrical airfoil No. 460. Both airfoils had the length c = 20 cm. The airfoils were systematically truncated in 2 cm steps, Fig. 6.2. Lift-drag coefficient curves and lift-moment coefficient curves were measured.

8 In

[3] a detailed discussion of the fluid mechanical aspects of this case can be found.

134

6 About the Kutta Condition

Fig. 6.2 Truncation of the airfoils No. 508 and 460 [15]

We discuss the results for the first 2 cm truncation. The airfoil No. 508 then has the ratio ‘trailing-edge thickness’ to ‘local chord length’ h/c ≈ 2.75 per cent, and the No. 460 airfoil h/c ≈ 5.5 per cent. We note that both values are much larger than those given above.  The experimental data show an increase of the drag at zero lift, C D0 , of about 20 per cent for the No. 508 airfoil, and about 63 per cent of the No. 460 airfoil.9  The maximum lift, C L max , is reduced by about 9 percent for the No. 508 airfoil, and about 3.5 per cent for the No. 460 airfoil. The center of pressure is moved forward as to be expected. Regarding the drag increase, generally the symmetrical airfoil is more affected by the truncation than the asymmetrical one. This is due to the larger trailing-edge angle θ of the airfoil 460. The measurements were made in a low-speed wind tunnel. Both airfoil models had a span of 1 m. In the report the Reynolds number and the state of the boundary layer are not given, but laminar-turbulent transition can be expected to have occurred at or shortly downstream of the maximum thickness locations of the airfoils. Coefficients also are given related to the actual shortened airfoil lengths. Some of the conclusions in [15] are not acceptable, because then not the coefficients must be considered, but the actual forces. Nevertheless, the data show the expected changes of the aerodynamic properties, if a trailing edge has a finite thickness. Even if manufacturing and practical demands lead to a certain thickness, it should be as small as possible in order to avoid, in particular, drag penalties. The topic of finite trailing-edge thickness of course must be seen in the general frame of airfoil and wing design, see, e.g., [10–12, 16]. The influence of the trailing-edge properties on the aerodynamic performance of an airfoil or wing is one topic. The other topic is the flow at and behind a blunt trailing edge. In general it can be assumed that the two boundary layers, which flow off the surface at the respective edge, have thicknesses much larger than the edge 9 The

prime indicates that the coefficient is related to the original airfoil length of 20 cm.

6.3 Geometric Properties of Trailing and Leading Edges of Actual Wings

135

thickness. Nevertheless, an embedded wake will be present, which, due to the vortex shedding, has an unsteady character. For two-dimensional cases several authors report the presence of a von Kármán vortex street, for instance [17]. Respective data are not known for three-dimensional lifting-wing cases, viz., cases with a shear of the upper and the lower external inviscid flow, the wing flow-fields shear, see Fig. 4.17 on Sect. 4.3.2. Related to the vortex shedding is noise generation due to a blunt trailing edge. This in general appears not to be a very critical issue. The reason is that the related frequencies are quite high. This can be seen from a consideration with the help of the Strouhal number. Vortex shedding behind a cylinder (two-dimensional case) has— Reynolds number dependent—Strouhal numbers Sr ≈ 0.2 (Re ≈ 104 ) to Sr ≈ 0.3 (Re ≈ 107 ). These numbers are generally observed for the vortex shedding behind blunt bodies, see, e.g., [18]. (The Reynolds number is defined as Re = ρuh/μ.) We write Sr =

fh , u

(6.1)

f being the shedding frequency, h the diameter of the cylinder, in our case the thickness of the trailing edge, u the free-stream velocity, in our case the external inviscid velocity at the trailing edge (upper and lower side). The relation for the frequency then reads f =

Sr u . h

(6.2)

With typical aircraft flight data we get frequencies in a narrow band at the upper bound of the human hearing domain. This “beep” is clouded by the noise coming from the airframe and its components, and the propulsion system. The blunt trailing-edge noise concerns less the cruise configuration of an aircraft, but mainly the take-off and the landing configuration. The low-noise aircraft is a big topic and the blunt trailing-edge noise is part of that. For a more recent study on this noise and possible reduction concepts see [19]. The situation is similar with wind turbine rotors, but also helicopter rotors. There potentially so called blunt “flatback” airfoils can be employed at the innermost part of the rotor blades. The benefit would be lighter and stronger blade structures. The flatback airfoil is found by opening the trailing edge, not by simply truncating the airfoil as is the rule with trailing edges of aircraft wings. The structural benefit, however, is bought with a possible extra noise generated by the vortex shedding at the blunt trailing edge, see, e.g., [20, 21]. Also here usually only the two-dimensional case is studied. This is permitted because the flatback airfoil is only employed at the inboard section of the rotor.

136

6 About the Kutta Condition

6.4 When Can an Edge Be Considered as to Be Aerodynamically Sharp? This question regards modeling issues in both theoretical/numerical and experimental work. One can imagine that the ratio ‘trailing-edge thickness’ to ‘upper and lower boundary-layer displacement thicknesses’ h/δ1 plays a role. Detailed investigations in this regard and—more so—criteria, are not known to the authors. The approaches of numerical aerodynamics today, Model 10 of Table 1.3, permit a proper grid resolution at blunt trailing or leading edges. The turbulence modeling problem, of course, remains. For pre-design activities Euler methods, Model 8, are in use. With them, a high resolution of, for instance, trailing edges, is not the rule. The situation is different at highly swept leading edges, where it is to ask whether for a given case a somewhat rounded edge can be considered and modeled as being sharp, see below. Experimental work in high Reynolds-number ground facilities should aim for a true representation of the geometrical properties of trailing and leading edges of the aircraft model. This is necessary, because then the ratio h/δ1 can be met. However, the resolution of geometrical details is a general problem, because usually in such tunnels the aircraft models are not very large. In low Reynolds-number facilities the ratio h/δ1 will not be met and the obtained drag data, for instance, will be flawed. Regarding the question when the swept round leading edge of a delta wing can be regarded as aerodynamically sharp, VFE-2 work has shown that the cross-flow bluntness parameter pb at least permits qualitative considerations, Sect. 10.2.4: pb (x) =

rle (x) , b (x)

(6.3) 

where locally rle is the leading edge radius and b (x) the span width, both being defined in the wing cross-section at the wing’s location x. In order to support the matter of the cross-flow bluntness parameter, we try a gedankenexperiment. We assume that it is possible, like for linear (Model 4) potential flow, to split the flow field past the delta wing into one part parallel to the wing’s xaxis and into one part normal to the x-y-plane, parallel to the z-axis. The free-stream component normal to the wing (z-direction) then is vn = sin α u ∞ , that in x-direction vt = cos α u ∞ . Accordingly we get Mn = sin α M∞ and Mt = cos α M∞ . We consider the flow normal to the wing in the form of the flow past the elliptical cylinder sketched in Fig. 6.3. This cylinder, sufficiently flattened, is to represent the cross-section of the delta wing. We define δ = b/d as the inverse thickness ratio of the cylinder. The circular cylinder is given, if δ = 1, the cross section of a delta wing approximately, if δ is very large.

6.4 When Can an Edge Be Considered as to Be Aerodynamically Sharp?

137

Fig. 6.3 Geometrical parameters of the elliptical cylinder. The inverse thickness ratio is δ = b/d. ϕ is the polar angle

If the flow is incompressible, linear potential theory, [22], yields for the flow velocity vc around the cylinder contour (1 + δ)vn . vc =  1 + δ 2 cot 2 ϕ

(6.4)

The surface pressure coefficient then reads cp = 1 − (

vc 2 ) . vn

(6.5)

For the circular cylinder we have δ = 1. Hence we obtain the familiar result at ϕ = 90◦ and 270◦ with the velocity vc,max = 2 vn , and the pressure coefficient c p,min = −3. If the elliptical cylinder is flattened more and more, the maximum velocity at the edges ϕ = 90◦ and 270◦ rises accordingly and with it the favorable pressure coefficient ahead of these locations and the adverse one behind them Fig. 6.4, left part, shows this for small δ numbers at ϕ = 90◦ .

Fig. 6.4 Surface pressure coefficients c p (ϕ) [23]. Left part: small inverse thickness ratios δ, c p in the interval 0◦  ϕ  180◦ . Right part: large inverse thickness ratios δ, c p in the interval 80◦  ϕ  100◦

138

6 About the Kutta Condition

Fig. 6.5 Gradients of the surface pressure coefficient dc p /dϕ(ϕ) [23]. Left part: small inverse thickness ratios δ, dc p /dϕ in the interval 0◦  ϕ  180◦ . Right part: large inverse thickness ratios δ, dc p /dϕ in the interval 80◦  ϕ  100◦ . The locations of the maxima of the adverse gradients are marked and their magnitudes are given

Note that the pressure distribution is symmetric around the “leading edges” at ϕ = 90◦ and 270◦ . For large δ numbers we obtain very steep pressure peaks, Fig. 6.4, right part. For the small thickness ratios the ϕ-gradient of the pressure coefficient as function of the polar angle ϕ is given in Fig. 6.5. We see that the maximum of dc p /dϕ rises with increasing δ and that it moves closer to the very leading edge at ϕ = 90◦ , left part of the figure. For us of interest is the maximum of the adverse gradient, which lies above the “leading edge”, i.e. at ϕ > 90◦ . For the large inverse thickness ratios very high gradients are present, which lie very close to the “leading edge”, right part of the figure. It appears that at least for δ ≈ 200 the leading edge comes close to act as an aerodynamic sharp one.10 If we now consider the inverse thickness ratio δ as being representative for the— along the leading edge varying—bluntness parameter pb of a round-edged delta wing pb |loc =

rle d/2 1 , |loc =  |loc ≡ b b 2δ

(6.6)

we possibly could derive from the above results criteria where ordinary flow separation would occur and where the leading edge indeed could be considered as to be aerodynamically sharp, which would imply flow-off separation.

10 We

gratefully acknowledge that the results shown in the following figures were provided by C. Weiland [23].

6.4 When Can an Edge Be Considered as to Be Aerodynamically Sharp?

139

Concluding we have to note that it is the gradient of the surface pressure, which is the most deciding parameter regarding separation, not the gradient of the surfacepressure coefficient. With dc p dp ρ∞ = sin2 α u 2∞ , dϕ 2 dϕ

(6.7)

we see that the angle of attack plays a role, too. The higher this angle, the higher is the actual maximal pressure gradient. At this point we finish our gedankenexperiment. To continue we would have to introduce a separation criterium. For two-dimensional flow the empirical criterion of B. Thwaites for laminar flow, and that of B. S. Stratford for turbulent flow, see Sect. 2.1, would be candidates. Our flow fields, however, are highly threedimensional, such that these criteria cannot simply be employed. In addition we must remember that our considerations so far were based on linear potential theory. When looking at the flow of reality, we at least must take into account that the surface pressure coefficient cannot become negative beyond all limits. The limit is given by the vacuum pressure coefficient c pvac , Appendix A.1. Figure 6.6 shows this for the general free-stream situation. In our case, however, with Mn = sin α M∞ the vacuum pressure coefficient can reach quite large—negative—numbers. Anyway, we must not overstress our gedankenexperiment.

Fig. 6.6 Vacuum pressure coefficient c pvac as function of the free-stream Mach number M∞

140

6 About the Kutta Condition

6.5 Implicit and Explicit Kutta Condition, Modeling and Grid Generation Issues At acute edges, where flow-off separation is present, we speak of an implicit Kutta condition. If the flow is forced away from the surface by an adverse pressure field, we have an explicit Kutta condition. The explicit Kutta condition has been employed—with mixed success—in order to enforce secondary separation in Euler solutions (Models 7 and 8) of inviscid flow past delta wings at angle of attack, see, e.g., [24, 25]. In general such approaches are not to be recommended. An explicit Kutta condition—enforced by a Kutta panel at the trailing edge of the airfoil—was employed successfully in order to model the shock-wave decambering in solutions of the conservative full potential equation (Model 5). In this approach the purpose of the Kutta panel was to force the trailing-edge flow into the Kutta direction, Fig. 6.1, Sect. 6.1.2. A particular approach is that with a free-floating and deformable, i.e., force-free Kutta panel. With the help of such a panel the grids of discrete numerical methods (Models 8 and 9) can locally be adapted to the flow pattern. That leads in wakes of all kinds to a more appropriate grid orientation and also to a more convenient implementation of turbulence models. We show two examples, which made use of such a panel [26]. However, the prerequisite for this approach is the self-organization of the grid with the help of suitable sensors [27]. The basic situation is given in Fig. 6.7. The left side shows the typical O-grid, which can be a structured or an unstructured grid. On the right side the grid adapted to the Kutta direction is given. It is a typical C-grid, in this case a quasi-prismatic grid. Important is the smooth transition of the grid from the wing surface into the near wake.

Fig. 6.7 Generic trailing edge discretization [26]. Left side: O-grid. Right side: adapted C-grid

6.5 Implicit and Explicit Kutta Condition, Modeling and Grid Generation Issues

141

Fig. 6.8 Flow past the RAE-2822 airfoil (M∞ = 0.73, α = 2.79◦ , Re = 6.5·106 ) [26]. Upper part: initial grid oriented at the bi-sector of the trailing edge. Lower part: final grid oriented at the free-floating deformable Kutta panel

A realistic example is given in Fig. 6.8. The configuration is the RAE-2822 airfoil in a transonic (supercritical) condition. The airfoil has a supersonic flow pocket on the suction side, which is terminated by a shock wave, like shown in Fig. 4.6 on Sect. 4.2.3. The grid is a hybrid Cartesian grid, with an embedded quasi-prismatic grid for the resolution of boundary layers and wakes [27]. The upper part of Fig. 6.8 shows the typical embedded C-grid, oriented along the bisector of the airfoil’s trailing edge. At the lower part with the help of a free-floating Kutta panel the embedded grid is oriented along the center line of the wake. In this case the wake is much better captured. We point in particular to the effect of the combined shock-wave and boundarylayer decambering, Sect. 6.1. The former results in the deflection of the Kutta direction out and upwards of the bi-sector of the trailing edge, the latter in the thicker boundary layer at the upper side of the airfoil and also at the upper side of the wake. The result is reflected in the lower part of Fig. 6.8 by the (self-organized) grid resolution with a thicker quasi-prismatic grid. The final example, shown in Fig. 6.9, demonstrates the versatility of the approach. In the upper part the initial grid for a three-element airfoil in high-lift state is given. At the sharp trailing edges of both the slat and the main airfoil straight free-floating deformable Kutta panels are located, oriented at the bi-sectors of the edges. The Kutta panel at the trailing edge of the main airfoil is short in order not to interfere with the possible boundary-layer separation over the flap, Sect. 9.2. The flap has a blunt trailing edge. No Kutta panel is placed there.

142

6 About the Kutta Condition

Fig. 6.9 Flow past a three-element airfoil (M∞ = 0.2, α = 16◦ , Re = 9·106 ) [26]. Upper part: initial grid with straight free-floating deformable Kutta panels. Lower part: grid after three iterations

The lower part of the figure shows the grid and the two Kutta panels after three iterations. The Kutta panels now indicate the center lines of the respective wakes. The results of the converged solution agrees well with the available experimental data. For details see [26, 27].

6.6 Problems Problem 6.1 What is the difference between boundary-layer decambering and shock-wave decambering? When does which of them appear, what are the reasons? Consider the airfoil case. Problem 6.2 Consider an aircraft flying with M = 0.1 at the altitude H = 1 km. Assume that at the wing’s trailing edge with the thickness h = 10 mm the velocity is u = u ∞ and also density and viscosity have the free-stream values. How large approximately is the Reynolds number and hence the Strouhal number? What shedding frequency is to be expected? Is it at the upper bound of the human hearing domain?

6.6 Problems

143

Fig. 6.10 Equivorticity lines of a wing with a no-slip boundary conditions and b with perfect-slip boundary conditions

Fig. 6.11 Starting vortex pair with secondary vortices behind a suddenly moved cylinder for Re = 550

Problem 6.3 Why is the angle of attack α a factor regarding dp/dϕ in Eq. 6.7? Problem 6.4 Derive the formula of the vacuum pressure coefficient. Problem 6.5 A hypothetical glider with perfect-slip boundary conditions on the wing can produce a starting vortex and lift just as a glider does with nonslip conditions. Discuss the reason for this behavior with the aid of Fig.6.10. Problem 6.6 Explain why at higher Reynolds numbers secondary vortices can develop in the wake of circular cylinders and spheres as shown in Fig. 6.11. Problem 6.7 The Kutta condition is a hypothesis, necessary in potential-flow (Model 4) theory, to determine airfoil circulation. In viscous fluid flow, such a condition is superfluous. Discuss the starting process for viscous flows and compare it with classical airfoil theory.

144

6 About the Kutta Condition

References 1. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014) 2. Lighthill, J.: On displacement thickness. J. Fluid Mech. 4, 383–392 (1958) 3. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011) 4. Hirschel, E.H., Lucchi, C.W.: On the Kutta Condition for Transonic Airfoils. MBB-UFE122AERO-MT-651, Ottobrunn, Germany (1983) 5. Zierep, J.: Der senkrechte Verdichtungsstoß am gekrümmten Profil. ZAMP, vol. IXb, pp. 764– 776 (1958) 6. Lucchi, C.W.: Shock correction and trailing edge pressure jump in two-dimensional transonic potential flows at subsonic uniform Mach numbers. In: 6th Computational Fluid Dynamics Conference, Danvers, MA. Collection of Technical Papers (A83-39351 18-02), AIAA, pp. 23–29 (1983) 7. Lucchi, C.W.: Ein Subdomain-Finite-Element-Verfahren zur Lösung der Konservativen vollen Potentialgleichung für Transsonische Profilströmungen (A Sub-Domian Finite-Element Method for the Solution of the Conservative Full Potential Equation for Transonic Airfoil Flow). Doctoral Thesis, Technical University München, Germany (1984) 8. Klopfer, G.H., Nixon, D.: Non-Isentropic Potential Formulation for Transonic Flows. AIAAPaper 83–0375 (1983) 9. Hirschel, E.H.: Basics of Aerothermodynamics, 2nd, revised edition. Springer, Cham (2015) 10. Küchemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford: also AIAA Education Series, p. 2012. Va, AIAA, Reston (1978) 11. Vos, R., Farokhi, S.: Introduction to Transonic Aerodynamics. Springer, Dordrecht (2015) 12. Obert, E.: Aerodynamic Design of Transport Aircraft. IOS Press, Delft (2009) 13. Abbott, I.H., Von Doenhoff, A.E., Stivers, Jr., L.S.: Summary of Airfoil Data. NACA Report No. 824 (1945) 14. Hoerner, S.F., Borst, H.V.: Fluid-Dynamic Lift. Hoerner Fluid Dynamics, Bricktown (1975) 15. Ackeret, J.: Versuche an Profilen mit abgeschnittener Hinterkante. Vorläufige Mitteilungen der Aerodynamischen Versuchsanstalt zu Göttingen, Heft 2, 18ff. (1924), also NACA Technical Meomorandum No. 431 (1927) 16. Henne, P.A.: Innovation with computational aerodynamics: the divergent trailing-edge airfoil. In: Henne, P.A. (ed.), Applied Computational Aerodynamics. AIAA Educational Series. pp. 221–262. AIAA, Washington, D.C. (1990) 17. Lawaczeck, O., Bütefisch, K.A.: Geplante Untersuchungen über v. Kármánsche Wirbelstraßen als eine mögliche Ursache für Buffet-Onset. In: Probleme der experimentellen transsonischen Aerodynamik, W. Lorenz-Meyer (ed.). DFVLR Bericht 251-77 A45, Göttingen, Germany, 6-1–6-8 (1977) 18. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006) 19. Herr, M.: Trailing-Edge Noise—Reduction Concepts and Scaling Laws. Doctoral Thesis, Technical University Braunschweig, Germany, also DLR-FB 2013-32 (2013) 20. Berg, D.E., Zayas, J.R.: Aerodynamic and Aeroacustic Properties of Flatback Airfoils. AIAAPaper 2008–1455 (2008) 21. Bangga, G.S.T.A., Lutz, Th., Krämer, E.: Numerical investigation of unsteady aerodynamic effects on thick flatback airfoils. In: Proceedings of the 12th German Wind Energy Conference DEWEK, May 2015, Bremen (2015) 22. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, Vol. 1 and 2, Springer, Berlin/ (1959). Also: Aerodynamics of the Aeroplane, 2nd edition (revised). McGraw Hill Higher Education, New York (1979)

References

145

23. Weiland, C.: Personal Communication (2018) 24. Newsome, R.W.: A Comparison of Euler and Navier-Stokes Solutions for Supersonic Flow Over a Conical Delta Wing. AIAA-Paper 85–0111 (1985) 25. Eberle, A., Rizzi, A., Hirschel, E.H.: Numerical Solutions of the Euler Equations for Steady Flow Problems. Notes on Numerical Fluid Mechanics, vol. 34. Vieweg, Braunschweig Wiesbaden (1992) 26. Deister, F., Hirschel, E.H.: Self-Organizing Hybrid-Cartesian Grid/Solution System for Arbitrary Geometries. AIAA-Paper 2000–4406 (2000) 27. Deister, F.: Selbstorganisierendes hybrid-kartesisches Netzverfahren zur Berechnung von Strömungen um komplexe Konfigurationen (Self-Organizing Hybrid-Cartesian Grid System for the Computation of Flows Past Complex Configurations). Doctoral Thesis, University Stuttgart, Germany, Fortschrittsberichte VDI, Reihe 7, Strömungstechnik, Nr. 430 (2002)

Chapter 7

Topology of Skin-Friction and Velocity Fields

The topological analysis of velocity and skin-friction fields potentially is a useful tool for flow-field interpretations and also for problem diagnosis. In practice, however, normally not much use is made of topological analysis, maybe because it is often treated in a rather formalistic way. In view of the topic of this book we intend to accentuate the practical rather than the theoretical side of the field of topology. Even this will be made only in a rather sketchy way. In the introductory Sect. 7.1 we first have a short look at the pertinent literature. It follows a general examination of the phenomena of interest in the flow fields, which pose the background of our book: flow attachment and separation, Sect. 7.1.2, as well as flow detachment, Sect. 7.1.3. In Sect. 7.1.4 finally Lighthill’s separation concept will be confronted with the concept of open-type separation and also open-type attachment. An introduction to singular points of skin-friction fields follows in Sect. 7.2. The classical approach to that topic is not considered in detail, the objective is the discussion of practical issues. Phase portraits—the flow patterns around a singular point— are addressed as well as off-surface flow-field portraits. In Sect. 7.3 the topic of singular lines, i.e. attachment and separation lines, is introduced. Classical flow topology only looks at the pattern—the phase portrait (see Sect. 7.2)—of the velocity, respectively the skin-friction field, at and around a singular point. Regarding singular lines, we take a broader view and concentrate on the particular flow-field properties connected to attachment and separation lines. Section 7.4 treats topological rules, again with emphasis on application. The last section, Sect. 7.5, is devoted to the structural stability of flow fields. Also here we do not look at it in the classical way. Our objective is to demonstrate practical issues. In the whole chapter we throughout assume steady flow—though with a few exceptions. Body surfaces may be flat or curved, the flow may be compressible or incompressible, laminar or—time-averaged—turbulent. Regarding the coordinate convention we note that the surface-parallel coordinates are x and z, with the velocity © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_7

147

148

7 Topology of Skin-Friction and Velocity Fields

components accordingly u and w, the surface-normal coordinate is y and the surfacenormal velocity component v. Exceptions are noted explicitly. In the Chapters devoted to the discussion of results of simulations of the flow fields past large aspect-ratio wings, Chap. 8, and small aspect-ratio delta-type wings, Chap. 10, we will use results of the present chapter in order to further elucidate the findings there.

7.1 Introduction 7.1.1 General Remarks Classical flow topology mainly treats singular points and in their vicinity the pattern of skin-friction lines, the phase portraits. When studying flow separation issues at body surfaces, we also have to look at flow attachment issues. We follow the classical approach, but also treat singular lines, i.e. attachment and separation lines. The reader is warned that we do not go into the details of definitions and derivations as can be found in the literature. We note in particular the classical NASA Technical Memorandum, later AGARDograph, of D. J. Peake and M. Tobak [1], the proceedings of a IUTAM-Symposium [2], two reports of U. Dallmann [3, 4], and the more recent book of J. Délery [5]. All works should be consulted, if a deeper study of flow topology is desired.

7.1.2 Three-Dimensional Attachment and Separation Flow attachment is generally understood as the impingement of the inviscid free stream on a body surface. The body surface itself is covered by a thin sheet of viscous flow, the attached viscous flow, which generally is of boundary-layer type (weak interaction with the external inviscid flow, Sect. 2.2). However, in a separation region (strong interaction with the external inviscid flow, see Sect. 2.2, too) also separated viscous flow can (re-)attach on the body surface. A prominent example is the separation bubble which can appear in two-dimensional form—for instance the separation bubble over an airfoil with a “peaky” pressure distribution—and also in three-dimensional form. We first consider the attachment process assuming inviscid flow throughout. The free stream impinges in the form of a streamline at, for instance, the forward attachment point, i.e., the nose point, of a fuselage. At this point—the primary stagnation point—the surface velocity is zero. The flow at the body surface then evolves— with non-zero velocity along the inviscid surface streamlines—exclusively from this

7.1 Introduction

149

attachment point. No other streamlines than the stagnation-point streamline impinge on the body surface.1 An attachment line is a location along the body surface where the arriving flow diverges to the left and the right side of it without impinging on the body surface. The attachment line itself is a streamline which may originate at the forward stagnation point, or elsewhere, see Sect. 7.1.4, and is part of a dividing surface which separates the two diverging flow parts. Examples are the attachment lines at the leading edge of swept wings and at the lower symmetry line of (round or nearly round) fuselages at angle of attack, Fig. 1.10. The attachment point and the attachment line in inviscid flow can well be prescribed by solutions of the Euler equations, the potential equation or the linearized potential equation (panel methods), Models 8, 4, 5 in Table 1.3. We loosely call such attachment lines primary attachment lines. If the flow on the body surface is viscous, the inviscid flow, impinging or arriving at the body surface, impresses its pressure field and streamline pattern almost fully on the boundary layer (weak interaction). The role of the surface streamlines of the inviscid flow is now taken over by the skin-friction lines. (At the attachment point the boundary layer has a finite thickness [6].) A (primary) attachment line usually begins at a singular point. At a round fuselage at angle of attack only one primary attachment line appears along the lower symmetry line, Sect. 7.5. However, the primary attachment line can be split into two primary attachment lines. This is typical for the windward side flow past a flat or almost flat surface at angle of attack, being the characteristic of a delta wing, Sect. 7.5. However, a primary attachment line must not necessarily begin at a singular point, then it is an open-type attachment line, Sect. 7.1.4. Attachment lines can appear also at other locations of a body surface, usually in connection with separation phenomena. Then we call them embedded attachment or reattachment lines. If they appear in a regular pattern, we may call them secondary, tertiary and so on attachment lines. Examples can be found in Chaps. 8 and 10. Often these lines are open-type attachment lines. The separation line in three-dimensional flow is the location, where the viscous flow, actually two converging boundary layers, separates from the body surface. This is the case of ordinary separation. The separation line is part of a dividing surface—a separatrix—, which separates the converging boundary layers on each side of it.2 The classical theory assumes that such a separation line always begins at a singular point, Sect. 7.1.4. Since long, however, it is accepted that open-type separation is a reality. In general very complex separation patterns can be present. Like in the case of attachment lines, secondary, tertiary and so on separation lines can be observed. Somewhat different is the case of flow-off separation which happens at acute (sharp) edges. Note that both “sharp” trailing edges and swept leading edges in

1 Note,

however, that more than one primary attachment point can be present, depending on the overall shape of the aircraft. 2 A very detailed discussion of the distinction between attached and separated flows is given in [1].

150

7 Topology of Skin-Friction and Velocity Fields

reality are not sharp in the exact sense of the word. They usually have a finite, though small thickness, and can only be aerodynamically sharp, Sect. 6.4. In general it can be assumed that the two boundary layers, which flow off the surface at the respective edge, have thicknesses much larger than the thickness of the edge. In reality hence flow-off separation with a thin embedded wake due to ordinary separation can be assumed to be present. The thin wake may even have an unsteady character, resembling a von Kármán vortex street. How the geometry of such a trailing edge is to be modeled in a discrete numerical computation method, has not yet been satisfactorily established. See in this regard the considerations of the Kutta condition, Chap. 6. Of interest when considering separated and vortical flow past bodies of finite extent are the situations at attachment points and lines, at separation points and lines, and also at detachment points and lines. The latter are a useful concept in order to distinguish between the viscous and the inviscid picture of flow leaving a finite body, see the next sub-section. This is to be recommended in view of the fact that separation of three-dimensional flow is not defined as simply and unambiguously as that of twodimensional flow, Sect. 1.3.3. However, in the literature the term detachment can have different meanings. The concept of limiting streamlines was introduced by W. R. Sears [7]. We mention it here, because it is often used in the literature and point to [6].

7.1.3 Detachment Points and Lines Regarding detachment points and lines we note that in (sub-critical) inviscid flow— other than in viscous flow—the flow leaves the surface of a finite body without transporting kinematically active or inactive vorticity away from the body surface, see also [6]. The surface streamlines, however, form similar patterns as we find them in viscous flow for the skin-friction lines. This all holds as long as no Kutta condition, either explicit or implicit is present. The term detachment line was introduced in [8]. We consider the inviscid flow past a wing-like thin ellipsoid at angle of attack, Fig. 7.1. The flow field and the streamlines were computed with exact potential-flow theory [9] (Model 4). We look from above at the ellipsoid.3 The forward stagnation point and the forward dividing streamline—the attachment line—lie, because of the positive angle of attack, at the lower side of the ellipsoid. The streamline pattern is symmetrical around the lateral axis (which is the major axis). Hence we find both the rear stagnation point— which we designate detachment point—and the rear dividing streamline—which we designate detachment line—at the upper side of the ellipsoid. In applied aerodynamics less pronounced detachment lines typically appear at the lee side of fuselages at small to moderate angles of attack. There a thickening of the

3 See

also Sect. 4.3.2.

7.1 Introduction

151

Fig. 7.1 Streamlines of the inviscid velocity field past a wing-like 3:1:0.125 ellipsoid at angle of attack α = 15◦ [8]. View toward the upper side

viscous layer happens without separation of the flow. This might be, for instance, of interest in design considerations.

7.1.4 Lighthill’s Separation Definition and Open-Type Separation and Attachment Before we come to the topic of singular points, we note a matter, which for a time was much discussed. In [10] M. J. Lighthill treats the definition of separation. He states as necessary condition for separation that the particular skin-friction line to which the other skinfriction lines converge, must originate from a singular point, which is a saddle point, see, e.g, S1 in Fig. 8.26 in Sect. 8.4.3. Likewise an attachment line must begin at a singular point, a nodal point, see, e.g, N2 in Fig. 8.26. Consider the two-dimensional flow past a wing section in Fig. 7.2a. The separation streamline emerges from a half-saddle point and then re-attaches, forming a closed separation bubble. The separation streamline decomposes the flow field into the bubble flow and the flow above it. The separation streamline hence is called a separatrix or separator. It separates portions of the flow field. The separatrix emerges from the half-saddle point on the surface and also ends in such a point. Two-dimensional separation is structurally unstable, Sect. 7.5. A small perturbation then leads to the flow field shown in Fig. 7.2b. Lighthill’s separation condition, however, is not violated.

152

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.2 Topological structure of two-dimensional flow separation at a wing section [3]: a structure with a closed separation bubble, b structure after a disturbance

In three-dimensional flow fields with separation the separatrices are stream surfaces which separate flow portions with different origin and behavior. On the body surface accordingly both attachment and separation lines, i.e., skin-friction lines with different origin, are separatrices. In Sect. 7.3 we treat them as singular lines. In any case, Lighthill’s statement, which J. Délery puts it as “A flow is separated if its surface flow pattern contains at least one saddle point” [5], for a long time was considered to be the valid definition. However, in the 1970s K. C. Wang did show that this is not necessarily true. He described the phenomenon of ‘open-type separation’ [11]. In such a case the separation line does not begin at a singular point, but at some location in the skinfriction field, see for example Fig. 8.32 in Sect. 8.4.3. Peake and Tobak call such a separation pattern ‘local separation’ [1]. However, not only open-type separation can be observed, but also open-type attachment [6]. This holds for both primary and higher-order attachment lines. An attachment line may not only begin without a singular (saddle) point, Fig. 8.27 on Sect. 8.4.3, it also may end without a singular point, Fig. 8.31 in Sect. 8.4.3. Whether the latter also holds for separation lines has not yet been established. In some instances they are seen as simply tapering off. In some of the following chapters we demonstrate and discuss both open-type attachment and separation lines. We note, though, that open-type attachment to our knowledge so far has been observed only in numerical realizations of flow fields. For them holds that reservations are due regarding turbulent flow because of the employed turbulence models. Regarding laminar flow, this does not hold.

7.2 Singular Points 7.2.1 Flow-Field Continuation and Phase Portraits The pattern of the skin-friction lines—also that of inviscid streamlines—on the surface of a body can be considered as a continuous vector field. Orthogonal to the skin-friction lines are surface vortex lines [10].4 Important is that through each point on the body surface passes one and only one skin-friction line. 4 H.

J. Lugt differentiates between vorticity lines and vortex lines [12]. What is called here vortex lines is in his nomenclature vorticity lines.

7.2 Singular Points

153

But on the body surface always locations are present, where singular points and singular lines (attachment and separation lines) exist.5 Singular points are locations in the skin-friction field where the skin friction as well as the surface vorticity become zero. The number of skin-friction lines passing through such a point generally is different from one. Along attachment and separation lines the skin friction is non-zero. Along attachment lines an infinite number of skin-friction lines diverges from them. In contrast to this along separation lines an infinite number of skin-friction lines converges to them. Of interest for us is what happens at singular points, i.e., what is the flow pattern in the vicinity of a singular point. (Singular lines will be treated in Sect. 7.3.) The classical approach begins with a flow-field continuation around a surface point P0 (x0 , y0 , z 0 ). We do not show this, but refer the reader instead to, for instance, [6], where a very general formulation of the wall compatibility conditions is employed. The basic assumptions are steady flow and a flat or curved body surface. The flow may be compressible or incompressible, laminar or—time-averaged—turbulent, heat flux may be present, the body surface is non-permeable and the no-slip condition holds. Important is that the flow-field continuation is made with the Navier-Stokes equations and not with the boundary-layer equations. For convenience, however, usually the derivation is made in Cartesian coordinates on a flat surface and for incompressible flow. This does not impair the general validity of the results. Our coordinate convention is as follows: x and z are the surface-parallel coordinates, y is the surfaces-normal coordinate, accordingly u and w are the surfaceparallel velocity components and v is the surface-normal component. With the wall shear-stress components τw x = μ

∂u ∂w | y=0 , τwz = μ | y=0 , ∂y ∂y

(7.1)

we obtain with a Taylor expansion for the velocity components at the point P(x, y, z) in the vicinity of the surface point P0 (x0 , y0 , z 0 )   ∂τwx 1 ∂p 2 1 ∂τwx τw x y + xy + yz + y + ... , (7.2) u| P = μ ∂x ∂z 2 ∂x P0   ∂τwz ∂τwz 1 ∂p 2 1 τw z y + xy + yz + y + ... , w| P = μ ∂x ∂z 2 ∂z P0

v| P = −

11 2μ



∂τwz ∂τwx + ∂x ∂z



 y 2 + ...

= P0

  1 1 ∂p y 2 + .... 2 μ ∂ y P0

(7.3)

(7.4)

5 We do not always mention detachment points and lines. It is self-evident what also holds for them

in the following discussion.

154

7 Topology of Skin-Friction and Velocity Fields

The systematic identification of singular points then is made with the help of the phase-plane analysis [13]. That analysis permits to find possible geometrical configurations of the patterns of the skin-friction lines—or the streamlines—in the neighborhood of the singular point.6 The final result is a vector and matrix arrangement of the first-order terms of the equations, see, e.g., [3, 5, 15, 16] 1 1 V (P) = A(P0 ) X + B(P0 ). y μ

(7.5)

Here X = (x, y, z)T is the location vector of point P and A a Jacobian matrix which contains the expansion terms of lowest order at point P0 : ⎛ ∂τ

wx

⎜ ∂x ⎜ ⎜ A=⎜ ⎜ 0 ⎜ ⎝ ∂τ

wz

∂x

1 ∂ p ∂τwx ⎞ 2 ∂x ∂z ⎟ ⎟ ⎟ 1 ∂p 0 ⎟ ⎟ . 2 ∂y ⎟ 1 ∂ p ∂τwz ⎠ 2 ∂z ∂z P0

(7.6)

The matrix B contains small terms of higher order. The eigenvalues of the matrix A read:

λ1,3

1 = 2



∂τwz ∂τwx + ∂x ∂z



1 ± 2



λ2 =

∂τwz ∂τwx − ∂x ∂z

2 +4

∂τwx ∂τwz , ∂z ∂x

1 ∂p . 2 ∂y

(7.7)

(7.8)

The matrix A is further analyzed by investigating the trace T , the Jacobian determinant J , and the discriminant  of it, see, e.g., [13]. The trace, the Jacobian determinant, and the discriminant read: 1 ∂ p ∂τwz ∂τwx + + , ∂x 2 ∂y ∂z

(7.9)

∂τwz 1 ∂ p ∂τwx ∂τwx 1 ∂ p ∂τwz − , ∂x 2 ∂ y ∂z ∂x 2 ∂ y ∂z

(7.10)

T =

J=

 = T 2 − 4J.

6 In

[14] these are called “phase portraits” of the surface shear-stress vector field.

(7.11)

7.2 Singular Points

155

Fig. 7.3 Patterns of skin-friction lines—phase portraits—in the neighborhood of P0 (x0 , y0 , z 0 ) [6]: basic singular points in the chart of trace T and Jacobian determinant J , with the discriminant  of the Jacobian matrix A as parameter

The combination and signs of these parameters determine the pattern of the skinfriction lines in the immediate neighborhood of a singular point P0 (x0 , y0 , z 0 ). The resulting singular points are collected in Fig. 7.3. These are the basic singular points. Other singular points or combinations of them (merged points) are also possible. They are not discussed here. The interested reader instead is referred to, e.g., [3, 5, 16]. Two classes of singular points are distinguished: saddle points S (J < 0) and nodal points N (J > 0,   0), both for attaching flow: T > 0, right-hand side of Fig. 7.3 and separating/detaching flow: T < 0, left-hand side of Fig. 7.3. In topological rules focus points F (J > 0,  < 0) are counted as nodal points. This also holds for center points C (J > 0, T = 0).

156

7 Topology of Skin-Friction and Velocity Fields

A nodal point is the common point of an infinite number of skin-friction lines, which for attaching flow—upper right quadrant of Fig. 7.3—are directed away from it, and for separating/detaching flow toward it—upper left quadrant. At that point all skin-friction lines except one—which is normal to it—are tangential to a given skin-friction line. The star node is an exception. An infinite number of skin-friction lines is also associated with a focus point, although without a common tangent line, upper quadrants of Fig. 7.3. No skin-friction line is associated with the singular point of a center point (T = 0, J > 0). For both attaching and separating flow two single skin-friction lines are associated with a saddle point, each toward and away from it—lower left and right quadrant of Fig. 7.3. All other skin-friction lines in the neighborhood of a saddle point are deflected from that point in the directions of the single skin-friction lines. Important is the fact that only the phase portraits lying in the quadrant J > 0, T < 0 are stable ones, see, e.g., [17]. For the phase portraits lying in the other quadrants it holds that small changes of flow or geometrical parameters change them. This is the matter of structural instability of singular points to which we come back in Sect. 7.5.

7.2.2 Off-Surface Flow-Field Portraits With the term off-surface flow-field portrait we denote the off-surface flow pattern in the vicinity of a singular point P0 (x0 , y0 , z 0 ). This of course is a matter of the pattern of the attaching or separating flow. We discuss the flow portraits of a few cases, which we deem to be most important. But we do this also in order to demonstrate the versatility of the flow-field continuation approach. Much more and more detailed material can be found, e.g., in [5, 16, 18]. • Two-Dimensional Separation We ask for the angle λ, the separation angle, at which the separation line leaves the surface.7 This concerns both plane and axissymmetric flow, assuming zero or negligible streamwise surface curvature. The obvious criterion for separation is the vanishing of the wall shear stress τwx , actually it is the change of sign of τwx in the separation point, Fig. 1.8 on Sect. 1.3.1.8 If in flow (x−) direction both the functions τwx (x) and pw (x) are known, we can determine the angle λ, under which the separation streamline leaves the surface.9 In Fig. 1.8 this streamline is the full line emanating from the separation point, which is the singular point P0 (x0 , y0 , z 0 ). (That point is indicated in the figure by the circle at y = 0.) We follow the derivation given by K. Oswatitsch in 1957 [18].

7 The reader should note that λ is not one of the eigenvalues of the matrix A, Eq. 7.6. The designation

λ for the separation angle is used here, because it was used in [6]. 8 Experimental evidence of separation usually is given by changes of the wall-pressure distribution compared to that of the unseparated case. A pressure plateau may be formed, or at the aft of a two-dimensional body the recompression is severely suppressed. 9 We keep the coordinate convention of the preceding sections.

7.2 Singular Points

157

With Eqs. (7.2) and (7.4) reduced to two-dimensional flow, we find at the separation point P0 , where τx = 0, with tanλ =

v y = x u

(7.12)

∂τwx ∂x ∂p ∂x

(7.13)

after some manipulations tanλ = −3

| P0 .

The pressure gradient ∂ p/∂x is positive, because it is the adverse gradient which leads to separation. In such a situation of course the gradient of the wall shear stress ∂τwx /∂x is negative. Hence λ is positive, as is to be expected. In Fig. 1.8 on Sect. 1.3.1 the broken line emanating from the separation point— the circle—is the location of the turning of the streamlines in the region beneath the separation line. This line is defined by the disappearance of the tangential velocity component: u(x, y) = 0. The elevation angle λ∗ of this line is found from Eq. 7.2 to be tanλ∗ =

2 tan λ. 3

(7.14)

Remains to ask how the separation line behaves away from the surface. In general it is to be expected that it turns more or less strongly toward the body surface. Moreover, it is obvious that the above result cannot be obtained by employing the boundary-layer equations. That is possible only with the help of solutions of the Navier-Stokes or RANS equations (Models 9 and 10 in Table 1.3). • Stagnation Point We consider the primary stagnation point at a body surface. There we find in the axis-symmetric case the star nodal point as an attachment node as shown in the upper right quadrant of Fig. 7.3. This is an isotropic node. At a general stagnation point, for instance at a blunt body at angle of attack, the star nodal point is non-isotropic [5]. A cut through the node, orthogonally to the surface, gives the picture like shown in Fig. 7.8b for the axis-symmetric case: the nodal point becomes a half-saddle point. At the primary attachment point we have an absolute pressure maximum, the stagnation pressure. The lateral pressure gradient is zero (we write it for the positive x-direction only): ∂ p/∂x = 0. Downstream of the attachment point we have a negative pressure gradient: ∂ p/∂x < 0, i.e., the flow gets accelerated away from the stagnation point. The skin-friction has an absolute minimum at the primary attachment point: τx = 0. The lateral gradient is zero: ∂τx /∂x = 0. Downstream of the attachment point we have a positive skin-friction gradient: ∂τx /∂x > 0, i.e., the skin-friction increases. We emphasize that at the stagnation point the boundary layer has a finite thickness δ [6].

158

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.4 Sketch of a streamline assumed to leave the body surface along a separation line [6]

How does the stagnation-point streamline impinge on the surface? The relation for the angle λ at the two-dimensional separation point, Eq. (7.13), also holds for this case. With ∂ p/∂x negative, and ∂τx /∂x positive in its vicinity downstream, when approaching the attachment point, tanλ approaches infinity. The result is that the attaching streamline at right angles impinges to the surface in that point. This is the same result which potential theory gives for the stagnation point. Generalizing this we state that both in inviscid and viscous flow the streamline impinges on the primary stagnation point at a right angle. In the upper left quadrant of Fig. 7.3 a separation node is shown as star node. Such a nodal point is shown at the aft end of the body in Fig. 7.8a. In reality such a node will not appear there, because in viscous flow we find separation upstream of the aft end of the body. Instead we may have a picture like shown in Fig. 7.9. There we have an attachment nodal point at the aft end of the body, which in the cut shows up as a half-saddle point. The separation star node of Fig. 7.3 generally appears to be possible only in the inviscid flow picture (detachment). • The Three-Dimensional Separation Line Above it was stated that a streamline only leaves the body surface at a singular point. What happens along a threedimensional separation line? Indeed, no streamline leaves that line. We show this like it was done in [18]. We assume that locally the direction of a hypothetical separating skin-friction line is the same as that of the separation line itself, Fig. 7.4. The angle λ2 , under which the streamline is assumed to leave the surface, is tanλ2 =

v y = . x u

(7.15)

Putting Eqs. (7.2) and (7.4) into Eq. (7.15) we obtain at the surface (z = 0): z x + ∂τ )y ( ∂τ 1 ∂x ∂z tanλ2 = − |P . 2 τx + ∂τx x + 1 ∂ p y 0 ∂x 2 ∂x

(7.16)

With tanλ2 → y/x for y → 0, x → 0, this equation can be rearranged to yield

7.2 Singular Points

159

Fig. 7.5 Illustration of a vortex filament which leaves the body surface in a focus point [18]. Thin lines are skin-friction lines, full lines streamlines

tanλ2 = −

x 3 ∂τ + ∂x

∂τz ∂z ∂p ∂x

+ 2 τxx

| P0 .

(7.17)

We see immediately that no meaningful result can be obtained from this equation for λ2 unless we have τx | P0 = 0. This would mean, because in P0 by definition τz = 0, that P0 is a singular point. The result is that indeed along the separation line no streamline can leave the surface. That is possible only in the singular point P0 . In the two-dimensional case ∂τz /∂z ≡ 0, and we are back to the result of Eq. (7.13). • The Focus Point A very particular singular point is the focus point, which in the upper quadrants of Fig. 7.3 is indicated for both separating and attaching flow. In the first case a vortex filament leaves the surface, in the other case such a filament impinges on the surface. Figure 7.5 shows such a vortex filament, which was obtained by flow-field continuation [18]. Neither a separation line nor a separation surface is involved, instead an infinite number of skin-friction lines and streamlines converges toward the axis of the filament. Oswatitsch in [18] notes that it may be a question of definition, whether to call this separation or not. In any case, vortex filaments of this type can be observed in reality, see, e.g., Fig. 8.29 in Sect. 8.4.3. • Flow-Off Separation At the sharp trailing edge of an airfoil potential theory (Model 4) yields a singularity. Actually in viscous flow we have flow-off separation there. From the very edge a streamline leaves the airfoil, constituting the center line of the wake. Even if it may be against our intuition we must consider the edge point as a singular point. Actually it is a half-saddle point, Sect. 7.4, like the stagnation point at the nose of the airfoil. In the reality of an aircraft’s wing the flow situation at the trailing edge is complex, Chap. 6. The trailing edge has a finite thickness, we have decambering effects and, moreover, in the case of the lifting wing, we have a shear between the upper and

160

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.6 Sketch of a separation line and the separating stream surface leaving the body surface [6]

lower side flow, Sect. 4.4. This all holds too for the flow-off separation at the sharp leading edge of a delta wing. • The Separating Stream Surface We consider now ordinary separation, and there the—dividing—stream surface between the two boundary-layer streams, which squeeze each other off the surface, Sect. 1.3.3. Its intersection with the body surface is the separation line, Fig. 7.6. With the help of Eq. (7.3) it is possible to estimate the angle λ1 of the separating surface with respect to the z-direction at the point P0 on the separation line. At that point the separating surface is defined by τz = 0. For z  0, if the elevation angle of the flow at the surface is small, we have w ≈ 0. Thus we find for the angle λ1 with w = 0 from Eq. (7.3): tanλ1 = −2

∂τz ∂z | . ∂ p P0 ∂z

(7.18)

Our result so far is that streamlines do not depart upwards from a separation line (see above “The Three-Dimensional Separation Line”). The separating stream surface, which emanates from it, is formed by the two boundary-layer streams squeezing each other off the body surface. The question now is, what is the situation at attachment lines? • The Attaching Stream Surface With the reasoning which we used for the separation line, it can be shown that no streamline impinges on the attachment line along its length. From Eq. (7.18) we deduce also that the attachment stream surface stands at a right angle to the surface, if ∂ p/∂z → o for z → 0, whereas ∂τz /∂z is finite. • The Boundary-Layer Flow Being Parallel to the Body Surface? Above it was stated and also shown that only in a few—singular—points on the body surface streamlines actually impinge on or leave the body surface. This implies that in attached viscous flow very close to a non-permeable wall the boundary-layer flow is parallel to the body surface. To prove this we consider a two-dimensional boundary layer. The result also holds for three-dimensional boundary layers.

7.2 Singular Points

161

We again use the result of the Taylor expansion in terms of Eqs. (7.2) and (7.4). We find for small distances y from the surface u ∼ y, v ∼ y 2 ,

(7.19)

and hence the streamline elevation angle θ in relation to the surface tan θ =

v ∼ y. u

(7.20)

The result is that when the surface is approached in attached boundary-layer flow, the flow in the limit becomes parallel to it: y → 0 : θ → 0.

(7.21)

We keep in mind that in a boundary layer y and v are small compared to x and u. Actually it is the

essence of boundary-layer theory that y and v have the order of magnitude O(1/ Rer e f ) [6]. Hence the above result tells us further that the whole viscous flow in the boundarylayer flow limit Rer e f → ∞ becomes parallel to the surface. A purely inviscid flow at the body surface of course anyway is parallel to that. Our result plays also a role in hydrodynamic stability theory. The derivation of the Orr-Sommerfeld equation assumes that the flow is exactly parallel to the surface, see, e.g., [6].10 For not so large Reynolds numbers “non-parallel effects” as well as “surface-curvature effects” come into play and are a topic of stability theory.

7.2.3 Singular Points in Off-Surface Velocity Fields On a body surface singular points can change their character, for instance in a plane normal to the surface through a singular point. Singular points on the surface then become, for instance, half-nodes N or half-saddles S , Sect. 7.4. Away from the body surface the patterns of streamlines around singular points can be very different. This holds for both viscous and inviscid flow. (This implies that singular points are not necessarily connected only to skin-friction line patterns.) In certain cases we can clearly observe streamlines, which connect singular points on the body surface with singular points in the flow field away from the surface. This even holds if we have unsteady flow due to vortex shedding. Then instantaneous patterns can be observed. In other cases the situation needs to be considered with care. This holds, for instance, when we consider the flow pattern in a surface orthogonally to the center line of a delta wing with lee-side vortices. Here, like above, streamlines appear to connect 10 The Orr-Sommerfeld equation describes when a two-dimensional laminar boundary layer becomes unstable, triggering laminar-turbulent transition, see, e.g., [6].

162

7 Topology of Skin-Friction and Velocity Fields

singular points on the body surface with singular points away from the surface. However, the streamlines are only apparent ones. Actually they are the streamlines traces, which are projected into the observation plane, the Poincaré surface [19]. In [5] they are called pseudo-streamlines. We will meet these cases again in Sect. 7.4, where we treat topological rules.

7.3 Singular Lines Attachment and separation lines generally are not a big topic in flow-field topology. They connect—in Lighthill’s picture, Sect. 7.1.4—singular points, or—a geometrical view—they separate flow domains, hence they are called also separators (see above). Here we call these lines singular lines, because certain geometrical and flow-field properties along them behave in a singular way compared to the properties of the main flow domains over the body surface. Attachment and separation lines are always present on an aircraft’s surface and some of the flow properties to be observed along these lines are of considerable practical interest. In order to avoid misconceptions we note that an attachment line is not a stagnation line. The only exception is the case of an unswept wing of infinite span, and that is the two-dimensional case. The same holds for a separation line. Along both attachment and separation lines the flow velocity and the wall-shear stress are non-zero. We study now these properties following closely the presentation given in [6]. However, we do that in a descriptive way only, without the proofs given there. The reader interested in the proofs is referred to that publication. Singular lines in the frame of this section are primary, secondary etc. attachment and separation lines. Primary attachment lines as a rule are attachment lines also of the inviscid flow past a body. An attachment line may have its origin in a singular point, Fig. 7.8a. That can be a nodal point—forward (primary) stagnation point—or a saddle point. However, like open-type separation, also open-type attachment is possible, Sect. 7.1.4. In later chapters examples are given. At an ordinary airplane configuration with or without swept wings, only one primary stagnation point is present. That is located at the nose of the fuselage.11 If the airplane has a forward swept wing, three primary attachment points are found. We remember, see Sect. 7.2.2, that only in a singular point—attachment point—a streamline impinges on the body surface. Along an attachment line this does not happen. This holds for both viscous and inviscid flow. It also holds that a streamline never becomes a skin-friction line. Separation lines are present only in viscous flow past a finite body, either as ordinary or as flow-off separation lines. Only in a singular point—separation point— a streamline leaves the body surface. Along a separation line this does not happen. A skin-friction line never becomes a streamline which leaves the body surface. In inviscid flow the picture is similar with detachment lines and detachment points. 11 We neglect possible forward stagnation points at the propulsion units and at antennas and the like.

7.3 Singular Lines

163

Fig. 7.7 Schematic of general singular lines of both inviscid and viscous flow: a attachment line, b ordinary separation/detachment line [6]

The reader is asked to contemplate these statements. Open attachment or separation lines do not have a singular point at their beginning, partly also not at their end. We indeed have a singular behavior of the flow in such cases. The above holds quite in general for every configuration and the flow past it. The results of our investigations in principle apply to any kind of attachment and separation lines. Attachment and separation/detachment lines appear in two canonical forms as shown in Fig. 7.7. Typical for an attachment line (a) is that an infinite number of skin-friction lines diverges from it. This holds also for the surface streamlines of the related inviscid flow. However, the inviscid and the viscous attachment lines generally do not coincide with each other. The reason is that the skin-friction lines of three-dimensional attached viscous flow are more strongly curved than the surface streamlines of the related (external) inviscid flow [6]. The two attachment lines coincide only if they lie, for instance, on the surface generator of an infinite swept wing (ISW).12 Typical for a ordinary separation line (b) is that an infinite number of skin-friction lines converges toward it. This holds also for the streamlines at a detachment line of inviscid flow. We look now at the interesting flow-field properties of attachment and ordinary separation/detachment lines and single out the following five items: 1. Relative maximum of the surface pressure At a curved inviscid attachment line, the surface pressure has a relative maximum in direction normal to the inviscid attachment line, 1 - · - 1 in Fig. 7.7a. In general, the pmax -line lies close to the attachment line (also to the viscous one), and only on it, if the ISW situation is given. The location of the relative pressure maximum can be of interest in view of laminar flow control. 2. Points-of-inflection line On the convex sides of the singular lines shown in Fig. 7.7, the stream lines or the skin-friction lines have a point of inflection. A 12 This

surface generator is a geodesic, the boundary layer is a quasi-two-dimensional one [6].

164

7 Topology of Skin-Friction and Velocity Fields

points-of-inflection line is present at every curved attachment or separation line. In general it lies close to the respective attachment or separation line. In the ISW situation it disappears. The point-of-inflection line is a purely geometrical property, interesting as visual indicator of three-dimensional attachment or separation lines. 3. Characteristic thickness of the viscous layer At an attachment line the characteristic thickness of the viscous layer c has a relative minimum in the direction orthogonal to it, 1 - · - 1 in Fig. 7.7a.13 At a separation line c has a relative maximum in the direction orthogonal to it, 2 - · - 2 in Fig. 7.7b. This thickness property is determining in view of property 5. 4. Relative minimum of |τw | Along attachment and separation lines the skin-friction is non-zero. Normal to a viscous attachment line, 1 - · - 1 in Fig. 7.7a, or to a separation line, 2 - · - 2 in Fig. 7.7b, the absolute value |τw | of the skin-friction vector has a relative minimum. The minimum lies close to the respective line. This property is interesting also as indicator of an attachment or separation line. 5. Extrema of the thermal state of the surface Along attachment and separation lines the wall temperature Tw and the heat flux in the gas at the wall qgw are non-zero. Normal to an attachment line, 1 - · - 1 in Fig. 7.7a, a relative maximum, and normal to a separation line, 2 - · - 2 in Fig. 7.7b, a relative minimum exist. These extrema lie close to the respective line. Extrema of the thermal state are of utmost importance for hypersonic flight vehicles. There along attachment lines hot-spot situations are present and along separation lines cold-spot situations [20]. We emphasize that at attachment lines the distances of the extremum lines or the points-of-inflection lines to the attachment lines as such in general are very small. The reason for this is that an attachment line usually is only weakly curved. In the ISW situation, the pmax -line, the points-of-inflection line and the |τw |-minimum line lie on it. At separation lines, the situation is different. At ordinary separation lines the distances between the separation line and the extremum lines of the skin-friction vector and the thermal state of the surface are not necessarily very small. This holds also for points-of-inflection lines. The reason for this is that ordinary separation in general is not related directly to curvature maxima of the surface. This does not hold for flow-off separation lines at sharp edges, for instance wing trailing edges, Fig. 1.10a, or sharp leading edges of highly swept wings, Fig. 1.10b. There the distances can be very small. The proofs of the listed properties are given in [6]. The results in principle are valid for incompressible and compressible, laminar and turbulent flow. However, for convenience, the presentations are sometimes simplified. Detachment lines are not treated as separate topic. What applies for them of the above five items is more or less self-evident. In closing this section, we ask how to recognize three-dimensional attachment and in particular separation lines on a body surface. Visually the respective patterns 13 For

the meaning and the definition of the characteristic thickness c see Appendix A.5.4.

7.3 Singular Lines

165

are easily to recognize, see the many examples in the book. But there is no simple separation criterion like in two-dimensional flow with τw = 0. In [21] the following indicators have been proposed to detect separation in computed data: 1. Local convergence of skin-friction lines. 2. Occurrence of a |τw |-minimum line. 3. Bulging of the boundary-layer thickness and the displacement thickness. For attachment lines we note accordingly: 1. Local divergence of skin-friction lines. 2. Occurrence of a | pw |-maximum line. 3. Occurrence of a |τw |-minimum line. 4. Indentation of the boundary-layer thickness and the displacement thickness.

7.4 Topological Rules 7.4.1 Introduction Topological rules give the relationship of singular points on a body surface or in a general observation surface. We give a short introduction to them. We basically follow the presentation given in [6]. For a deeper study see, e.g., [1] or [5]. General assumptions are that the flow is steady, the body is simply connected, that the velocity field and the skin-friction field past the body are continuous, and that the body is immersed in a uniform upstream flow field.14 Nodal points and focus points are topologically equivalent, hence are counted together. Only a few illustrations are given, more are to be found in the following section and in later chapters.

7.4.2 Surface Rules Two basic rules are considered: • Rule 1, due to A. Davey [22] and also M. J. Lighthill [10], concerns the inviscid surface-velocity field or the skin-friction field on a three-dimensional body. It says that on the body surface the number of nodal points N (focus points are counted as nodal points) is larger than the number of saddle points S by two:

14 Time-dependent

flow can be treated by applying the rules to the instantaneous flow field [5].

166

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.8 Schematic of steady inviscid flow past an axis-symmetric body. a Streamlines on the surface: two nodal points N . b Streamlines seen in the two-dimensional plane cutting the body through its axis: two half-saddle points S



N−



S = 2.

(7.22)

This rule holds for simply connected body surfaces. For non-simply connected surfaces the complexity p is taken into account by   N− S = 2 − 2 p. (7.23) For simply connected surfaces we have p = 0. If one hole is present in the surface p = 1, for two holes p = 2 and so on [5]. • Rule 2, due to J. C. R. Hunt et al. [23], concerns skin-friction lines and streamlines in a two-dimensional plane cutting a three-dimensional body. The sum of nodal points N plus one half of the number of half-nodal points N is one less than the sum of saddle points S plus one half of the number of half-saddle points S : (



N+

 1

1

N )−( S ) = −1. S+ 2 2

(7.24)

We illustrate these two rules with the flow past an axis-symmetric body. Figure 7.8a shows the streamlines of the inviscid flow at the body surface with the forward (attachment) and the rearward (detachment) stagnation point. Both are nodal points, Rule 1 is fulfilled because saddle points are absent. In Fig. 7.8b the body is cut by a two-dimensional plane through its axis. The nodal points of Fig. 7.8a now become half-saddle points. Rule 2 is fulfilled.

7.4.3 Off-Surface Rules If the flow near the surface is viscous, we see separation at some aft location of the body, Fig. 7.9. We assume a steady separation region.15 The forward and the rearward

15 In reality this is given only for very small Reynolds numbers, see, for instance, Fig. 7.17 and also

the flow visualizations in [24].

7.4 Topological Rules

167

Fig. 7.9 Schematic of steady viscous flow past an axis-symmetric body. Skin-friction lines and streamlines seen in the two-dimensional plane cutting the body through its axis: one saddle point S, four half-saddle points S , two focus points F, counted as nodal points

stagnation point are half-saddles. Although inviscid flow attaches at the forward stagnation point, S1 , the flow there is viscous, i.e. the viscous layer or boundary layer at that point has a finite thickness. Note that the rearward stagnation point, S2 , now is an attachment point, too. The shear layers emanating from S3 and S4 are merging and split in S1 and then move partly, with a wake-like appearance, toward S2 . Actually it is the circumferential separation line which shows up in the cutting plane as the two half-saddle points S3 and S4 . The separation region is a toroid, which shows up as two center points (degenerated focus points) F1 and F2 . The separation region is closed by a saddle point, S1 . Rule 2 is fulfilled. Quarter-saddle points were introduced in [25] in order to treat the topology of the flow past a delta wing, independent of the type of leading edge, sharp or rounded. If the lower side of a delta wing—or of a fuselage—is flat or nearly flat, two primary attachment lines lie near the side edges of the lower surface, Sect. 7.5. Between them the flow is nearly or fully two-dimensional. Consider now Fig. 7.10. The skin-friction lines at the lower side of the wing leave the primary attachment lines like sketched in Fig. 7.21 on Sect. 7.5. In a plane twodimensional cut A-A, the Poincaré surface, then only the traces of the outward flow are seen and the primary attachment lines appear as quarter-saddle points S

.16 • Rule 2 then changes into Rule 2’: (



N+

 1  1 

1

N )−( S + S ) = −1, S+ 2 2 4

(7.25)

which in our case results in 1 1 4 − (1 + 7 + 2) = −1. 2 4

(7.26)

If the lower side of the body or wing has a convex shape—like shown in Fig. 7.20 on Sect. 7.5—the attachment line lies at the lower apex of it. Then, instead of the

16 Note that with the attachment lines—the two primary lines, and also (!) with the two secondary lines and the tertiary line—inviscid flow attaches. However, the attachment-line flows themselves are viscous, i.e. the viscous layers or boundary layers at those lines have finite thicknesses, Sect. 7.3.

168

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.10 Sketch of steady viscous flow past a delta wing with primary and secondary lee-side vortices [25]. Skin-friction lines and streamlines seen in the plane two-dimensional cut A-A, i.e. the Poincaré surface: one saddle point S (•), seven half saddle points S (◦), two quarter-saddle points S

(), four focus points F (×), counted as nodal points Fig. 7.11 Sketch of the open lee-side flow field in the Poincaré surface. Possible secondary and higher-order vortical structures were omitted

two quarter-saddle points present at the lower side of the delta wing in Fig. 7.10, one half-saddle point is present and Rule 2 is fulfilled. Finally we note two principally different lee-side flow topologies, which can be observed in a Poincaré surface. The first one, which we call open lee-side flow field, is sketched in Fig. 7.11. It appears at the lee side of delta wings, depending on the shape of the wing and the angle of attack, Sect. 7.5. However, we observe this kind of topology also at large aspect-ratio wings, there regarding the wing-tip vortices. In later chapters examples are shown. The other lee-side flow topology is the closed lee-side flow field, Fig. 7.12. That is observed primarily above delta wings at high angles of attack. A problem, which sometime occurs, is the interpretation of a particular flow pattern. Consider in Fig. 7.13 the inviscid flow streamlines at the upper surface of the thin ellipsoid from Sect. 7.1.3. The angle of attack now is α = 30◦ . The flow solution again was made with potential theory (potential flow, Model 3).17

17 We

gratefully acknowledge that the following figures were provided by C. Weiland [26].

7.4 Topological Rules

169

Fig. 7.12 Sketch of the closed lee-side flow field in the Poincaré surface. Possible secondary and higher-order vortical structures were omitted

Fig. 7.13 Streamlines at the upper surface of a 3:1:0.125 ellipsoid at the angle of attack α = 30◦

We look from above at the flow field and see the detachment point and the detachment line. (The streamlines coming from the lower side to these entities are not plotted.) Attachment point and attachment line lie at the lower side of the ellipsoid. Along the centerline the flow appears to be completely in x-direction. What we expect to see in the Poincaré surface is the trace of the center streamline toward the lower side of the ellipsoid and away from the upper side, Fig. 7.14. This would demand—at the upper side—in Fig. 7.13 a recognizable flow component toward the centerline. This more or less hidden flow pattern can be visualized by considering two particular solutions of the potential equation, one for α = 0◦ and one for α = 90◦ . In [26] this was achieved by setting in the first case u ∞ = sin 30◦ and in the second u ∞ = cos 30◦ . Because we deal with potential theory, the computed velocity fields can be superimposed. The resulting flow field is that shown in Fig. 7.13.

Fig. 7.14 The Poincaré surface at x = 0.5

170

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.15 Streamlines of the two particular solutions at the upper surface of the ellipsoid. Left side: solution for α = 0◦ . Right side: solution for α = 90◦

Indeed the α = 90◦ solution clearly shows the flow turning toward the centerline. Moreover, the dot in the right picture indicates the upper detachment point where the flow leaves the surface. With all due reservations this is a possible explanation. Of course with that we have a special case, which cannot be generalized. In the reality of three-dimensional compressible viscous flow other considerations may be helpful.

7.5 Structural Stability and Changes of Flow Fields Above we have noted that only the phase portraits lying in the quadrant J > 0 and T < 0 of Fig. 7.3 are stable. The flow field past a body depends on the parameters Reynolds number, Mach number, thermal state of the surface, see, e.g., [20], as well as on the angles of attack, of side-slip, and on the body geometry and its aero(thermal)elastic deformation. If one or more of these quantities are changed, the flow field past the body changes. If then the topological structure of the flow field, i.e., the number and the kind of the singular points and their connections remain unchanged, the structure is topologically stable. If, however, small changes of one or more of the above quantities lead to a new topological structure of the flow field, the original topological structure is unstable. In that case (a) new singular points may appear, original ones may disappear, and (b) the connections of the singular points may change. In case (a) we have a local bifurcation, that is the flow field is changed only locally. In case (b) we speak of a global bifurcation, because the whole flow field is changed. Global bifurcation is to be expected, if, for instance, saddle point-saddle point structures are given, like shown in Fig. 7.2a. The separation bubble there is beginning and ending with a half-saddle point each.

7.5 Structural Stability and Changes of Flow Fields

171

Fig. 7.16 Structural change of the flow at the aft of a blunt body [27]. Local bifurcation: A → B, steady separation; global bifurcation: B → C, periodic separation (vortex shedding)

Consider the situation shown in Fig. 7.16. Schematically the flow field at a blunt afterbody is given with the Reynolds number Re as bifurcation parameter (from the thesis of B. Schulte-Werning, a doctoral student of the first author of this book [27]). From case A, attached viscous flow at very small Reynolds number Re (two halfsaddle points, one is located at the forward stagnation point, Rule 2 is fulfilled), the flow field changes with increasing Re to case B, where we see at the rear stagnation point a steady symmetric separation pattern. This is considered to be a local bifurcation. We count now one saddle point, four half-saddle points, and two focus (nodal) points. Also with that Rule 2 is fulfilled, i.e., here, too, we have a valid topological structure. If Re is increased further, global bifurcation leads to a break-up of the connections between the original saddle points and a periodic vortex shedding is present, case C. At a real body a further increase of Re then will lead to three-dimensional unsteady separation and vortex shedding. We illustrate the above and in particular the latter observation with the flow past a sphere as function of increasing Reynolds-number domains, Fig. 7.17. The figure is based on a compilation in [27] of experimental and theoretical data given in the literature. We observe the following separation and wake phenomena as function of the Reynolds number Re: (1) Re < 20. Creeping flow, no separation behind the sphere. (2) 20 < Re < 500. Steady annular vortex behind the sphere. For Re > 130 the free stagnation point behind the sphere begins to oscillate and the wake becomes unstable. (3) 500 < Re < 1.5·103 . Periodic shedding of annular vortices and appearance of a “vortex chain”. (4) 1.5·103 < Re < 104 . Discrete annular vortices are forming an “annular vortex street”, which, however, disintegrates shortly behind the sphere. Periodic vortex clusters appear. (5) 104 < Re < 3.7·105 . Rotating separation of two vortex filaments at the back side of the sphere and formation of a helix-like wavy wake. (6) 3.7·105 < Re. Turbulent boundary layer over the sphere (the figure indicates a transition triggering device). Separation happens now far behind the spheres

172

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.17 Structures of wakes of a sphere in steady motion as function of the Reynolds number [27]

equator. A strong drag decrease results. In the time-average a horse-shoe like separation of a vortex sheet appears. It must be noted that this classification is a very crude one [27]. It is not unambiguous whether the wake development always is an unsteady three-dimensional phenomenon and whether the structural changes of the flow topology are as shown. Much depends on the setup of an experiment or a numerical investigation. Therefore the different experimental and theoretical/numerical investigations may have led to different flow-field realizations. The sphere is geometrically a very simple body. The flow past it can be very complex as we just have seen, with several structural changes. A strong tendency to structural changes of the separation topology, however, is seen also on hemispherenosed cylindrical configurations, which represent the simplest form of an aircraft’s fuselage. In [28] the interested reader will find experimental results obtained with such a configuration for Mach numbers in the subsonic and the transonic regime, with laminar, forced and natural turbulent flow, at angles of attack up to α = 40◦ . The flow topology is analyzed, also cases with open-type separation are identified. Further discussions of these experimental data are contained in [3, 4]. Structural changes do not necessarily happen instantly. We look at Fig. 7.18 where the topology of the skin-friction lines on the upper side of a delta wing and the pseudostreamlines in the Poincaré surface are sketched for different angle-of-attack ranges [29]. First we study the lee-side picture, looking at the left side of the wing, see also the discussion of Fig. 10.15 on Sect. 10.2.2.

7.5 Structural Stability and Changes of Flow Fields

173

Fig. 7.18 Possible flow structures at a slender delta wing with sharp leading edges [29]. Upper part: small angle of attack, accordingly only a small separation bubble along the leading edge, open lee-side flow field. Middle part: medium angle of attack, lee-side vortices, secondary vortices, open lee-side flow field. Lower part: large angle of attack, lee-side vortices, secondary vortices, closed lee-side flow field

174

7 Topology of Skin-Friction and Velocity Fields

• Leeward side At the beginning of our consideration we note that in general over a slender deltawing leading-edge separation phenomena begin to appear at the rear of the wing. With increasing angle of attack they move forward in direction of the wing tip. • Small angle of attack At small α a small three-dimensional separation bubble is present along the leading edge. We see a kind of an open lee-side flow field with parallel or conical flow at the center domain. • Medium angle of attack A fully developed lee-side vortex system—with secondary vortices—is present. The flow field is of open lee-side type. At the center domain again parallel or conical flow is present. • High angle of attack The lee-side flow field is of closed type. The vortex system is like that at medium angle of attack. Along the center line now lies an attachment line. Skin-friction lines diverge from it to both sides. How the change from one structure to the other happens, is an open question. Criteria for the appearance of either flow structure are not known to the authors of this book. Deciding parameters are the wing shape, the leading-edge shape, angle of attack, free-stream Mach and Reynolds numbers. The structure changes basically also appear at delta wings with round leading edges. At transonic and supersonic free-stream Mach number embedded cross-flow shocks are present. Additionally we note that at very slender wings, in general slender configurations, the lee-side vortex system with increasing angle of attack becomes asymmetric with a strong influence on forces and moments, Fig. 7.19, see also Sect. 10.2.5. This phenomenon is discussed in detail in [1] from the side of structural stability/instability. • Windward side Whereas the lee-side flow field with their vortex phenomena has found widespread interest, the flow at the windward side almost none. As mentioned in Sect. 7.4.3 we can distinguish between two classes of configurations: (1) more or less round fuselage, (2) flat or nearly flat lower surface of a wing or a fuselage, in particular a delta wing with round or sharp leading edges. For class (1) we get the picture shown in Fig. 7.20: at angle of attack the primary attachment line lies on the windward surface centerline of the forebody. There it will remain with increasing angle of attack unless at very high angle the whole flow field becomes asymmetrical and even unsteady. But what happens, if we continuously flatten the body until we have the shape of a delta wing? Obviously at some station of the process a bifurcation will occur and we get the two attachment lines typical for the flat delta wing. In class (2) the situation is quite different. A gedankenexperiment shows us that at zero angle of attack of for instance a symmetrically round leading-edged delta wing the primary attachment lines are located on the generatrices of the leading edges. This is exactly the situation, which we have at a large aspect-ratio wing at zero angle of attack. If the angle of attack is increased, the attachment lines move a little distance below the generatrices. Even at high angles of attack this distance will be small. At very high angles of attack the two attachment lines probably will merge into a single attachment line at the center line of the wing, which can be considered as a reverse bifurcation.

7.5 Structural Stability and Changes of Flow Fields

175

Fig. 7.19 Asymmetric flow past a slender delta wing at high angles of attack at M∞ = 2.8 [1]. At the right interpreting sketches from D. J. Peake and M. Tobak of the vapor-screen photographs at the left from [30] Fig. 7.20 Sketch of the skin-friction line pattern at the lower side of a circular forebody at angle of attack [31]

176

7 Topology of Skin-Friction and Velocity Fields

Fig. 7.21 Sketch of the skin-friction line pattern at the flat lower side of a forebody at angle of attack [31]

What we see in the lower part of Fig. 7.18 probably in reality happens only if already an asymmetrical lee-side vortex arrangement is present. All this will depend to a degree on the actual shape of the windward side, it also holds for sharp leading edges. In the angle-of-attack range considered here, the primary attachment line will never lie in the center line of the lower surface, as indicated in Fig. 7.18. For all angles of attack we get the picture shown in Fig. 7.21. Important is that between the two primary attachment lines the flow is more or less two-dimensional, depending on the degree of flatness of the surface. The two-dimensionality is a very welcome flow property, because it leads to a favorable onset flow, for instance of that approaching an engine air inlet or an aerodynamic control surface [31]. Precise angle of attack ranges as discussed with Fig. 10.15 on Sect. 10.2.2 are not known, because class (2) has sofar found only little attention in the field of flow topology. • Concluding Remarks We have discussed structural stability and flow-field changes by looking at the flow past a sphere and past a slender delta wing. We have seen that a host of possible topological structures can evolve, depending on the range of flow parameters and geometrical properties. However, we did hardly cover the wide range of actual flow and geometrical parameters, which we may encounter in real flight vehicle operation. Our aim was to show qualitatively the high complexity, which can be present and with which the flight vehicle designer is confronted. No general theory and no criteria exist to predict what one gets topologically in reality. Of course, basic flow structures and their changes can be described with the help of topological considerations, structural stability and bifurcation theory as is shown, for instance, very detailed in a DLR report by U. Dallmann [4]. In design work and in problem diagnosis, however, one has to be aware of the possibility of multiple phenomena, which have a more or less large influence on the separated and vortical flow field past the flight vehicle and hence on the behavior and the performance of the latter. It helps to study—actually that is necessary— what was found in earlier ventures and in experimental and in theoretical/numerical research, investigations and vehicle shape development. Even then surprises, either unwelcome or welcome ones, cannot be avoided.

7.6 Problems

177

7.6 Problems Problem 7.1 Open-type attachment and separation lines are present in some of the figures in Sect. 8.4. Identify these figures and give short descriptions. Problem 7.2 Which topological rules apply for the two flow fields sketched in Fig. 7.2? Problem 7.3 Consider the flow past the circular forebody of Fig. 7.20. If you look at the cross section ahead of the inlet, what topological structure is present, if the flow would be inviscid? Problem 7.4 The eigenvalues of the matrix A, Eq. (7.6), are found with det (A − λI), where I is the n × n identity matrix, and λ (n = 1, 2, 3) are the eigenvalues. Derive the eigenvalues of the matrix A. Problem 7.5 What is the prerequisite for plane-of-symmetry flow? Give a verbal definition. Problem 7.6 Assume a curved inviscid attachment line. Inflection points appear in the approaching streamlines on one side of the attachment line. On what side do they appear and why? Problem 7.7 What are off-surface flow portraits? Give a verbal account. Problem 7.8 In Sect. 7.3 several flow characteristics associated with singular lines, i.e., attachment and separation lines, are discussed. Give a summary of these characteristics. Problem 7.9 How is separation defined in two-dimensional flow? What are the indicators of separation in three-dimensional flow?

References 1. Peake, D.J., Tobak, M.: Three-dimensional interaction and vortical flows with emphasis on high speeds. NASA TM 81169 (1980) and AGARDograph 252 (1980) 2. Moffat, H.K., Tsinober, A. (eds.): Topological Fluid Mechanics. Proceedings of the IUTAM Symposium, Cambridge, GB, 1989. Cambridge University Press (1990) 3. Dallmann, U.: Topological structures of three-dimensional flow separations. DLR Rep. 221-82 A 07 (1983) 4. Dallmann, U.: On the formation of three-dimensional vortex flow structures. DLR Rep. 221-85 A 13 (1985) 5. Délery, J.: Three-Dimensional Separated Flow Topology. ISTE, London and Wiley, Hoboken (2013) 6. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014)

178

7 Topology of Skin-Friction and Velocity Fields

7. Sears, W.R.: The boundary layer of yawed cylinders. J. Aeronat. Sci 15, 49–52 (1948) 8. Hirschel, E.H., Fornasier, L.: Flowfield and Vorticity Distribution Near Wing Trailing Edges. AIAA-Paper 84–0421 (1984) 9. Schwamborn, D.: Boundary layers on finite wings and related bodies with consideration of the attachment-line region. In: Viviand, H. (ed.) Proceedings of the 4th GAMM-Conference on Numerical Methods in Fluid Mechanics, Paris, France, October 7 - 9, 1981. Notes on Numerical Fluid Mechanics, vol. 5, pp. 291–300. Vieweg, Braunschweig Wiesbaden (1982) 10. Lighthill, M.J.: Attachment and separation in three-dimensional flow. In: Rosenhead, L. (ed.), Laminar Boundary Layers, pp. 72–82. Oxford University Press, Oxford (1963) 11. Wang, K.C.: Boundary layer over a blunt body at high incidence with an open type of separation. Proc. R. Soc., Lond. A 340, 33–55 (1974) 12. Lugt, H.J.: Introduction to Vortex Theory. Vortex Flow Press, Potomac (1996) 13. Kaplan, W.: Ordinary Differential Equations. Addison-Wesley Publishing Company, Reading (1958) 14. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Qualitative Theory of SecondOrder Dynamic Systems. Wiley, New York (1973) 15. Hornung, H., Perry, A.E.: Some Aspects of Three-Dimensional Separation, Part I: Streamsurface Bifurcations. Z. Flugwiss. und Weltraumforsch. (ZFW) 8, 77–87 (1984) 16. Bakker, P.G., de Winkel, M.E.M.: On the Topology of Three-Dimensional Separated Flow Structures and Local Solutions of the Navier-Stokes Equations. In: Moffat, H.K., Tsinober, A. (eds.), Topological Fluid Mechanics. Proceedings of the IUTAM Symposium, Cambridge, GB, 1989, pp. 384–394. Cambridge University Press, Cambridge (1990) 17. Tobak, M., Peake, D.J.: Topology of three-dimensional separated flows. Ann. Rev. Fluid Mech., Palo Alto 14, 61–85 (1982) 18. Oswatitsch, K.: Die Ablösebedingungen von Grenzschichten. In: H. Görtler (ed.), Proceedings of the IUTAM Symposium on Boundary Layer Research, Freiburg, Germany, 1957, pp. 357– 367. Springer, Berlin (1958). Also: The Conditions for the Separation of Boundary Layers. In: Schneider, W., Platzer, M. (eds.) Contributions to the Development of Gasdynamics, pp. 6–18. Vieweg, Braunschweig Wiesbaden, Germany (1980) 19. Dallmann, U., Hilgenstock, A., Riedelbauch, S., Schulte-Werning, B., Vollmers, H.: On the footprints of three-dimensional separated vortex flows around blunt bodies. In: Attempts of Defining and Analyzing Complex Vortex Structures. AGARD-CP-494, 9-1–9-13 (1991) 20. Hirschel, E.H.: Basics of Aerothermodynamics. 2nd, revised edn. Springer, Cham (2015) 21. Hirschel, E.H.: Evaluation of Results of Boundary-Layer Calculations with Regard to Design Aerodynamics. AGARD R-741, 5-1–5-29 (1986) 22. Davey, A.: Boundary-layer flow at a saddle point of attachment. J. Fluid Mech. 10, 593–610 (1961) 23. Hunt, J.C.R., Abell, C.J., Peterka, J.A., Woo, H.: Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization. J. Fluid Mech. 86, 179–200 (1978) 24. Van Dyke, M.: An Album of Fluid Motion. The Parabolic Press, Stanford (1982) 25. Hirschel, E.H.: Viscous Effects. Space Course 1991, RWTH Aachen, Germany, 12-1 to 12-35 (1991) 26. Weiland, C.: Personal Communication (2017) 27. Schulte-Werning, B.: Numerische Simulation und topologische Analyse der abgelösten Strömung an einer Kugel (Numerical Simulation and Topological Analysis of the Separated Flow Past a Sphere). Doctoral Thesis, Technical University München, Germany (1990) 28. Bippes, H., Turk, M.: Oil flow patterns of separated flow on a hemisphere cylinder at incidence. DLR Rep. 222-83 A 07 (1983) 29. Werlé, H.: Apercu sur les Possibilités Expérimentales du Tunnel Hydrodynamique a Visualization de l’O.N.E.R.A. ONERA Tech. Note 48 (1958)

References

179

30. Fellows, K.A., Carter, E.C.: Results and Analysis of Pressure Measurements on Two Isolated Slender Wings and Slender Wing-Body Configurations at Supersonic Speeds. Vol. 1, Analysis, ARA Rep. 12 (1969) 31. Hirschel, E.H., Weiland C.: Selected Aerothermodynamic Design Problems of Hypersonic Flight Vehicles. In: Progress in Astronautics and Aeronautics, AIAA, Reston, Va., vol. 229. Springer, Heidelberg (2009)

Chapter 8

Large Aspect-Ratio Wing Flow

This chapter is devoted to the discussion of the flow—mainly the trailing vortex layer and the pair of trailing vortices—past lifting large aspect-ratio wings in view mostly of the results and insights gained in Chap. 4. Considered always is the cleanwing situation, selected topics of the real-wing situation (high-lift system, integrated engines etc.) are presented in Chap. 9. We begin in the introduction with general considerations of wing-wake properties as well as wing shapes and their influence on the wing flow. Unit Problems are treated in the following three sections. In Sect. 8.2 it is shown that—and how—the flow-vorticity properties are present in panel method solutions (Model 4 in Table 1.3) of lifting wing flow. The same is made in Sect. 8.3 for Euler solutions (Model 8). In Sect. 8.4 finally the flow past the Common Research Model (CRM) is studied with the help of a RANS/URANS solution (Model 10). Besides the wake flow and its vorticity properties also issues of the flow at the wing root and at the wing tip are considered.

8.1 Introduction 8.1.1 Five Flow Domains at and Behind the Lifting Wing The topic of the chapter is separated and vortical flow past clean lifting large aspectratio wings.1 Generic wing forms and also the wing of a modern transonic transport aircraft configuration are considered. We first have a look at the separation and vortex phenomena being present over and behind a real aircraft in cruise condition, Fig. 8.1. The aircraft is considered to be in straight and level flight, in steady longitudinally stable motion and in a trimmed state. 1 Regarding

differentiation aspects of large and low aspect-ratio wings see Sect. 8.4.4.

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_8

181

182

8 Large Aspect-Ratio Wing Flow

Fig. 8.1 Schematic of the development of the wake system of a large aspect-ratio aircraft in cruise flight [1]. Five Flow Domains: Domain 0: Wing Flow (not indicated), Domain 1: Near field, Domain 2: Extended near field, Domain 3: Mid field/Far field, Domain 4: Decay region

Four flow domains of the wake are indicated in the figure [1]. There are no sharp boundaries between them. Depending on the view one takes considerable overlaps are present. We aim for a qualitative description of the separated and vortical flow over and behind the aircraft. We add a fifth domain, Domain 0, to also treat the flow phenomena found over the wing, in particular at its trailing edge. – Flow Domain 0 Wing: On the wing the upper and lower side flow-fields shear is present. At the upper wing side the flow generally is directed slightly toward the wing’s root, at the lower side toward the wing’s tip. Engine and flap track installations locally influence the flow fields. Because the trailing edge is swept backward, except maybe for a kink,2 the combined overall flow direction is outward, resulting in a positive vortex-line angle (y), Fig. 4.11. The trailing-edge flow shear angle ψe (y) increases from the wing’s root toward its tip. The resulting near-wake structure, Fig. 4.12, however is somewhat distorted due to the actual shape of the trailing edge, which generally has a finite thickness, Sect. 6.3. The change of ψe (y) in y-direction is connected to the spanwise distribution of the circulation Γ (y), reflected by the compatibility condition Eq. (4.20) in Sect. 4.4. Accordingly the wake—the trailing vortex layer—carries kinematically active vorticity Ωs , whose strength increases toward the wing tip. The kinematically inactive vorticity Ωt locally reflects the chordwise boundary-layer properties. The flow leaves the wing’s trailing edge in the form of flow-off separation. To be added then are three further—ordinary—separation phenomena, Fig. 8.2. Present in any case are the tip vortices which come into being by the flow around the wing tips and the subsequent separation. Their sense of rotation is that of 2 This

kink near the wing root is called a Yehudi break after its inventor at Boeing [2].

8.1 Introduction

183

Fig. 8.2 Schematic of ordinary and flow-off separation at the CRM configuration, [3], topic of Sect. 8.4

the trailing vortices, into which they finally are merged. Usually at the wing tip a secondary separation is present, which leads to an additional weak, but counterrotating secondary vortex, tertiary vortices are possible, too. Hence we have a tip-vortex system. At engine pylons and nacelles separation phenomena may lead to discrete vortices, which interact directly or indirectly with the near wake. No such vortices should be due to the flap tracks. At the wing roots finally horse-shoe vortices are present, if no proper wing-root fairing is applied.3 In general a wing-root fairing is present, with different layout at the upper and the lower side of the wing root. – Flow Domain 1 Near field: At the beginning of Domain 1 the near wake is established as a trailing vortex layer, containing kinematically active and inactive vorticity. Domain 1 extends from the wing’s trailing edge downstream over a distance of approximately the reference chord length of the wing lμ or the half span b/2. Often it is considered to extend up to the beginning of the afterbody of the fuselages. The roll-up process of the trailing vortex layer is in the beginning, which eventually leads to the pair of counterrotating trailing vortices at the end of the domain, see the schematic in Fig. 4.18. Besides the roll-up of the trailing vortex layer, other effects occur, in particular the appearance of the wing-tip vortex system, as well as possible wing-root vortices. – Domain 2 Extended near field: Due to self-induction, Sect. 3.12.1, the trailing vortex layer begins to curl up into the pair of counterrotating trailing vortices. The tip-vortex system and possible other vortices are merged into the vortex layer 3 These

vortices sometimes are called necklace vortices.

184

8 Large Aspect-Ratio Wing Flow

Fig. 8.3 The wake of an aircraft in flight [4]

during the curl-up process. Because the aircraft flies in a trimmed state, a weaker trailing vortex layer with a kinematically active vorticity of opposite sign leaves the horizontal stabilizer. It also curls up into a pair of trailing vortices which each counter-rotate to the main trailing vortex pair. They finally are merged into that pair, however with particular mechanisms. From the fuselage as well as from the vertical stabilizer mainly kinematically inactive vorticity joins the aircraft’s wake. The influence of the propulsion jets is more or less restricted to the extended near field and usually without larger importance. Figure 8.3 gives an impression of the aircraft’s wake in the extended near field. – Flow Domain 3 Mid field/Far field: The far field sees the fully established trailing vortex pair more or less unchanged for a distance of up to one hundred wing spans. The vortex diameters increase slightly due to diffusion processes. The ratio of vortex spacing to vortex length of course shrinks and the susceptibility increases for reciprocal induction processes. – Flow Domain 4 Decay region: In the decay region finally the vortex pair breaks up due to the so-called Crow instability, Sect. 3.11.3. Atmospheric disturbances are taken up, their intensities governing the extent of the decay process.

8.1 Introduction

185

8.1.2 Influence of the Trailing Vortex Layer and Vortices on the Wing’s Performance Of interest now is the influence of the trailing vortex layer/vortices on the wing’s lift and drag. L. Prandtl described with his lifting-line model, Fig. 3.3, the situation in the frame of what now is called circulation theory (Model 4) [5]. The basic result is that the kinematically active vorticity in the trailing vortex layer, respectively the trailing vortices, induces an extra wing drag, the induced drag. Besides the skin-friction drag and the form drag this is the third part of the total drag, which can amount to be the dominant part of it. In the transonic and the supersonic regime the wave drag and the interference drag appear, Sect. 2.4. We do not reproduce Prandtl’s theory, instead we point to the literature, e.g., [6]. We state the three important results for a finite-span lifting wing without aerodynamic or geometrical twist at a subsonic flight speed such that the flow can be considered as being incompressible: – (1) If the wing has an elliptical planform, the circulation distribution Γ (y) is elliptical, the induced drag Di is minimum, – (2) If the circulation distribution is elliptical, the induced downwash velocity wi at the wing is constant in span direction, as is the induced angle of attack αi . – (3) If an elliptic circulation distribution is given, the induced drag coefficient is inversely proportional to the aspect ratio Λ, Sect. 3.16: C Di =

C L2 . πΛ

(8.1)

With this result the amount of induced drag Di of two wings at equal lift L 2 = L 1 and equal reference area Ar e f2 = Ar e f1 , but different aspect ratios Λ, can be compared in the frame of potential theory (Model 4). Hence neglecting viscous forces and compressibility effects we obtain in terms of the force coefficients   1 C2 1 . (8.2) − C Di,2 − C Di,1 = L π Λ2 Λ1 Aircraft wing design, also in the transonic flight regime, always makes use of these insights, however, not by following them exactly. Only a few aircraft actually had or have a wing with elliptical planform. A wing with such a planform is more expensive to manufacture than a wing with straight leading and trailing edge. Examples were seen, where the wing originally had an elliptical planform, which was changed to a tapered planform, once quantity production was demanded. An approximation to an elliptical planform to a degree results from wing tapering. The taper ratio of a wing is λt = ct /cr , where ct is the wing-tip chord length, and cr that of the wing’s root. The aspect ratio Λ—in the literature also denoted with A R—always is an important parameter, because it governs the lift-to-drag ratio L/D—the aerodynamic effi-

186

8 Large Aspect-Ratio Wing Flow

ciency of the wing—and hence the flight performance. Values range from Λ ≈ 6 for small general aviation aircraft and up to Λ ≈ 50 for high-performance gliders. In general the aspect ratio should be as large as possible. Limiting are aerodynamic shape demands, structural issues, weight, maneuverability, sub-system issues, and airfield restrictions. Special aerodynamic shape demands arise beyond the subsonic flight regime. Regarding transonic aircraft we have seen in Sect. 2.4 that due to drag issues either a swept wing or a thin wing must be employed. The thin wing concept is not a choice for transonic transport aircraft, because of structural and fuel carrying demands. The rule now is the backward swept wing with supercritical chord sections, although in principle the wing also can be swept forward. Modern backward swept-wing aircraft, for example the Airbus A350 and the Boeing 787 have an aspect ratio Λ = 9.5. In the supersonic flight regime, Chap. 10, we generally find aircraft with small aspect-ratio wings, for instance the unswept but tapered thin-wing F-104 Starfighter with Λ = 2.45, the delta wing (ogive wing) Concorde with Λ = 1.55, the Eurofighter Typhoon fighter aircraft with Λ = 2.205, the F-35A with Λ = 2.663, and in the hypersonic flight regime the Space Shuttle Orbiter with a strake-delta planform and Λ = 2.265. Wing design still strives for a spanwise elliptical circulation distribution. The circulation around a wing chord section of course is not the line integral of the velocity along a streamline. The inviscid streamlines already at an elliptic wing do not lie in chord direction, see, e.g., Fig. 7.1. Instead of the span-wise distribution of the circulation usually that of the equivalent local lift coefficient Cl is considered, Sect. 3.16. The desired local lift distribution is found—for the given design lift of the wing— with the help of changes of the wing’s taper, the (static) geometrical twist distribution, and the camber distribution.4 Also playing a role is the aerodynamic twist due to the elastic deformation of the wing under load, in particular if the wing is a swept wing, see, e.g. [7]. In the frame of this book we cannot go into the details of wing design. We point to the publications of D. Küchemann 1978/2012 [8], H. Schlichting and E. Truckenbrodt 1959/1979 [9], E. Obert 2009 [2], R. Rudnik in the Handbuch of Luftfahrzeugtechnik (in German) 2014 [10], R. Vos and S. Farokhi 2015 [7], and of A. Rizzi and J. Oppelstrup 2020 [11]. In the following sections we demonstrate with a few Unit Problems the capabilities of three kinds of computational flow models of Table 1.3. In Sect. 8.2 results of several panel methods (Model 4) are checked with flow computations past lifting wings (Flow Domain 0). It is shown that the overall results are in reasonable agreement. Regarding the flow fields near the trailing edges of the wings, however, most of the methods yield improper results. Only a higher-order formulation gives the proper result, fulfilling at the trailing edge the compatibility condition, Eq. (4.20).

4 The

decades old concept of in-flight camber changes now seems to approach reality thanks to the potential of carbon-fiber structures.

8.1 Introduction

187

An Euler solution is applied to a large aspect-ratio wing in Sect. 8.3 (Flow Domains 0 and 1). Demonstrated is that the near-wake properties are computed in a proper way with a Model 8 method. At the trailing edge the compatibility condition is fulfilled. In Sect. 8.4 finally the flow past the wing of the Common Research Model (CRM) is investigated (Flow Domains 0, 1 and 2). Applied was a Model 10 method, i.e., a method with the highest modeling level. All the considered lifting large aspect-ratio wings have in common a trailing vortex layer, which originates at the trailing edge of the wing. Tip-vortex systems play a certain role, too. This is shown only for the CRM case. There also happens an interaction with the trailing vortex layer of the horizontal tail plane. Our aim is to provide and foster the understanding of the basic properties of the trailing vortex systems and their behavior. We further want to demonstrate the capabilities of numerical simulation methods of such flows and the interpretation of the results from a fluid-mechanical point of view. Generally we do not deal with detailed configurational issues of large aspect-ratio aircraft. In the following chapter, Chap. 9, we sketch some practical problems.

8.2 Panel Method (Model 4) Solutions—Proper and Improper Results in Flow Domain 0 This section is devoted to a short presentation of results of potential-flow theory for two different lifting wings in Flow Domain 0. Results are shown regarding the trailing-edge flow shear angle ψeu , the kinematically active vorticity content Ωs , the vortex-line angle , the circulation Γ and its derivative in spanwise direction dΓ /dy, as well as skin-friction line patterns of boundary-layers, which for one of the wings were obtained with boundary-layer computations. We discuss examples of flows found with panel methods, i.e., Model 4 methods. The wings are the Kolbe wing, [12], and a forward swept wing, which was studied in [13]. The results were first published in [14]. All figures in this section are from that paper. In [14] also properties of panel methods were treated, which are of importance when considering Model 4 methods. In the following figures we give references to panel method 1 and to other panel methods (2, 3, etc.). Panel method 1 is the HISSS method of L. Fornasier [15].5 This higher-order panel method has linear source distributions and quadratic doublet distributions in both chord-wise and span-wise directions. We will see that only this method yields proper trailing-edge flow results. The other methods are lower-order methods. They have stepwise constant doublet distributions—or equivalent—in both chord- and span-wise directions. Such approaches lead to an erroneous determination of the v-component of the velocity vector near the wing’s trailing edge. A similar problem was reported in [16], where 5 HISSS

method = Higher-Order Subsonic-Supersonic Singularity method.

188

8 Large Aspect-Ratio Wing Flow

Table 8.1 Geometrical and flow parameters of the Kolbe wing example [14] ϕ0 [◦ ] Λ λt M∞ α [◦ ] Recm cm [m] Boundary-layer state 45

3

0.5

0.25

8.5

18·106

1

Fully turbulent

Fig. 8.4 Planform of the Kolbe wing [14]

the failure of such low-order panel methods to compute the flow past thin wings was attributed to an inadequate doublet distribution in the chord-wise direction. Earlier boundary-layer studies based on such improperly computed inviscid flow fields even led authors come to the conclusion that the boundary-layer flow over swept wings is predominantly two-dimensional. This of course is not true.

8.2.1 The Kolbe Wing The Kolbe wing is a swept wing with a large sweep angle ϕ0 of the leading edge [12]. This angle, the aspect ratio Λ, the taper ratio λt , the free-stream Mach number M∞ , the angle of attack α, the Reynolds number Recm , the reference (mean) chord length cm , and the state of the boundary layer regarding the results of a boundary-layer computation are given in Table 8.1. The planform of the wing is shown in Fig. 8.4. The chord section normal to the leading edge is a NACA 64-010 airfoil.6 Consider now Fig. 8.5. It shows the dimensionless span-wise circulation distribution Γ /(cm u ∞ ), its derivative d Γ /d y, and the distribution of the kinematically active vorticity content Ω ≡ Ωs , all computed with the panel methods 1 and 2. 6 The

large leading-edge sweep of the wing combined with a large angle of attack—see below— makes it necessary to investigate whether leading-edge separation can be present.

8.2 Panel Method (Model 4) Solutions—Proper and Improper Results in Flow Domain 0

189

Fig. 8.5 Kolbe wing [14]: comparison of the results of panel method 1 and 2: circulation distribution Γ , its derivative d Γ /d y, and the vorticity content Ω = Ωs /u ∞ as functions of the half-span coordinate 2y/b

The Γ (y) results of method 1 and the lower-order method 2 do not differ much, likewise the results regarding lift and induced drag (not shown). The point-wise investigation of the velocity components at the trailing edge with Eq. (4.20) yields values of Ω (=Ωs /u ∞ ), which for the higher-order method 1 agree very well with d Γ /d y, which means that the compatibility condition, Sect. 4.4, is fulfilled. However, for method 2 with Ω ≈ 0 they are wrong throughout. The trailing-edge flow shear angle ψ, given in Fig. 8.6 reflects this result. Panel method 1 shows the expected rise of ψl (=ψe /2) toward the wing tip, whereas method 2 yields a nearly zero shear angle.7 In contrast to that, the vortex-line angle ε is the same for both methods. The vortex line for the back-swept trailing edge is deflected in wing-tip direction by ε ≈ 5◦ . At the root and at the tip ε is approximately zero. Regarding the matter of properly or improperly computed inviscid velocity fields and the boundary-layer computations based on them, we study Fig. 8.7. Results of three-dimensional boundary-layer computations are shown, performed with the external inviscid flow fields found with panel method 1 and 2. The boundary-layer method was the integral method of Cousteix-Aupoix in the MBB version [17]. We look at the streamlines of the external inviscid flow and the skin-friction lines of the fully turbulent boundary layer at both the suction (upper) and the pressure (lower) side of the wing. Upstream of and at the trailing edge on both sides the inviscid streamlines found with method 1 show directions according to the vortexline angle ε and the shear angle ψl in Fig. 8.6. The inviscid streamlines found with panel method 2, however, approach the trailing edge nearly in chord direction. This holds in particular for the pressure side. 7 For the definition of the trailing-edge flow shear angle ψ

e

and the vortex-line angle ε see Fig. 4.11.

190

8 Large Aspect-Ratio Wing Flow

Fig. 8.6 Kolbe wing [14]: comparison of the results of panel methods 1 and 2: trailing-edge flow shear angle ψl and vortex-line angle ε as functions of the half-span coordinate 2y/b

Note that only method 1 yields the points of inflection of the inviscid streamlines near approximately 75 per cent chord length at almost the whole pressure side, Fig. 8.7b. The error in the inviscid flow field and therefore in the boundary-layer solution found with method 2 extends over approximately 50 per cent chord length upstream of the trailing edge of both the suction and the pressure side of the wing.

8.2.2 The Forward-Swept Wing The application of the higher-order panel method 1 and of some lower-order panel methods to a forward-swept wing, which was studied in [13], gives a similar result as above. The leading edge sweep angle ϕ0 of the wing, the aspect ratio Λ, the taper ratio λt , the free-stream Mach number M∞ (incompressible flow), and the angle of attack α are given in Table 8.2. The planform of the wing is given in Fig. 8.8. All methods agree rather well regarding the span-wise circulation distribution Γ (y). However, only for method 1 the derivative d Γ /d y(y) and the vorticity content Ωs (y) are compatible, Fig. 8.9. The trailing-edge flow shear angle ψl (= ψ/2) in Fig. 8.10 again shows the expected trend only for method 1. The vortex-line angle is negative with ε ≈ −7 ◦ , i.e., the vortex lines are deflected in wing-root direction. Actually it is the whole flow field at both sides of the wing which is deflected toward the wing’s root.8 The results of the lower-order methods 8 This

deflection is the reason for the well observed unwelcome accumulation of boundary-layer material at the wing root (and the fuselage) of forward swept wings. This can lead to adverse separation phenomena and, with rear-mounted engines at the aft end of the fuselage, makes special measures necessary regarding the position of the engines. The general deflection of course is also a property of the trailing vortex layer which leaves the wing’s trailing edge.

8.2 Panel Method (Model 4) Solutions—Proper and Improper Results in Flow Domain 0

191

Fig. 8.7 Kolbe wing: comparison of the results of panel methods 1 and 2 and a three-dimensional boundary-layer method [14]. Streamlines of the external inviscid flow and skin-friction lines of the turbulent boundary layer: a suction side, b pressure side of the wing

192

8 Large Aspect-Ratio Wing Flow

Table 8.2 Geometrical and flow parameters of the forward-swept wing example [14] ϕ0 [◦ ] Λ λt M∞ α [◦ ] −35.1

4.5

0.35

0

4

Fig. 8.8 Planform of the forward swept wing [14]

Fig. 8.9 Forward swept wing: comparison of results of several panel methods [14]. Circulation distribution Γ , its derivative d Γ /d y, and the vorticity content Ω as functions of the half-span coordinate 2y/b

show much scatter, depending on where the kinematic flow condition is implemented, either on the skeleton plane of the wing, or on the true wing surface.

8.3 Creation of Lift in an Euler Solution (Model 8) for a Lifting …

193

Fig. 8.10 Forward swept wing [14]: comparison of results of several panel methods. trailing-edge flow shear angle ψl and vortex-line angle ε as functions of the half-span coordinate 2y/b

Fig. 8.11 Planform of the considered wing [18]

8.3 Creation of Lift in an Euler Solution (Model 8) for a Lifting Large-Aspect Ratio Wing—Proof of Concept in Flow Domain 0 and 1 The considered large, rather medium aspect-ratio wing is a trapezoidal wing with small leading-edge sweep ϕ0 = 25◦ , Fig. 8.11. The results were presented first at the Symposium on the International Vortex-Flow Experiment on Euler Code (Model 8 of Table 1.3) Validation in 1986, FFA Bromma, Sweden [18]. All figures in this section are from that presentation. The results prove the viability of Model 8 solutions for lifting high aspect-ratio wings.

194

8 Large Aspect-Ratio Wing Flow

Table 8.3 Geometrical and flow parameters of the trapezoidal wing example [18] ϕ0 [◦ ] Λ λt M∞ α [◦ ] 25

3.75

0.3

0.3

5

Table 8.4 Computed force and moment coefficients of the trapezoidal wing example [18] α = 5◦ Euler solution thick wing, Linear theory thin wing, M∞ = 0.3 M∞ = 0.01 CL C Di CM

0.342 0.0103 −0.155

0.315 0.0085 −0.187

8.3.1 The Computation Case and Integral Results The leading edge sweep angle ϕ0 of the wing, the aspect ratio Λ, the taper ratio λt , the free-stream Mach number M∞ (weakly compressible flow), and the angle of attack α are given in Table 8.3. The wing has 12 per cent thickness, the chord-wise section is the NACA 64-012 airfoil. The computation was made with the Euler code described in [19]. The computed lift (C L ), induced drag (C Di ) and pitching moment (C M ) coefficients are given in Table 8.4. They are compared with the results of a linear method (Model 4). Because that method did not take into account the wing’s thickness and moreover was made for M∞ = 0.01, the lift and induced drag results are about 10 per cent lower. This is the right order of magnitude. The results further give proof that Euler methods (Model 8) can be viable design tools.

8.3.2 Details of the Computed Flow Field of Domain 0 and 1 In Fig.8.12 the computed inviscid streamlines on the surface of the wing and in the near wake are given. The streamlines at the wing surface show well the expected pattern, which was sketched in Fig. 4.17. The trailing-edge flow shear angle ψe increases from nearly zero close to the symmetry plane of the wing to about 25◦ at about 90 per cent half span. This behavior is reflected by the distribution of Ωs and dΓ /dy in Fig. 8.19. The vortex-line angle  is nearly zero, because the trailing edge is unswept. Downstream of the trailing edge the streamlines lie in the skeletal plane of the wake, hence no shear is discernible there. A roll-up tendency of the trailing-edge vortex layer is not discernible, either. At the wing’s tip two streamlines indicate the flow around the edge from the lower to the upper side. Whether the Euler solution here approximates

8.3 Creation of Lift in an Euler Solution (Model 8) for a Lifting …

195

Fig. 8.12 Right-hand side of the wing and the near wake with the computed inviscid surface streamlines (view from above) [18]

Fig. 8.13 Plot of the flow vectors over the rear part of the wing and in the near wake at 40 per cent half-span [18]

a tip vortex, cannot be decided. At the trailing edge the said streamlines lie well away from the tip region. In Sect. 5.1 it was argued regarding the wing’s wake—and this is our major point— that the discontinuity layer of potential theory is widened up in the Euler solution due to the numerical diffusive transport. The Euler wake hence represents the trailing vortex layer of the wing. This Euler wake is sketched in the right part of Fig. 5.1. Regarding the Euler boundary layer and the Euler wake in the present solution we look at Fig. 8.13, which at 40 per cent half-span shows a velocity-vector plot. On the wing’s surface the Euler boundary layer is barely discernible. (A computation with a coarser grid, however, had a pronounced Euler boundary layer.) The kinematically inactive wake part of the Euler wake is also non-existent, as is demanded, Fig. 5.1. The kinematically active part of the Euler wake at x/c = 1.05 behind the wing— i.e. at five per cent chord behind the unswept trailing edge—extends over roughly four cells in z-direction, Fig. 8.14. The details of the wake are unclear. Anyway, in terms of the velocity profiles the demanded features of the Euler wake are present. The Euler wake is more pronounced in terms of the total-pressure loss, as seen in the following Fig. 8.15. Shown are the profiles 1 − pt / pt∞ as functions of z

196

8 Large Aspect-Ratio Wing Flow

Fig. 8.14 Cross-flow vector plot in the plane x/c = 1.05 behind the right half of the wing (c is the mid-wing chord length) [18]

Fig. 8.15 Distribution of the total pressure loss 1 − pt / pt∞ (z) at four span stations downstream of the trailing edge of the wing at x/c = 1.05 [18]

above and below the trace of the chord plane at the four stations 0.25, 0.5, 0.75, and 0.9 half-span of the right-hand side of the wing at the mid-wing chord location x/c = 1.05 behind the trailing edge. Unexplained is the apparent total-pressure gain present mainly below the Euler wake at the three inner stations. At the outer station (0.9 half-span) obviously grid-resolution problems are present, causing a significant total-pressure loss outside the wake. In the following three figures the dimensionless velocity components u(z), v(z), and w(z) are given at the four span stations, again at x/c = 1.05. The components u(z) and v(z) are not the components shown in Fig. 5.1 in the local wake coordinate system. The vortex-line angle  is almost zero at all span stations, therefore for convenience the velocity components are approximated by the components found in the Cartesian x, y, z-reference coordinate system shown in the lower left part of Fig. 4.17. In Fig. 8.16 the kinematically inactive part of the Euler wake u(z) indeed is nearly uniform. In Fig. 8.17 the shear of the flow between the upper and the lower side of the Euler wake is well discernible. It increases from the wing’s root toward the wing tip according to the increase of the trailing-edge flow shear angle ψe seen in Fig. 8.12. The asymmetry of the kinematically active v-profiles at the inner stations is due to the small negative vortex-line angle . Again grid-resolution problems are visible.

8.3 Creation of Lift in an Euler Solution (Model 8) for a Lifting …

197

Fig. 8.16 Distribution of the velocity component u(z) at four span stations downstream of the trailing edge of the wing at x/c = 1.05 [18]

Fig. 8.17 Distribution of the velocity component v(z) at four span stations downstream of the trailing edge of the wing at x/c = 1.05 [18]

Fig. 8.18 Distribution of the velocity component w(z) at four span stations downstream of the trailing edge of the wing at x/c = 1.05 [18]

The w-components are stronger positive below the Euler wake than above, Fig. 8.18. At the location of the Euler wake they are nearly zero, as is to be expected.

198

8 Large Aspect-Ratio Wing Flow

Fig. 8.19 Spanwise circulation distribution Γ of the Euler solution and of linear theory, the spanwise derivative dΓ /dy, and the local vorticity content Ωs [18]

8.3.3 The Circulation and the Kinematically Active Vorticity Content Figure 8.19 shows the distribution of the circulation Γ in spanwise direction found with the Euler solution.9 It is compared with the result of the linear Model 4 method. In Fig. 8.19 also the compatibility condition, Sect. 4.4, is evaluated. The spanwise derivative dΓ /dy and the local kinematically active vorticity content Ωs agree quite well with each other from approximately 30 per cent span up to approximately 95 per cent span. Hence in that span interval the compatibility condition, Eq. (4.20), is very well fulfilled. In the wing’s mid-section that is not the case, like near the wing tip, where the flow around it disturbs the picture. In Fig. 8.20 finally the magnitude of the vorticity ω(z) is plotted at the four spanwise stations. Actually it is the vorticity component ωx connected to the kinematically active part of the Euler wake. The structure of its distribution is in accordance with the hypothesis put forward here.

9 In

the mid-section of the wing, the Euler solution shows an irregularity, which seems to be due to the sharp apex of the wing.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

199

Fig. 8.20 Distribution of the magnitude of the vorticity ω(z) (=ωx ) at four span stations downstream of the trailing edge of the wing at x/c = 1.05 [18]

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2 8.4.1 Introduction The trailing vortex layer and the wake-vortex system of a transport aircraft configuration are investigated in the Flow Domains 0, 1 and 2. The section’s content is based on the master thesis of S. Pfnür, [3], a student of the third author of this book. Most of the figures are from his thesis. The considered aircraft configuration is the NASA/Boeing Common Research Model (CRM), [20], which serves as the test configuration of the AIAA CFD Drag Prediction Workshop series, see, e.g., [21, 22]. The CRM shape is a generic wing/body/horizontal tail configuration with transonic supercritical wing design, Fig. 8.2. The airfoil sections and the twist of the wing correspond to the 1-g wing shape at cruise condition.10 The CRM shape was designed for CFD validation purposes, hence no propulsion units were added. The wing, moreover, has a constant trailing-edge thickness over the whole span. The clean wing is attached without a wing-fuselage fairing. Its trailing edge shows a kink, the Yehudi break, mentioned already on Sect. 8.1. The CRM planform is given in Fig. 8.21. Indicated is the wing-tip reference point (WRP) and the x ∗ , y ∗ , z ∗ - reference coordinate system, with z ∗ being normal to the x ∗ –y ∗ plane. The WRP-location is x ∗ = x/b = 0, y ∗ = y/(b/2) = 1, z ∗ = z/(b/2) = 0. The geometrical parameters of the CRM computation case are given in Table 8.5. Note that the reference area Ar e f is the Wimpress area, which Boeing introduced, Sect. 3.16. The flight parameters, Table 8.6, correspond to that of “Case 1b” of the workshop, with the setting angle of the horizontal tail plane α H T P = 0◦ . Despite α H T P = 0◦ , due to the downwash induced by the lifting wing, a negative vertical force is present 10 The 1-g shape is the shape of the elastic wing due to the aerodynamic load at nominal level flight,

see, e.g., [7].

200

8 Large Aspect-Ratio Wing Flow

Fig. 8.21 Planform of the Common Research Model [3]. WRP is the wing-tip reference point (x ∗ = 0, y ∗ = 1, z ∗ = 0) Table 8.5 Geometrical parameters of the CRM computation case [3] ϕ L E [◦ ] ϕ25 [◦ ] λt b [m] lμ [m] Λ 37.5

35

0.275

58.763

7.00532

Table 8.6 Flight parameters of the CRM computation case [3] M∞ Relμ T∞ [K] Tw [K] α [◦ ] 0.85

5·106

310.93

310.93

2

9

Ar e f (m2 ) 383.69

β [◦ ]

Viscous flow portions

0

Fully turbulent

on the horizontal tail plane, like a trim force. Accordingly the horizontal tail plane first has a trailing vortex layer and then a trailing vortex pair. The force and moment coefficients for the longitudinal motion of this case were found to be C L = 0.43, C D = 0.0245, and C M = 6.8 × 10−5 , with the reference point at 33.67 m from the fuselage nose.

8.4.2 Computation Method and Grid Properties The computation method employed was the Reynolds-averaged Navier-Stokes (RANS) method TAU (Model 10), the code for unstructured grids developed at the DLR [23]. The code as unsteady RANS (URANS) code also permits an unsteady computation approach. Two turbulence models were employed, the Spalart-Allmaras S A/k − ω model (original publication [24]) in the S A − neg version, and the Menter − SST model

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

201

Fig. 8.22 The inserted PS grid at x ∗ = 4 [3]. Indicated are the spanwise and vertical numbers of elements

(original publication [25], SST stands for Shear-Stress Transport). Initial turbulence model and grid studies were made with both turbulence models in the RANS mode, the final investigations with the Menter − SST model in the URANS mode. This was made because of a certain unsteady solution behavior. Of course all results obtained in domains of strong ordinary separation must be seen with some reservations. This more so because in some cases a strong dependence on local discretization properties was observed. The basic grid is an unstructured one, determined following the gridding guideline given in [20]. After some adaptations the final unstructured grid had 30.17·106 nodes and 132.3·106 elements. Despite this final resolution it was necessary to insert a partly structured grid (PS grid) in order to correctly capture the wake-flow properties. It extended from the mid wing chord location downstream to x ∗ = 9. Again some adaptations were made. The final medium-sized PS grid had 45.2·106 nodes and 114.2·106 elements. Figure 8.22 shows the element distribution of the PS grid at the cross-section x ∗ = 4, a location approximately halfway along the extended near field. We show a selection of results, highlighting separation and vortex phenomena. We discuss these results in much detail in order to cover all aspects of separated and vortical flow over the CRM wing. A detailed study of aspects of the skin-friction line topology of this case can be found in [26]. With reference to the introduction, Sect. 8.1 and Fig. 8.1, we consider the flow features in the Domains 0, 1 and 2, i.e. from over the wing to the extended near field.

202

8 Large Aspect-Ratio Wing Flow

Fig. 8.23 Schematic of a horse-shoe vortex at the wing/fuselage intersection [10]

8.4.3 Flow Domain 0: The Flow over the Wing, We first study the phenomena at the wing’s root, then at the wing’s tip and finally at its trailing edge. • Wing Root If a wing is attached to the fuselage without a wing-root fairing, the flow separates ahead of the wing’s leading edge and a horse-shoe vortex is created, Fig. 8.23.11 This vortex is the cause of a drag increment and further it can lead to buffet at high angles of attack. Work in the 1930s by Th. von Kármán and co-workers led to the introduction of smooth wing-root fairings (fillets) [28]. Such fairings effectively eliminate these horse-shoe vortices. Wing-root fairings are the rule today for large transport airplanes. Farther below we give an example. At the CRM configuration no root fairing was introduced. The root area hence permits to study separation and vortex phenomena. Figure 8.24 gives a view toward the wing-root area from above. In the left part we see the large pressure coefficient c p at the wing’s leading edge, which leads to the root separation. Over the wing the strong flow acceleration in chord direction reduces the static pressure (blue color), which is impressed also on the side of the fuselage. The cross-section extent of the horse-shoe vortex is small. The absolute value of the skin-friction coefficient |c f | in the right part of the figure indicates separation at the fuselage side ahead of the wing’s root area. Along the leading edge the strong flow acceleration in chord direction is reflected by high skin friction. Toward the trailing edge we see a reduction of the absolute value of skin friction. Figure 8.25 shows the view toward the wing-root area from below. On the left side the c p -distribution indicates a much weaker flow acceleration in chord direction compared to that at the upper side of the wing. The horse-shoe vortex now has a much larger spanwise extent than at the upper side of the wing. This in particular holds toward and in the vicinity of the trailing edge. The amount of the skin-friction coefficient |c f | shows a large extent of the separation area below the wing’s root. At the fuselage generally the skin friction has low values. 11 In

[27] this situation is called flow obstacle, although there the swept obstacle is not a topic.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

203

Fig. 8.24 Wing-root area from above (suction side of the wing) [3]. Left part: pattern of the skin-friction lines and distribution of the surface-pressure coefficient c p . Right part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient c f ≡ |c f |

Fig. 8.25 Wing root area from below (pressure side of the wing) [3]. Left part: pattern of the skin-friction lines and distribution of the surface-pressure coefficient c p . Right part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient c f ≡ |c f |

A closer look at what happens at the wing’s leading edge in the vicinity of the intersection with the fuselage is given in Fig. 8.26. The flow arriving from the front part of the fuselage is forced—due to the presence of the wing root—to separate. We observe then a succession of singular points and associated singular lines. The appearance of the singular lines happens first at the saddle point S1 , from which to the left (lower wing side) and to the right (upper wing side) the primary separation line originates. Next then lies the nodal point N1 . Originating from it we see to the left (lower wing side) an attachment line (secondary attachment line), which tapers off after a short distance (open-type ending). To the right the attachment line is not well developed and is shortly squeezed between the primary and the secondary separation lines.

204

8 Large Aspect-Ratio Wing Flow

Fig. 8.26 Detail wing root topology [3]. Upper part: pattern of the skin-friction lines and distribution of the surface-pressure coefficient c p . Lower part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient c f ≡ |c f |

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

205

Fig. 8.27 LTA leading edge with root fairing [26]. Left part: pattern of skin-friction lines and distribution of the surface-pressure coefficient c p . Right part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient |c f |

Next comes the saddle point S2 . From it a secondary separation line originates. To the left it is well visible until further away below the wing, where it becomes indistinguishable from the primary separation line, Fig. 8.25. To the right we have a similar picture. The secondary separation line above the wing seems to join the primary separation line, too, Fig. 8.24. Finally we come to the nodal point N2 , which represents a nice star nodal point. In the picture to the left from it the attachment line at the wing’s leading edge is well discernible. Along it the skin friction clearly has a finite value, as is to be expected. Its amount also is constant or nearly constant in spanwise direction up to the wing tip. Oddly enough, below N2 to the left an open-type attachment line appears to be present. Probably only the skin-friction lines concentrate along the cutting curve of the wing surface with the fuselage. The pattern of singular points at the wing root discussed in [26] is much simpler. Only the saddle point S1 is present and the nodal point N2 . We note that the simulations there were made with a not—further specified—SA-turbulence model. Pfnür in his grid refinement studies found a similar pattern for a coarse PS grid. That result was obtained with the S A − neg turbulence model [3]. Anyway, the different results show that here a problem is hidden. Experimental data, which could clear up the situation, are not available. Even then one would have to be careful to draw conclusions, because it is a matter of the range of similarity parameters, which also have an influence. All these issues disappear, if a proper wing-root fairing is employed, Fig. 8.27. The figure, taken from [26], shows for a large transport aircraft (LTA) with similar geometrical and flow parameters a smooth transition from the flow past the fuselage to that at the wing’s leading edge. The wing’s attachment line begins as open-type attachment line! Back to the present CRM case: in Fig. 8.28 the dimensionless axial vorticity component ζ=

ωx b 2u ∞

(8.3)

206

8 Large Aspect-Ratio Wing Flow

Fig. 8.28 Dimensionless axial vorticity component ζ around the wing/fuselage intersection [3]. Left part: leading-edge region. Right part: upper trailing-edge region

is plotted at several downstream locations in the wing-root domain. The left part shows in the leading-edge area at the suction (upper) side of the wing that the vortex apparently has a rather small cross-section. The blue color indicates that it rotates in clockwise direction. There is no evidence of a secondary vortex. On the pressure (lower) side we have counter-clockwise rotation. The crosssection is larger and the secondary vortex is clearly indicated. In the trailing-edge region of the suction side we have no indication of the vortex, except for the last stations, where separation is indicated, right part of Fig. 8.28. On both the wing and the fuselage surface the boundary layer thickening in streamwise direction is visible. At the upper side of the wing near the trailing edge the pattern of the skin-friction lines shows a larger separation event in form of a focus point F, Fig. 8.29. In this focus point a vortex filament leaves the surface, Sect. 7.2.2. It is to be expected that at the side of the fuselage a counter-rotating vortex filament leaves the surface. That is well indicated by the skin-friction line pattern there. The experimental investigations did not indicate either phenomenon. • Wing Tip At the well rounded wing tip of the CRM configuration the pattern of the skin-friction lines indicates a smooth flow around it from the lower side to the upper side of the wing, Fig. 8.30. The attachment line’s end is of open-type, Fig. 8.31. The line simply fans out. At the left side of the figure we observe the expected pressure distribution: relative pressure maximum along the attachment line, Sect. 7.3, fast expansion in chordwise direction on the suction side, less strong expansion on the pressure side, all with only small or even zero gradients along the leading edge. At the outer end of the leading edge, however, we too see a strong expansion toward the outer side of the wing tip. The skin-friction line pattern, right part of the figure, shows the relative minimum of |c f | along the attachment line. It lies very close to the attachment line, because

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

207

Fig. 8.29 Pattern of the skin-friction lines at the upper side of the trailing-edge area of the wing with a focus point separation [3]. Such a point also is present at the side of the fuselage

Fig. 8.30 Wing tip seen from below: pattern of the skin-friction lines and distribution of the surface-pressure coefficient c p [3]

all gradients in x-direction are small. Toward the upper side of the wing and to the tip we see maxima of |c f |, which are due to the flow acceleration there. Figure 8.32 illustrates the flow around the wing tip to the upper side (suction side) of the wing. The pressure first decreases around the wing tip, accelerating the flow toward the upper wing side. There a general pressure rise toward the trailing edge is present. The result is that downstream of mid-chord an open-type separation line begins to form along the upper side of the wing tip. The round wing tip becomes sharp toward the wing’s trailing edge, which, however, has a finite thickness. The sharp-edged part locally causes a strong drop of the surface pressure (blue color) ahead of and at the sharp tip portion, Fig. 8.33, upper

208

8 Large Aspect-Ratio Wing Flow

Fig. 8.31 End of the attachment line at the wing tip without singular point (open-type) [3]. Left part: pattern of the skin-friction lines and distribution of the surface-pressure coefficient c p . Right part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient c f ≡ |c f | Fig. 8.32 The wing tip seen from above: pattern of the skin-friction and distribution of the surface-pressure coefficient c p [3]

part. Obviously this leads to a large transport of kinematically active vorticity content into the tip vortex. The flow field beneath the tip vortex becomes rearranged similar to what happens at the lee side of a delta wing at angle of attack, Chap. 10. The pressure field leads to a strong outward bending of the skin-friction lines near the trailing edge on the upper side (there also the surface pressure is very low). The bending is terminated by a pressure rise and an open-type secondary separation line comes into being. All is reflected also in the skin-friction field in the lower part of the figure. The secondary vortex is formed just inboard of the wing tip. Topological considerations demand the presence of an attachment line between the two separation lines, which is clearly indicated. This secondary attachment line is of open type, too. Moreover we see that a tertiary separation line is formed together with the topologically necessary—tertiary—attachment line. We conclude that below the primary tip

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

209

Fig. 8.33 Detail of the wing tip skin-friction line topology [3]. Upper part: pattern of the skinfriction lines and distribution of the surface-pressure coefficient c p . Lower part: pattern of the skin-friction lines and distribution of the absolute value of the skin-friction coefficient c f ≡ |c f |

210

8 Large Aspect-Ratio Wing Flow

Fig. 8.34 Levels of the dimensionless axial vorticity component ζ in the domain of the tip-vortex system [3]

Fig. 8.35 Topological schematic of the open lee-side flow-field in the Poincaré surface just upstream of the right wing’s trailing edge. The view is in positive x-direction—right wing from front too back

vortex both a secondary and a tertiary vortex are present, which means that we have a tip-vortex system. The levels of the dimensionless axial vorticity component ζ are reflecting the tip-vortex system, Fig. 8.34. Note that ζ > 0 means that the—primary—tip vortex turns clockwise, when looking at the right wing in positive x-direction. This is the same direction as with the trailing vortex behind the right wing of the aircraft. ζ < 0 means counter-clockwise turning, which the secondary vortex does. The tertiary vortex again turns clockwise. The net vorticity content is that of the whole tip-vortex system. The sketch of the open lee-side flow field in Fig. 8.35 shows how the arrangement of the vortex phenomena in the Poincaré surface can be understood. The wing’s contour is highly simplified. The fuselage is completely omitted.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2 Table 8.7 Figure 8.35: the singular points and their meaning Singular point Type F1 , F2 , F3     S1 , S3 , S5 , S7     S2 , S4 , S6 , S8

Focus Half-saddle Half-saddle

211

Kind of flow Vortex center Attachment line Separation line

Fig. 8.36 Domain of the wing-tip vortex system: downstream development of the axial velocity u/u ∞ (left side) and of the turbulence intensity T u (right side) [3]

In Table 8.7 we collect the singular points from Fig. 8.35 and their type and meaning. In order to apply the topological rule for the Poincaré surface, Rule 2 from Sect. 7.4, we have to look at the whole wing, i.e., also at the left wing. Therefore, taking into account also the singular points present at that wing side, we arrive at      1  14 1  S+ N − S = (6 + 0) − 0 + = −1. 2 2 2 (8.4) The fulfillment of Rule 2 tells us that the topology is a valid one. This, of course, is a necessary, but not a sufficient condition. The surface skin-friction line pattern seen in Fig. 8.33, however, makes the observation valid. Still remaining are the reservations regarding the employed turbulence model. Two general properties of the tip-vortex system are given in Fig. 8.36: the development in downstream direction of the mean axial velocity u/u ∞ (left side) and of the turbulence intensity T u (right side). Regarding the axial velocity we first observe its expected reduction within the boundary layer. In the evolving—primary—tip vortex the velocity defect still is there, but an increase of u/u ∞ is visible in the vortex core. The free-stream velocity with u/u ∞ = 1, however, is not reached within the vortex. 

N+

212

8 Large Aspect-Ratio Wing Flow

Due to the employed turbulence model, the turbulent kinetic energy k is a quantity determined during the solution process. The—isotropic—turbulence intensity T u is found with √ 2k Tu = . (8.5) u∞ In Fig. 8.36, right side, we observe the well developed turbulence intensity. In the evolving primary tip vortex a distinct increase of T u is visible. The values of T u are almost twice as high as in the boundary layer.

8.4.4 Excursion: The Wing-Tip Vortex System and Non-linear Lift So far we have discussed the phenomena present in the wing-tip—or wing sideedge—flow regime. Three questions arise: (1) what does the tip-vortex system effectuate regarding the wing’s performance as such, (2) are there similarities to the flow past other wing configurations than the large aspect-ratio wing of the CRM configuration? (3) how is the near-field/extended near field affected? We consider the situation at the right wing only. 1. The tip-vortex system, as a whole turning in the same direction as the trailing vortex, downstream of the trailing edge and the wing tip is merging with the latter. In this process extra circulation is added to the trailing vortex, regarding the amount see below. This, of course, leads to an increase of the induced drag. When looking at the surface pressure in the tip-vortex regime, Fig. 8.32, we see a distinct low-pressure area there (blue color). Such a phenomenon leads to what is called “non-linear lift”, compared to the “linear” lift, found in potential theory of lifting wings (Model (4) Of course, the respective surface area is very small compared to the total wing’s surface. Nevertheless, an increment is added to the wing’s lift (and pitching moment). However, this non-linear lift generally is not a topic with large aspect-ratio wings, as the contribution is negligible. The non-linear lift increment of course adds an increment to the root bending moment, too, and locally also to the wing’s torsion (nose down in this example). 2. The above increments of induced drag, lift, pitching moment and structural forces and moments obtain another importance when looking at wing configurations other than the large aspect-ratio wing. (a) Non-linear lift due to the low pressure area generated below the tip-vortex system first of all is found on wings with small aspect ratio. L. Prandtl in 1921 gave experimental data for rectangular wings, which show a non-linear behavior of the lift coefficient for Λ < 3, see in this regard the discussions in [6, 9]. The strength of the non-linear lift also depends on the shape of the wing’s side edge, its sweep, and whether it is sharp or rounded.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

213

(b) The effect also is seen at aerodynamic trim and control surfaces as well as at lift-enhancement surfaces like slats, Fowler flaps, etc. It is to surmise that at such surfaces, too, side-edge (tip) vortex phenomena lead to first-order effects regarding the forces and moments, when the surface has a low aspect ratio. An early discussion can be found in the doctoral thesis of B. Göthert from the year 1940 [29]. (c) The really prominent example with non-linear lift is that of the delta wing at angles of attack where lee-side vortices appear, Chap. 10. These vortices appear at swept leading edges when a critical combination of the normal angle of attack α N and the normal leading-edge Mach number M N is exceeded, Sect. 10.2.6.12 The swept leading edge then can be seen to act like a side edge. The fluid-mechanical mechanisms present at delta wings basically are the same as we find them at the wing side edges of large aspect-ratio wings, regardless of whether sharp or round leading edges are present. Basically the same also is true for the flow-field topology in terms of the pattern of the skin-friction lines over the wing’s surface and the pattern found in the Poincaré surface, Fig. 8.35. 3. The tip-vortex system obviously affects the roll-up process of the trailing vortex layer. In the case of the large aspect-ratio wing, the influence is rather weak. The stronger the tip-vortex or side-edge system is, the more it will dominate the roll-up process. This is reflected by the length of the roll-up process. We have noted in Sect. 4.4 that in the case of large aspect-ratio swept or unswept wings the roll-up process practically is completed a few half-span distances behind the wing. In the case of low-aspect ratio wings that location can be already at a chord length or less behind the wing. Hence the extent and the flow properties of the near field and the extended near field are affected by the strength of the tip vortex system. In particular the magnitude of the downwash behind the wing depends to a degree on whether the trailing vortex layer is rolled up or not. This then influences the effectiveness of the horizontal tail surfaces [9]. We summarize: non-linear lift appears to be present at every finite-span lifting wing, regardless of its aspect ratio. It contributes the more to the overall lift, the smaller the aspect ratio of the wing is. At large aspect ratio wings its amount is negligible. At aerodynamic trim, control and lift enhancement surfaces, which often have a low aspect ratio, the flow around the side edges is similar to that around a wing tip. Therefore sizeable non-linear effects can be expected. For delta wings and strake configurations their non-linear lift effects are an important aerodynamic property not only in view of their influence on the overall lift, but also in terms of the maneuverability and agility of the aircraft.

12 Note that at the leading edge the vortex separation is beginning at the rear and then moves forward

with increasing angle of attack.

214

8 Large Aspect-Ratio Wing Flow

Fig. 8.37 Calculated skin-friction lines (black) and external (boundary-layer edge) streamlines (red) at the upper side of the wing [3]. The angle β between the two lines in the projection on the wing’s surface is color-coded (insert)

8.4.5 Flow Domain 0 Contd.: Trailing-Edge Flow and the Compatibility Condition Figure 8.37 shows flow patterns on the upper (suction) side of the wing. We recall that if an external inviscid streamline locally is curved, the skin-friction line below it is more strongly curved [26]. A point-of-inflection in the external streamline, of course, accordingly alters the skin-friction line pattern. Figure 8.37 also shows—indicated by the color strip—the presence of a shock wave roughly at mid-chord and in spanwise direction between y ∗ ≈ 0.25 and y ∗ ≈ 0.7. The wing’s shape was designed for the nominal 1-g cruise case with camber and twist, the outboard airfoil sections being supercritical ones. For y ∗  0.25 and y ∗  0.7 the airfoil sections have a thickness to chord length ratio small enough that no shock wave is present.13 What does the shock wave indicated in Fig. 8.37 effectuate? First of all the preshock boundary-layer edge Mach number appears to be below Me  1.3 − 1.35. If

13 Note

that along the attachment line the flow velocity does not change much. This is indicated, too, by the pressure distribution along it, see the figure of the attachment line’s end, Fig. 8.31, left side. Hence the consideration of the local chord properties—without regard to their spanwise location—is permitted, though not in an exact quantitative sense.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

215

this is given, the boundary layer does not separate due to the presence of the shock, it only thickens, Sect. 6.1.2. The shock wave locally obviously is not orthogonal to the upstream flow. Hence both the external inviscid streamline and all streamlines in the boundary layer, including the skin-friction line, are deflected. The figure shows a rather small deflection of the external inviscid streamlines, but a large one of the skin-friction lines, indicated by the black color. Closer to the trailing edge a general increase is seen of the local angle β between the lines. The shock wave appears to influence that patterns only weakly. The angle β in general has a maximum along the trailing edge, which, however is diminished toward the wing tip. Nevertheless, the presence of the shock wave must be considered in view of the compatibility condition at the trailing edge, Sect. 4.4. The shock wave in the inviscid flow field leads to a velocity defect and an, if only slight, deflection of the external streamlines. (The velocity defect also manifests itself in a shock-decambering effect, Sect. 6.1.2.) Both effects appear to be small as we will see below. Of special interest is the appearance of the overlaid patterns of the external inviscid streamlines at the upper and the lower side of the wing, Fig. 8.38. At the trailing edge the angle between them, the local trailing-edge flow shear angle ψe , Sect. 4.2.6, is the decisive element of the compatibility condition at the trailing edge. If the external inviscid velocities at the trailing edge are constant or nearly constant in spanwise direction, which is the case here, it is only ψe , which governs the compatibility condition.

Fig. 8.38 The inviscid external streamlines at the upper and the lower side of the wing [3]

216

8 Large Aspect-Ratio Wing Flow

The visual inspection of the streamline pattern at the trailing edge shows that the angle ψe increases toward the wing tip, as demanded by the compatibility condition, Sect. 4.4. Important properties of the near wake, i.e., the trailing vortex layer, just behind the wing’s trailing edge at x/c = 1.005 are plotted in Fig. 8.39. In this figure the location y ∗ = y/(b/2) in a sense follows the trailing edge, i.e., it is not function of x ∗ = const. The evaluation is made in the local wake coordinate system, Fig. 4.11, where the trailing-edge flow shear angle ψe is defined as well as the vortex-line angle ε. Along the trailing edge the Kutta direction is not taken into account. The nomenclature is somewhat different from that used in Fig. 4.12. The lateral edge velocity components there, veu and vel , are now ve2u and ve2l , the thicknesses δu and δl are z e∗u and z e∗l , and the upper trailing-edge flow shear angle ψeu is ψu .14 The dimensionless circulations σ(ψ) and σ(Cl ), as well as the derivative dσ(Cl )/dy ∗ are defined in Sect. 3.16, the vorticity content Ω 1 in Sect. 4.2.6. The lateral velocity component ve2 (y ∗ ) = veu cos ψu = vel cos ψl in Fig. 8.39a at the upper side of the wing is negative, i.e., it is directed toward the wing root. At the lower side it is directed toward the wing tip. As was to be expected, too, both components increase toward the wing tip, hence also the difference between them. At y ∗ = 0.37 the kink in both curves indicates the location of the Yehudi break. Near the wing root, both curves change sign. The graphs in Fig. 8.39b indicate the spanwise distribution of the wake thicknesses at the upper and the lower side. The wing’s chord length decreases from the root to the tip. Hence the boundary-layer running length decreases, too. This is the reason why the upper and lower wake thicknesses decrease toward the wing tip.15 The sharp rise of the wake thickness at the wing tip’s upper side is due to the tip-vortex system. The slightly thicker wake leaving the upper side reflects the presence of a larger portion of an adverse pressure gradient. All this confirms what was found in an earlier investigation of such flows [30]. The upper side trailing-edge flow shear angle ψu (y ∗ ) and the vortex-line angle ε(y ∗ ) in Fig. 8.39c show the expected behavior. The (negative at the wing’s upper side) shear angle ψu increases from the root to the tip, indicating the increase of the strength of the wake’s kinematically active vorticity content in that direction. At the left the influence of the fuselage is visible, and also that of the Yehudi break. The vortex-line angle ε is positive because of the backward sweep of the trailing edge, Sect. 4.2.6. That sweep from the root up to the Yehudi break is small, Fig. 8.21, but then quite large. Hence from that break on, ε is nearly constant until the wing tip is approached. 14 Note

that the upper indices ’2’ of the velocity components do not indicate that they are squared velocities. They simple indicate the lateral direction in the local wake coordinate system, t in Fig. 4.11. 15 This observation assumes a more or less two-dimensional behavior of the wing’s boundary layers. The justification is twofold: (1) the thickness of the boundary layer along the attachment line is more or less constant, and with this the initial conditions for the chordwise flow, (2) a larger threedimensionality is present only in the immediate vicinity of the attachment line. Hence for qualitative considerations and also for crude estimations the assumption of two-dimensionality is permissible.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

217

Fig. 8.39 Properties of the trailing vortex layer and the compatibility condition at the location x/c = 1.005 immediately behind the trailing edge [3]: a lateral velocity component ve2 (y ∗ ) at the upper and the lower side, b wake thickness z e∗ (y ∗ ) at the upper and the lower side, c trailing-edge flow shear angle ψu (y ∗ ) at the upper side and vortex-line angle ε(y ∗ ), and d non-dimensional circulation σ(y ∗ ), its derivative in y ∗ -direction and the local vorticity content Ω(y ∗ )

Figure 8.39d gives the vorticity content Ω1 (solid black line)—computed with the flow properties at the trailing edge—and the dimensionless circulation distribution dσ(Cl )/dy ∗ (blue dashed line)—found from the computed circulation distribution. Both are connected via the compatibility condition Eq. (4.20), Sect. 4.4. Because we consider the wing’s right-hand side, both entities are negative. In order to help the reader in the understanding of the result, we discuss that now in terms of the magnitudes |Ω1 | and |dσ(Cl )/dy ∗ |.

218

8 Large Aspect-Ratio Wing Flow

Fig. 8.40 Spanwise circulation distribution obtained with the compatibility condition σ(ψ) and with the Kutta-Joukowsky theorem σ(Cl ), and comparison with the elliptical distribution σ ∗ [3]

Ω1 at the wing root is positive and changes sign at y ∗ ≈ 0.15. |Ω1 | then increases and at the Yehudi break reaches a local maximum with |Ω1 | ≈ 0.08. A further weak increase is present up to y ∗ ≈ 0.8. Beyond that a strong increase happens up to |Ω1 | ≈ 0.35. The spanwise integration of Ω1 , beginning at the wing tip, results in the circulation distribution, here given as dimensionless distribution σ(ψ) (blue dashed line). The value rises from σ(ψ) = 0 at the wing tip to the maximum σ(ψ) ≈ 0.071 at y ∗ ≈ 0.2. The root circulation at y ∗ = 0.13 is σ0 (ψ) = 0.0697. Sofar we have dealt with the vorticity content Ω1 found from flow properties at the wing’s trailing edge and derived from it the circulation distribution σ(ψ). Now we check these results with the help of the Kutta-Joukowsky theorem. This is made by relating with Eq. (3.31) locally the circulation Γ (y) to the computed pressure distribution, respectively the local lift coefficient Cl (y). The resulting dimensionless circulation σ(Cl ) (green dash-dotted line) is in reasonable agreement with σ(ψ). The former has somewhat higher values at the outboard and lower at the inboard side. The root circulation is almost equal. Also a reasonable agreement shows the differentiated circulation dσ(Cl )/dy ∗ with Ω1 . At the Yehudi break the difference is larger and even larger at the root. In Fig. 8.40 the two obtained circulation distributions are compared with the elliptic distribution σ ∗ , Sect. 3.16. σ(Cl ) and σ ∗ show a good agreement for y ∗ > 0.6, whereas σ(ψ) is smaller by up to 15 per cent in this region. For y ∗ < 0.6 the elliptic distribution is smaller, while the other two agree rather well, except for the root region, see above. The root circulations are σ0∗ = 0.061, σ0 (ψ) = 0.0697 and σ0 (Cl ) = 0.068. The two computed values are well above the elliptic one, and differ themselves only by about 2.5 per cent.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

219

Our conclusion is that the concept of kinematically active and inactive vorticity content, which culminates in the formulation of the compatibility condition, is a viable one. The differences in the spanwise circulation distributions are due to some properties of the flow fields at the upper and the lower wing surface. These are the (inviscid) flow deflection and the velocity defect due to the shock wave at the wing’s upper surface as well as local separation effects at the trailing edge due to its finite thickness. Further we must keep in mind that the vorticity-content concept assumes Re → ∞, whereas here we have a Reynolds number of finite magnitude.

8.4.6 Flow Domain 1: The Trailing Vortex Layer in the Near Field The wake profiles in the near field behind the trailing edge—Domain 1, see Fig. 8.1—are studied, as well as the wake properties at x ∗ = 0.5. • The Profiles of the Trailing Vortex Layer Close to the Trailing Edge. We begin with an investigation of the wake profiles just behind the trailing edge of the wing, Fig. 8.41. The three dimensionless velocity components v i are those indicated in Fig. 4.12, except for v 3 .16 The coordinate system in vertical position is located with z ∗ = 0 lying at the local trailing-edge position. In spanwise direction the locations are with y ∗ = 0.13 close to the wing root, with y ∗ = 0.4 just outboard of the Yehudi break, and with y ∗ = 0.99 very close to the wing tip. The x/c-locations are related to the local chord length with x/c = 1.03 lying very close to the trailing edge and the two others at x/c = 1.17 and x/c = 3.35. The streamwise v 1 /u ∞ -profile represents the boundary-layer profile with kinematically inactive vorticity content leaving the wing’s trailing edge. The extent of the profiles declines in z ∗ -direction, like that of the others, from the wing root to the wing tip because the chord length is reducing in that direction, see the discussion of Fig. 8.39. Despite the shock wave at the upper side of the wing, the velocities at the upper and the lower side are nearly the same. The profiles in stream direction are filling up quickly. At x/c(y) = 3.35 they are nearly vanished. At the same time the wake is sinking. This is most pronounced at y ∗ = 0.13. The lateral v 2 /u ∞ -profile is the wake profile with the kinematically active vorticity content. The shear is well discernible, with the local shear angle ψ increasing toward the wing tip. Only at y ∗ = 0.13ψ is negative due to the wing-body interference, see also the spanwise circulation distribution in Fig. 8.40. The downwash, represented by v 3 /u ∞ , is stronger at the upper side of the wing than at the lower side. At y ∗ = 0.99 the downwash is positive at the lower side, i.e. it is an upwash. This appears to be induced by the combination of the evolving trailing vortex and the tip-vortex system. 16 The superscript index notation does not mean that the velocity components are contravariant ones

as they are defined for instance in [26].

220

8 Large Aspect-Ratio Wing Flow

Fig. 8.41 Wake profiles at three different spanwise and axial positions [3]. The dimensionless velocity components are given in the local wake coordinate system, Fig. 4.11

The maximum of the dimensionless vorticity ωb/2u ∞ increases toward the wing tip. Its profile also mirrors the decrease of the wake thickness in that direction, like that of the v 1 /u ∞ -profile. Quantitatively all compares well with the results of the Euler simulation in Sect. 8.3. There of course the u(z)-profile, the equivalent to v 1 /u ∞ , is unity throughout, Fig. 8.16. The reader is asked to study also the results in [30].

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

221

Fig. 8.42 Distribution of the dimensionless axial vorticity ζ at x ∗ = 0.5 [3]

• The Properties of the Trailing Vortex Layer at x ∗ = 0.5. Next we study nearfield properties at the location x ∗ = 0.5. For a visualization of this location see Fig. 8.50. We consider the right side of the flow field, y ∗ > 0. At x ∗ = 0.5 the roll-up process of the trailing vortex layer already has resulted in a distinct trailing vortex, Fig. 8.42. In the trailing vortex layer the dimensionless axial vorticity ζ, eq. (8.3), increases from ζ ≈ 2 near the wing root to ζ ≈ 6 near the wing tip. The Yehudi break has a weak influence, visible in the vortex layer. The inset in Fig. 8.42 shows the trailing vortex in more detail. In its core we have ζ ≈ 25. Some wiggles present in the picture are due to grid properties. The merging of the tip-vortex system into the trailing vortex is vaguely indicated. The figure also shows, at and below the horizontal stabilizer, a domain with weak negative vorticity. This is due to a downward force, which is present at the stabilizer, like a trim force. A trailing vortex layer leaves the trailing edge and rolls up into a pair of trailing vortices, counter-rotating to the wing’s trailing vortices. The downward force is present, although the stabilizer nominally has zero angle of attack. However, because trailing vortex layer and vortices of the wing induce a downwash velocity component w, see Sect. 8.1, the stabilizer effectively has a negative angle of attack. Figure 8.43 at y ∗ = 0.15 visualizes the downwash velocity component w/u ∞ behind the wing and in particular at the horizontal stabilizer. Note that the stabilizer itself induces an upwash velocity. Of interest are also the velocity distributions and the turbulence intensity in the trailing vortex layer and the vortex. Particularly interesting is the axial velocity u/u ∞ , Fig. 8.44. We observe a distinct deficit of the axial velocity both in the vortex layer

222

8 Large Aspect-Ratio Wing Flow

Fig. 8.43 Dimensionless vertical velocity w/u ∞ behind the wing at y ∗ = 0.15 [3] Fig. 8.44 The dimensionless velocity u/u ∞ at x ∗ = 0.5

and the vortex. The vortex layer shows minimum velocities of u/u ∞ ≈ 0.93, whereas in the vortex core we have u/u ∞ ≈ 0.97. At x ∗ = 0.5 the roll-up of the trailing vortex layer is not yet completed and the trailing vortex is not yet fully developed. (The vortex is young with its age τ ∗ = 0.0039.) In Fig. 8.45 we compare the vortex development at this location with two analytical vortex models. The computed circumferential velocity vθ,num (red symbols) is compared to the analytically found velocities of models, which represent fully developed vortices: Rankine vortex vθ,Rankine (green symbols), Eq. (3.16) and the Lamb-Oseen vortex vθ,Lamb−Oseen (blue symbols), Eq. (3.17), both to be found in Sect. 3.10.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

223

Fig. 8.45 Comparison of numerical and analytical circumferential velocities vθ (r/(b/2)) at x ∗ = 0.5. The vortex age is τ ∗ = 0.0039 [3]

Two numbers are needed for the latter, the viscous core radius rc and the root circulation Γ0 . The first one was determined from the numerical solution, see Fig. 3.9, the second one is taken in non-dimensional form from Fig. 8.40. Figure 8.45 indicates that at the location rc the numerically found value of the circumferential velocity is much below the analytical ones. The latter also differ much. In the vortex core, r < rc the linear velocity distributions are well obtained, outside of it the agreement at least is reasonable.

8.4.7 Flow Domain 2: The Trailing Vortices Appear We look at the roll-up process of the wake and at the wing’s circulation in Domain 2, see Fig. 8.1. • The Roll-Up Process in Terms of the Dimensionless Axial Vorticity ζ. In the extended near field we follow the development of the trailing vortices. First we have a look at their interaction—in terms of the axial vorticity ζ—with the trailing vortices of the horizontal tail plane. That surface is located between x ∗ = 0.15 and 0.30. We only describe the situation behind the right wing. The trailing vortex grows in diameter and moves inwards. The trailing vortex layer weakens, grows in thickness and is sinking faster than the trailing vortex. (The author of [3] notes that the choice of the turbulence model has an influence on this development.) At x ∗ = 5 we see that a part of the vortex layer interacts with that of the horizontal tail plane, Fig. 8.46. This part finally will not be ingested into the trailing vortex. The roll-up process is not finished at x ∗ = 9, Fig. 8.47. The separated part of the trailing vortex layer—in a vortex pairing process with the trailing vortex of the horizontal tail plane, which acts as a trim surface—now is counterrotating to the wing’s trailing vortex. The vortex age of the trailing vortex τ ∗ = 0.071 still is low. In

224

8 Large Aspect-Ratio Wing Flow

Fig. 8.46 Distribution of the dimensionless axial vorticity ζ at x ∗ = 5 [3]

Fig. 8.47 Distribution of the dimensionless axial vorticity ζ at x ∗ = 9 [3]

[31] it is reported that for a high-lift transport aircraft the roll-up process was finished at τ ∗ = 0.25 − 0.3. We again compare now the numerical and analytical circumferential velocities vθ (r/(b/2)) as we did it at x ∗ = 0.5, Fig. 8.45. Now the agreement of the numerical and the Lamb-Oseen data is very good, Fig. 8.48. The Rankine vortex of course approximates well the circumferential velocity only away from r/(b/2) = 0.1 and below r/(b/2) = 0.05. However, it must be expected that further downstream the agreement will be less good due to the assumptions in the analytical models.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

225

Fig. 8.48 Comparison of numerical and analytical circumferential velocities vθ (r/(b/2)) at x ∗ = 9 [3]

The motion of the—still not fully developed—trailing vortex is represented by the vortex trajectory, Fig. 8.49. In Sect. 4.4 we have noted that the distance b0 between the fully developed trailingvortex pair increases with increasing aspect ratio. The ratio b0 /b, which also is called the load factor s, Sect. 3.16, for the elliptical circulation distribution is s = s∗ =

b0 π = = 0.7854, b 4

(8.6)

only for wings with very large aspect ratio s → 1. In the y ∗ -z ∗ plane, Fig. 8.49, the vortex center is seen to move from the initial position at the WRP, Fig. 8.21, toward the wing’s root, first upward and then strongly downward.17 In the x ∗ − y ∗ plane the position of the vortex center appears to reach its final spanwise position at x ∗ > 10 with y ∗ = s < 0.8. We summarize: (1) the vortex is still young at x ∗ = 10 (the roll-up process is not yet finished), (2) the solution becomes erroneous for x ∗ > 9 (see the graph in the z ∗ -x ∗ plane and below), and (3) the circulation is the original one minus that of the counterrotating trailing vortices of the horizontal tail plane, Fig. 8.50. If in [3] the final value of the load factor is found to be s = 0.715 < s ∗ = 0.7854, these three items must be taken into account as possible cause for this small value. • The Wing’s Circulation in Flow Domain 2. The dimensionless circulation G(x ∗ ) = Γ (x ∗ )/Γ0 in the extended near field is given in Fig. 8.50. The upper blue line is the isolated wing’s circulation. It is nearly constant up to x ∗ = 9. Up to this location the inserted PS grid, Sect. 8.4.2, is present. The evaluated data at x ∗ = 10 show that the original grid resolution is insufficient there, remember the remarks in Sect. 5.3. 17 The

WRP was chosen as initial position, which, however, is a makeshift, because the actual flow situation at that location is highly complex, Fig. 8.33.

226

8 Large Aspect-Ratio Wing Flow

Fig. 8.49 Trajectory of the right-hand side trailing vortex [3]. Upper part left: y ∗ -z ∗ plane. Upper part right: x ∗ -y ∗ plane. Lower part: x ∗ -z ∗ plane

The isolated wing’s circulation up to x ∗ ≈ 3 is slightly above G = 1. This probably is due to inaccuracies in the determination of the root circulation caused by the wing/fuselage interference. The lower curve represents the net circulation. The opposite circulation of the horizontal tail plane was determined once at x ∗ = 0.4 and then assumed to be constant. Even if the accuracy of the approach is not fully satisfying, the result shows that the influence of the horizontal tail plane, which in effect is a trim surface, is rather large. This points to the fact that the aircraft’s trim needs attention in order to minimize the trim drag. The development of the isolated trailing vortex as it grows in the roll-up process in x-direction is given in Fig. 8.51. The development is shown in terms of the dimensionless circulation G(x ∗ ) = Γ (x ∗ )/Γ0 . G increases in downstream direction because the trailing vortex layer successively is absorbed by the trailing vortex.

8.4 RANS/URANS Solution (Model 10) for the CRM Case: Flow Domains 0, 1, and 2

227

Fig. 8.50 The dimensionless circulation G(x ∗ ) in the extended near field [3]. The upper blue curve is G of the wing alone, the lower black one that with the opposite circulation of the horizontal tail plane taken into account Fig. 8.51 The development of the dimensionless circulation G(x ∗ ) of the trailing vortex [3]

Figure 8.51 contains two curves: Gωx ) and G(vθ ), both as function of x ∗ . The first one was found by applying Stokes theorem by integrating ωx over the cross-section of the vortex, Sect. 3.3, the second one by computing the line integral around the vortex with vθ .

228

8 Large Aspect-Ratio Wing Flow

Fig. 8.52 Development of the characteristics of the trailing vortex [3]. Left: the viscous core radius rc (x ∗ ). Right: the maximum circumferential velocity component vθ (x ∗ )

The results in a sense are qualitative ones, because of the difficulty to define the integration domains. The trailing vortex is not a circular one, it is moreover connected to the trailing vortex layer. Both curves initially grow with a large gradient in x ∗ direction, which at x ∗  3 becomes smaller. The values at x ∗ = 9 clearly show that there the roll-up process is not finished. Interesting for us is the value of G(ωx ) close to the WRP. There it represents the net circulation contained in the tip-vortex system. This circulation is G(x ∗ = 0) = 0.068. This means that the net circulation of the tip-vortex system is about seven per cent of the circulation of the trailing vortex. In closing this section we have a look at some of the characteristic properties of the evolving trailing vortex. We have to note that the magnitudes of the obtained data to a degree depend on the applied turbulence model. All quantities considered change strongly in the region up to two wing spans downstream of the trailing edge (x ∗  2)and then rather slowly. In Fig. 8.52 the viscous core radius rv , Sect. 3.16, increases very fast throughout (left part of the figure). The maximum circumferential velocity has a rather large magnitude in the beginning with vθmax ≈ 0.27 and then decreases fast to vθmax ≈ 0.07 (right part of the picture). This strong reduction must be seen as the consequence of the fast growth of the viscous core radius. (Beyond x ∗ = 9 all results have deficits because of the insufficient grid resolution there (see above).) The axial velocity defect (u ∞ − u min )/u ∞ (x ∗ ) in Fig. 8.53 initially has a value of about 10 per cent, but then decreases fast to around one per cent (left part of the figure). A similar drop is seen of the maximum dimensionless axial vorticity component ζ = ωx b/2u ∞ (right part of the figure).

8.5 Concluding Remarks

229

Fig. 8.53 Development of the characteristics of the trailing vortex [3]. Left: the axial velocity defect (u ∞ − u min )/u ∞ (x ∗ ). Right: the maximum dimensionless axial vorticity component ζmax (x ∗ )

8.5 Concluding Remarks We have demonstrated by means of Unit Problems the capabilities of three kinds of mathematical flow models regarding the simulation and the analysis of the vortical and separated flow past large aspect-ratio lifting wings. In Sect. 8.2 the concept of the local vorticity content was employed in order to investigate the capabilities of several panel methods (Model 4) in the Flow Domain 0. It turned out that even if the overall results are in reasonable agreement, the flow fields near the trailing edges of the investigated wings were not computed correctly by most of the methods. This concerned methods with lower-order singularity formulations. Their solutions did not fulfill the trailing-edge compatibility condition, Eq. (4.20). Only a higher-order formulation gave the proper results. The vortex-line angle  in general was computed correctly. The application of a discrete numerical solution of the Euler equations (Model 8) to the flow past a large aspect-ratio wing in Sect. 8.3 yielded satisfactory results in the Flow Domains 0 and 1. Usually the results of such simulations are considered in terms of the forces and moments only. Here the properties of the Euler wake and partly also of the Euler boundary layer were investigated. The near-wake properties were found to be qualitatively correct. At the trailing edge the compatibility condition is fulfilled. In Sect. 8.4 the flow past the wing of the Common Research Model was investigated with a RANS/URANS method (Model 10). Considered were the flow and wake properties in the Flow Domains 0, 1 and 2. In Domain 0 also the flow at the wing root, along the leading edge and at the wing tip was studied. Because the CRM has no wing root/fuselage fairing, a complex separation pattern was found there, similar as at the wing tip. The result must be considered with some reservation, because the choice of the turbulence model and

230

8 Large Aspect-Ratio Wing Flow

the discretization approach influence the outcome [3]. The pattern of the skin-friction lines revealed that both the classical and the open-type separation, Sect. 7.1.4, were present. The end of the attachment line in the vicinity of the wing tip was found to be of open-type. The wing-tip separation completely happens at the rear upper side of the wing. Besides the primary vortex, a secondary and even a tertiary vortex was found. Therefore we speak of the wing-tip vortex system. Wing-tip separation always was found to be of open-type. The distribution of the surface pressure coefficient reveals a distinct low pressure area near the trailing edge of the wing tip. Such an area is the cause of the so-called non-linear lift. Hence this kind of lift is present also on large aspect-ratio wings. However, it has such a small magnitude that it does not play a role. The trailing-edge compatibility condition was investigated, too. It was found that the concept of kinematically active vorticity permits a good explanation of the lifting wing’s flow situation at the trailing edge. The roll-up process of the wing’s trailing vortex layer into the pair of trailing vortices in Flow Domain 1 and 2 was visualized. It was found that the wing tip vortex system substantially contributes to the wake’s circulation. Also visualized was the interaction of the wing’s wake with the wake of the horizontal tail plane, which effectively acts like a trim surface. It was further shown that beyond the end of the inserted PS grid the original grid is sof coarse that the circulation of the wake/trailing vortex system is not preserved downstream. This is a matter, which always needs attention.

8.6 Problems Problem 8.1 Check with the correlations given in Fig. 10.23 whether at the Kolbe wing at α = 8.2◦ the possibility of leading edge separation and hence lee-side vortices exists. How is the situation, if the wing would have a sharp leading edge? Problem 8.2 Make the same check for the forward-swept wing. Assume M∞ = 0.25. Problem 8.3 In Table 8.4 the lift found with the Euler solution for M∞ = 0.3 is compared with that found with linear theory for M∞ = 0.01. Apply the PrandtlGlauert rule for M∞ = 0.3 to the result of linear theory. How do now the results compare? Problem 8.4 The lift coefficient of the CRM case was found to be C L = 0.445. Assume an elliptical circulation distribution and compute the induced drag coefficient C Di . How large in per cent is it in relation to the total drag coefficient C D = 0.025? Problem 8.5 For the CRM case three root-circulation values were obtained, Sect. 8.4. Assume elliptical circulation distributions for all of them and compute C L∗ and C D∗ i .

8.6 Problems

231

Compare with the data given on Sect. 8.4 and in Problem 8.4. Choose the Oswald factor to be e = 0.8. Problem 8.6 From Fig. 8.50 it can be concluded how much the overall circulation is reduced due to the horizontal tail surface. How are the lift and the induced drag affected? Problem 8.7 Consider an aircraft flying at constant speed at the beginning of the cruise segment, Fig. 1.3, of a long-distance flight. The wing span is b = 60 m, disregarding the fuselage, the aspect ratio is Λ = 9. The mass of the aircraft is 250,000 kg, the flight speed is u ∞ = 800 km/h, the flight altitude H = 10 km. Assume that Model 4 holds and that a spanwise elliptic circulation distribution is present. (a) How large is the circulation Γ0 of the trailing vortices? (b) How large is Γ0 , if the wing span would only be b = 30 m, and the aspect ratio Γ = 4.5? (c) How large is the wing loading Ws for the two cases? (d) What do the results mean regarding the strength of the trailing vortices? (e) How are the coefficients of lift and induced drag affected? Problem 8.8 An elliptic wing of aspect ratio Λ = 10 flies at M∞ = 0.8. What is the lift curve slope? Use the lifting-line model (low speed) in Appendix A.4 and correct the value.

References 1. Breitsamter, C.: Wake vortex characteristics of transport aircraft. Prog. Aerosp. Sci. 47(1), 89–134 (2011) 2. Obert, E.: Aerodynamic Design of Transport Aircraft. IOS Press, Delft (2009) 3. Pfnür, S.: Numerical Analysis of the Trailing-Edge Vortex Layer and the Wake-Vortex System of a Generic Transport-Aircraft Configuration. Master’s Thesis, Institut für Luft- und Raumfahrt, Technical University München, Germany (2015) 4. Schrauf, G., Laporte, F.: AWIATOR—wake-vortex characterization methodology. In: KATnet Conference on Key Aerodynamic Technologies, Bremen, Germany (2005). Accessed 20–22 June 2005 5. Prandtl, L.: Tragflügeltheorie, I. und II. Mitteilung. Nachrichten der Kgl. Ges. Wiss. Göttingen, Math.-Phys. Klasse, 451–477 (1918) and 107–137 (1919) 6. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011) 7. Vos, R., Farokhi, S.: Introduction to Transonic Aerodynamics. Springer Science+Business Media, Dordrecht (2015) 8. Küchemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford. Also AIAA Education Series, p. 2012. AIAA, Reston, V (1978) 9. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, Vol. 1 and 2, SpringerVerlag, Berlin/Göttingen/Heidelberg (1959), also: Aerodynamics of the Aeroplane, 2nd edition (revised). McGraw Hill Higher Education, New York (1979) 10. Rudnik, R.: Transportflugzeuge. In: Rossow, C.-C., Wolf, K., Horst, P. (eds.) Handbuch der Luftfahrzeugtechnik, pp. 83–113. Carl Hanser Verlag, München, Germany (2014) 11. Rizzi, A., Oppelstrup, J.: Aircraft Aerodynamik Design with Computational Software. Cambridge University Press (2020)

232

8 Large Aspect-Ratio Wing Flow

12. Kolbe, D.C., Boltz, F.W.: The Forces and Pressure Distributions at Subsonic Speeds on a Plane Wing Having 45◦ of Sweepback, an Aspect Ratio of 3, and a Taper Ratio of 0.5. NACA RM A51G31 (1951) 13. Hirschel, E.H., Sacher, P.: A Comparative Theoretical Study of the Boundary-Layer Development on Forward Swept Wings. In: R.K. Nangia (ed.), Proceedings of International Conference on Forward Swept Wings, Bristol, 1982. University of Bristol, U.K. (1983) 14. Hirschel, E.H., Fornasier, L.: Flowfield and Vorticity Distribution Near Wing Trailing Edges. AIAA-Paper 1984–0421 (1984) 15. Fornasier, L.: HISSS–A Higher-Order Subsonic/Supersonic Singularity Method for Calculating Linearized Potential Flow. AIAA-Paper 1984–1646 (1984) 16. Sytsma, H.S., Hewitt, B.L., Rubbert, P.E.: A Comparison of Panel Methods for Subsonic Flow Computation. AGARD-AG-241 (1979) 17. Hirschel, E.H.: Das Verfahren von Cousteix-Aupoix zur Berechnung von turbulenten, dreidimensionalen Grenzschichten. MBB-UFE122-AERO-MT-484, Ottobrunn, Germany (1983) 18. Hirschel, E.H., Rizzi, A.: The Mechanism of Vorticity Creation in Euler Solutions for Lifting Wings. In: A. Elsenaar, G. Eriksson (eds.), Proceedings Symposium on the International VortexFlow Experiment on Euler Code Validation, FFA Bromma, Sweden, pp. 127–162 (1987). Accessed 1–3 Oct 1986 19. Rizzi, A., Eriksson, L.-E.: Computation of flow around wings based on the Euler equations. J. Fluid Mech. 148, 45–71 (1984) 20. Vassberg, J.C., DeHaan, M.A., Rivers, S.M., Wahls, R.A.: Development of a Common Research Model for Applied CFD Validation Studies. AIAA-Paper 2008–6919 (2008) 21. Vassberg, J.C., Tinoco, E.N., Mani, M., Rider, B., Zickuhr, T., Levy, D.W., Brodersen, O.P., Eisfeld, B., Crippa, S., Wahls, R.A., Morrison, J.H., Mavriplis, D.J., Murayama, M.: Summary of the fourth AIAA computational fluid dynamics drag prediction workshop. J. Aircr. 51(4), 1070–1089 (2014) 22. Levy, D.W., Laflin, K.R., Tinoco, E.N., Vassberg, J.C., Mani, M., Rider, B., Rumsey, C.L., Wahls, R.A., Morrison, J.H., Brodersen, O.P., Crippa, S., Mavriplis, D.J., Murayama, M.: Summary of data from the fifth computational fluid dynamics drag prediction workshop. J. Aircr. 51(4), 1194–1213 (2014) 23. Gerhold, T.: Overview of the Hybrid RANS Code TAU. In Kroll, N., Fassbender, J. (eds.) MEGAFLOW—Numerical Flow Simulation for Aircraft Design. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, NNFM 89, pp. 81–92. Springer, Berlin (2005) 24. Spalart, P.R., Allmaras, S.R.: A One-Equation Turbulence Model for Aerodynamic Flows. AIAA-Paper 1992–0439 (1992) 25. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32(8), 1598–1605 (1994) 26. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014) 27. Délery, J.: Three-Dimensional Separated Flow Topology. ISTE, London and Wiley (2013) 28. Kármán, Th, von, : Aerodynamics-Selected Topics in the Light of their Historical Development. Cornell University Press, Ithaca, New York (1954) 29. Göthert, B.: Systematische Untersuchungen an Flügeln mit Klappen und Hilfsklappen (Systematic Investigations at Wings with Flaps and Servo-Flaps). Doctoral Thesis, Technical University Braunschweig, Germany (1940), also Jahrbuch 1940 der Luftfahrtforschung, pp. 278–307 (1940) 30. Wanie, K.M., Hirschel, E.H., Schmatz, M.A.: Analysis of Numerical Solutions for ThreeDimensional Lifting Wing Flows. Z. f. Flugwissenschaften und Weltraumforschung (ZFW) 15, 107–118 (1991) 31. Breitsamter, C.: Nachlaufwirbelsysteme großer Transportflugzeuge - Experimentelle Charakterisierung und Beeinflussung (Wake-Vortex Systems of Large Transport Aircraft— Experimental Characterization and Manipulation). Inaugural Thesis, Technische Universität München, 2007, utzverlag, München, Germany (2007)

Chapter 9

Particular Flow Problems of Large Aspect-Ratio Wings

The topics of this book are the basic principles and Unit Problems of separated and vortical flow in aircraft wing aerodynamics. This chapter is devoted to short considerations of application-oriented topics. Complete literature reviews are not intended, we give compact accounts of selected topics and provide references for further reading. In the following Sect. 9.1 the shock-wave/boundary-layer interaction occurring at the supercritical airfoil is considered. It follows a sketch of the flow and separation phenomena at a high-lift system, Sect. 9.2. The wing in high-lift configuration is the topic of Sect. 9.3. The effect of the nacelle-strake vortex is discussed in Sect. 9.4. Section 9.5 treats the topic of wing-tip devices and Sect. 9.6 finally the problem of aircraft-wake control.

9.1 Supercritical Airfoil—Shock-Wave/Boundary-Layer Interaction The subject of this section is transonic shock wave/boundary-layer interaction, see, e.g., [1]. Why is this of importance? Breguet’s range formula in its simplest form relates the flight range R with the parameters flying speed v∞ , aerodynamic quality (lift to drag ratio) C L /C D , specific impulse Isp , as well as the structure parameters empty mass m e , payload mass m P and fuel mass m F 1 : R = v∞

  CL mF . Isp ln 1 + CD me + m P

(9.1)

This formula can also be written in the form

1 The relation of the specific impulse

Isp to the specific fuel consumption b [kg/s N] of the propulsion

system is Isp = 1/(b g). © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_9

233

234

9 Particular Flow Problems of Large Aspect-Ratio Wings

R = M ∞ a∞

  CL mF . Isp ln 1 + CD me + m P

(9.2)

Of interest in our context is the combination of the terms M∞

CL , CD

which tells us that the flight Mach number should be as high as possible and at the same time also the aerodynamic quality, the lift to drag ratio. In Sect. 2.4.2 it was shown that a phenomenon exists, the drag divergence, which manifests itself as a strong drag increase once a certain flight Mach number, the drag-divergence Mach number Mdd , is reached. The thinner the airfoil, the higher is Mdd . Wings of transonic transport aircraft are swept and have supercritical airfoils, Sect. 2.4.2. The present section shortly discusses and illustrates flow and separation phenomena up to transonic buffet, which are of relevance for the drag-divergence Mach number of a supercritical airfoil, respectively wing. Even if in reality the flow past a swept wing is three-dimensional, the two-dimensional case is a fair approximation to the reality, of which an example is shown in Fig. 8.37. Consider Fig. 9.1. It gives in more detail the flow field past the lifting transonic airfoil, which was considered in Sect. 4.2.3, Fig. 4.6. The sonic line—broken line— is indicated also in the boundary layer below the supersonic flow pocket at the suction side of the airfoil. The terminating shock wave is slightly curved, as is to be demanded, and at the airfoil’s surface in the inviscid picture is to end normal to it, for both propositions see Sect. 4.2.3.

Fig. 9.1 Schematic of the flow field of a supercritical airfoil, after [2]. The thickness δ of the boundary layer is not to scale

9.1 Supercritical Airfoil—Shock-Wave/Boundary-Layer Interaction

235

Fig. 9.2 Schematic of weak shock-wave/turbulent boundary-layer interaction [2]

The sonic line is shown to curve around the tip (1) of the shock wave and attaches to it in (2). The interpretation is that the shock wave above (2) is oblique to the local flow direction. Therefore the flow behind it is supersonic and then decelerates isentropically to sonic and subsonic flow. The shock wave interacts with the boundary layer in the interaction domain. Two cases are to be distinguished, weak and strong shock-wave/boundary-layer interaction, in our example with turbulent flow. The interaction is considered to be weak, if the boundary layer does not separate due to the shock wave, and strong, if it does. The critical pre-shock Mach number at the shock foot is M f oot ≈ 1.3–1.35, [1], see also Sect. 2.2. • Weak shock-wave/boundary-layer interaction. First we consider the weak interaction case, Fig. 9.2. The external inviscid flow is supersonic. The flow in the boundary layer of course also is supersonic down to a distance from the wall, which depends on the overall boundary-layer flow properties. In the subsonic part of the boundary layer information about the outer pressure field can travel upstream. A fan of isentropic compression waves results above the sonic line and the pressure rise due to the shock wave is smeared out. No separation occurs, only a thickening of the boundary layer. The shock wave just above the boundary layer and in its upper part is slightly oblique to the flow. That leads to weak supersonic flow behind the shock wave with isentropic recompression. The result is the “supersonic pocket” behind the shock wave, marked in the figure with M > 1. Downstream of the interaction domain the boundary layer is subsonic and the wall shear stress is reduced. Overall the boundary layer is at risk to separate early at the rear of the wing. That would change the flow-off separation there to a complex trailing-edge separation pattern.

236

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.3 Schematic of strong shock-wave/turbulent boundary-layer interaction [2]

• Strong shock-wave/boundary-layer interaction. Once the pre-shock Mach number is large enough, the shock wave will lead to separation of the boundary layer. A separation bubble is formed, Fig. 9.3. The upstream displacement effect in the subsonic part now has such a strength that the shock wave at the outer edge of the boundary layer is split into two oblique shock waves, the lambda shock. At the foot of the upstream shock wave an isentropic compression fan is present. The supersonic flow pocket is much larger than that of the weak interaction case. From the triple point shown in the figure a slip line ensues, in the form of a vortex sheet like shown in Fig. 4.5. As in the case of weak interaction, downstream of the interaction domain the boundary layer thickness has increased and that to a higher degree. The wall shear stress is reduced more severely, the boundary layer increasingly is more prone to suffer separation at the rear of the airfoil, i.e. trailing-edge separation. Increasing the pre-shock Mach number would move forward this separation until it directly begins at the interaction domain. Regarding the thickening of the boundary layer downstream of the interaction domain we note that another interesting effect exists. All boundary-layer thicknesses depend on the inverse of some power of the Reynolds number: δi ∝ Re−n , Appendix A.5.4. In [3] it is shown that across a normal shock wave the unit Reynolds number always decreases. Behind the interaction domain this effect alone would lead to an increase of the boundary-layer thickness. Whether and how this effect in our case plays a role is not known. Summarizing we note that shock-wave/boundary-layer interaction leads to several adverse effects. They are, on the one hand the increase of drag due to the increased form drag and in addition the wave drag, on the other hand the reduction of lift due to boundary-layer and shock-wave decambering, Sect. 6.1.

9.1 Supercritical Airfoil—Shock-Wave/Boundary-Layer Interaction

237

Fig. 9.4 Lift coefficient C L : schematic of separation and interaction effects as function of the flight Mach number M∞ [2]

If trailing-edge separation occurs, respectively total separation behind the shock wave, drag divergence happens, Sect. 2.4.2. For the laminar wing weak and strong interactions are of particular importance, because the laminar boundary layer does not accept as much adverse pressure gradient as the turbulent one. Figure 9.4 schematically shows how the lift coefficient is affected in the different Mach-number domains.2 In the subsonic domain the maximum lift coefficient is reached when total separation, or wing stall happens. In the transonic domain first the lift is reduced and once total separation behind the shock wave is present, the strong transonic lift dip occurs, which goes along with unsteadiness of the flow field: transonic buffet. Interaction with aeroelastic effects of the wing then leads to the transonic buffeting. In aircraft operations transonic buffet and in particular transonic buffeting limit the flight Mach number. The buffet onset boundary in the figure is indicated as design point. The supercritical airfoil is to reduce shock-wave/boundary-layer interaction effects by minimizing the overspeed—the increment of speed over the freestream speed—in particular at the suction side of the wing. Weak and strong interaction effects in this way are pushed to higher flight Mach numbers. For the basics of interaction effects see, e.g., [1], for the implication for the transonic wing design, e.g., [2]. The reader interested in the respective early German developments will find ample information in [5]. 2 The

figure goes back to F. Thomas [4].

238

9 Particular Flow Problems of Large Aspect-Ratio Wings

Both the experimental and the numerical simulation of shock-wave/boundarylayer interaction and transonic buffet are highly demanding. The experimental investigation needs both Mach-number and Reynolds-number similarity. Wind tunnel test-section effects must be overcome. The numerical simulation faces the problem of turbulence modeling in unsteady flow and in the presence of shock waves.3 Numerical studies in particular of the buffet onset have shown that obviously simulations based on the RANS or URANS equations (Model 10 of Table 1.3) are not suited well. In [6] it is demonstrated that zonal detached-eddy simulation (Model 11), rather than RANS and URANS and the standard detached-eddy simulation, permits to reproduce the self-sustained flow oscillations on the airfoil. In [7] results of delayed detached-eddy simulation (Model 11) of transonic buffet and URANS simulation are presented. The former successfully simulates the unsteady shock-wave/boundary-layer interaction. Alternate vortex shedding and a spanwise undulation are present. Near the trailing edge the flow unsteadiness is overestimated. Results of experimental investigations in order to provide test cases for numerical simulation approaches are reported in [8]. The authors observe that the experimental results suggest that the transonic buffet essentially is a two-dimensional mean-flow phenomenon, although three-dimensional effects are present. URANS simulations of transonic buffet on different wing shapes are published in [9]. The phenomenon is studied on two-dimensional, infinite swept and finite-span wing configurations. For small wing sweep angles the mechanisms are like those for two-dimensional flow. For moderate sweep angles a propagation of lateral pressure disturbances was found, which is not present in two dimensions. At high-sweep wings, when the wing becomes stalled, shock buffet has vanished. Wing tip effects are shown to be large at low-aspect ratio wings. Much effort was and is spent on ways and devices to influence and reduce shock-wave/boundary-layer interaction effects on supercritical wings. Passive and active means in particular for laminar wings were investigated for instance in the EUROSHOCK projects of the European Union in the late 1990s [10, 11]. The bump, a small distensible local contour modification in the shock region mainly to reduce the strength of the shock wave, was found to be the most effective device when drag reduction is the main driver. Benefits are present also regarding transonic buffet. The problems come in when the bump must be adaptive to changing flight conditions. The potential for turbulent flow is to be explored. The laminar wing still is an important topic. Increasing ecological pressure will lead to more efforts in research and industry. The adaptive bump in turbulent flow with the function of both wave-drag reduction and buffet control is a challenging subject, too. In [12] URANS investigations are reported on the effect of adaptive two-dimensional bumps and of “smart vortex generators” in the form of three-dimensional bumps. On a supercritical, unswept wing section the effect of bump flow conditions, crest height, and streamwise positioning 3 If

laminar-turbulent transition would be involved, the situation would be much more challenging.

9.1 Supercritical Airfoil—Shock-Wave/Boundary-Layer Interaction

239

of two- and three-dimensional shock control bumps was analyzed regarding buffet behavior and overall performance. Two-dimensional shock-control bumps were found to improve buffet behavior thanks to the shock strength reduction with positive effects on flow separation. Threedimensional bumps yield the same buffet-affecting mechanisms. The finite spanwise extent makes them less dominant. Also shown was that the strength of the vortical wake of three-dimensional bumps can be tuned by appropriate bump shaping. The tuned strength then correlates positively with delayed buffet onset. For both laminar and turbulent flow control smart structures and materials are required. Moreover the overall problem of drag reduction needs to be addressed: sub-boundary layer devices, mass-less air jets, vortex generators as well as reversedflow flaps and mini flaps for the control of the flow at the trailing edge, which always is of finite thickness, as was shown in Sect. 6.3. In view of practical applications installation penalties and actual benefits are to be identified. This is of large importance in order to chose the right course in the efforts, see, e.g., [13].

9.2 Flow Past a High-Lift System High-lift systems in the form of lift-enhancement surfaces are indispensable devices in particular on transport aircraft wings. They permit to achieve the necessary airfield performance, i.e., the needed high-lift capabilities for landing but also for take-off. The overall demands are low approach speed, low take-off drag, all to be achieved with an in total small and simple system with low weight and low complexity as formulated to the point in [14, 15]. A large variety of leading edge and trailing edge devices is in use, see, e.g., [16, 17] and also [18]. A high-lift system usually has a slat in the front and a single-slotted—or even double- or triple-slotted—Fowler flap at the rear. The deployed flaps increase the camber of the wing and hence the lift, which goes along with a reduction of the lift-to-drag ratio. Other nose devises than the slat are the Krueger flap and the hinged leading-edge droop-nose device (DND), see, e.g., [14, 15]. We are interested in the flow and separation phenomena, which are present at a high-lift configuration with slat. Consider Fig. 9.5, which shows the basic configuration of a two-dimensional high-lift airfoil and the arising phenomena. The gap or slot between the slat and the main wing, as well as that between the main wing and the Fowler flap are the devices, which make the system a viable one. The flow through them, driven by the pressure difference between the lower (pressure) and the upper (suction) side, prevents boundary-layer separation at the upper side of both the main wing and the Fowler flap. At the upper side of the slat due to the strong flow acceleration a supersonic flow pocket arises despite the typical low free-stream/flight Mach number at which the high-lift system is activated. The supersonic flow pocket is terminated by a nearly normal shock wave with the associated shock-wave/boundary-layer interaction.

240

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.5 High-lift configuration and schematic of the flow phenomena (after [14]). Laminarturbulent boundary-layer transition is not indicated everywhere, but always is present

At the lower side of the slat a geometry-induced (cove-) separation is present. From the trailing edge of the slat a shear layer or wake with kinematically inactive vorticity content arises via flow-off separation. The situation is the same at the end of the main wing, again with cove separation. The flow over the Fowler flap at the rear is even more complex, because of the flow through the rear gap. The critical location is that at the rear of the upper side of the flap, where the flow is prone to ordinary, i.e., adverse pressure-gradient induced separation. In all cases confluence of the different boundary layers leads to complex flow situations.4 The two-dimensional flow picture shown in Fig. 9.5 changes for a real wing, Fig. 9.6. Three-dimensional effects due to the finite wing span, the sweep of the wing, spanwise segmentation and nacelle interactions reduce the efficiency of a nominally two-dimensional high-lift system. We illustrate with results of numerical simulations some of the flow and separation phenomena arising at a high-lift system [19]. The results were obtained in the frame of a research project of the European Union [20]. The topic of this project was the accuracy and reliability of hybrid RANS-LES (Model 11) methods, in particular the grey-area mitigation. The test case was a three-element airfoil at M∞ = 0.15 and Rec = 2.094 · 106 , with c = 0.6 m being the length of the airfoil with retracted slat and flap. The angle of attack was α = 6◦ . Figure 9.7 illustrates the expected complex separation phenomena and simulation challenges. Experimental data was obtained, too. Experimental uncertainties regarding flow three-dimensionality, side-wall effects, laminar-turbulent transition and angle-of-

4 The

shear layers leaving the trailing edges of all elements of the high-lift system are also noise sources due to the finite thickness of the edges, Sect. 6.3.

9.2 Flow Past a High-Lift System

241

Fig. 9.6 The wing movables of the A350 XWB-900 [15]. DND: hinged leading-edge droop-nose device

Fig. 9.7 Geometry and salient flow features of the three-element airfoil [19]

attack corrections precluded the use of the obtained data as strict validation data. The data is used rather as demonstration data. Five European partners applied their zonal or embedded hybrid RANS-LES (Model 11) methods to the test case.5 A few figures are shown in order to illustrate the obtained results. Mean flow-field data was obtained by time averaging and by spatial averaging in the homogeneous spanwise direction. Figure 9.8 shows the streamlines past the configuration resulting from the different simulation approaches. The streamline topologies show the same overall global behavior. Each of the three wing elements has its own attachment point of the external inviscid flow. (We note that the boundary layer at an attachment point has a finite thickness [21].) Below 5 We do not present the details of the numerical approaches. The interested reader is asked to consult

the original publication [19].

242

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.8 Mean flow-field streamlines achieved with non-zonal (left) and embedded (right) numerical simulations [19]

Fig. 9.9 Mean surface-pressure coefficient distributions achieved with non-zonal (left) and embedded (right) numerical simulations [19]

the wing elements the boundary layers are attached, as is the case also at the upper side of the slat and the main wing. At the upper side of the Fowler flap all solutions indicate separation and show recirculation areas of different extensions. It must be kept in mind that the flow patterns represent time-averaged data. The reality sees vortex shedding from the upper side of the flap. Surface-pressure coefficient distributions found by the partners, again time- and span-averaged, are given in Fig. 9.9. Included too are experimental results. The comparison with them is affected by the uncertainties mentioned above. Nevertheless, it appears that each of the simulations yields acceptable results. At the pressure sides of all wing elements the computed pressure distributions agree well with each other and the experimental ones. At the upper sides generally

9.2 Flow Past a High-Lift System

243

Fig. 9.10 Schematic of sub boundary-layer vortex generators (SBLVGs) [22]

the non-zonal approaches appear to be closer to each other and the experimental values. This is not the case with the results of the embedded simulations, although the differences are not large altogether. The in reality to be expected periodic vortex shedding from the upper side of the Fowler flap of course is not reflected in the data. Interesting is the question, how that flap separation can be influenced and, in general, how the performance of a high-lift system can be enhanced. Several passive and active means in this regard were investigated in the research and technology project AWIATOR (Aircraft Wing with Advanced Technology Operation) of the European Union and in the research project IHK (Innovative High-Lift Configurations) of the German Aerospace Research Program [22]. Purely passive means of flow control are sub-boundary-layer vortex generators (SBLVGs). They have a height of 0.8–0.9 of the boundary-layer thickness δ, i.e., they are located in the ‘outer layer’ or ‘turbulent layer’ of the turbulent boundary layer. Their height hence is smaller than that of ordinary vortex generators. Their location is just ahead of the largest thickness of the Fowler flap, Fig. 9.10. When the flap is retracted, they are below the shroud in the cove. The SBLVGs basically are slender triangles. Typical dimensions, spacings and orientations are given in Fig. 9.10. Like ordinary vortex generators, but with less eigen drag, they produce streamwise coherent vortices of small scale. In this way, fluid from the outer layer with large momentum is transported toward the wall, effectively reducing the separation disposition of the boundary layer. SBLVGs, being devices with low complexity, weight and cost penalty, have been tested in wind tunnels and full-scale also in flight with an Airbus A340. The flight tests have shown an increase of lift due to the larger flap angle—35◦ instead of 32◦ — without separation and without a reduction of the aerodynamic efficiency L/D at take-off. If separation is shifted to a higher flap angle, also buffet at the horizontal tail plane is shifted. The devices as add-on devices can increase the performance of a given underperforming flap system. On the other hand SBLVGs have the potential to create smaller and lighter Fowler flap systems. They are studied, too, as means for flow control of engine inlets, see, e.g., [23].

244

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.11 Schematic of an adaptive Mini-TED device at a Fowler flap [22]

Active means to enlarge the lift of a take-off configuration are adaptive miniature trailing-edge devices (Mini-TEDs), Fig. 9.11. They were tested on an A340 with a length of 2 per cent clean-wing chord, i.e., 90–150 mm length [22]. A Mini-TED located at the very trailing edge with 90◦ deflection actually is a Gurney flap. Such a flap enlarges the camber of an airfoil or wing. In [22] it is reported that adaptive Mini-TEDs were shown to improve the performance of the high-lift configuration of a large aspect-ratio wing, and to reduce drag and buffet onset in cruise condition. Mini-TEDs can be used to adapt wing twist in cruise, and, depending on the spanwise location, for roll and spanwise load control. Wind-tunnel tests have shown that also lift and the lift-to-drag ratio can be improved for a canard-delta fighter aircraft configuration at low speed [22]. Adaptive elements and devices like Mini-TEDs, adaptive drooped hinge flaps (ADHF), pulsed blowing—also at the wing’s leading edge—have been proven to be effective devices to improve the performance of high-lift systems and to be means for load control and other interesting applications. However, compared to the passive SBLVGs they lead to a higher system complexity, as well as mass, volume and energy increments. Their application on aircraft depends on a number of requirements besides their actual effects.

9.3 The Wing in High-Lift Condition We treat, following—very abbreviated—the discussion by R. Rudnik in [24], the major vortex phenomena occurring at a swept, large aspect-ratio wing together with the underwing engine in high-lift configuration. The configuration is typical for a modern transport aircraft designed for transonic flight, Fig. 9.12. Separation and vortex phenomena occurring at the clean lifting large aspect-ratio wing were treated in Chap. 8. The flow past the isolated high-lift system was the

9.3 The Wing in High-Lift Condition

245

Fig. 9.12 Transport aircraft: locations, which are prone to ordinary separation [24]

topic of the preceding Sect. 9.2. Now we discuss the phenomena present at a swept, tapered wing’s root, its tip and in particular at the nacelle of the underwing mounted engine, all in presence of the employed high-lift system. At the high-lift configuration in Fig. 9.12 four locations are indicated, where separation and vortex phenomena can be and are indeed present: • • • •

(a) the wing-root area, (b) the region around the engine, (c) the trailing edge of the Fowler-flap system, (d) the wing tip.

Also in the background of our topic are the stalling characteristics of swept, tapered wings. The wing’s sweep leads to an extra thickening of the upper-side boundary layer at the outer wing section and there to a higher separation tendency. This tendency is due to the wing upper and lower flow-fields shear, and is increased due to the wing’s taper. If we have a parallel isobar pattern, or one near to it, [25], the surface pressure gradients, both favorable and adverse, become the larger, the smaller the local chord length is. This mainly concerns the upper side of the wing, where the boundary layer then becomes prone to separation. If the stall begins at the outer wing, the roll control by means of the aileron is no longer possible. The stall characteristics of swept wings and the aerodynamic demands regarding take-off and landing have led to the presently employed high-lift systems, which permit a safe and controllable low-speed flight. A typical high-lift system consists of slats or Krueger flaps along the wing’s leading edge and single-slotted or multiple-

246

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.13 Schematic of the vortex phenomena present at a large aspect-ratio wing in high-lift configuration [24]

slotted Fowler flap systems along the wing’s trailing edge. The principle of a high-lift system is shown in Fig. 9.5.6 We look now at the effects present at the full-wing configuration, which have a significant influence on its high-lift performance. High-lift performance means the achievable maximum lift coefficient and the stalling characteristics, which both are strongly influenced by the engine installation. Also playing a role are the flow phenomena at the wing root and the wing tip, and in particular also those at the lateral edges of elements of the high-lift system. Both ordinary and flow-off separation lead to vortex sheets and vortices, which at and downstream of their location of origin influence the flow pattern, see the considerations in Sect. 2.2. We now investigate the occurrence and effects of separation and vortices at the wing shown in Fig. 9.13. The high-lift system is assumed to consist of a leading-edge flap and a trailingedge flap, the former being a slat and the latter a Fowler flap. These high-lift elements in span direction are sub-divided into several elements, as shown in Fig. 9.6.7 Because of the small gaps between the elements in span direction, they can be considered as 6 At

propeller-driven subsonic aircraft with no or only small wing sweep usually no leading-edge devices are employed. Also playing a role there is the favorable interference between the propeller wake and the wing flow. 7 In our discussion we assume that instead of the droop nose device (DND) inboard of the engine also a slat is present, like outboard of it.

9.3 The Wing in High-Lift Condition

247

uninterrupted. This does not hold for the slat at the position of the engine. Because present-day bypass engines are mounted close to the wing, see the following Sect. 9.4, the slat has a cutout at the location of the pylon. Regarding the trailing-edge flap system, present-day aircraft usually have no cutout behind the engine. This became possible with the single-slot flap system, which did replace the older two-slot flap system. The former does not extend as far downward as the latter. The end-to-end Fowler flap system of course has a partition at the Yehudi break. In Fig. 9.6 this is indicated as inboard flap and outboard flap. An important aspect regarding the trailing-edge flap system is that no interference with the engine jet takes place. This demands a sufficient vertical downward location of the engine and a proper setting angle of the engine against the horizontal. We discuss the phenomena present at the four locations, and begin with the wingroot area, location (a) in Fig. 9.12. As was shown in Sect. 8.4.3, at the clean wing’s root separation is present, leading to a horse-shoe vortex. This is only the case, if the wing is attached to the fuselage without a fairing. With a properly designed fairing the flow does not separate at the wing’s leading edge and a horse-shoe vortex is not present, Fig. 8.27. In the high-lift configuration the situation is different, Fig. 9.13. Two vortices are indicated at the wing’s root, where the extended slat leads to an interruption of the leading-edge contour: the clockwise turning slat-horn vortex and the weaker counterrotating onglet vortex. The slat horn has the purpose to create a strong slat-end vortex. Both vortices reduce the separation inclination of the boundary layer along the wing root. At larger angle of attack large scale separation will begin at the rear of the wing as indicated in Fig. 9.12, location (a). That separation is considered to be rather uncritical, compared to separation beginning at the outer wing. The induced roll moment is small, moreover the associated pitch-down moment reduces the angle of attack. This is a good-natured stall behavior, which, however, should not occur too early. Then the maximum lift capacity would be limited. The complex flow situation at location (b) is due to the interference of the nacelle flow with the wing flow in the high-lift configuration. At the edges of the slat cutout two counterrotating vortices evolve. At the left-hand wing shown in Fig. 9.13 the vortex at the inboard slat edge turns counterclockwise, at the outboard slat edge clockwise. The slat cutout leads to a reduction of the maximum lift of the wing. This reduction is only partly compensated by a lift increment due to the flow past the nacelle. That flow results in the so-called nacelle vortices, which are counterrotating, too. They come into being because of boundary-layer separation at the sides of the nacelle. With close-coupled engines at high angles of attack only the inboard nacelle vortex passes over the wing, the other below it. The inboard nacelle vortex induces an upwind above the wing and hence enhances the separation tendency there. The separation zone is beginning behind the slat cutout, grows laterally and moves downstream with an inboard tendency, as indicated in Fig. 9.13. In that figure also a nacelle-strake vortex is indicated. It runs over the slat cutout and is an effective means to eliminate the lift loss due to the close coupled high-

248

9 Particular Flow Problems of Large Aspect-Ratio Wings

bypass ratio engine. The mechanism of the nacelle-strake vortex is discussed and visualized in the following Sect. 9.4. Location (c) in Fig. 9.12 denotes the possible separation problem at the Fowler flap. If the high-lift system works properly, flow-off separation happens at the trailing edge of the flap. As discussed in the preceding Sect. 9.2, at the highly loaded flap ordinary separation can happen upstream of the flap’s trailing edge. In that case of course the effect of the high-lift system is reduced. Means to reduce the flap separation were sketched in that section, too. Location (d) finally indicates the problem area at the wing tip. The tip-vortex system at a lifting large aspect-ratio wing is discussed in Sect. 8.4.3. Wing-tip devices are the topic of Sect. 9.5. The problem at location (d) is due to the fact that the slat generally ends—due to structural constraints of the wing—at a location somewhat ahead of the wing-tip section. Consequently the effect of the slat is missing there and separation already occurs at a smaller angle of attack than at the main part of the wing. At the outer edge of the deployed slat a vortex is present, which influences the separated flow regime and furthermore interacts with the tip-vortex system. We have given only an overall discussion of the separation phenomena occurring at a wing in high-lift condition. At what angle of attack and at what location of a wing separation happens is a consequence of the layout of the high-lift system and the engine installation of the aircraft under consideration, as well as the desired stall behavior of that aircraft [24].

9.4 The VHBR Engine and the Nacelle-Strake Vortex The still increasing bypass ratio of jet engines leads to increasing nacelle diameters.8 The underwing arrangement of the engines is limited by the needed ground clearance. In order to avoid a mass critical extension of the landing gear the jet engines are becoming ever closer coupled to the wing, see, e.g., [26] and also Fig. 3.18. Consider the left part of Fig. 9.14. Schematically it shows a very high bypass-ratio engine (8  VHBR  12) closely mounted below the wing. The ground clearance is indicated. The slat is lowered and because of the engine’s proximity to the wing, above the nacelle the slat has a cutout. The right part of the figure shows that this cutout leads to the so-called nacelle-wake separation, see, e.g., [27]. This separation is due to complex interacting flow phenomena at the nacelle and the slat cutout. The result is a local wing stall, which reduces the high-lift effect and leads to an oversizing of the high-lift system. The currently employed remedy for VHBR engines is a rather simple device, the nacelle strake, which induces the nacelle-strake vortex, see e.g., [28]. The nacelle strake is a geometrically small device, placed at the inboard side of the nacelle, where below the wing and between the fuselage and the nacelle a channel effect is present. With a proper location the strake induces at and above the 8 The

higher the bypass ratio, the lower are the fuel consumption and the noise emission.

9.4 The VHBR Engine and the Nacelle-Strake Vortex

249

Fig. 9.14 Schematic of an engine wing arrangement with lowered slat (after [29]) Left part: VHBR engine and the slat cutout. Right part: the resulting nacelle-wake separation at the main wing part

required angle of attack the nacelle-strake vortex with its location behind the slat cutout and above the wing. There the vortex induces a suction-pressure field, like a lee-side vortex—or more in general, a side-edge vortex—, which in turn suppresses the nacelle-wake separation. With a correctly placed nacelle strake the induced vortex leads to the recovery of a large part of the lift loss due to the engine mounting. The interaction of the nacelle/pylon/nacelle-strake flow with that of the highlift system, in particular the slat, is a topic of high importance. It was treated, for instance, in the frame of the EUROLIFT II project of the European Union. In [27] it was demonstrated for a high angle of attack case that RANS (Model 10) simulations today are well suited to deal with the existing flow problems. Suitable mathematical models, accurate numerical solutions and comprehensive experimental data are the requirements for advances in wing and nacelle stall simulations [30]. The flow situation at a close-coupled flow-through nacelle with the nacelle-strake vortex is shown in Fig. 9.15. The sense of vortex rotation is made visible with the amount of the vorticity component ωx : red indicates the clockwise direction, blue the opposite, as indicated at the upper left of the figure. Several longitudinal vortices arise at the slat cutout. They interact with the flow at the upper side of the wing, which at and downstream of the leading edge is highly three-dimensional. The nacelle-strake vortex at the inboard side of the nacelle is well discernible. In EUROLIFT II project the flow past the KH3Y configuration—representing a modern transport aircraft—was studied [27], see also [31]. The configuration complexity was increased in three steps from Stage 1 to Stage 3, Fig. 9.16. The Stage 1 configuration is the baseline high-lift configuration with an onglet and a slat horn, but without engine. The slat deflection angle is δs = 26.5◦ and the Fowlerflap angle δ F = 32◦ . In Stage 2 a through-flow nacelle together with slat cutout is added. The nacelle represents a modern high by-pass engine. Stage 3 finally includes the nacelle strake. RANS simulations and wind-tunnel measurements were made for a low and a high Reynolds-number case below and at maximum lift conditions. We first look at results of the low Reynolds-number case. The free-stream Mach number is M∞ = 0.176 and the Reynolds number Rel = 1.33 · 106 .

250

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.15 Vortices at a close-coupled flow-through nacelle at pre-stall condition [29]

Fig. 9.16 Extension stages of the KH3Y configuration (right-hand wing) from a baseline high-lift configuration (Stage 1) to the realistic full aircraft configuration (Stage 3) [27]

Figure 9.17 shows both the computed skin-friction lines and the distribution of the wall shear-stress component c f x for the configurations Stage 2 and 3 at three angles of attack. When c f x becomes negative—red color in the figure—, the flow is considered to be separated. At the lowest of the angles of attack no major separation phenomena are present. At α = 15.92◦ separation develops at the inboard slat close to the pylon. With increasing angle of attack this separation finally extends in spanwise direction and down to the end of the main wing. When the nacelle strake is present, Stage 3, no slat separation occurs at the three angles of attack and up to maximum lift. The beginning of lift breakdown is then dominated by the separation at the 6th slat track. These RANS results are in accordance with the wind-tunnel results. We look now at some results for the high Reynolds-number case. The free-stream Mach number is M∞ = 0.204, the Reynolds number Rel = 25 · 106 , and the angle of attack α = 17.5◦ . The slat deflection angle and the Fowler-flap angle are the same as for the low Reynolds-number case.

9.4 The VHBR Engine and the Nacelle-Strake Vortex

251

Fig. 9.17 Low Reynolds-number case (right-hand wing): numerically predicted wall shear-stress distributions and skin-friction line patterns for Stage 2 (left) and Stage 3 (right) at three angles of attack [27]:  0.00 (red)  c f x  0.011 (blue)

The nacelle-strake vortex effect, visualized with the skin-friction line patterns and the c f x -distribution computed for Stage 2 and Stage 3, is shown in Fig. 9.18. We look at the nacelle at the right-hand wing. The c f x -distribution is coded such that red color indicates negative values, which means that the flow is separated. Blue color indicates a high wall-shear stress. The visualization of the associated vortex phenomena is presented in Fig. 9.19. The color denotes the amount of the vorticity component ωx . Red indicates clockwise, blue counter-clockwise rotating, see also Fig. 9.15. The combined look at the two figures for Stage 2 shows at the inboard side of the nacelle the convergence of the skin-friction lines and a strong counter-clockwise rotating vortex. A vortex filament, clockwise rotating, emerges from the upper side of the inboard slat. At the right-side end—looking at the figure—of the outboard slat a counter-clockwise rotating vortex emerges. On the main-wing element from the left to the right in the figure the skin-friction lines indicate a separation line, bending away to the left, an attachment line, which tapers off soon, and in the large red patch possibly also a vortex filament leaving the surface. The red patch indicates the large separation region, where the high surface

252

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.18 The effect of the nacelle-strake vortex on the skin-friction line pattern and the c f x distribution for the high Reynolds-number case at the right-hand wing [27]:  0.00 (red)  c f x  0.011 (blue). Left part: Stage 2 (no nacelle strake), right part: Stage 3 (with nacelle strake)

Fig. 9.19 The vortex visualizations for the high Reynolds-number case at the right-side wing [27]: Left part: Stage 2 (no nacelle strake), right part: Stage 3 (with nacelle strake)

pressure leads to the reduction of the high-lift effect. Also seen on the upper side of the pylon is a short separation line, leading to the small counter-clockwise rotating vortex above the pylon. For Stage 3 the strong counter-clockwise rotating strake vortex already changes the skin-friction line pattern at the inboard side of the nacelle. The vortex filament separation at the upper side of the inboard slat has disappeared. Only small changes are present at the right-side end of the outboard slat. The pylon vortex still is indicated. On the main wing, due to the presence of the nacelle-strake vortex, the skinfriction line pattern has changed completely. It now indicates, together with the c f x distribution, that the flow basically is attached. From the left to the right in the figure we see a separation line, an attachment line, and a second separation line. The color of the c f x -distribution in each case qualitatively confirms this: low values along

9.4 The VHBR Engine and the Nacelle-Strake Vortex

253

the separation line, high values along the attachment line, there also present—not shown—a relative maximum of the wall pressure.9 Unfortunately no surface pressure distributions are available, so that the effect of the nacelle-strake vortex in this regard cannot be demonstrated. With current VHBR engines the nacelle-strake vortex is sufficient to suppress the nacelle-wake separation. This passive means is no more effective for ultra high bypass-ratio engines (UHBR  12), which presently are beginning to come into service. In order to achieve the needed ground clearance, such engines are even closer coupled to the wing than the VHBR engines. The slat cutout then becomes too large for the passive nacelle strake to be effective. Active means, like sealing elements, continuous or pulsed slot blowing etc. are being studied or being introduced, see for example the discussion and reporting in [29]. From the side of aircraft design of course system complexity problems arise with such means, which are not present with passive means.

9.5 Wing-Tip Devices In the introduction to Chap. 8 we have discussed the matter of induced drag. The larger the aspect ratio Λ of a wing is, the smaller is the induced drag Di . The induced drag is part of the total drag of an aircraft. Its magnitude depends on the given flight phase and in cruise flight at C L /C D |opt can be up to 50 per cent of the total drag [32]. The span and hence the aspect ratio of the wing of a transport aircraft at least is restricted due to the ramp or apron size at an airport, for instance (length × span) 60 m × 60 m. This then is an incentive to use wing-tip devices in order to compensate the restricted wing span. Another incentive would be to improve the aerodynamic performance of a given aircraft by adding wing-tip devices. Wing-tip devices in their earliest forms were the end-plates, located at the tips of the wing or of the horizontal tailplane. The idea behind it was—as one can still find in the literature—that the device hinders the relief of the high pressure at the underside of the wing around the wing tip to the upper side with its low pressure. In this picture the corresponding flow around the wing tips is the cause of the “tip vortices”, which on their part are the cause of the induced drag. In the background obviously Prandtl’s lifting-line model prevails. Even today one can find in widely used text books that no clear difference is made between the trailing vortex and the tip vortex. The trailing vortex evolves due to the wing upper and lower flow-fields shear, Sect. 4.3.2. At the lifting large aspect-ratio wing it emerges out of the trailing vortex layer behind the wing, being present in full only some wing half-span distances behind the wing’s trailing edge, see for example Fig. 4.18, which shows the situation as it appears in the frame of potential-flow (Model 4) theory.

9 Attachment

and separation lines are singular lines, for the related flow properties see Sect. 7.3, respectively [21].

254

9 Particular Flow Problems of Large Aspect-Ratio Wings

The tip vortex, or the tip-vortex system, as we have seen in Sect. 8.4.3, comes into being due to ordinary separation at the upper side of the wing-tip area. The flow there of course passes from the lower side of the wing around the wing tip to the upper side. There, depending on the given pressure field the separation happens. A wing-tip device indeed influences this process as well as the trailing vortex layer behind the wing. As mentioned above, the induced drag of a wing decreases as the wing’s aspect ratio increases. Ground-operational demands restrict wing spans. A wing-tip device in form of a winglet or a sharklet is a means to compensate a span restriction.10 Other applications are the adaptation/extension of a given wing in order to improve the aerodynamic performance. The shape of a winglet or sharklet hence is designed to reduce the induced drag and at the same time changes the strength of the tip vortex, or tip-vortex system. In Sect. 8.4 it was seen for the investigated case that the circulation of the tip vortex is about seven per cent of the total wake circulation. Hence a reduction of the strength of the tip-vortex system would have an effect, but whether positive or negative is not clear over the large range of operating conditions. A wing-tip device of course leads to an increase of the viscous drag of the wing, i.e., the skin-friction drag and the form drag increases. If the extension leads to a corner flow situation, interference effects and shock-wave drag do occur in the transonic flight domain. The device further increases the wing’s mass. Hence the sought effect of a wing-tip device is reduced by these effects, see below. The purpose of this section is to give the reader an overview of some aircraftrelevant issues associated with wing-tip devices. No review is intended of the approaches and the vast literature regarding wing-tip devices. In-depth studies of the fluid-mechanical situation at such devices, like that for the CRM wing in Sect. 8.4.3 regarding the wing tip-vortex system and its interaction with the trailing vortex layer, are not available. A wing-tip device must not only be seen in view of a reduction of the induced drag. For the aircraft it has other effects, too. In order to obtain an overall picture, A. Büscher in his doctoral thesis uses the concept of the “equivalent drag reduction” relative to a wing with planar wing tip [34, 35]. The equivalent drag reduction coefficient ΔC Dequiv as an objective function for design work has five terms: ΔC Dequiv = ΔC Daer o + ω · ΔC Dwr bm + ΔC Dmass + ΔC Dtrim + ΔC Dmech .

(9.3)

ΔC Dequiv being negative indicates a reduction of the equivalent drag.

10 The classical winglet is a vertical surface at the wing tip, whereas now wing-tip extensions evolve smoothly out of the wing’s surface. Sharklet so-called are winglets solely at the aircraft of the A320 family of Airbus, see, e.g., [33].

9.5 Wing-Tip Devices

255

Basically it is assumed that for a given aircraft, respectively wing configuration, a wing-tip device is added. For some of the terms the reference configuration of the wing tip is the Küchemann wing tip. The five terms are summarily presented. 1. ΔC Daer o represents the effect of the wing-tip device as such, i.e., the net aerodynamic drag reduction. It consists of three or four parts: due to the wing-tip device the reduction of the induced drag (negative sign), due to the device’s surface, trailing edge, and volume the increase of skin-friction drag, form drag, and at transonic flight wave drag (all three with positive sign). 2. ω · ΔC Dwr bm : if a wing-tip device is added, it leads to an increase of the bending moment along the wing and hence also of the wing-root bending moment (wrbm). This leads to an adjustment of the wing structure and with that to an increase of the wing’s mass. Hence the empty mass of the aircraft is increased. If the maximum take-off weight is to be kept, despite the reduced fuel demand limitations of either the disposable load or the operational range of the aircraft are the consequence. The mass-increase effect summarily is taken into account by the term ω. This term is discussed below. 3. ΔC Dmass is due to the mass of the wing-tip device and a possible extra wing mass increase due to its attachment on the wing. 4. ΔC Dtrim is an increment of the trim drag due to the change of the pitching moment of the aircraft. It consists of two parts. The first represents the change of the pitching moment due to the wing-tip device, the second its change due to the changed induced drag. 5. ΔC Dmech is due to the extra mass, if movables at the wing-tip device are used at low speed to improve the high-lift performance of the device. The factor ω (0  ω  1) is a weighting factor between the empty mass of the aircraft and the drag, called ‘wing-root bending moment weighting factor’. Its value depends on the chosen design objective of the aircraft layout: (a) direct operating costs (DOC), or (b) maximum take-off weight (MTOW), or (c) flight range (RANGE). Hence three scenarios can be considered in which two of the factors are fixed and the third one is variable. The determining component of the aircraft is the wing. It is the only aircraft component, which directly reacts on the design modifications with drag and mass changes. In the first case the range is the variable parameter. If the MTOW is not changed, only a small drag reduction is necessary in order to reduce the fuel consumption so much that the DOC do not rise. Hence ω = ω R AN G E is small. The resulting design has a high aerodynamic quality. If, however, due to an increase of the empty mass the MTOW can be increased, the situation is different. Due to the direct coupling to the DOC at constant RANGE the MTOW increase can only be restricted, if the drag reduction is much larger than in the first case. Consequently ω = ωmtow is larger for this case. The third scenario of a drag reduction demands constant MTOW and RANGE. In order to reach this despite the rise of the empty mass a large reduction of fuel consumption is necessary, which only can be achieved with a large drag reduction.

256

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.20 The long-range aircraft reference configuration [34]

Fig. 9.21 Schematic of the long-range configuration and the spanwise partition in LIDCA [34]

The resulting reduction of the DOC is ensured with a high ω = ω D OC , which leads to a more mass-optimized design. The considered wing-tip devices in Büscher’s thesis are based on a high ω. Hence the factor is referred to as ω D OC . Part of the parameter study was made with the smaller factor ω M T O W . The change of the factor implies a reduction of the rootbending moment of about 40 per cent. We look now at some overall results. The study was made for a long-range research configuration with Küchemann wing tip, but without nacelles at high-speed conditions and also for a high-lift take-off configuration with nacelles and extended slats, flaps and ailerons. The long-range research configuration is shown in Fig. 9.20. The flight Mach number was M∞ = 0.85, the Reynolds number Re L = 54.2·106 , with L = 11.5 m, and the lift coefficient C L = 0.5, equivalent to cruise flight. The study was based on the lifting-line method LIDCA (Model 4), which includes a database module for airfoil data. The method was validated with RANS (Model 10) computations. The LIDCA configuration is shown in Fig. 9.21. The aspect ratio of the wing is Λ = 8, the semispan 39.5 m. Changes of the wing geometry due to the wing-tip device were restricted to outside of 95 per cent of the wing’s semispan. That location is indicated in the lower left part of Fig. 9.21. The

9.5 Wing-Tip Devices

257

Fig. 9.22 Visualization of the most effective wing-tip devices together with the Küchemann wing tip and the Large Winglet [34]

resulting semispan of the wing was restricted to 42.5 m, and the extension of anhedral devices in downward direction to 2.75 m, measured from the attachment location. The following results concern the four most effective wing-tip devices for the long-range case, which were found with LIDCA: concept 1, 2, 4, and 5 [34]. RANS computations were made in order to check the results. Figure 9.22 shows the concepts 1, 2 and 5, the last one being a sharklet concept. Concept 4 is the same as concept 1, but with a smaller sweep angle of the 25 per cent line—40◦ instead of 45◦ —and a reduced wing-tip area. Both are anhedral devices with the angle ν = −20◦ . Concept 2 has the dihedral angle ν = 10◦ . The figure in addition illustrates the Küchemann wing tip of the reference concept and the Large Winglet, which also was considered in the study. The results in terms of the equivalent drag reduction obtained for the four concepts are given in Fig. 9.23. We note that the results for all concepts were found with the high wing-root bending moment weighting factor ω D OC . The figure contains two basic results: (1) concept 4 leads to the largest equivalent drag reduction; (2) the RANS solutions, which were made for the validation of the LIDCA results, confirm all LIDCA results and in particular also that for concept 4. The RANS solutions throughout yield a higher drag reduction for all concepts, with a delta of about 0.3 per cent. The decomposed LIDCA results in Fig. 9.24 yield the following outcome: (1) for concept 1 and 2 each of the increments of ΔC Daer o , ΔC Dwr bm and ΔC Dmass have almost the same magnitude. The slightly higher aerodynamic increment of concept 1 is due to the higher absolute value of ν and the larger effective span width; (2) concept 4 has the smallest increments of all concepts. The somewhat smaller aerodynamic increment is compensated by the smaller wing-root bending moment increment and the smaller mass increment; (3) concept 5 has the highest aerodynamic increment, but also the highest increments of the wing-root bending moment increment and the device mass, hence it has the smallest equivalent drag reduction coefficient, which, however, is near to that of concept 2. These results, which in a highly condensed manner present the total results of [34], show the merit of wing-tip devices as non-planar wing tip shapes. Their advantages

258

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.23 The equivalent drag reduction coefficient ΔC Dequiv for the four most effective wing-tip concepts [34]

Fig. 9.24 Decomposition of ΔC Dequiv for the four concepts [34]

are the simple geometry and the rather unproblematic integration at a new or an existing wing in the sense of a retrofit solution without a degradation of the wing’s performance at other conditions. It appears, however, that if the wing span is not restricted, a wing-tip device has no advantage over a planar wing extension. If, however, the wing’s span is restricted, or the performance of a given wing is to be improved, non-planar wing tip shapes definitely have benefits for the aircraft. The wing-tip device can have a positive or a negative dihedral, i.e., an anhedral, both show the desired effect. The devices today usually have the positive dihedral, since that does not lead to restrictions regarding the ground clearance. Restrictions in span due to the ramp dimensions can be overcome in another way, which, however, up to date was not a viable one. But now Boeing with its 777X

9.5 Wing-Tip Devices

259

series introduced the—upward—foldable wing extension.11 On ground the span of the aircraft is 64.8 m, in flight 71.8 m. The folding mechanism with its actuator of course has its own mass, which acts against the benefit of the extension. The energy demand certainly is small, but the overall system complexity of the aircraft is increased.

9.6 The Wake-Vortex Hazard: The Problem and Means to Control It As we have shown and discussed in Chap. 8, the trailing vortex layer of the lifting wing rolls up into the pair of trailing vortices. In this process it absorbs the two wingtip vortex systems, the vortex sheets and vortices originating from the fuselage, the tail unit, the engines, flap tracks etc. and last but not least the counterrotating trailing vortex layer from the trim surface. The resulting flow field summarily is called the aircraft vortex wake. This wake can pose the so-called wake-vortex hazard for other, following aircraft [36, 37]. Potential flow theory (Model 4 of Table 1.3) can be used to achieve basic insight regarding this hazard. The theory yields for an elliptic wing-circulation distribution that the strength of the trailing vortex Γ0 is proportional to the wing’s lift, Sect. 3.16, and hence to the mass of the aircraft. This result basically holds also outside of the range of validity of potential flow theory. Behind the aircraft we see the situation sketched in Fig. 9.25. The vortex wake is marked by the two trailing vortices, a downwash between them and an upwash at their outer sides. When another aircraft enters the aircraft wake or flies in it, it meets the wake-vortex hazard. That manifests itself in increased air turbulence, which leads to increased dynamic structural loads, as well as passenger discomfort. Further happens a loss of lift, and a rolling moment may be induced, too. The hazard may arise everywhere on the flight path, but in particular during take-off from the runway, during approach to and landing on the runway. The wake-vortex hazard became a larger topic when the Boeing 747 entered service in the beginning of the 1970s. With a maximum take-off mass (MTOM) of up to 448 t it was much heavier than the Boeing 707 with at most 142 t MTOM. About 35 years later, when the Airbus A380 with about 570 t MTOM was to enter service, the wake-vortex hazard again attracted much attention. Starting already in the early 1970s comprehensive investigations were dedicated to reduce the wake-vortex hazard in terms of diminishing the wake field impact on a follower aircraft, in particular the induced rolling moment. Strategies to minimize the wake vortex hazard can be divided into two categories [38]. In the first category, the focus is on a Low Vorticity Vortex wing (LVV) design, which reduces the wake vortex hazard by enhancing the dispersion of the vorticity field. After the roll-up of the trailing vortex layer is completed, the resulting trailing 11 Airbus

studies such and alternative foldable solutions.

260

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.25 Schematic of the aircraft’s wake with the trailing vortex pair and the related downwash and upwash [37] Fig. 9.26 Wake alleviation concepts [39]: change from the high to the low vorticity vortex (LVV)

vortices should feature a larger core size and smaller swirl velocities at the core radius, Fig. 9.26 [39]. Consequently, the induced rolling moment is diminished. In the second category, the focus is on a Quickly Decaying Vortex (QDV). An enhanced wake vortex decay may be achieved by promoting three-dimensional instabilities by means of passive or active devices. Particularly, the growth rates of the long wave “Crow” instability, [40], have to be increased significantly, Fig. 9.27, e.g.,

9.6 The Wake-Vortex Hazard: The Problem and Means to Control It

261

Fig. 9.27 Wake alleviation concepts [39]: the quickly decaying vortex (QDV). t ∗ : non-dimensional time, T ∗ : non-dimensional time up to vortex linking

by the interaction of multiple vortex systems or active excitation. The Crow instability is a sinusoidal vortex-pair instability due to an initial disturbance, Sect. 3.11.3. The vortex trajectories are deflected laterally and finally connect, forming a chain of vortex rings, which typically happens in Domain 4 behind the aircraft, Fig. 8.1. A variety of configurative measures has been investigated for the implementation of these concepts [39]. Considering the LVV strategy, wing control surfaces, such as spoilers, and modified variants, e.g., flap edge elements, are used to create areas of highly turbulent flow aimed to expand the core size of the trailing vortices [41]. Also, a modified wing-load distribution may minimize the induced rolling moment for a following aircraft. The alteration of the circulation distribution of the wakegenerating wing can be obtained using for instance differential flap or spoiler settings. It has been shown that a wing with an outboard partially deflected flap and an inboard fully deflected flap produces—at least in the extended near field—a smaller induced rolling moment than a wing with a standard flap setting [39]. Different measures can also be attributed to the QDV approach. Because a multiple vortex system shows instabilities, which can grow more rapidly, passive devices aim to promote these kinds of instabilities through the deliberate production of distinct vortices in addition to those coming from the wing tip and the flap edge. The production of additional distinct vortices can also be achieved by differential flap setting. The efficiency of these concepts depends on the persistence of such additional vortices, which is determined by configurational details of the aircraft. Particularly, active devices are considered as a possible powerful means to amplify

262

9 Particular Flow Problems of Large Aspect-Ratio Wings

Fig. 9.28 Examples of passive means for vortex control—spoilers and flap elements [39]: a four engine large transport aircraft in approach configuration, b delta spoiler at the wing’s upper side at the outboard nacelle station, c differential flap setting

wake vortex instabilities. The operational implementation, however, would run into the problem of heightened system complexity. Passive and active means to alleviate the wake-vortex hazard are now sketched in some detail. Generally it must be observed that the system complexity connected to alleviation means, is an obstacle for the implementation of such means. • Passive Means. Viscous and convective mechanisms are used to spread the vorticity field of the trailing vortex over a wider spatial area, reducing peak values in axial vorticity and circumferential velocities, Fig. 9.26. The radial transport of vorticity and thus the expansion of the vortex core is supported by increasing the turbulence intensity of the rolling up vortex layer and/or in the formation area of concentrated single vortices. Flap edge devices, spoiler elements and wing fins are applied to produce turbulent wakes enhancing the turbulent mixing in specific regions of the wake-vortex near field [42, 43]. As a first example, the effect of delta-shaped spoiler elements has been studied on a four engine large transport aircraft model (scale 1:20) for the high-lift approach configuration, compare Table 1.1. Wind and water tunnel experiments were performed, Fig. 9.28a and b, [39, 44]. Regarding the full-scale case this configuration belongs to the category of a heavy leader aircraft (>136,000 kg take-off weight).

9.6 The Wake-Vortex Hazard: The Problem and Means to Control It

263

The spoiler elements are used to create a highly turbulent flow in the region of outboard flap-vortex shedding, and help the merging of the outboard flap and outboard nacelle vortex. These vortices contribute significantly to the formation of the rolledup, final trailing vortex. The concentrated region of highly turbulent flow, created by the delta-spoiler burst leading-edge vortices and the turbulent wake emanating from the spoiler trailing edge, affects the merging area of the outboard flap and outboard nacelle vortices. It is aimed to enlarge the radial vorticity distribution, also influencing the region of the free circulation center. During the roll-up process, with the merger of the outboard flap and outboard nacelle vortices, the highly turbulent flow is fed into the core region of the main trailing vortex. Relative to the reference case (no spoiler deflection), a marked dispersion of the vorticity field is achieved in the wake-vortex extended near flow field— Fig. 8.1—resulting in a reduction of the axial peak vorticity. A significant expansion of the viscous core by 50–90 per cent occurs accompanied by a reduction in maximum induced velocities of about 50 per cent. The alleviation in the maximum induced rolling moment acting on a follower aircraft of the light weight category ( αbd even small changes of the free-stream properties and the geometry of the wing can lead to an erratic behavior of the wing’s forces and moments. With increasing α the breakdown location moves upstream toward the wing’s tip until a complete destruction of the lee-side vortex system happens.

Fig. 10.20 Flat, sharp-edged delta wing: correlation of the influence of leading-edge sweep angle ϕ0 and angle of attack α on the behavior of the lee-side vortex system [9]

294

10 Small Aspect-Ratio Delta-Type Wing Flow

We note that the breakdown boundary ranges from rather small angles of attack at leading-edge sweep angles of about ϕ0 ≈ 50◦ to α ≈ 33◦ at ϕ0 ≈ 77◦ . Above ϕ0 ≈ 77◦ the lee-side vortex system becomes asymmetric (vortex overlapping), with a strong influence on forces and moments. This phenomenon is visualized in Fig. 7.19. For higher sweep angles the critical angle of attack becomes smaller. With leading-edge sweep and angle of attack outside of the “healthy” domain— without a distance producing fuselage—maximum lift is reduced, roll instability appears for increasing ϕ0 at always smaller angles of attack, even for the symmetric flight attitude (β = 0◦ ), see also Sect. 11.1. The same phenomenon also is present at slender fuselage noses and rocket configurations.

10.2.6 Correlations of Lee-Side Flow Fields Of large interest in particular for the aircraft designer is the knowledge of fundamental dependencies of the lee-side flow field on the wing’s geometry, flight parameters and vehicle attitude. This then leads to a first appraisal of the required aerodynamic properties of the aircraft. Main influencing factors from the side of the wing geometry are the leading-edge sweep angle, the leading-edge nose radius—down to the sharp leading edge—and from the free-stream side the angle of attack and the flight Mach number. A first comparatively coarse correlation was given by A. Stanbrook and L.C. Squire [69]. They recognized that the flow properties at flat, thin wings with high leading-edge sweep depend on the flow relationships normal to the leading edge in terms of the “normal angle of attack α N ” and the “normal leading-edge Mach number M N ”: α N = arctan(tan α/ cos ϕ0 ),

(10.5)

M N = M∞ cos ϕ0 (1 + sin2 α tan2 ϕ0 )0.5 .

(10.6)

The authors of [69] proposed criteria for the existence of lee-side vortex systems, now called the Stanbrook/Squire boundary, Fig. 10.21. For sharp-edged delta wings the relation for that boundary approximately is (α N in degree) (10.7) M N = 0.75 + 0.94 · 10−3 α1.63 N , and for round-edged wings, for α N  6◦ , M N = 0.2 + 0.375 · 10−3 (α N + 23.3)2 .

(10.8)

That above M N ≈ 1 not only attached flow exists, but that, due to shock interferences, also vortex flow develops, was shown by W. Ganzer, H. Hoder and J. Szodruch, [70], see also Sect. 10.7.

10.2 Non-linear Lift Exemplified with the Plain Delta Wing

295

Fig. 10.21 Formation of lee-side vortex systems, the Stanbrook/Squire boundaries [9]: a thin wing/sharp leading edge, b thick wing/round leading edge

We do not intend to give a review of the literature about correlations. In [9] several correlations of experimental data can be found (selection): • Vortex position αV relative to the wing’s lee-side surface as function of the angle of attack α and the lateral position ϕV . • Vortex location αV as function of the angle of attack α. • Correlation of the location of the vortex axis αV with the Reynolds number Re L and the chord thickness. • Vortex-axis angle ϕV /ϕ0 as function of M N for turbulent secondary separation. • Influence of the angle of side-slip β on the lateral position ϕV . In Figs. 10.22 and 10.23 summarizing correlations from [9] are presented of possible flow patterns at sharp- and round-edged delta wings, see also [55]. The limiting curves (hatched) are transition regions between the flow patterns. Deviations up to ±10 percent are possible due to the spectrums of the experimental data. In order to minimize Reynolds-number effects, only experimental data with

296

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.22 Possible flow patterns at wings with sharp leading edges [9]

turbulent secondary separation were included. The indicator M N at the abscissa denotes the upper limit for the formation of leading-edge vortex systems. Again a scatter of up to ±10 percent is possible due to configuration particularities. The MILLER/WOOD legend refers to [71]. Comparing Figs. 10.22 and 10.23, the Mach-number influence with regard to attached flow (0) and vortex development (1) is much stronger for round-edged, compared to sharp-edged wings. Region (2), in which fully developed and stable vortex systems are present, is large for the sharp-edged, and small for the roundedged wings. In the domain of higher angles of attack (3), nearly identical flow patterns are present for both sharp and rounded leading edges. Differences are due to effects of vortex breakdown, viz. adverse pressure gradients at the trailing edge and/or shock-wave interactions. Large differences are present in the regions (4)–(7).

10.2 Non-linear Lift Exemplified with the Plain Delta Wing

297

Fig. 10.23 Possible flow patterns at wings with round leading edges [9]

10.2.7 Flow-Physical Challenges The flow-physical challenges when dealing with lee-side vortex systems of small aspect-ratio delta-like wings are manifold. In research this regards analytical, experimental and numerical investigations, and in design of aircraft with such wings in addition, for instance, the treatment of elastic properties of the whole airframe, surface features, and unsteady flight conditions. Here we look at the challenges, which already some basic flow-field features pose. We follow the considerations of J.M. Luckring and O.J. Boelens in [72], where they look at the phenomena present at the simplest possible lee-side vortex field over a flat blunt-edged swept wing, Fig. 10.24. That was made in view of the phenomena present at a diamond wing with the leading-edge sweep angle ϕ0 = 53◦ . We aim for

298

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.24 The flow-physical challenges of lee-side vortex systems of small aspect-ratio delta-like wings [72]

a broader look at the challenges, and also have added two more challenges, no. 6 and 7. 1. Incipient separation at the blunt leading edge is a phenomenon where the understanding still is insufficient. The flow toward the leading edge evolves at the windward side of the wing from the primary attachment line, see Figs. 7.21 and 7.10. The properties of this flow—streamline pattern and flow momentum, depending to a degree on the shape, in particular the radius of the leading edge—play a role and whether the flow is laminar or turbulent. The most complicated case would be, if laminar-turbulent transition plays a role. Transition is not only governed by the properties of the flow field as such, but also by the surface properties, in particular the surface roughness and also by the level of free-stream turbulence and the thermal state of the surface, see, e.g., [73]. This all must be regarded, and that in particular, if experimental data or flight-test data are in the background of an investigation. 2. The properties of the evolving primary vortex depend on the separation location and the flow properties present there. In any case the vortex is different from a vortex emanating from a sharp leading edge, also whether that is sharp in the sense of the word, or whether it is to be considered as aerodynamically sharp, Sect. 6.4. 3. The secondary vortex is induced by the primary one. It evolves from a separation line—the secondary separation line, in the figure the full line between (3) and (5)—and again it is to ask, whether laminar or turbulent flow is present along this line. Laminar-turbulent transition at some location on this line—embedded transition—would very much complicate the situation. In any case the evolution of the secondary vortex has a repercussion on the primary vortex.

10.2 Non-linear Lift Exemplified with the Plain Delta Wing

299

4. The attached flow field portion of course is very strongly affected by the presence of the primary and also the secondary vortex. A role plays the separation location at the leading edge. Again also possible laminar-turbulent transition comes in. The overall flow field topology moreover plays a role: open or closed lee-side flow field, Sect. 7.4.3. 5. The matter of a possible inner vortex pair was considered above in Sect. 10.2.3. It can be seen as a tertiary or as a second primary vortex pair. A question is, whether it is a phenomenon appearing only at low Reynolds numbers and on top of that is connected to laminar-turbulent transition. 6. The issue of the wind-tunnel environment is not restricted to the question of the free-stream turbulence. In principle it is the test-section environment, which matters. Ideally it should have no influence on the experimental outcome. If it does, numerical investigations of the case—which anyway should be made in the frame of a collaborative approach—must treat the model-in-test-section situation. Aeroelastic deformation under aerodynamic load of both the model and its support must be known and reduced as much as possible. Otherwise a numerical investigation must take it into account, too. 7. Flow unsteadiness has two main aspects regarding the simulation with discrete numerical methods. One is the large-scale massive separation, marked by unsteady vortex shedding, also the unsteady strong shock-wave/boundary-layer interaction, the other is the inherent small-scale unsteadiness of turbulent flow. The second author of this book and J.M. Luckring recently gave a comprehensive review in particular regarding the first aspect—separated flow simulation in view of flight vehicle aerodynamics [63]. Problems of the first aspect cannot be simulated successfully with Model 10 methods, RANS methods, even with unsteady RANS, i.e., URANS methods. For such problems scale-resolving methods, Model 11, appear to be the methods of choice, with regard to computational effort in combination with RANS/URANS methods as hybrid approaches. The second aspect regards the fact that turbulent vortices, as well as turbulent feeding layers, do not have, also like turbulent boundary layers, smooth edges, see the remark on Sect. 1.5. The rugged unsteady edge-flow properties are the source of dynamic loads, vibration excitation, etc. Whether they influence the large-scale flow phenomena over the wing, is not known. In any case, if they have an influence, numerical simulation must employ scale-resolving, Model 11, methods. This listing of flow-physical challenges is not intended to scare off the reader. It is meant to show that a number of challenges still exists. The discussion of several Unit Problems in the following sections shows that these challenges have been identified, and partly have been overcome. The former vortex-flow experiments VFE-1 and VFE-2 and the follow-up AVT task groups had their successes not least because of their combination of experimental and theoretical/numerical work. Such collaborative approaches are the key to further improvement of the understanding of the

300

10 Small Aspect-Ratio Delta-Type Wing Flow

flow physics of lee-side vortex systems and their description for flight-vehicle design purposes.

10.2.8 Manipulation of Lee-Side Flow Fields, an Overview In the shape-definition process of a flight vehicle means to influence the vortex system in relevant flight domains can be of interest [9]. Here we just note a few of them, in Chap. 11 several sections can be found in this regard: • • • •

Wing-planform shaping and optimization. Fuselage forebody strakes. Spanwise blowing over the wing. Geometrical alignment of the wing design.

10.3 Vortical Flow Past the Sharp-Edged VFE-1 Delta Wing—Different Models The first concerted international, now classical approach to the problem of vortical flow past sharp-edged delta wings is the Vortex-Flow Experiment I (VFE-1), the first Unit Problem of lee-side vortices, which we discuss. We have cited reports on VFE-1 in Sect. 10.1.2: [52–54]. A concise look at this type of vortical flow is given in [58]. The VFE-1 configuration is a cropped delta wing with sharp leading edges, short sharp side edges, and nearly flat upper and lower surfaces, the latter except for the support sting, Fig. 10.25. Force and surface-pressure measurements, as well as flow-field surveys, all for a number of flow conditions were performed in wind tunnels worldwide. Summarized are the experimental results in [54]. The general features of the flow field found at a sharp-edged delta wing at geometrical conditions, which lead to a lee-side vortex field, Fig. 10.11, are • flow-off separation at the leading edges, • the resulting vortex layers, containing both kinematically active and inactive vorticity, roll up under eigen-induction, Sect. 3.12.1, • and thus form the primary lee-side vortex pair, • which on the lee side of the wing induces, via ordinary separation, a pair of secondary vortices (even tertiary vortices are possible). • This lee-side vortex system is connected to a strong lee-side surface pressure reduction, the suction pressure, which generates the non-linear lift, see also Sect. 8.4.3.

10.3 Vortical Flow Past the Sharp-Edged VFE-1 Delta Wing—Different Models

301

Fig. 10.25 The VFE-1 configuration [58] Table 10.1 Geometrical and flow parameters of the cropped delta wing example ϕ0 (◦ ) L (m) b (m) M∞ T∞ (K) Tw Re L 65

0.6

0.476

0.4

300

Adiabatic

3.1 · 106

α (◦ ) 9

Selected results of numerical simulations of such flow fields with different flowphysical and mathematical models are now presented. We show and discuss results of discrete numerical solutions of the Euler equations (Model 8, Table 1.3) and of the Navier–Stokes/RANS equations (Model 9 and 10), the latter with two different turbulence models. The geometrical and flow parameters are given in Table 10.1. For this sub-sonic case we find the normal angle of attack to be α N = 20.5◦ and the normal Mach number to be M N = 0.18. For these numbers the correlation in Fig. 10.22 yields that, region (2), a stable and fully developed leading-edge vortex system is to be expected. Vortex breakdown is expected to happen around α = 20◦ , Fig. 10.20, an angle much higher than the present one. The results presented in Fig. 10.26, generated by W. Fritz at EADS, München, Germany, show some of the manyfold aspects—and pitfalls—to be considered when using discrete numerical methods.

302

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.26 VFE-1 configuration without support sting: surface-pressure coefficient c p (η) at x/c = 0.6 (above) and 0.8 (below), experimental data with support sting [58]. Influence of describing equations (Euler, RANS), grid resolution and turbulence model

In the upper left and the lower left part of the figure we immediately note that the— grid-independent—Euler solution yields the (primary) suction peak more outboard and much higher than that experimentally obtained. At the very leading edge we also see a much higher value. The secondary separation and hence the secondary suction peak, of course, is not present in the Euler solution. The RANS solutions give an acceptable agreement with the experimental data for the 193×129×81 “fine” grid, for which grid-independence can be assumed. The primary suction peak agrees well with that found in the experiment. At the location of the secondary vortex the suction peak is much more pronounced than in the experimental data. Two different turbulence models have been employed: (a) the Baldwin–Lomax model with the Degani–Schiff modification, and (b) the Wilcox k − ω model. The computations were made fully turbulent, i.e., possible laminar flow portions were neglected.

10.3 Vortical Flow Past the Sharp-Edged VFE-1 Delta Wing—Different Models

303

In the upper right and the lower right part of Fig. 10.26 the results of the simulations with these two models are shown. The differences are small, but visible and significant. At the lower side of the wing all results are nearly identical. The differences to the experimental data are due to the influence of the support sting, neglected in the computations. Interesting differences become evident when looking at the computed totalpressure loss and the eddy viscosity at the location x/c = 0.8, Figs. 10.27 and 10.28. With both turbulence models the computed total-pressure loss indicates well the boundary layers at the upper and the lower side of the wing, and the primary and the secondary vortex at the upper side. In Fig. 10.28 with the Wilcox model the contours are somewhat fuller than in Fig. 10.27. There the secondary vortex appears to be larger and even a tertiary vortex seems to be indicated. The feeding layers, at least of the primary vortex, are indicated only weakly. The picture is quite different when looking at the computed eddy viscosity, each in the lower part of the figures. With the Baldwin–Lomax model the eddy viscosity appears, with limitations, only in the boundary layers, whereas in the vortex area it is barely indicated. The picture is quite different with the Wilcox model, Fig. 10.28. There all appears to be well pronounced, inboard of the vortex area, however, exaggerated.

Fig. 10.27 Baldwin–Lomax model: total-pressure loss (upper part) and eddy viscosity (lower part) at x/c = 0.8 [58]

304

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.28 Wilcox k − ω model: total-pressure loss (upper part) and eddy viscosity (lower part) at x/c = 0.8 [58]

These results tell us two things regarding the flow in the non-linear lift domain of sharp-edged, low aspect-ratio lifting wings, here delta-type wings: • Turbulence modeling in RANS (Model 10) solutions is of large importance. A sufficiently exact simulation of the crucial vortex phenomena may be possible in certain cases, in the present case (Unit Problem) to a degree even with the Baldwin– Lomax model. If complex vortex ensembles, typical for aircraft configurations, are present, a careful investigation is necessary in order to choose an adequate turbulence model. If laminar-turbulent transition is present, in general and in particular embedded along attachment lines, it must be determined, whether it has a non-negligible influence on the important flow features or not. If yes, then the topic of transition prediction becomes important, if not, this is a “lucky situation”. • In the present case, the two RANS solutions—also the one with the Baldwin– Lomax Model even with a very erroneously computed eddy viscosity—yield rather good results compared to the experimental data. Regarding the overall forces and the pitching moment, even the Euler (Model 8) solution gives results with discrepancies, which are acceptable in predesign work, see the next section. In that section we also discuss in detail the flow field past the VFE-1 wing. The reason for this reasonably good Euler solution is that to first order the properties of the inviscid flow field are the deciding ones, not the properties of the viscous flow, the boundary layers. In Sect. 4.3 we have shown for large aspect-ratio wings, how the appearance of lift can be understood: the second break of symmetry. We have seen that at angle of attack overall a shear—representing the kinematically

10.3 Vortical Flow Past the Sharp-Edged VFE-1 Delta Wing—Different Models

305

active vorticity—is present in the streamline pattern of the external inviscid flow, the upper and lower wing flow-fields shear. This shear, although it cannot always be shown in the way as it is possible for the large aspect-ratio wing, is present at any lifting wing, also in the present case. It appears at the trailing edge and also at the leading edges. In the following section we show this for the VFE-1 configuration in detail and give proof, that the mechanism, which leads to the lee-side vortices, is the same that leads to the trailing vortex layer of large aspect-ratio wings.

10.4 Creation of Lift in the Euler Solution (Model 8) for the Sharp-Edged VFE-1 Delta Wing—Proof of Concept With this second Unit Problem we give proof that at non-linearly lifting sharp-edged delta wings the generation of lee-side vortices, hence the generation of vorticity and the rise of entropy, follows the same mechanism as we have seen at the trailing edge of lifting large aspect-ratio wings, Sect. 8.4. Moreover, this is a proof that discrete Euler (Model 8) solutions of such cases in principle are viable solutions. We consider the flow past the small aspect-ratio VFE-1 delta wing, which we already have treated in the last section. Figure 10.29 gives the coordinate convention. The results were obtained in the frame of the doctoral thesis of R. Hentschel—a doctoral student of the first author of this book—on three-dimensional self-adaptive grid generation [51, 74]. Except for one figure, all figures in this section are from that thesis.

Fig. 10.29 Planform of the considered sharp-edged cropped delta wing [51]. The lengths are nondimensionalized with the length of the wing, L = 0.6 m

306

10 Small Aspect-Ratio Delta-Type Wing Flow

10.4.1 The Computation Case and Integral Results The case goes back to the International Vortex-Flow Experiment on Euler Code Validation (VFE-1). The wing is the AFWAL configuration of NLR [75], see also Fig. 10.25. The geometrical and flow parameters for this transonic case are given in Table 10.2. The results discussed in the previous section were obtained with different parameters. Basically it was one of the objectives of the doctoral work [51] to show how the vorticity and the lee-side vortices arise in a discrete modeled Euler (Model 8) solution. The creation of the vorticity was studied by employing the local vorticitycontent concept, Chap. 4. A Navier–Stokes/RANS solution (Model 9/10) was made for comparison. The Baldwin–Lomax turbulence model was used, however only at the upper (leeward) side of the wing, due to indications from the experiment. The flow at the lower (windward) side was treated as laminar flow. The combination of the two characteristic parameters, the normal angle of attack α N = 22.64◦ and the normal leading-edge Mach number M N = 0.38, is such that in this case stable and fully developed leading-edge vortices as well as embedded cross-flow shocks can be expected, according to the correlations given in Sect. 10.2.6. Vortex breakdown for the present leading-edge sweep and angle of attack is of no concern, because it is expected to happen around α = 20◦ , Fig. 10.20. That the results of the Euler and the Navier–Stokes/RANS simulations are reliable is demonstrated in Table 10.3 with the comparison of experimental and computed force and moment coefficients. The data are selected from the detailed data given in [51]. The maximum deviations are around 10 per cent. We further show computed and measured surface pressure coefficients in the cross-section at the location x = 0.6, Fig. 10.30. The data at this location are more or less representative for the whole wing.

Table 10.2 Geometrical and flow parameters of the cropped delta wing example [51] ϕ0 (◦ ) L (m) b (m) M∞ T∞ (K) Re L α (◦ ) Tw 65

0.6

0.476

0.85

300

Adiabatic

9 · 106

Table 10.3 Measured and computed force and moment coefficients [51] Source CL CD CM Experiment NLR Euler simulation Navier–Stokes/RANS simulation

0.4573 0.5022 0.4765

0.0846 0.0831 0.0890

–0.271 –0.3030 –0.2846

10

L/D 5.405 6.047 5.354

10.4 Creation of Lift in the Euler Solution (Model 8) ...

307

Fig. 10.30 Comparison of computed and measured distributions of the pressure coefficient c p (y/s) at the cross-section at x = 0.6 [51]. Open and full circles are experimental data, the full lines are the interesting computed ones. Left: comparison with the Euler (Model 8) solution, right: comparison with the Navier–Stokes/RANS (Model 9/10) solution

At the lower side of the wing we see a very good agreement of the computed and the measured pressure coefficients. At the upper side we find the pressure distribution typical for Euler simulations of such flow fields: the pressure peak is too large and its location is too much to the outboard. All this is due to the missing secondary vortex. Well discernible is the cross-flow shock. The agreement of the Navier–Stokes/RANS data with the experimental data is much better. The suction peak nearly has the right position. The peak value also is too high, but not as much as that of the Euler simulation. The shock wave has a better resolution. The data show at the first glance, why lift, drag and the moment found with the Euler solution are in a rather reasonable agreement with the experimental data. At the lower side the surface pressure fields show a very good agreement, at the upper side the deviations in the Euler data compensate each other to a degree. If this is the case, in conceptual design work Model 8 simulations can be used without too big risks. This, however, only holds for a given configuration class, which can be very narrow. We ask now what is to be expected regarding the flow field found with the Euler simulation, in particular regarding the appearance of the lee-side vortices.

308

10 Small Aspect-Ratio Delta-Type Wing Flow

10.4.2 Details of the Computed Flow Field In the reality (Model 1) besides the primary lee-side vortex pair generally a secondary vortex pair is present, each vortex pair with the associated feeding (vortex) layers. (Even tertiary vortices can appear.) This feature cannot be expected to be found in an Euler simulation (Model 8). Figure 10.31 visualizes this schematically with crosssection views of two virtual flow realizations for an open lee-side flow field. At the left side the situation present in reality is sketched, at the right side that to be expected from the Euler simulation. Possible cross-flow shocks are not indicated. A detailed picture of the flow field found with the present Euler simulation is given in Fig. 10.32. It regards the cross section at x ≈ 0.86—just at the beginning of the straight side edge—in the span interval 0.23  2y/b  0.42. The side edge of the wing is located at 2y/b = 0.3964. Shown in the upper left part of the figure is the final self-adapted grid. The grid was adapted in the computational space with the entropy gradient |∇s|. This sensor yields very good results in the recognition of boundary layers, vortex layers, contact discontinuities and shock waves. We see that, beginning at the sharp side edge, the trace of the vortex layer (feeding layer of the lee-side vortex) is well captured (see also the iso-lines of the total-pressure loss, lower right figure). The vortex layer first curves away from the edge and then comes around toward the center of the wing. The core area of the lee-side vortex lies at about 80 per cent half-span. The grid there is refined down to the wing’s surface. Located in this domain is also the expected cross-flow shock. The vortex layer is well discernable too in the Mach number plot (lower left) and in the total-pressure loss plot (lower right). A spiraling of the vortex layer is not observable. At 2y/b ≈ 0.251 the total-pressure loss plot indicates the location of the secondary attachment line A2 , Fig. 10.33, right side. The—inviscid—surface flow at 2y/b  0.251 is in outward direction, showing a vortex-layer structure.

Fig. 10.31 Schematics, not to scale, of vortex systems (open lee-side flow fields) in Poincaré surfaces of delta wings at higher angles of attack (view from behind onto the right-hand side of the wing). Left: perceived reality (Model 1) as well as expected result of Navier–Stokes/RANS simulations (Model 9/10), right: expected result of Euler simulations (Model 8)

10.4 Creation of Lift in the Euler Solution (Model 8) ...

309

Fig. 10.32 Cross section of the flow field at x ≈ 0.86 [51]. Upper left part: the self-adapted grid; upper right part: iso-pressure coefficient lines (–1.6  c p  0.02); lower left part: iso-Mach number lines (0.7  M  2.02); lower right part: iso-lines of total-pressure loss (0  1 − pt / pt∞  0.35)

The cross-flow shock is located at 2y/b ≈ 0.35. It lies below the lee-side vortex and does not appear to extend far upwards. Outward its location no vortex-layer structure is visible. Hentschel notes that details of the computed flow pattern can depend much on the grid’s topology and fineness. Surface streamlines of the computed inviscid flow field are shown in Fig. 10.33. (Note that we show only one half each of the lower and the upper side.) The primary attachment lines A1 at the lower side of the wing have a quasi-conical appearance. They are located close to the leading edges. Between them the flow is only nearly two-dimensional because the lower side of the wing is not fully flat.12 In the small domains between the primary attachment lines and the leading edges the flow is in outward direction with slightly curved streamlines. The primary separation lines S1 lie at the very leading edges, because these are sharp. We have flow-off separation there. We have indicated the location of the primary separation line S1 at the sharp leading edges. The lee-side vortex leads to a secondary attachment line A2 . Between the secondary attachment lines we see a more or less two-dimensional flow toward the wing’s trailing edge. This points to an open lee-side flow field. The secondary attachment line tapers off at about 90 per cent of the wing’s length. This indicates that there the lee-side vortices lift off from the surface. 12 A more or less truly two-dimensional pattern inboard of the primary attachment lines is only to be expected at the fully flat windward side of a configuration. Such a pattern is always desired in view of the onset flow into inlets of propulsion systems and over aerodynamic trim and control surfaces.

310

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.33 Surface streamlines of the Euler simulation [51]. Left: lower (windward) side, right: upper (leeward) side (open lee-side flow field). Primary attachment line: A1 , primary separation line (sharp leading edge): S1 , secondary attachment line: A2 . The flow-off directions at the leading edges and the sharp side edges are denoted by ν

Between A2 and the leading edge with S1 the surface flow is directed outboard. The cross-flow shock, Fig. 10.32, leads to a kink in the surface streamlines. At the leading edge finally the flow-off from the upper surface evidently has a larger angle than at the lower side: ν2 > ν1 . The resulting angle ν2 − ν1 between the flow at the upper and the lower side is the leading-edge flow shear angle ψeL E . In Fig. 10.37 we come back to the shear-angle distribution. At this point we have to remark that it has not been possible to explain the flowfield situation as we did it for the lifting large aspect-ratio wing, Sect. 4.3.2. There we could show that a shear exists between the inviscid flow fields at the pressure and the suction side of the wing, the upper and lower flow-fields shear. The break of symmetry of the surface streamline pattern (the second break) then could be seen as the origin of the wing’s lift. Back to our wing: at the side edge of the wing the flow-off angles are the other way around: ν4 < ν3 . This means that the side-edge flow shear angle ψeS E is negative, Fig. 10.37. Hence at the wing’s straight side edges each a counterrotating secondary vortex develops. (This is not to be confused with the secondary vortices arising below the primary ones, if viscous flow is considered.) The phenomenon of counterrotating vortices at straight side edges was reported also in [76], where numerical and experimental results are presented. At the inner part of the trailing edge the shear angle ψe is nearly zero. This means that there only little kinematically active vorticity content is leaving the wing’s trailing edge. Regarding the flow field at and downstream of the trailing edge of delta wings see also Sect. 3.12.3. From the Navier–Stokes/RANS solution in [51] we discuss only the skin-friction line pattern at the wing’s lower and upper surface, Fig. 10.34. At the lower side—left

10.4 Creation of Lift in the Euler Solution (Model 8) ...

311

part of the figure—the pattern is quite similar to that found with the Euler simulation. The primary attachment line A1 again lies close to the leading edge. The skin-friction lines toward the leading edge and in the domain between the primary attachment lines, however, show a pattern slightly different to that given by the Euler solution in Fig. 10.33. As long as no fundamental difference exists between the Euler and the Navier–Stokes/RANS simulation, this must be expected. Even if the boundary-layer flow is only weakly three-dimensional, the skin-friction line pattern in principle is different from the streamline pattern of the appendant external inviscid flow [73]. At the upper side of the wing again we have got an open lee-side flow field. The secondary attachment line A2 has a location quite similar to that of A2 found with the Euler simulation. However, it tapers off much earlier, which seems to indicate that the lift-off of the lee-side vortex happens earlier than with the Euler simulation. At this time, though, we must allow for the possibility that this result is due to the applied turbulence model. Anyway the question arises whether turbulent attachment happens along A2 or laminar-turbulent transition. (The influence of laminar or turbulent flow on the strength of the lee-side vortex pairs of delta wings is discussed in the Concluding Remarks, Sect. 10.8.) If we accept the solution in this regard, we observe now the fundamental difference to the Euler result: the—primary—lee-side vortex pair induces a pair of small secondary lee-side vortices. The feeding layer leaves the surface at the secondary separation line S2 . Of course then between S2 and the primary separation line S1

Fig. 10.34 Skin-friction lines of the Navier–Stokes/RANS simulation [51]. Left: lower (windward) side, right: upper (leeward) side. Nomenclature see Fig. 10.33

312

10 Small Aspect-Ratio Delta-Type Wing Flow

at the sharp leading edge the—tertiary—attachment line A3 must be present. The flow from the tertiary attachment line toward the primary separation line S1 at the sharp leading edge is well discernible. The presence of a cross-flow shock as well of counter-rotating side-edge vortices cannot be deduced from the skin-friction line pattern.

10.4.3 The Circulation and the Kinematically Active Vorticity Content in the Euler Simulation We give a very compact summary of the results of [51] regarding the connection of the circulation and the kinematically active vorticity content in the Euler simulation (Model 8) of the flow past the cropped delta wing. The circulation Γ in the lee-side vortices of the wing is found by the integration of the vorticity in surfaces F normal to the longitudinal axis—the x-axis—of the wing (theorem of Stokes, Sect. 3.3):  

 Γ =

v ds =

ω d F.

(10.9)

The result is given with the upper curve in Fig. 10.35. The circulation Γ increases along the x-axis—represented here by the (minus) i-coordinate—due to the vorticity fed into the flow field by the two vortex layers, which originate at the sharp leading edges of the wing. The circulation increases until the straight side edge is reached (location 3). There the secondary vortices due to the straight side edges, which counter-rotate to the primary vortices, introduce negative circulation. Hence the total circulation decreases until the constant value behind the wing is reached (left of location 2 at i = 20). That value is only approximately constant, because of the coarse grid there, see in this regard also Sect. 5.3. Regarding the vorticity content we first consider locally the geometry of the vortex layer, which leaves the sharp leading edge of the wing, Fig. 10.36. Sketched is an element of that layer with the width ds. The layer has the thickness dn. The grey surface lies approximately parallel to the wing’s surface, cutting the vortex layer at right angles. Indicated on it is the decomposed wake structure in the idealized reality, Fig. 4.12, right part. At the right in Fig. 10.36 the broken line, designated “inviscid shape”, would be the uniform u(z)-profile of the idealized inviscid flow. Together with the other profile it represents the ideal Euler wake shown in Fig. 5.1, right part. Due to the properties of the present vortex layer, which are more complex than those of the shear layers treated in Chap. 4, the vorticity content was found with the integration of the vorticity  Ω=

ω dn

(10.10)

10.4 Creation of Lift in the Euler Solution (Model 8) ...

313

Fig. 10.35 Circulation Γ , its derivative dΓ /di and the vorticity content Ω ds (circles) found with the Euler solution [51]. The flow is from the right to the left. Note that the coordinate i decreases in downstream direction

Fig. 10.36 Schematic of a sharp leading edge with emanating vortex layer [51]

314

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.37 Leading-edge flow shear angle ψ (= ψe ), vorticity content Ω/|U | and the absolute value |U | as function of the location i [51]

in the following way. In the case of the Euler wake ω results from the vortex-resembling velocity profile lying in the grey surface element cutting the vortex layer in Fig. 10.36. The change of the circulation between location i 2 and i 1 is then connected to the integral of the vorticity in that surface element according to the compatibility condition Eq. (4.20): dΓ = di

  ω dn ds.

(10.11)

In Fig. 10.35 the result is shown. Note that the coordinate i decreases in downstream direction, hence the derivative dΓ /di initially goes negative. Negative is also ds. The wiggly development of Γ , reflected in dΓ /di, is due to different refinement levels of the grid. Nevertheless, the agreement between dΓ /di and Ω ds is satisfactory. Close to the maximum of Γ the vorticity content decreases and the sign of dΓ /di changes properly. The general development and the minimum of dΓ /di are well reproduced by Ω. The compatibility condition, Eq. (4.20), holds for sub-critical flow, i.e., shock-free flow with the same absolute value of the velocity vector at the upper and the lower side of the leading edge. In our case a cross-flow shock is present at the upper side of the wing. Its influence on the velocity at the leading edge appears to be relatively weak as Fig. 10.37 indicates. The values 2 tan ψ/2 and Ω/|U | are in satisfactory agreement. Along the swept leading edge the leading-edge shear angle results to ψe ≈ 30◦ to 40◦ . At the location of the straight side edge the angle becomes negative, pointing to the presence of the side-edge counterrotating vortex. This short summary in particular shows how in the discrete modeled Euler solution (Model 8) for the sharp-edged delta wing the creation of the lee-side vortex pair can

10.4 Creation of Lift in the Euler Solution (Model 8) ...

315

be understood. A kinematically active vorticity content is introduced into the flow by the shear of the flow directions at the upper and the lower sides of the sharp leading edges. The two resulting vortex layers curl up by self induction and are forming the pair of lee-side vortices. These then are responsible for the non-linear lift increment. With ordinary separation at regular surfaces the introduction of kinematically active vorticity content into the separating shear layer cannot be demonstrated as simply as in this case. In principle, however, the mechanism should be the same.

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing In the year 2001 D. Hummel and G. Redeker proposed a new vortex-flow experiment, the VFE-2 [58]. The configuration was to be chosen such that the following flow regimes were covered: (i) (ii) (iii) (iv)

Attached flow without vortex formation (0◦  α  4◦ ). Separated vortical flow without vortex breakdown (4◦  α  20◦ ). Separated vortical flow with vortex breakdown (20◦  α  40◦ ). Separated deadwater-type flow (40◦  α  90◦ ).

10.5.1 The Wing and the Subsonic Computation Case An appropriate configuration was a plain delta wing with a leading-edge sweep of ϕ◦ = 65◦ . The wing should have a flat inner portion and interchangeable leading edges. These demands were fulfilled by the NASA NTF delta wing configuration, which is shown in Fig. 10.38. The aspect ratio of the wing is 1.85, three rounded and one sharp leading edge were available. The general aim of VFE-2 was to provide flow-field-data for comparisons with numerical results [77]. Experimental topics were laminar-turbulent transition, surface-pressure measurements in the onset-of-separation domains of configurations with round leading edges, boundary-layer measurements, skin-friction line patterns in view of the determination of secondary and tertiary separation lines, flow-field measurements regarding the primary and secondary vortices (components of velocity and vorticity, turbulent energy and eddy viscosity), vortex-breakdown flow fields including surface-pressure fluctuations caused by the spiral mode of breakdown. Numerical topics were validation and improvement of the existing codes, codeto-code comparisons on common structured and unstructured grids, investigations of turbulence models in RANS (Model 10) simulations, assistance related to the setup and evaluation of new wind-tunnel experiments, synergistic effects through test runs on the simple VFE-2 configuration prior to full-scale aircraft investigations.

316

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.38 The NASA NTF delta-wing configuration [59]. Mean aerodynamic chord c = 2/3· c R , 1 in. = 2.54 cm

The work on VFE-2 took place from 2003 to 2008 within the RTO Task Group AVT-113. In the special issue “VFE-2” of Aerospace Science and Technology,13 edited by D. Hummel and R.M. Cummings, in eight papers reports on that work were given. Summarizing articles are—besides others—those by J.M. Luckring and D. Hummel [78], and W. Fritz and R.M. Cummings [79]. We discuss selected results of the investigations. The flow parameters are given in Table 10.4. Note that experimental parameters can be slightly different. The normal angle of attack and the normal leading-edge Mach number of this case are α N = 29.22◦ and M N = 0.188. This case hence lies in the regime (2) “development of the vortex system” in Fig. 10.23 and therefore is covered by the experimental data underlying the correlations there. The critical angle of attack regarding vortex breakdown is α = 20◦ , i.e., well above of that of the present case.

Table 10.4 Flow parameters of the subsonic VFE-2 case M∞ Re∞,c c (m) T∞ (K) Twall 0.4

13 Volume

3 · 106

0.4358

297.4

Adiabatic

24, Issue 1, pp. 1–294 (January−February 2013).

α (◦ )

Viscous flow

13

Fully turbulent

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

317

10.5.2 Two Pairs of Primary Vortices Experimental investigations at DLR Göttingen with the parameters given in Table 10.4 yielded for the medium-radius rounded leading edge an interesting picture of vortex flow: co-rotating vortices, i.e., a “doubled primary” vortex system [80]. In the upper part of Fig. 10.39 the surface pressure is given, obtained with Pressure Sensitive Paint (PSP) measurements, together with the velocity field, found with Particle Image Velocimetry (PIV) measurements in cuts at x = const. locations [81]. In the lower part the result of a numerical simulation (RANS with the k-ω model) of the flow field at EADS Munich is in a convincing agreement with the experimental result [82]. The figure gives total-pressure loss contours at x = const. locations. Regarding the effect of laminar-turbulent transition in the actual flow field see the concluding remarks in Sect. 10.8. Overall the experimental and the numerical simulation give the following picture: at about x = 0.4 an inner vortex pair appears, at about x = 0.5 an outer vortex pair appears, around x = 0.7 both vortex pairs appear to have the same size, further downstream the outer vortex pair becomes stronger, as is to be expected, whereas the inner pair decays. A look at the computed skin-friction line pattern provides deeper insight, Fig. 10.40. The RANS computation was made with DLR’s TAU code, fully turbulent, employing the Spalart–Allmaras turbulence model [83]. We note that the computation was made with a slightly larger angle of attack than before: α = 13.3◦ . The figure gives the top view at the left-hand side of the wing. The free-stream is in direction from the left lower corner of the figure upward along the center line of the wing. In Sect. 10.2.4 we have noted that at a wing with round leading edge the formation of the (primary) lee-side vortex begins at the rear of the wing and with increasing angle of attack moves forward toward the wing’s apex. In our case the angle of attack is such that the primary vortex originates at about 45 per cent of the wing’s length. Hence at the apex of the wing and downstream of it up to x/c R ≈ 0.4 the pattern of the skin-friction lines indicates that the flow from the lower side passes without separation around the round leading edge to the upper side of the wing.14 In the VFE-2 literature this is attributed to the rather large value of the cross-flow bluntness   parameter pb = r L E /b , with b being the local span, Sect. 6.4. In our case at x/c R = 0.1, for instance, we have for the medium-radius leading edge pb = 0.0107. For this value the inverse thickness ratio of the flattened elliptical cylinder of our gedankenexperiment in Sect. 6.4 is δ = 46.7. Figure 6.4, right part, shows for δ = 50 a peak value c p = −2,500 at ϕ = 90◦ . The magnitude of the adverse pressure gradient, Fig. 6.5, right part, is dc p /dϕ = 1,471 at ϕ = 90.7◦ behind the ‘leading edge’. At x/c R = 0.7 we have for the medium-radius leading edge pb = 0.00153. The inverse thickness ratio then is δ = 326.48. Figure 6.4, right part, now shows for 14 This is similar to the flow pattern at the blunt delta wing, Sect. 10.7: the laminar flow goes around the blunt leading edge and separation and vortex formation happens at the upper side of the wing.

318

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.39 Pressure (surface color), velocity (vectors), and vorticity (vector color) distributions of the VFE-2 configuration with medium-radius rounded leading edges. Upper part: PSP and PIV measurements [81]. Lower part: numerical solution [82]

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

319

Fig. 10.40 Surface pressure and skin-friction lines at the left upper side of the VFE-2 delta wing [83]. The dashed red lines indicate the locations of the vortex axes

δ = 300 a peak value c p = −90,000 at ϕ = 90◦ . The magnitude of the adverse pressure gradient is dc p /dϕ = 278,940 at ϕ = 90.2◦ , behind, but very close to the ‘leading edge’. If we project these results on our case, we see that with constant radius r L E the leading edge at the front part of the wing indeed acts as the round one, which it is.  But at the rear of the wing the increasing b leads to an effective decrease of the bluntness parameter. The leading edge becomes relatively sharper, fixing there the primary separation location like the aerodynamically sharp leading edge does. The flow around the effectively blunt leading edge is present up to x/c R ≈ 0.4, however, at x/c R  0.2 apparently with a small bubble separation directly at the leading edge, where a thin strip of blue color indicates low pressure. In any case up to x/c R ≈ 0.4 the skin-friction lines first point away from the leading edge and then curve around into a direction first parallel to the leading edge and then toward it. At the same time from the center line of the wing skin-friction lines converge toward those coming from the leading edge eventually forming a pattern of converging lines, which typically is found upstream of open-type separation, see for example Fig. 7.21 in [73]. All that can be considered as the domain of incipient separation.

320

10 Small Aspect-Ratio Delta-Type Wing Flow

At x/c  0.45 on the wing’s surface the outer broken red line depicts the location of the center of the outer primary vortex. Below it we see in blue the induced low pressure domain. The outer primary vortex originates immediately at the leading edge. When following a path from the leading edge of the wing toward its center line, we find a succession of singular lines, however, being not in all cases unambiguous: • • • • • • •

the primary separation line (barely visible), S1 , leading to the outer primary vortex, the primary attachment line, A1 , the secondary separation line, S2 , the secondary attachment line, A2 , the tertiary separation line, leading to the inner primary vortex, S3 , the tertiary attachment line, A3 , the fourth separation line, S4 .

Figure 10.41 schematically shows at x/c R ≈ 0.7, with some reservations, the flowfield in the Poincaré surface. All indicators point to an open lee-side flow field. At the  lower side of the wing A0 denotes the (left-hand) attachment line, and at the upper side F1 the outer primary vortex, F2 its secondary vortex, and F3 the second (inner) primary vortex. The attachment lines A show up as quarter-saddle points S  at the lower side of the wing and as half-saddle points S  at its upper side. We have not sketched a possible secondary vortex due to the inner primary vortex, because we see no clear indication for that. Neither the skin-friction line pattern nor the surface pressure distribution point to a secondary inner vortex. To check the topological structure shown in Fig. 10.41, we apply Rule 2’ from Sect. 7.4. Of course we also have to take into account the singular points at the other side of the wing. Rule 2’ then reads

Fig. 10.41 Left side of the wing: topological schematic of the open lee-side flow-field in the Poincaré surface at about x/c R ≈ 0.7. The view is in positive x-direction. One prime denotes a half-saddle point, two primes denote a quarter-saddle point

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

321

Fig. 10.42 Surface pressure and streamlines (particle traces) at the left upper side of the VFE-2 delta wing [83] 

N+

     1  1   1   13 2 S+ N − S + S = (6 + 0) − 0 + = −1, + 2 2 4 2 4

(10.12) telling us that the topological structure is a viable one. The formation of the inner primary vortex in this case to a degree seems to be coupled to the formation of the outer primary vortex, Fig. 10.42. Particle traces from within and from above the boundary layer show that, see also Sect. 10.8. The convergence of the skin-friction lines, noted above, is seen here to lead to a thickening of the boundary layer ahead of the separation area. This of course indicates a rise of the kinematically active and inactive vorticity content. The particle traces, as to be expected, mainly enter the inner primary vortex, a few of them also the outer one. Downstream of this area, due to the decreasing bluntness parameter, the outer primary vortex strongly is fed with kinematically active vorticity, which is not the case for the inner primary vortex. Hence downstream the latter decays due to the always present viscous effects. An experimental investigation of a larger VFE-2 configuration at a lower Reynolds and Mach number than that discussed here was performed by A. Furman and the third author of the present book [84]. We shortly discuss an aspect of that case. The flow parameters are given in Table 10.5.

322

10 Small Aspect-Ratio Delta-Type Wing Flow

Table 10.5 Flow parameters of the selected low sub-sonic case VFE-2 case of [84] M∞ Re∞,c c (m) T∞ (K) Twall α (◦ ) Viscous flow 0.14

2 · 106

0.98

292.5

Adiabatic

13

lam./turb. transition

Whereas in the case discussed above possible bubble-type laminar-turbulent transition was present at x/c R ≈ 0.3, [78], in the present case at the upper side of the wing laminar flow was observed from the apex downstream to x/c R ≈ 0.3, Fig. 10.43. Again an inner and an outer primary vortex is observed (right-hand side of the wing). The inner vortex originates from ordinary laminar separation (blue color). Further downstream, after laminar-turbulent transition of the boundary layer has happened, the inner vortex is fed by turbulent flow (red color). The outer primary vortex is sketched as initially originating via laminar, but then mainly via turbulent separation. Basically we see the same skin-friction line pattern and the same open lee-side flow field as shown in the Poincaré surface in Fig. 10.41. The outer primary separation line S1 lies well at the upper side of the wing.

Fig. 10.43 Surface oil-flow pattern (left) and sketch of the flow topology (right) of the VFE-2 configuration with medium-radius rounded leading edges [84]

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

323

10.5.3 Vortex Breakdown In Sect. 10.2.5 we have shown that limits exist of wing slenderness and angle of attack because of the possibly occurring phenomena of vortex breakdown and vortex overlapping. Vortex breakdown was an important topic of the VFE-2 work. Following the summary given in [78], as well as the discussions in other VFE-2 papers, we give a short overview. Important first of all is the observation that at high angles of attack the flow past the wing may become unsteady. This, of course, has consequences for both experimental and numerical investigations. Regarding the latter, we note that unsteady Reynoldsaveraged Navier–Stokes (URANS) solutions (Model 10) should be employed, possibly also scale-resolving methods like Large-Eddy Simulation (LES) as well as its derivatives Detached-Eddy Simulation (DES) and Delayed Detached-Eddy Simulation (DDES), all belonging to the class of Model 11 methods. – Subsonic Case, M∞ = 0.4 For the VFE-2 configuration with the leading-edge sweep angle ϕ0 = 65◦ the correlation in Fig. 10.20 yields the critical angle of attack αbd ≈ 20◦ .15 Vortex breakdown begins in a region of strong streamwise deceleration, located commonly at the wing’s trailing edge. With increasing angle of attack its onset moves upstream. At the VFE-2 wing vortex breakdown of spiral type, Sect. 3.13, of the outer primary vortex was observed. In the subsonic domain (M∞ = 0.4) it began around α = 20◦ . For the sharp and the medium-rounded leading edge it had progressed over the whole wing already at α = 23◦ . In a numerical study—RANS with the k-ω turbulence model—by W. Fritz it was found that weak vortex breakdown can appear at a lower angle of attack than αbd ≈ 20◦ [82]. The computation was made for the medium rounded leading edge at a higher Reynolds number than that given in Table 10.4, viz. Re∞,c = 6 · 106 , the angle of attack was α = 18◦ . The outer and the inner primary vortex were found to be located close to the apex (again with coupled origins), the secondary vortex is not visualized, Fig. 10.44. The inner primary vortex appears to be weak with little influence on the pressure field. The core of the outer primary vortex is compact up to x/c R ≈ 0.8. Downstream of this location a spiral-type vortex breakdown is present. Behind the wing’s trailing edge the vortex appears to be re-configured, Sect. 3.13. The pink-colored bubble represents an iso-surface of zero axial (x-direction) velocity. Inside the bubble the axial velocity is negative, which is an indicator of vortex breakdown, being in this case a weak one. Ahead and behind the pink bubble the axial velocity is positive. At the wing’s thick rounded trailing edge, Fig. 10.38, the pink color indicates ordinary separation, the kink at the outside probably being due to the secondary vortex, which locally suppresses separation. The inner primary vortex appears to be healthy throughout. 15 Note that the VFE-2 wing in this case has a blunt trailing edge, Fig. 10.38, whereas the correlation

is based on wings with sharp trailing edges. The result hence must be seen with a certain reservation.

324

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.44 Computed flow field at α = 18◦ . Right-hand side of the wing: surface pressure, left-hand side: stream traces with 3-D volume ribbons [82]

Why was vortex breakdown found at a lower angle of attack than expected? The author attributes it to uncertainties regarding the location and the influence of laminarturbulent transition and to deficits in the turbulence modeling, in his simulation being the k-ω model. The comparison with experimental data, not shown here, reveals differences in the flow-field data at the front of the wing, whereas the agreement is good at the rear of the wing. In any case, Fig. 10.44 tells us why vortex breakdown is of concern for the aerodynamic design of flight vehicles with delta wings. Vortex breakdown usually begins at the rear of the wing, in our case at around 70 per cent of the wing’s length. The suction effect of the lee-side vortex system—low surface pressure—and with that the non-linear lift, is reduced. Hence a loss of lift occurs. Critical is that this only happens at the rear of the wing. Therefore the lift loss goes together with a pitch-up of the wing, which can be rather strong. If the breakdown occurs asymmetrically, a sideslip and a rolling moment results. If with increasing angle of attack the breakdown has moved up to the wing’s apex, the non-linear lift has gone completely. This all may go together with effects of unsteadiness. – Transonic Case, M∞ = 0.85 In the summarizing paper [78] it is discussed how in a sense premature vortex breakdown in the transonic regime (M∞ = 0.85) can be understood. The onset of breakdown was found to happen suddenly at some place in the middle of the wing. It possibly was triggered by an interaction of the vortex

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

325

Table 10.6 Flow parameters of the transonic sharp-edged VFE-2 case of [85] M∞ Re∞,c c (m) T∞ (K) Twall α (◦ ) 0.85

6 · 106

0.436

281.4

Adiabatic

23

Viscous flow Turbulent

with the terminating shock wave of the local supersonic flow regime formed by the sting mount. A detailed account of shock-wave effects on the vortex breakdown at the VFE-2 wing with sharp leading edges was given by S. Crippa, a doctoral student of the second author of this book [85], see also [86]. Crippa performed Detached Eddy Simulations (DES, Model 11), and we discuss a few of his findings. The flow parameters are given in Table 10.6. For this case we find the normal angle of attack and the normal leading-edge Mach number to be α N = 45.13◦ and M N = 0.47. This case lies in the regime (3) “vortices fixed in span direction, …, embedded shock waves possible” in Fig. 10.22 and hence is covered by the experimental data underlying the correlations there. The critical angle of attack regarding vortex breakdown, now with the valid correlation for sharp-edged delta wings, is as quoted already above α = 20◦ . Figure 10.45 gives the computed vortex-breakdown location as function of the time for the last computation cycle. The breakdown location is defined as the foremost chordwise location, where fully-reversed flow in the outer primary vortex is present. An inner primary vortex was not detected at that angle of attack. The figure demonstrates the high degree of unsteadiness of vortex breakdown. The vortex breakdown location rather suddenly moves within ≈7 ms downstream

Fig. 10.45 Vortex-breakdown location as function of the time [85]. Time steps denoted by red dots are those of which details of the flow field are presented in Figs. 10.46 and 10.47

326

10 Small Aspect-Ratio Delta-Type Wing Flow

from x/c R ≈ 0.55 to x/c R ≈ 0.73. The subsequent upstream motion then is rather gradual. The reason for the different gradients of the upstream and the downstream motion can be found by an inspection of the flow field in the symmetry plane of the wing. Note that the tip of the support sting lies at x/c R = 0.63, Fig. 10.38. We do the inspection in the following Figs. 10.46 and 10.47 for the red-dot locations shown in Fig. 10.45. Of the two figures the first one, Fig. 10.46, shows the flow-field properties at the time step t = 0.1492 s for a better inspection in an enlarged mode. We see a single shock wave at x/c R ≈ 0.51, which moves upstream (indicated by the black arrow in the lower left frame). The associated surface-pressure coefficient (c p ) distribution is shown in the upper left frame. (This frame lies at and near the center line of the wing, see the upper right frame.) The flow comes from the left, the nose of the support sting is located at the lower right corner, the foremost part of the reversed-flow isosurface (grey indicates zero axial velocity) is at the upper right corner. The low c p ahead of the shock wave is in green, yellow and red. The bundling of the iso-c p -lines indicates the location of the shock wave, behind it the high pressure is in light blue. The pressure rises across the local wing span toward the center line, though to different degrees, the

Fig. 10.46 Sub-frames showing instant flow-field properties at the time step t = 0.1492 s [85]. Upper left frame: view from above at the suction side of the wing ahead of the support sting with the surface-pressure distribution, flow comes from the left. Upper right frame: frontal-isometric view of the right-hand half-span suction side with the surface pressure distribution and the reversed-flow isosurface (grey). Lower left frame: surface-pressure coefficient c p (x/c R ) at the intersection of the symmetry plane and the wing surface, location corresponds to the lower edge of the figure above it. Lower right frame: normal view along the symmetry plane with the local Mach-number range, blue = subsonic, green = sonic, red = supersonic

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

327

Fig. 10.47 Sub-frames showing instant flow-field properties at different time steps [85]. Upper left part: t = 0.1547 s, upper right part: t = 0.1560 s, lower left part: t = 0.1586 s, lower right part: t = 0.1650 s. For the contents of the sub-frames see the legend of Fig. 10.46

suction effect of the lee-side vortex is gone. Unsteady lift loss and upward pitching moment increments are the consequence. The lower right frame shows that in the symmetry plane shortly downstream of the wing’s apex a supersonic flow pocket is present, terminated by a curved shock wave, which impinges nearly orthogonal on the wing’s surface.16 At the support sting the flow again is accelerated to supersonic speed. That flow pocket then is terminated close to the trailing edge of the wing. The vortex breakdown at x/c R ≈ 0.57 is associated with the first shock wave, as indicated in the upper left frame. The presence of two shock waves needs a closer inspection. The first shock wave, whose footprint crosses the whole span of the wing (upper left frame), is the primary one.17 The second shock wave appears to be solely due to the support sting, which is a half-ogive followed by a half-cylindrical part. The half-ogive induces by its displacing effect a flow expansion right from its blunt nose—like a blunt body does, in this case in transonic flow. The flow speed becomes supersonic (red color) and then is terminated by the nearly normal shock wave. That occurs at the tip of the half-ogive shaped forebody of the sting, at the intersection with the cylindrical rear part of the sting. There a discontinuity of the surface 16 For inviscid two-dimensional flow over a regular surface theory demands orthogonal impingement, Sect. 4.2.3. 17 How and whether it interacts with the to be conjectured cross-flow shocks is not clear. In [87] it is shown for a similar case that the embedded cross-flow shock wave lies very close to the wing’s surface between the primary and the secondary vortex, see also Fig. 10.32.

328

10 Small Aspect-Ratio Delta-Type Wing Flow

curvature is present, barely detectable in the side view (lower right part) of Fig. 10.38. That discontinuity induces the terminating recompression shock wave, which in the respective frames in Fig. 10.47 is also seen to occur—approximately—at the same location. The shock wave hence is the trace of the stand-up collar-like shock wave surface, standing around the tip of the half-ogive shaped forebody of the support sting. However, whether and how it possibly interacts with the vortex-breakdown evolution is not clear. At the next time step, t = 0.1547 s, upper left part in Fig. 10.47, in the symmetry plane the rear supersonic pocket forms a kind of a bridge to the flat wing portion ahead of the tip of the support sting. Accordingly a double-shock system lies ahead of the sting tip, indicated by the two horizontal bars in the lower left frame of the upper left part of the figure. The first, now weaker shock has moved upstream to x/c R ≈ 0.48, the—also weak—second shock is located at x/c R ≈ 0.6 close to the sting’s tip. The vortex breakdown has jumped back—conjectured to be due to this shock arrangement, see above—from the position behind the first shock to a position behind the second shock, upper right frame of the upper left part of the figure. Possibly the weakening of the first shock wave and the forward motion of the second one induces this kind of abrupt pass-over of the breakdown to the rear position. The frame in the upper right part of Fig. 10.47, 0.0013 s later, shows that the two supersonic regions in the symmetry plane have merged. The downstream motion toward the sting tip of the first shock wave slowly continues. At the time step t = 0.1586 s the first shock wave has reached its furthermost downstream location at x/c R ≈ 0.61. That also holds for the vortex breakdown position, which lies at its furthermost downstream position at x/c R ≈ 0.73, Fig. 10.45. The shock wave has ceased much of its strength and does no more push back the vortex breakdown location. After that the first shock wave begins to move upstream again, which too moves upstream the vortex breakdown position. At t = 0.1650 s, lower right part of Fig. 10.47, the strength of the shock has increased, the vortex breakdown has moved forward to x/c R ≈ 0.64. Shortly after that time step the next cycle is beginning when the shock has reached its most forward position, which is coupled to a decrease of the shock strength. We have seen hence that the flow field in conjunction with the vortex breakdown is highly unsteady and for the transonic case is coupled to the presence of an embedded highly volatile shock-wave system. A conclusion drawn in [85] is that a time-dependent scale-resolving solution permits to recognize disturbances in the region between the shock wave(s), the sting tip and the primary vortex. These disturbances travel upstream toward the first shock wave. The frequencies of the upstream moving disturbances and the spiral motion of the destructed vortex core after the breakdown were found to be very similar. Also reported in [85] is a consideration of axial vortex-core flow properties. Figure 10.48 shows a result for M∞ = 0.8, Re∞,c = 2 · 106 , and α = 26◦ . We present the result in order to illustrate the breakdown situation in terms of the core-flow velocity.

10.5 Vortical Flow Past the Round-Edged VFE-2 Delta Wing

329

Fig. 10.48 Axial core velocity u/u ∞ as function of x/c R , sharp-edged case [85]. Blue line: result of the numerical simulation, black dots: experimental data (PIV)

Computed and measured data is compared. The flow field was considered to be steady. The Reynolds number in the experiment was larger with Re∞,c = 3 · 106 , but this was considered to be acceptable, because the sharp-edged wing case was treated. There was a certain problem, because in the experiment vortex breakdown was found to occur between x/c R = 0.6 and 0.7, whereas in the computation it was further upstream at x/c R = 0.4. Hence the differences in the figure. Nevertheless, the comparison reveals the typical behavior of the axial core velocity during vortex breakdown. From the wing’s apex the axial vortex-core velocity is building up fast to an overspeed of u/u ∞ ≈ 1.5 at x/c R = 0.05. The velocity rises further and ahead of the computationally found breakdown location at x/c R = 0.4 we see the high axial velocity u/u ∞ = 1.88. The experimental value is somewhat higher with u/u ∞ = 1.962 at x/c R = 0.5, with smaller values downstream of that location. The vortex breakdown at x/c R = 0.4 is marked by a steep drop of the axial velocity to u/u ∞ = −0.6. Behind that location the velocity stays negative, but rises slowly to a small positive value at the end of the wing. A re-configuration of the vortex over the wing does not happen.

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case Selected results from the thesis of a doctoral student of the third author of this book, A. Hövelmann, are presented as Unit Problem. Hövelmann investigated both experimentally and numerically the vortex-flow phenomena over the moderately swept, low aspect-ratio AVT-183 configuration with round leading edges, and over the SAGITTA configuration with varying leading-edge contour [88]. We discuss

330

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.49 The SAGITTA configuration [88]. Left part: plan view. Right part: planform parameters

some of the results found for the latter configuration. All figures in this section are from [88]. The SAGITTA (Latin: arrow) diamond wing is relatively thick and with basically round leading edges. Signature requirements, however, led to a sharp leading-edge contour up to 20 per cent in spanwise direction of the configuration, Fig. 10.49, left part. The leading-edge sweep angle ϕ L E = ϕ0 = 55◦ is rather small. SAGITTA’s geometrical planform parameters are given in Fig. 10.49, right part. Note that SAGITTA is a slightly cropped diamond wing. The overall length is cr = 1.2 m and the small wing-tip chord ct = 0.025 cr . The moment-reference point is denoted by xmr p . The aspect ratio is Λ = 2.001, the taper ratio λ = 0.025, and the mean aerodynamic chord lμ = 0.801 m. The original wing over the full span has a symmetric NACA 64A012 airfoil section. We present selected results of numerical simulations of the flow past the original SAGITTA configuration (Geo 1), the configuration with completely sharp leading edge (Geo 5), and the configuration with completely round leading edge (Geo 6). The URANS (Model 10) computations were made assuming fully turbulent flow, the Spalart–Allmaras one-equation turbulence model was applied. The flow parameters are given in Table 10.7. The wing of the SAGITTA configuration consists of four segments, Fig. 10.50.18 Segment I of the Geo 1 configuration has a sharp leading edge, the other three segments have rounded ones with r L E /cr = 0.99 per cent, which amounts to δ = 52. This means that even at the rear an aerodynamically sharp leading edge is not yet present, Sect. 6.4. Geo 5 has a sharp leading edge throughout, Geo 6 a round leading edge throughout.

Table 10.7 Flow parameters of the sub-sonic SAGITTA case [88] M∞ Re∞,lμ lμ (m) T∞ (K) Twall 0.13

18 In

2.3 · 106

0.801

288.15

Adiabatic

α (◦ )

Viscous flow

s. Table 10.8 Turbulent

the following figures segment I is denoted with IB (inboard), segments II and III are denoted with MB (midboard).

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case

331

Fig. 10.50 Segments and leading-edge contours of SAGITTA’s right wing [88]. Left part: Geo 1 configuration with partly sharp leading edge. Middle part: Geo 5 configuration with fully sharp leading edge. Right part: Geo 6 configuration with fully round leading edge

In order to get a first impression of which lee-side vortex-flow patterns are to be expected for either a sharp or a round leading edge, we look at the correlations in Figs. 10.22 and 10.23. The results together with the normal angle of attack α N and the normal leading-edge Mach number M N are listed in Table 10.8. A look at Fig. 10.20 tells us that—at sharp-edged delta wings—the angle of attack above which vortex breakdown happens, is very low with αbd ≈ 9◦ .

Table 10.8 Possible lee-side vortex-flow patterns at SAGITTA with fully sharp and with fully round leading edge (LE) for M∞ = 0.13, ϕ0 = 55◦ , and three angles of attack α (◦ ) α N (◦ ) MN LE Possible vortex-flow pattern 8

16

13.76

26.36

0.076

Sharp

0.080

Round Sharp

Round 24

37.82

0.086

Sharp

Round

Development of vortex system No vortex system Stable and fully developed vortex system Development of vortex system Stable and fully developed vortex system Development of vortex system

332

10 Small Aspect-Ratio Delta-Type Wing Flow

From the correlation data in Table 10.8 we can expect that for the wing with round leading edge not much of non-linear lift will evolve, in particular also because vortex breakdown is beginning already at a low angle of attack. Although for the sharp-edged wing the situation looks better, also for it the early beginning of vortex breakdown will limit the extent of non-linear lift. First we discuss selected computational results of the flow past the Geo 1 configuration.19 Considered is the upper-side flow field of the right-hand side of the wing for three angles of attack, Fig. 10.51. (It appears that the flow field, like those of the other cases, is an open-type lee-side flow field.) In the figures time-averaged surface-pressure coefficients c p are shown as well as streamlines, which are started close to the leading edge. In the figures, also in the corresponding ones for Geo 5 and Geo 6, the main flow direction is always from the lower right to the upper left. At segment I of Geo 1 already at α = 8◦ a leading-edge vortex is present, upper part of Fig. 10.51 (IB = inboard). At the intersection of segment I and II the vortex detaches and moves inward. (At segment II the flow still is attached.) The segment I vortex grows with increasing angle of attack, the suction level rises, too, the timeaveraged axial vorticity level becomes high, but decreases downstream. At the other, round-edged segments, the flow is attached, at segment IV, however, a first indication of separation is visible. It is possibly due to the small cropped part of the wing. At α = 16◦ the situation has changed, middle part of Fig. 10.51.20 The flow past segment IV now is separated, irregular recirculation is indicated, and the small suction peak present there at lower angle of attack has disappeared. The flow past the segments II and III is still attached. Finally at α = 24◦ , lower part of Fig. 10.51, a very complex flow pattern is present. The midboard (MB) lee-side vortex, which came into being at segment III by ordinary separation, dominates the flow field downstream. Underneath it a suction peak is not present, vortex breakdown now clearly is present. Secondary separation exists. At segment II incipient separation is indicated. The lee-side vortex coming from segment I, the inboard (IB) vortex, has formed secondary separation already at a lower angle of attack. The skin-friction line pattern and the surface-pressure coefficient c p , for this angle of attack, both time-averaged, are shown in Fig. 10.52. At segment I (IB) the primary separation line PSL appears to lie above the sharp leading edge, not exactly at its edge. In direction toward the center line of the wing follow the secondary attachment line SAL, the secondary separation line SSL and the primary attachment line PAL, the latter strongly curved into the direction of the wing’s center line. (The reader should notice that we usually denote with ‘primary’ the attachment line or lines at the windward side of a configuration, see Fig. 7.21. Here PAL is the attachment line of the primary vortex emanating from PSL.)

19 The reader should note that the Geo 1 configuration resembles the by H.A. Wilson and J.C. Lovell manipulated leading edge contour of the DM-1 glider, Fig. 10.7. 20 In [88] the lee-side flow development is visualized in Δα = 4◦ steps. Due to space restrictions we discuss only in 8◦ steps.

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case

333

Fig. 10.51 Upper-side flow field at the right-hand side of the SAGITTA Geo 1 wing: time-averaged surface-pressure coefficient c p and field streamlines [88]. Upper part: α = 8◦ , middle part: α = 16◦ , lower part: α = 24◦

334

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.52 Top view at the right-hand side of the SAGITTA Geo 1 wing at α = 24◦ showing the time-averaged skin-friction lines and the time-averaged surface-pressure coefficient c p [88]. LEV: leading-edge vortex, PSL: primary separation line, SSL: secondary separation line, PAL primary attachment line, SAL: secondary attachment line. The freestream direction is upward from below

Overall we see an open lee-side flow pattern. Whether the separation and attachment lines are of open-type beginning, cannot be assured. An open-type ending is evident. More detailed than in Fig. 10.51 we see that the suction peak underneath the IB leading-edge vortex is well developed, but very soon ceases to exist due to the vortex breakdown. At the following MB segments we find a well discernible flow past the round leading edge and the primary separation line PSL located away from the leading edge at the wing’s upper side. Downstream it comes closer to the leading edge. There a secondary separation line SSL and a secondary attachment line SAL appear. Beginning and ending of the singular lines appears to be of open type. The suction peak underneath the MB leading-edge vortex is seen to be well developed, and also very soon to cease to exist due to the vortex breakdown. We contrast now these results first with those found for the sharp-edged Geo 5 wing, and then with those found for the round-edged Geo 6 wing, both sketched also in Fig. 10.50. We begin with Geo 5. At α = 8◦ a lee-side vortex exists along the whole leading edge, upper part of Fig. 10.53. The correlation data in Table 10.8 suggest the beginning of the development of the vortex system for this angle of attack. Similar, but stronger than for the respective case of Geo 1, at segment IV an indication of separation or vortex breakdown is present. Influence of the small cropped part of the wing? Anyway, the amount of suction pressure along the leading edge is small, hence the non-linear lift at this angle of attack is small, too. Figure 10.53, middle part, shows that for α = 16◦ the lee-side vortex is fully developed with a larger amount of non-linear lift than at the smaller angle of attack. The non-linear lift, however, is degraded by the breakdown of the vortex, which

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case

335

Fig. 10.53 Upper-side flow field at the right-hand side of the fully sharp-edged SAGITTA Geo 5 wing: time-averaged surface-pressure coefficient c p and field streamlines [88]. Upper part: α = 8◦ , middle part: α = 16◦ , lower part: α = 24◦

336

10 Small Aspect-Ratio Delta-Type Wing Flow

already has reached segment III. At α = 24◦ finally the vortex has reached a large strength, indicated by the high suction pressure underneath it, lower part of Fig. 10.53. But vortex breakdown has reached segment II. Hence the adverse effects of vortex breakdown, loss of non-linear lift and potentially a pitch-up moment are present. We look now at the results found for the round-edged SAGITTA Geo 6 wing. At α = 8◦ attached flow is present along the whole leading edge, except for a small area at the cropped wing tip, upper part of Fig. 10.54. The correlation data in Table 10.8 suggest that no vortex system exists for this angle of attack. At the segments III and IV a small amount of suction pressure is present. For α = 16◦ attached lee-side flow is present up to segment IV. Sizeable suction pressure is found. At the aft part of the wing, segment IV, a vortex is present, middle part of Fig. 10.54. That vortex obviously underwent breakdown almost from its beginning. Flow reversal is indicated, the suction pressure is gone at almost the whole leading edge of segment IV. At α = 24◦ along the first half of the leading edge the flow still is attached, with a rather strong suction pressure visible, lower part of Fig. 10.54. Vortex breakdown has moved upstream to the beginning of segment II. The suction pressure mostly has disappeared beyond segment II. When comparing the results found for the three SAGITTA configurations Geo 1, Geo 5 and Geo 6, we assert that mostly rather weak lee-side vortex phenomena are present, even secondary ones, that suction pressure exists to a small extent, and that vortex breakdown happens already at small angles of attack. The results as a whole suggest that non-linear lift is small for all three configurations. Hence its influence on the aerodynamic properties of SAGITTA is small, too. In [88] computed aerodynamic coefficients and derivatives of the longitudinal motion are compared for the three configurations. We show the result for the lift and the pitching moment coefficient and their derivatives. In the figures also results for Geo 1 found with the Athena Vortex Lattice method AVL are plotted. The lift coefficient C L (α) and its derivative dC L (α)/dα are given in Fig. 10.55. In the whole angle-of-attack interval the lift results, left part of the figure, including the AVL results, indeed do not differ much from each other. SAGITTA’s lift almost fully is due to its plan form. That non-linear effects are present, even if only small, can be seen when looking at the lift-coefficient derivative in the right part of Fig. 10.55. The linear AVL method yields a lift slope, which evenly falls with increasing angle of attack. This is in contrast to the non-linear results. The derivative of the sharp-edge Geo 5 configuration increases to a maximum at α ≈ 11◦ . The reason for this is the existence of the full-span leading-edge vortex with its increasing suction level. Beyond that angle of attack vortex breakdown leads to a general reduction of the derivative. For the round-edged Geo 6 configuration we see a weak maximum of dC L (α)/dα at α ≈ 8◦ , followed by a weak continuous decrease. The Geo 1 configuration with the sharp-edged segment I and rounded leading edges in segments II, III, and IV shows the strongest non-linear effect with a maximum of dC L (α)/dα at around α ≈ 18◦ ,

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case

337

Fig. 10.54 Upper-side flow field at the right-hand side of the fully round-edged SAGITTA Geo 6 wing: time-averaged surface-pressure coefficient c p and field streamlines [88]. Upper part: α = 8◦ , middle part: α = 16◦ , lower part: α = 24◦

338

10 Small Aspect-Ratio Delta-Type Wing Flow

followed by a very steep decrease. That appears mainly to be due to the breakdown of the segment I vortex at the rear of the wing. The pitching-moment coefficient Cm (α) in the left part of Fig. 10.56 is negative— due to the choice of the reference point xmr p , see the right part of Fig. 10.49—hence SAGITTA in pitch is stable—depending on the center of gravity location versus the chosen reference point. The leading-edge contours play a minor role only above α ≈ 10◦ . Due to the symmetric airfoil of the configuration the pitching moment is zero at α = 0◦ . The result found with the linear AVL method agrees with the others up to α ≈ 13◦ . Strong deviations are found of the pitching-moment derivative dCm (α)/dα seen in the right part of Fig. 10.54. They are due to the diverse effects present, i.e., separation, vortex development with the associated suction pressure, and vortex breakdown with the associated pitch up. We highlight a few of the consequences of the effects appearing at the three configurations. The fully sharp-edged Geo 5 configuration shows a nearly constant derivative up to α ≈ 14◦ , where a small relative maximum occurs. With increasing α then it drops sharply to dCm (α)/dα = −0.375 at α ≈ 24◦ . The fully round-edged

Fig. 10.55 Longitudinal motion of configurations Geo 1, Geo 5 and Geo 6 [88]. Left part: lift coefficient C L (α). Right part: lift-coefficient derivative dC L (α)/dα

Fig. 10.56 Longitudinal motion of configurations Geo 1, Geo 5 and Geo 6 [88]. Left part: pitching moment coefficient Cm (α). Right part: pitching-moment derivative dCm (α)/dα

10.6 Partly Developed Swept Leading-Edge Vortices, the SAGITTA Case

339

Geo 6 configuration almost everywhere exhibits a steady drop of the derivative down to dCm (α)/dα = −0.43 at α = 25◦ . Geo 1 finally up to α = 15◦ has a steady drop of the derivative like Geo 6 has. Then a steep increase leads to a relative maximum at α = 21◦ , followed by a small decrease down to dCm (α)/dα = −0.28 at α = 24◦ . Hence due to the intricate flow phenomena along the leading edge with varying contours, Geo 1 shows an appreciably varying longitudinal stability behavior.

10.7 Laminar Hypersonic Flow Past a Round-Edged Delta Wing The wing of this Unit Problem is the Blunt Delta Wing (BDW), which was a study configuration in the European HERMES project, see, e.g., [2]. The BDW configuration is a very strongly simplified re-entry vehicle configuration flying at moderate angle of attack. The flow past the BDW was investigated experimentally and numerically by many authors. Here we discuss a few results from the thesis of a doctoral student of the first author of this book, S. Riedelbauch [89], see also [90]. Riedelbauch studied the BDW’s surface-radiation cooling, here we discuss the flow field and the possible non-linear lift.21 All figures in this section except two are from [89]. The configuration is a simple slender delta wing with a blunt nose and rounded leading edges, Fig. 10.57. The lower side has a dihedral (γ = 15◦ , lower part of Fig. 10.57) and therefore is only approximately flat. Navier–Stokes (Model 9) computations with perfect-gas assumption were performed with the parameters given in Table 10.9. Although in reality laminar-turbulent transition would occur with the flight parameters listed there, the flow was assumed to be laminar throughout. We note in passing that at the wing’s surface radiation cooling was modeled with the surface-emission coefficient being  = 0.85. The normal angle of attack and the normal leading-edge Mach number of this hypersonic case are α N = 38.08◦ and M N = 3. This case lies far out to the right in Fig. 10.23 and hence is not covered by the experimental data underlying the correlations there. Anyway, the leading edge is supersonic, and lee-side cross-flow shocks are to be expected.

Table 10.9 Computation parameters of the blunt delta wing (“flight” case) [89] u M∞ H (km) T∞ (K) Re∞ L (m) ϕ 0 (◦ ) α (◦ ) ε (m−1 ) 7.15

21 Aspects

30

226.506

2.69 · 106

14

70

15

0.85

Boundary layer Laminar

of the radiation cooling of the BDW are discussed in [91], the topic of flow properties in this regard along singular lines is treated in [73], see also our Sect. 7.3.

340

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.57 Configuration of the BDW and the coordinate convention [89]: a side view, b view from above, c cross-section B - B

The cross-flow bluntness parameter ranges from pb = 0.143 at x/L = 0.1 to pb = 0.074 at x/L = 1. Hence we are well away from an aerodynamically sharp leading edge. Experimental investigations were performed at FFA Sweden with a BDW model having a modified nose [92]: sharp in comparison to the blunt nose shown in Fig. 10.57. The parameters partly were different and are given in Table 10.10: The computation for the blunt-nosed BDW with the experimental data yielded the distribution of the surface pressure coefficient c p (y) at the location x/L = 0.5 shown in Fig. 10.58. The lift predominantly is created at the windward side of the configuration. At the lee side is to a degree present the hypersonic shadow effect [91]. That means that not much of a negative c p is existing. The classical suction peak of delta wings is only very weakly indicated, hence in this case one really cannot speak of non-linear lift being present.

Table 10.10 Experimental parameters of the sharp-nosed blunt delta wing [92] u M∞ T∞ (K) Re∞ L (m) Tw (K) ϕ 0 (◦ ) α (◦ ) (m−1 ) 7.15

74

39 · 106

0.15

288

70

15

Boundary layer Laminar

10.7 Laminar Hypersonic Flow Past a Round-Edged Delta Wing

341

Fig. 10.58 Distribution of the computed surface pressure coefficient c p (y) at x/L = 0.5 [89]. Note that the windward side is above and the lee side below

Fig. 10.59 Detail of the distribution of the computed [89] surface pressure coefficient c p (y) at x/L = 0.5 of the lee side and experimental [92] data ()

A detail of the lee-side pressure distribution is shown in Fig. 10.59. Indicated are a few measured data, the triangles [92]. The deviation from the computed data is less than 10 per cent. The absolute minimum of the pressure is c pmin = −0.02492 at y/c = 0.78, close to the vacuum pressure, see Appendix A.1, of c pvac = −0.02794.

342

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.60 Computed skin-friction line pattern (blunt-nosed wing, above [89]) and oil-flow picture (sharp-nosed wing, below [92]). The free-stream comes from the left

It usually is taken as granted that discrete numerical solutions of the Navier–Stokes equations (Model 9) do not suffer from modeling problems, like RANS solutions. The discretization, of course, must be correct, as well as the whole solution procedure. We present, besides the comparison of the surface pressure data above, a visual comparison between experimentally and numerically found skin-friction line patterns, Fig. 10.60. In the lower part of that figure the upper side of the BDW wind-tunnel model from [92] is shown. The sharp nose, compared to the blunt one of the actual BDW seen in the upper part, clearly is discernible. Well visible is the influence of the model support’s sting at the lower right. Apart from the nose and the aft region the agreement between the computed and the measured skin-friction line patterns is acceptable. The oil-flow picture by far does not contain the fine details, which we see in the computed picture. In the oil-flow picture recognizable are only the primary (S1 ) and the secondary (S2 ) separation lines. We now have a look at the topology of the skin-friction field computed for the “flight” case, Table 10.9. We wish to identify attachment and separation lines. We see at the lower (windward) side of the configuration in Fig. 10.61a the classical skin-friction line pattern present at the windward side of a delta wing. Because the lower side of the BDW is not fully flat, the flow exhibits a slight two-dimensionality between the two primary attachment lines A1 . The latter are marked by strongly divergent skin-friction lines. The forward stagnation point, which is a nodal point,

10.7 Laminar Hypersonic Flow Past a Round-Edged Delta Wing

343

Fig. 10.61 Selected computed skin-friction lines at the surface of the BDW [89]: a look at the lower side, b look at the upper side of the configuration. The free-stream comes from the left

Sect. 7.2, lies also at the lower side, at about 3 per cent of the body length. The primary attachment lines are almost from the beginning parallel to the leading edges, i.e., they do not show a conical pattern. At the upper (leeward) side of the wing the situation is quite different, Fig. 10.61b. The flow streams from the lower side around the blunt leading edge well onto the upper side, where it then separates. In the lower part of Fig. 10.61b we see along the vertical line—from the leading edge toward the symmetry line—a succession of separation and attachment lines: the primary separation line S1 , the secondary attachment line A2 , a secondary separation line S2 , and the tertiary attachment line A3 in the middle of the lower side, the latter indicating a closed lee-side flow field. All is mirrored on the upper part of the picture. Again a conical pattern is not discernible, except for a small portion near the nose. The secondary separation lines are almost parallel to the single tertiary attachment line along the upper symmetry line of the wing. Both the primary and the secondary separation lines are lines of “open-type separation”, i.e., the separation line does not begin in a singular point on the surface,

344

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.62 Selected computed skin-friction lines at the upper side of the BDW near the nose (detail of Fig. 10.61b) [89]

Fig. 10.63 Structure of the closed lee-side flow field in the Poincaré surface [89]

Sect. 7.1.4. Figure 10.62 shows this, and also that all attachment lines are of opentype, too. With these surface patterns we can qualitatively sketch a structure of the leewardside flow in the Poincaré surface, Fig. 10.63. The computed cross-flow shocks are indicated. They lie above the lee-side vortex system, quite in contrast to the findings in Sect. 10.4. The two primary attachment lines A1 show up as quarter-saddle points, whereas half-saddle points indicate the two primary separation lines S1 , the two secondary attachment lines A2 , the two secondary separation lines S2 , and the tertiary attachment line A3 . Above the wing the two focus points F1 of the primary lee-side vortices, the two focus points F2 of the secondary lee-side vortices, and the free saddle point S0 above the lee-side vortices (closed lee-side flow field) are indicated. We apply the topological rule 2’, Eq. (7.25). Because the focus points are counted as nodal points, we find

10.7 Laminar Hypersonic Flow Past a Round-Edged Delta Wing



   1 1 1 4+ 0 − 1 + 7 + 2 = −1, 2 2 4

345

(10.13)

and conclude that the topology is a valid one.

10.8 Concluding Remarks After the introduction and the short historical notes we presented some basics of the phenomenon of non-linear lift appearing at delta wings, Sect. 10.2. Correlations of empirical data were given showing when lee-side vortex systems are expected to be present, Sect. 10.2.6, and when vortex breakdown and overlapping can happen, Sect. 10.2.5. These correlations proved to give reasonable first insight regarding the appearance and behavior of the vortex systems in the Unit Problems under consideration. The first Unit Problem then illustrated the significance of different flow-physical and mathematical models for the computational simulation of lee-side vortex flow, Sect. 10.3. We have shown with simulations for the VFE-1 configuration with loworder turbulence models and even with the Euler equations (Model 8), that they can give reasonable results for pre-design work, provided that the delta wing has sharp leading edges.22 The second Unit Problem, Sect. 10.4, was devoted to the proof that at sharp-edged delta wings the creation of vorticity including the entropy rise and eventually the appearance of the—primary—lee-side vortices follow the same mechanism, which is present at large aspect-ratio wings. The proof made use of the concept of kinematically active vorticity content and the compatibility condition, both Chap. 4, which in Chap. 8 were applied to the wing of the CRM configuration, Sect. 8.4. We have shown and discussed only a small number of the results achieved with the round-edged vortex-flow experiment VFE-2, which is the third Unit Problem treated. Important is that the relative bluntness of the leading edge, i.e. the parameter  pb = r L E /b plays a major role: with constant leading-edge radius it diminishes from the front of the wing toward the rear. This means that the leading edge becomes relatively sharper in downstream direction. Accordingly the formation of the primary lee-side vortex is affected. In the present case the wing has—in terms of the nomenclature of the VFE-2 community—a second primary lee-side vortex system, located ahead of the first one and probably coupled to that, Sect. 10.5.2. This phenomenon was discussed and after that the phenomenon of vortex breakdown. Both a subsonic and a transonic case 22 Regarding

the numerical simulation of vortical flow fields we assume that always sufficient investigations were made in order to ensure grid independency. This holds for all simulations with Model 8, 9, and 10 methods. Regarding experimental simulations, similar requirements must be fulfilled with respect to the experimental set-up, the free-stream properties, the test section as test environment, the model properties—including the sharpness of the leading edges—and the measuring devices.

346

10 Small Aspect-Ratio Delta-Type Wing Flow

was considered. For the latter in particular the high degree of flow unsteadiness was demonstrated. The following Unit Problem was the flow past the SAGITTA configuration at the low subsonic Mach number M∞ = 0.13. The configuration has a sharp leading edge directly behind its apex and downstream a round one. Moreover with ϕ0 = 55◦ it has a smaller leading-edge sweep than the VFE-1 and the VFE-2 configuration, which both have ϕ0 = 65◦ . This difference appears not to be much, but the empirical correlations already show—also due to the small Mach number—that the lee-side vortex system is developed only weakly and that vortex breakdown is to happen already at a small angle of attack. Accordingly non-linear lift effects due to the lee-side vortex system are rather weak. The last Unit Problem, the laminar flow at hypersonic speed past a blunt-edged wing at high angle of attack, revealed a well established lee-side vortex system. The skin-friction line pattern shows well how primary and secondary singular lines govern the vortex system of the closed lee-side flow field. However, due to the hypersonic shadow effect, a non-linear lift is almost non-existent. In closing these remarks we list some of the general observations made during the VFE-1 and in particular the VFE-2 investigations, mainly following [78, 79]. The observations regard the influence of the angle of attack, Reynolds and Mach number, vortex breakdown, and laminar-turbulent transition and turbulence modeling. • Angle of attack At wings with sharp leading edges the lee-side vortex system above a threshold angle generally appears to be present along the whole leading edge. Experimental observations, however, challenge that to a certain degree. If the wing has a round leading edge, the primary lee-side vortex first appears at the rear of the wing and with increasing angle of attack moves forward. This effect appears to be connected to the bluntness parameter pb . The gedankenexperiment in Sect. 10.2.4 indicates that for a given bluntness the actual adverse pressure gradient also depends on sin2 α, i.e., increasing angle of attack increases the separation disposition at the very leading edge. • Reynolds number In experiments it frequently was observed that in low Reynoldsnumber delta wing cases with presumably laminar flow throughout, the lee-side vortices were stronger than with turbulent flow in high Reynolds-number cases, and hence also the non-linear lift. This holds for wings with round and with sharp leading edges, for the latter only regarding the secondary vortices. This observation can be commented on with a consideration of the—kinematically active—vorticity content of the involved boundary layers at separation. Figure 10.64 shows schematically the flow situation observed at a delta wing with round leading edges. Both the pairs of the primary and the secondary vortices are sketched. The possible cross-flow shocks are not indicated. On the left of the figure the situation arising with laminar flow is indicated, on the right that with turbulent flow. Because the laminar boundary layer cannot negotiate as much adverse pressure gradient as the turbulent one, it will separate earlier than the turbulent one. This

10.8 Concluding Remarks

347

Fig. 10.64 Schematic of ordinary separation in the cross-section of a delta wing with round leading edges and a closed-type lee-side flow field [55]. Pressure coefficient distribution (without cross-flow shock effects) and separation locations of a laminar flow, b turbulent flow. c Location of primary and secondary separation and vortices

holds for both the primary and the secondary separation and also for these locally highly three-dimensional boundary-layer flows. We assume now that the structure of separation, in particular the shear between the upper and the lower inviscid flow at the leading edges—and also at the location of secondary separation—is not completely different in the two cases (a) and (b). If this can be accepted, the kinematically active vorticity content at separation is larger in the laminar case than in the turbulent one. This is due to the fact that the separation in the laminar case happens at a considerably higher velocity level than in the turbulent case. This tells us two things. First, the matter of separation prediction with numerical methods (Model 10), in particular at wings with round swept leading edges, hinges on the suitability of the employed turbulence model. At wings with round leading edges and with regard to the secondary vortices, discrete modeled Euler solutions (Model 8) are not suited to describe these flows. Second, if embedded laminar-turbulent transition happens at the lee side of the wing, the properties of secondary separation and vortices hardly can be simulated properly. The reason is that today only empirical or semi-empirical transition criteria and models are

348

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.65 Onset of separation of the outer primary vortex as function of the angle of attack (AOA ≡ α) [93]. Upper part: Rec = 2 · 106 , lower left part: Rec = 6 · 106 , lower right part: Rec = 60 · 106

available, but no non-empirical ones [73]. Increasing the Reynolds number moves a vortex system downstream. This observation can be explained with the concept of the strength of a boundary layer, Sect. 2.1. For two-dimensional flow it was demonstrated by means of empirical separation criteria that for a given adverse pressure gradient an increase of the Reynolds number reduces the separation tendency of both laminar and turbulent flow. This result, with due reservations, can be generalized to hold for the present flow problems. Indeed, S. Crippa and A. Rizzi, [93], in this sense illustrate the Reynolds-number effect at the VFE-2 configuration at M∞ = 0.4, Fig. 10.65. We show the results for three Reynolds numbers. Of concern was how to determine the onset. In the numerical simulation two approaches were employed: (1) convergence of skinfriction lines at the separation location (blue color), and (2) tracing of the vortex core location (red color). The results from the experiment (black color) are also specified. All approaches more or less agree well with each other. In all cases the Reynolds-number effect is well illustrated. At small angle of attack (AOA ≡ α) the agreement of experimental and numerical data is questionable, with increasing angle it becomes better. Anyway, at small α for the highest Reynolds number the separation onset is at x/cr = 0.6. The location moves for-

10.8 Concluding Remarks

349

Fig. 10.66 Onset of separation of the outer and the inner primary vortex as function of the angle of attack (AOA ≡ α) [93]. Left part: Rec = 6 · 106 , right part: Rec = 60 · 106

ward with increasing α. For the lower Reynolds numbers it is found more forward at x/cr = 0.35 and 0.25. At the largest angles of attack the separation location seems to be no more affected by the magnitude of the Reynolds number. An interesting result is shown in Fig. 10.66: the location of the onset of separation of the inner primary vortex obviously is coupled to that of the outer primary vortex. Because of uncertainties in the data, the results are plotted only for the higher Reynolds numbers. The, to a high degree, constant distance between the onset locations of the outer and the inner primary vortex points to the fact that the two vortex pairs are closely coupled. This is the same with the secondary vortex of the outer primary vortex. Hence, as initially mentioned, one better should speak of the tertiary vortex pair instead of the inner primary vortex pair. • Mach number From the two Mach number cases considered of the VFE-2 case, M∞ = 0.4 (subsonic) and M∞ = 0.8 (transonic), it was seen that basically the flow-field topology remains the same. In the transonic case, the formation of the outer primary vortex begins at a smaller angle of attack. Its axis is shifted inboard, the inner primary vortex appears to be either very weak or not present at all. • Vortex breakdown Vortex breakdown at large angles of attack basically is a highly unsteady phenomenon. This means that computational simulation must be made with unsteady approaches, e.g., URANS or scale-resolving methods (Model 10 or 11 methods). Vortex breakdown first occurs in the vicinity of the trailing edge and with increasing angle of attack moves upstream. For the VFE-2 configuration with sharp and medium-radius leading edges vortex breakdown of the outer primary vortex was seen to be present along the whole leading edge at α = 23◦ . At the SAGITTA configuration with a smaller leadingedge sweep vortex breakdown begins to appear already around α = 9◦ . The correct prediction of the onset of breakdown by means of numerical simulations is difficult. In all such simulations vortex breakdown already occurred at smaller angles of attack than found experimentally. The reason for that probably is a too low prediction of the axial velocity of the vortex. Shortcomings of the

350

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.67 Schematic of the effect of laminar-turbulent transition on the separation behavior at delta wings with round leading edges [94]

employed turbulence models and/or the URANS approach as such are the causes for that. Scale-resolving methods appear to be the way out. • Laminar-turbulent transition and turbulence modeling The laminar-turbulent transition affects ordinary separation behavior. As explained in Sect. 2.1, a turbulent boundary layer can negotiate a larger adverse pressure gradient than a laminar one. The other way around this means that with a given adverse pressure field the turbulent boundary layer will separate at a more downstream location than the laminar one. The laminar-turbulent transition behavior at a given pressure field, Mach number, free-stream and body-surface conditions depends on the magnitude of the Reynolds number, see, e.g., [73]. Figure 10.67 from [94] visualizes the effect of increasing Reynolds number on the transition location and the separation behavior at a delta wing with round leading edge. At the lowest Reynolds number (left sketch) laminar primary (open-type) separation happens close to the leading edge. Once transition has occurred, the separation location of the now turbulent boundary layer moves to the upper side of the wing. The sketch in the middle shows that with increasing Reynolds number transition happens more upstream and most of the turbulent primary separation line now is located at the upper side of the wing. The further increase of the Reynolds number leads to transition upstream of the separation location and hence the beginning of

10.8 Concluding Remarks

351

the fully turbulent primary open-type separation effectively is shifted downstream with the separation line. Of course, as before, it is located at the upper side of the wing. If transition occurs in the attached boundary layer ahead of the location where the vortex emerges, simulation approaches have a problem, if the separation phenomena are influenced. That problem is increased, if embedded transition occurs, for instance along attachment lines, or even in the evolving or the fully evolved vortices. Transition criteria are not available for such flows, the popular en approaches, [73], are not viable. Comparisons of numerical RANS and URANS simulations with different turbulence models have shown that important differences in the obtained flow phenomena can occur. In this regard best-practice approaches would be helpful.

10.9 Problems Problem 10.1 With flight vehicles we observe the following general wing shapes: (a) more or less rectangular wings, (b) swept wings, (c) slender delta-like wings. (1) In what Mach-number domains are these shapes found? What is behind this? (2) Are there other wing shapes? (3) What is the difference between an hypersonic airbreather and a winged re-entry vehicle like the former Space Shuttle Orbiter? Problem 10.2 Two vortex phenomena limit the “healthy” flow regime above a delta wing at angle of attack. Give a short description of these phenomena and their influence on the aerodynamic properties of the wing. Problem 10.3 Consider the DM-1 glider in Fig. 10.7. Assume an angle of attack α = 30◦ and a flight Mach number M∞ = 0.3. What do the correlations in Sect. 10.2.6 say regarding the occurrence of lee-side vortices (a) for the configuration with the original round leading edge and ϕ0 = 60◦ and (b) for the manipulated configuration with sharp leading edge and ϕ0 = 64◦ ? Problem 10.4 Consider Fig. 10.41. Would the lee-side flow field as a closed one be viable? Apply Rule 2’. Problem 10.5 Sketch qualitatively the spanwise distributions of both the lift C L (y)l(y) and the lift coefficient C L (y) for a delta wing of about 65◦ leading-edge sweep at about 20◦ angle of attack. Which separation scenarios occur? What is the impact on the lift characteristics? Problem 10.6 Consider a planar delta wing of ϕ0 = 65◦ leading-edge sweep with nearly sharp leading edges and an aspect ratio of Λ = 1.9. Which pitching moment characteristics as function of angle of attack may be attributed to this wing? Problem 10.7 Consider a double-delta wing featuring sharp leading edges at angle of attack of about α ≈ 20◦ . The leading-edge sweep of the forward portion of the

352

10 Small Aspect-Ratio Delta-Type Wing Flow

Fig. 10.68 Delta wing at angle of attack α

Fig. 10.69 Double delta wing

wing (strake) is ϕ01 = 75◦ , the leading-edge sweep of the rearward portion of the wing is ϕ02 = 55◦ . (1) Provide a topological consistent sketch of a leading-edge vortex system with respect to primary and secondary vortex structures for a cross flow plane intersecting the wing in the rearward portion. (2) Which trends may be expected with respect to pitch-up and roll-reversal tendencies? Which flow-physics effects are associated with these tendencies? Problem 10.8 A flat delta wing with dimensions according to sketch below, Fig. 10.68, flies at an angle of attack of α. Where is the center of pressure located? Problem 10.9 A slender wing has the dimensions in Fig. 10.69. What is the slope of the lift curve, dC L /dα? Problem 10.10 Simplify the case of a wing-and-horizontal stabilizer to a twodimensional problem where the wing and stabilizer are a flat plate, Fig. 10.70. The angle of attack is zero. Using the lumped vortex method, what is the vortex strength of the wing, with dimensions according to the sketch.

References

353

Fig. 10.70 Two-dimensional wing-stabilizer configuration

References 1. Küchemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford (1978), also AIAA Education Series. AIAA, Reston (2012) 2. Hirschel, E.H., Weiland, C.: Selected Aerothermodynamic Design Problems of Hypersonic Flight Vehicles. Springer, Berlin, and Progress in Aeronautics and Astronautics, vol. 229. AIAA, Reston (2009) 3. Hirschel, E.H.: Towards the virtual product in aircraft design? In: Periaux, J., Champion, M., Gagnepain, J.-J., Pironneau, O., Stoufflet, B., Thomas, P. (eds.) Fluid Dynamics and Aeronautics New Challenges. CIMNE Handbooks on Theory and Engineering Applications of Computational Methods, Barcelona, Spain, pp. 453–464 (2003) 4. Edwards, C.L.W., Small, W.J., Weidner, J.P.: Studies of scramjet/air-frame integration techniques for hypersonic aircraft. AIAA-Paper 75-58 (1975) 5. Kuczera, H., Sacher, P.: Reusable Space Transportation Systems. Springer, Berlin, and Praxis Publishing, Chichester (2011) 6. Krämer, E.: Kampfflugzeuge. In: Rossow, C.-C., Wolf, K., Horst, P. (eds.) Handbuch der Luftfahrzeugtechnik, pp. 113–150. Carl Hanser Verlag, München (2014) 7. Modin, K.E., Clareus, U.: Aerodynamic design evolution of the SAAB JAS 39 Gripen aircraft. AIAA-Paper 91-3195 (1995) 8. Clareus, U.: Aerodynamic highlights of a fourth generation delta canard fighter aircraft. In: Proceedings of the 6th Annual International Symposium, 14–19 August 2001, Zhukovsky, Russia (1995) 9. Staudacher, W.: Die Beeinflussung von Vorderkantenwirbelsystemen schlanker Tragflügel (The manipulation of leading-edge vortex systems of slender wings). Doctoral thesis, University Stuttgart, Germany (1992) 10. Hitzel, S.M., Schmidt, W.: Slender wings with leading-edge vortex separation: a challenge for panel methods and Euler solvers. J. Aircr. 21(10), 751–759 (1984) 11. Winter, H.: Strömungsvorgänge an Platten und profilierten Körpern bei kleinen Spannweiten. Forschung im Ingenieurwesen 6, 67–71 (1935) 12. Busemann, A.: Aerodynamischer Auftrieb bei Überschallgeschwindigkeit. Proc. VoltaKongress, Rom 1935, 328–360 and Luftfahrtforschung 12, 210–220 (1935) 13. Ludwieg, H.: Pfeilflügel bei hohen Geschwindigkeiten. Aerodynamische Versuchsanstalt Göttingen, Bericht 39/H/18 (1939) 14. Meier, H.U. (ed.): German Development of the Swept Wing–1935-1945. Library of Flight. AIAA, Reston (2010) 15. Bollay, W.: A non-linear wing theory and its application to rectangular wings of small aspect ratio. ZAMM 19, 21–35 (1939) 16. Gersten, K.: Nichtlineare Tragflächentheorie, insbesondere für Flügel mit kleinem Seitenverhältnis. Ingenieur-Archiv 30(6), 431–452 (1961) 17. Pelletier, A.J.: Paper darts to deltas—The designs of Roland Payen. Air Enthu. Mag. 68, 33–44 (1997) 18. Wagner, W.: The History of German Aviation: The First Jet Aircraft. Schiffer Military/Aviation History, Atglen, PA (1998)

354

10 Small Aspect-Ratio Delta-Type Wing Flow

19. Jones, R.T.: Wing planforms for high-speed flight. NACA 863 (1945) 20. Wilson, H.A., Lovell, J.C.: Full-scale investigation of the maximum lift flow characteristics of an airplane having approximately triangular planform. NACA RM L6K20 (1946) 21. Nowarra, H.J.: Die deutsche Luftrüstung 1933-1945 (four parts). Bernard & Graefe Verlag, Bonn (1989) 22. Masters, D.: German Jet Genesis. Jane’s Publishing Company, London (1982) 23. Heinzerling, W.: Die Geschichte des Pfeilflügels. Teil 2: Die projektmäßige Anwendung des Pfeilflügels. DGLR-Jahrbuch 1981, 071-1–071-33 (1981) 24. von Gersdorff, K.: Transfer of German aeronautical knowledge after 1945. In: Hirschel, E.H., Prem, H., Madelung, G. (eds.) Aeronautical Research in Germany—From Lilienthal Until Today, pp. 325–344. Springer, Berlin (2004) 25. Polhamus, E.C.: Vortex lift research: early contributions and some current challenges. In: Campbell, J.F., Osborne, R.F., Foughner, J.T. (eds) Vortex Flow Aerodynamics, vol. 1, NASA CP-2416, pp. 1–30 (1986) 26. Friedrich, O., Herbst, W., Obermeier, E.: Evolution of the New European Fighter—A German Industrial Perspective. MBB Military Aircraft Division, München (1989) 27. Örnberg, T.: A note on the flow around delta wings. KTH-AERO-TN 38 (1954) 28. Marsden, D.J., Simpson, R.W., Rainbird, W.J.: An investigation into the flow over delta wings at low speeds with leading-edge separation. Cranfield College of Aeronautics, Rep. No. 114 (1958) 29. Werlé, H.: Sur l’Éclatement de Tourbillions d’Apex d’une Aile Delta aux Faible Vitesses. La Recherche Aeronautique No. 74 (1960) 30. Hummel, D.: Experimentelle Untersuchung der Strömung auf der Saugseite eines schlanken Deltaflügels. ZFW 13(7), 247–252 (1965) 31. Hummel, D.: Zur Umströmung scharfkantiger Deltaflügel bei großen Anstellwinkeln. ZFW 15(10), 376–385 (1967) 32. Hummel, D., Redeker, G.: über den Einfluß des Aufplatzens der Wirbel auf die Aerodynamischen Beiwerte von Deltaflügeln mit kleinem Seitenverhältnis beim Schiebeflug. WGLR Jahrbuch 232–240 (1967) 33. Legendre, R.: Ecoulement en Voisinage de la Pointe Avant d’une Aile à Forte Flèche aux Incidences Moyennes. La Recherche Aeronautique No. 30 (1952) 34. Brown, C.E., Michael, W.H.: Effect of leading-edge separation on the lift of a delta wing. J. Aeronaut. Sci. 21, 690–694 (1954) 35. Küchemann, D.: A non-linear lifting-surface theory for wings of small aspect ratio with edge separation. RAE Aero Report No. 2540 (1955) 36. Mangler, K.W., Smith, J.H.B.: A theory of the flow past a slender delta wing with leading-edge separation. RAE Aero Report No. 2593 (1957) 37. Polhamus, E.C.: A concept of the vortex lift of sharp edge delta wings, based on a leading-edge suction analogy. NASA TN D-3767 (1966) 38. Hall, M.G.: A theory for the core of a leading-edge vortex. J. Fluid Mech. 11(2), 209–228 (1961) 39. Hall, M.G.: A numerical method for solving the equations of a vortex core. RAE TR 65106 (1965) 40. Ludwieg, H.: Zur Erklärung der Instabilität der über angestellten Deltaflügeln auftretenden freien Wirbelkerne. ZFW 10, 242–249 (1962) 41. Benjamin, T.B.: Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14(4), 593–629 (1962) 42. Polhamus, E.C.: Applying slender wing benefits to military aircraft. J. Aircr. 21(8), 545–559 (1984) 43. Smith, J.H.B.: Theoretical modelling of the three-dimensional vortex flows in aerodynamics. AGARD CP-342, Paper 17 (1983) 44. Hoeijmakers, H.W.M.: Computational vortex flow aerodynamics. AGARD CP-342, Paper 18 (1983)

References

355

45. Hirschel, E.H., Krause, E. (eds.): 100 Volumes of ‘Notes on Numerical Fluid Mechanics and Multidisciplinary Design’. NNFM100. Springer, Berlin (2009) 46. Rom, J.: High Angle of Attack Aerodynamics. Springer, New York (1992) 47. Vos, J.B., Rizzi, A., Darracq, D., Hirschel, E.H.: Navier-Stokes solvers in European aircraft design. Prog. Aerosp. Sci. 38, 601–697 (1992) 48. Eriksson, L.-E., Rizzi, A.: Computation of vortex flow around wings using the Euler equations. In: Viviand, H. (ed.) Proceedings of the Fourth GAMM-Conference on Numerical Methods in Fluid Mechanics, Paris, 7–9 October 1981. Notes on Numerical Fluid Mechanics, vol. 5, pp. 87–105. Vieweg-Verlag, Braunschweig (1982) 49. Hoeijmakers, H.W.M.: Computational aerodynamics of ordered vortex flow. Doctoral thesis, Technical University Delft, The Netherlands (1989) 50. Hitzel, S.M.: Zur numerischen Simulation dreidimensionaler, kompressibler Flügelvorderkantenwirbelsysteme in einem Euler-Verfahren (About the numerical simulation of leading-edge vortex systems with an Euler Method). Doctoral thesis, University Stuttgart, Germany (1993) 51. Hentschel, R.: Entwicklung und Anwendung eines dreidimensionalen selbstadaptiven Verfahrens zur Simulation viskoser Strömungen auf der Basis strukturierter Gitter (Development and application of a three-dimensional self-adaptive method for the simulation of viscous flows on the basis of structured grids). Doctoral thesis, University Stuttgart, Germany (1996) 52. Elsenaar, A., Eriksson, G. (eds.): International vortex-flow experiment on Euler code validation. In: Proceedings of the Symposium Stockholm, 1–3 October 1986. FFA, Bromma, Sweden (1987) 53. Drougge, G.: The international vortex flow experiment for computer code validation. In: ICASProceedings 1988, vol. 1, pp. XXXV–XLI (1988) 54. Elsenaar, A., Hjelmberg, L., Bütefisch, K.-A., Bannink, W.J.: The international vortex flow experiment. Vortex Flow Aerodynamics, AGARD-CP-437, 9-1–9-23 (1988) 55. Eberle, A., Rizzi, A., Hirschel, E.H.: Numerical Solutions of the Euler Equations for Steady Flow Problems. Notes on Numerical Fluid Mechanics, vol. 34. Vieweg, Braunschweig (1992) 56. Schwarz, W.: IEPG-TA15 computational methods in aerodynamics, MBB contribution to the theoretical program of Phase 2. MBB-FE122-AERO-MT-833 (1986) 57. Eberle, A., Misegades, K.: Euler solution for a complete fighter aircraft at sub- and supersonic speed. AGARD CP-412, 17-1–17-12 (1986) 58. Hummel, D., Redeker, G.: A new vortex flow experiment for computer code validation. In: Proceedings of the RTO AVT Symposium on Vortex Flow and High Angle of Attack Aerodynamics. Loen, Norway, 7–11 May 2001. RTO-MP-069, Paper Nr. 8 (2002) 59. Chu, J., Luckring, J.M.: Experimental surface pressure data obtained on 65◦ delta wing across Reynolds number and Mach number ranges. NASA TM 4645 (1996) 60. Hummel, D.: The second international vortex flow experiment (VFE-2): objectives and first results. J. Aerosp. Eng. 220(6), 559–568 (2006) 61. Hummel, D.: The second international vortex flow experiment. In: Hirschel, E.H., Krause, E. (eds.) 100 Volumes of ‘Notes on Numerical Fluid Mechanics and Multidisciplinary Design’, NNFM100, pp. 231–240. Springer, Berlin (2009) 62. Luckring, J.M.: The discovery and prediction of vortex flow aerodynamics. Aeronaut. J. 123(1264), 729–804 (2019) 63. Rizzi, A., Luckring, J.M.: Evolution and use of CFD for separated flow simulations relevant to military aircraft. In: Proceedings of the AVT-307 Research Symposium—Separated Flow: Prediction, Measurement and Assessment for Air and Sea Vehicles. Trondheim, Norway, 7–9 October 2019. STO-MP-AVT-307, Paper No. 3 (2019) 64. Jones, R.T.: Properties of low-aspect-ratio pointed wings at speeds below and above the speed of sound. NACA Rep. 835 (1946) 65. Hirschel, E.H.: Neue strömungsphysikalische Aspekte in der Aerodynamik. Physik in unserer Zeit, Jahrg. 15(5), 145–155 (1984) 66. Hummel, D.: On the vortex formation over a slender wing at large angles of incidence. AGARD CP-247, 15-1–15-17 (1978)

356

10 Small Aspect-Ratio Delta-Type Wing Flow

67. Luckring, J.M.: Initial experiments and analysis of blunt-edge vortex flows for VFE-2 configurations at NASA Langley, USA. Aerosp. Sci. Technol. 24, 10–21 (2013) 68. Wentz Jr., W.H., Kohlmann, D.L.: Vortex breakdown on slender sharp-edged wings. J. Aircr. 8(3), 156–161 (1971) 69. Stanbrook, A., Squire, L.C.: Possible types of flow at swept leading edges. Aeronaut. Q. 15(1), 77–82 (1964) 70. Ganzer, W., Hoder, H., Szodruch, J.: On the aerodynamics of hypersonic cruise vehicles in off-design conditions. In: ICAS 1978, Proceedings, vol. 1, pp. 152–161 (1978) 71. Wood, R.M., Miller, D.S.: Fundamental aerodynamic characteristics of delta wings with leading-edge vortex flows. J. Aircr. 22(6), 479–485 (1985) 72. Luckring, J.M., Boelens, O.J.: A unit-problem investigation of blunt leading-edge separation motivated by AVT-161 SACCON research. RTO-MP-AVT-189, 27-1–27-27 (2011) 73. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014) 74. Hentschel, R.: The creation of lift by sharp-edged delta wings. An analysis of a self-adaptive numerical simulation using the concept of vorticity content. Aerosp. Sci. Technol. 2, 79–90 (1998) 75. Elsenaar, A., Hoeijmakers, H.W.M.: An experimental study of the flow over a sharp-edged delta wing at subsonic and transonic speeds. Vortex Flow Aerodynamics, AGARD-CP-494, 15-1–15-19 (1991) 76. Burggraf, U., Ehlers, T.: Aerodynamische Untersuchungen an Doppeldeltaflügelkonfigurationen mit Seitenkante. Strömungen mit Ablösung, DGLR-Bericht 94-04, pp. 185–190 (1994) 77. Hummel, D.: The international vortex flow experiment 2 (VFE-2): background, objectives and organization. Aerosp. Sci. Technol. 24, 1–9 (2013) 78. Luckring, J.M., Hummel, D.: What was learned from the new VFE-2 experiments. Aerosp. Sci. Technol. 24, 77–88 (2013) 79. Fritz, W., Cummings, R.M.: What was learned from the numerical simulations for the VFE-2. AIAA-Paper 2008-399 (2008) 80. Konrath, R., Klein, Ch., Schröder, A., Kompenhans, J.: Combined application of pressure sensitive paint and particle image velocimetry to the flow above a delta wing. In: Proceedings of the 12th International Symposium on Flow Visualization, DLR Göttingen, Germany, 10–14 September 2006, pp. 1–14. Optimage Ltd., Edinburgh, UK (2006) 81. Konrath, R.: Personal communication (2019) 82. Fritz, W.: Numerical simulation of the peculiar subsonic flow field about the VFE-2 wing with rounded leading edge. Aerosp. Sci. Technol. 24, 45–55 (2013) 83. Schütte, A., Lüdecke, H.: Numerical investigations on the VFE-2 65-degree rounded leading edge delta wing using the unstructured DLR TAU-Code. Aerosp. Sci. Technol. 24, 56–65 (2013) 84. Furman, A., Breitsamter, C.: Turbulent and unsteady flow characteristics of delta wing vortex systems. Aerosp. Sci. Technol. 24, 32–44 (2013) 85. Crippa, S.: Advances in vortical flow prediction methods for design of delta-winged aircraft. Doctoral thesis, Kungliga Tekniska Högskolan (KTH), Rep. TRITA-AVE 2008:30, Stockholm, Sweden (2008) 86. Schiavetta, L.A., Boelens, O.J., Crippa, S., Cummings, R.M., Fritz, W., Badcock, K.J.: Shock effects on delta wing vortex breakdown. AIAA-Paper 2008-0395 (2006) 87. Donohoe, S.R., Bannink, W.J.: Surface reflective visualizations of shock-wave/vortex interactions above a delta wing. AIAA J. 35(10), 1568–1573 (1997) 88. Hövelmann, A.: Analysis and control of partly-developed leading-edge vortices. Doctoral thesis, Technical University München, Germany, Verlag Dr. Hut, München (2017) 89. Riedelbauch, S.: Aerothermodynamische Eigenschaften von Hyperschallströmungen über strahlungsadiabate Oberflächen (Aerothermodynamic properties of hypersonic flows past radiation-cooled surfaces). Doctoral thesis, Technische Universität München, Germany, 1991. Also DLR-FB 91-42 (1991)

References

357

90. Riedelbauch, S., Hirschel, E.H.: Aerothermodynamic properties of hypersonic flow over radiation-adiabatic surfaces. J. Aircr. 30(6), 840–846 (1993) 91. Hirschel, E.H.: Basics of Aerothermodynamics, 2nd Revised edn. Springer, Cham (2015) 92. Linde, M.: Experimental test on a planar delta wing at high Mach number and high angle of attack. FFA TN 1988-59, Bromma, Sweden (1988) 93. Crippa, S., Rizzi, A.: Numerical investigation of Reynolds number effects on a blunt leadingedge delta wing. AIAA-Paper 2006-3001 (2006) 94. Hummel, D.: Effects of boundary layer formation on the vortical flow above slender delta wings. In: Proceedings of the RTO AVT Specialists’ Meeting on Enhancement of NATO Military Flight Vehicle Performance by Management of Interacting Boundary Layer Transition and Separation. Prague, Czech Rep., 4–7 October 2004. RTO-MP-AVT-111, 30-1–30-32 (2004)

Chapter 11

Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

The main topic of this book are the basics of separated and vortical flow in aircraft wing aerodynamics. However, like in Chap. 9, we present also here some applicationoriented topics. The implications of in particular vortex breakdown are important not only for the aerodynamic coefficients, but also for their derivatives. They are determining the vehicle’s stability and control characteristics, i.e., the flight properties and the handling qualities, and last, but not least, its flight envelope. These issues are shortly discussed in Sect. 11.1. Consequently we discuss some means to influence lee-side vortex systems by geometrical shaping of a wing, and also by other active and passive means. We concentrate on configurational design aspects when trying to draw benefits from the lee-side vortex induced lift with respect to performance and controllability of the complete configuration. Simultaneously introduced adverse effects—for example regarding the longitudinal stability—have to be analyzed and reduced/eliminated in the process of aircraft (pre-) design. Generation of leading-edge vortex systems at delta wings in many cases will go together with their manipulation. Geometrical measures may become visible as so-called trigger devices at various locations—fuselage, wing, tail unit—of the configuration under consideration. We give overviews in Sects. 11.2, 11.3 and 11.4. Besides the geometrical measures, concentrated and distributed spanwise blowing over the wing upper surface allows to generate vortex systems via “artificial” cross-flow variations and also to stabilize a geometrically introduced vortex system, Sect. 11.5. After that in Sect. 11.6 a step by step modification of a wing geometry is shown which leads to a design with the desired aerodynamic properties. In the review article [1] about control of leading-edge vortices, also rather exotic approaches are treated.

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_11

359

360

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.1 Lift characteristic of a delta wing with lee-side vortices

11.1 Lift and Stability Problems Connected to Lee-Side Vortex Systems of Delta-Type Wings With Figs. 11.1 and 11.2 we recapitulate main insight from the preceding Chap. 10. Figure 11.1 shows the basic phenomenon of lee-side vortices and the non-linear lift due to them. The latter arises already at rather small angles of attack. Figure 11.2 gives in detail, but idealized, the flow, respectively the lee-side primary vortex phenomena occurring at sharp and round-edged thin planar wings. Along the abscissa the leading-edge sweep angle ϕ0 ranges from about 50◦ to 85◦ . Along the ordinate the angle of attack α ranges from about 2◦ to 40◦ . Occurring in these intervals are the interesting vortex phenomena, shown here for the right-hand side of the wing. Indicated are five domains of interest. In domain 0 no lee-side vortex is present, the lee-side flow is attached. Domain 1 indicates the evolution of the lee-side vortex, which with increasing α is moving from the back of the wing—the trailing edge— toward its tip. (Note that indicated in the figure is only the evolution of the primary vortex.) The higher the sweep angle ϕ0 is, the smaller is the angle of attack, at which the lee-side vortex is beginning to appear. This, by the way, is true for all of the phenomena indicated in Fig. 11.2, except for the vortex breakdown and the maximum attainable angle of attack. Domain 2 sees the fully developed lee-side vortex moving inward with increasing angle of attack. In Domain 3 the span-wise location of the vortex is fixed. Domain 4, in reddish color, at its lower bound indicates the beginning of vortex breakdown— αvor tex bur sting —, and at its upper bound the maximum angle of attack αmax .1 Beyond this angle total stall is present, maximum lift is surpassed.

1 Vortex

breakdown is also denoted as vortex bursting.

11.1 Lift and Stability Problems Connected to Lee-Side Vortex Systems …

361

Fig. 11.2 Flow phenomena at sharp- and round-edged thin planar delta wings as function of the leading-edge sweep angle ϕ0 and the angle of attack α. The broken lines constitute the upper bounds of the sharp leading-edge domains 1 and 2

Important is that both the lower and the upper bound of domain 4 indicate that both, the beginning of vortex breakdown and its reaching the wing tip, start at the lowest angle of attack for the wing with smallest leading-edge sweep ϕ0 ≈ 50◦ . With increasing sweep angle the critical angles of attack move from α = 5◦ to approximately 37◦ , respectively from 24◦ to 40◦ . Regarding the vortex overlapping over the wing, Sect. 10.2.5, we note that the influence of the fuselage of a full configuration reduces the likelihood of its occurrence. The effect, of course, then can occur at the aircraft’s forebody. The general problem is that a straightforward transfer of data and behavior of the isolated wing to the full aircraft in general is not possible. Together with the non-linear lift comes an unwelcome effect, the pitch-up tendency of the wing, respectively the aircraft. Figure 11.3 in the upper part shows the coefficient of the pitching moment Cm (α), which builds up like the non-linear lift component, reaching its maximum just ahead of C L max . A proper sizing and positioning of the horizontal stabilizer can correct this tendency. Two other unwelcome effects are indicated in the lower part of that figure, the behavior of the coefficients of the rolling moment Clβ —concerning the dihedral or lateral stability—and the yawing moment Cn β —concerning the directional stability. The divergence of both the rolling moment and the yawing moment happens at the divergence angle of attack αdiv , which is smaller than αmax of the lift. The consequence of the two divergencies may be the loss of the controllability of the aircraft. This comes in terms of aileron reversal, reduced rudder efficiency, and spin

362

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.3 Wing-body-tail configuration: typical idealized curve characteristics of the lift coefficient C L , the pitching-moment coefficient Cm , the rolling-moment coefficient Cl , and the yawing-moment coefficient Cn as function of the angle of attack α. β is the sideslip angle. Lower right part: divergence behavior

danger. Furthermore asymmetries in breakdown between the two wing sides may cause additional moments, which are to be actively controlled. In addition strong unsteadiness of the flow is present, once vortex breakdown happens. The resulting dynamic structural loads also are of great concern. Hence the aerodynamical useful maximum lift cannot be exploited due to these issues. This is demonstrated in Fig. 11.4 for thin wings with sharp leading edge in the low-speed domain. The maximum lift coefficient C Lmax is obtained for a leading-edge sweep angel ϕ0 ≈ 66◦ , upper part of the figure. In the lower part it is shown that for the sweepangle interval ϕ0  50◦ the divergence angle αdiv is distinctly smaller than αmax of the lift. Hence the lift potential of the wing cannot be exploited due to the lateral stability and control problem. Regarding the pitch-up problem, Fig. 11.5 shows an empirical limiting curve for tolerable pitch-up. The curve is based on [3]. The correlation of J.A. Shortal and B. Maggin, [4], lies closely to the left of it. Below the yellow-shaded curve the pitchup of the delta-wing configuration is controllable, above not. Note that wings with

11.1 Lift and Stability Problems Connected to Lee-Side Vortex Systems …

363

Fig. 11.4 Thin delta wings with sharp leading edge in the low-speed domain: maximum lift coefficient C Lmax , maximum angle of attack αmax , and divergence angle of attack αdiv as function of the leading-edge sweep angel ϕ0 (based on [2])

Fig. 11.5 Delta-wing configuration: empirical limiting curve for tolerable pitch-up (based on [3])

364

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

the aspect ratio Λ below the broken line—Λ = 4 tan ϕ0 —are cropped delta wings, like the VFE-1 configuration, Fig. 10.25. Since a long time configurational and other measures are studied and also employed successfully in order to extend the useful angle-of-attack domain of aircraft. Flyability and controllability in the stall and also the post-stall domain are of major concern. Flight in the post-stall domain of course demands thrust-vectoring of the propulsion system, or even control thrusters, like those employed on the former Space-Shuttle Orbiter. In the following sections we discuss shortly some topics in this regard. Always only the longitudinal motion, and stability and control of the flight vehicle or the isolated wing is being considered.

11.2 Wing-Planform Shaping and Optimization Two topics are presented in a compact manner, first the effects of the geometry of a delta wing at low flight Mach number, and then that of small aspect-ratio wings of high-speed flight vehicles.

11.2.1 Effects of the Wing Geometry To generate and/or manipulate leading-edge vortex systems, the geometry of the wing provides the driving parameters. Besides the wing planform (aspect ratio/leadingedge sweep), the wing section characteristics (leading-edge radius/camber/thickness) are contributors. But note that the “father” of the design is the delta-wing (planform). In this context adjustment of the aircraft forebody cross section can play an important role in view of the directional/longitudinal stability at high angle of attack (“shark nose” type forebody, nose strakes). Regarding the wing planform and its modifications, we note that the driving parameter for the development and the stability of leading-edge vortex systems is the leading edge sweep angle ϕ0 . For the pure delta-wing the sweep is coupled to the aspect ratio via Λ = 4/tan ϕ0 . We note: – (a) To develop a “full” leading-edge vortex system over a (sharp-edged) delta-wing the sweep angle should exceed ϕ0 = 53◦ , hence the wing aspect ratio Λ should be less than 3. – (b) When the vortex breakdown is reaching—from the aft of the wing—the trailing edge, lift development is affected by a break. The respective angle is α L B , Fig. 11.6, low-speed data. We note that a break in this form does not appear in all cases. Therefore a break is not indicated in the respective figures in the foregoing section. – (c) Vortex breakdown then will move forward with increasing angle of attack α. Maximum lift is reached for an angle of attack approximately half of the leading-

11.2 Wing-Planform Shaping and Optimization

365

Fig. 11.6 Flat sharp-edged delta wings [2]: maximum angle of attack αmax and angle of attack of lift break α L B as function of the leading-edge sweep angle ϕ◦ . Low-speed wind-tunnel data of a wing-body configuration

Fig. 11.7 Flat sharp-edged delta wings [2]: location of vortex breakdown x/li at αmax as function of the leading-edge sweep angle ϕ◦ . Low-speed wind-tunnel data

edge sweep angle ϕ0 . This behavior is indicated in Fig. 11.6, too. Figure 11.7, again with low-speed data, shows the location of vortex breakdown as function of ϕ0 . The subsequent increase of the separation area over the wing raises the danger of longitudinal and lateral instabilities, both static and dynamic ones.

366

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

11.2.2 Wings of High-Speed Vehicles After having presented these results, relevant for low flight speed, say M∞  0.4, we now turn to higher Mach-number flight vehicles, which generally have wings with small aspect ratio. Two representative flight vehicles were examined in Sect. 10.1, the Concorde, Fig. 10.1 and the Space Shuttle Orbiter, Fig. 10.2. Lee-side vortex phenomena occurring over the wings of high-speed vehicles are correlated with the normal angle of attack α N and the normal leading-edge Mach number M N , Sect. 10.2.6. The two flight vehicles feature the typical sharp—Concorde—and round—Space Shuttle Orbiter—leading edges. We note that the Orbiter at hypersonic flight is outside of the scope of our considerations, but see for such configurations the Blunt Delta Wing case in Sect. 10.7. General aspects of the hypersonic flight regime can be found in [5]. To make use of additional nonlinear lift a basic wing planform can be modified by highly swept leading-edge extensions, positioned inboard at the wing-body juncture. Such “strakes” are found at the Concorde and at the fighter aircraft F-16 and F-18, Fig. 10.5. At the Concorde the ogive wing can be considered in the following way: a highly swept delta-type wing is modified—inboard and in the tip region—by continuously increasing the leading-edge sweep towards the body and the wing tip location. Thus strake-type planform parts are produced at these locations, which smoothly merge into the shape of the main wing, yielding in this way the ogive wing shape. Comparing the wings of the Concorde and the F-16 and F-18, two major geometrical differences are visible: 1. The basic wing of Concorde is a highly swept delta type with low aspect ratio. In contrast to that the fighters have trapezoidal wings with low leading-edge sweep and medium aspect ratios Λ > 3. 2. The inboard strakes differ in area and shape. Additional surface due to the strake is less than one per cent for the Concorde, but around 10 per cent for the fighters, and transition of the leading-edge sweep into the basic wing is abrupt for the fighters. The reason for these differences in wing shapes is found in their design points. The Concorde is a supersonic transport aircraft—without horizontal stabilizer—with a cruise Mach number M∞ = 2. The sweep of its wing is adapted to this Mach number and presents a subsonic leading edge and hence reduced wave drag. At subsonic Mach numbers the configuration benefits from the development of the stable leading-edge vortices and their high nonlinear lift contribution at take-off and landing. In turn the needed angle of attack is reduced there. Note that the ground-effect also increases with decreasing aspect ratio. For the fighters the strake alone is the origin of their high non-linear lift for subsonic/transonic high angle-of-attack maneuvering. The effect exemplarily is shown in Fig. 11.8 [6]. Compared is the maximum lift in terms of C L max and the buffet onset

11.2 Wing-Planform Shaping and Optimization

367

Fig. 11.8 Comparisons of the aerodynamic ranges of maximum lift C L max (left) and buffet-onset lift C L bo (right) of basic wing and basic wing with strake as function of the flight Mach number M∞ [6]. Low-speed wind-tunnel data

lift in terms of C L bo as functions of the flight Mach number M∞ for a trapezoidal wing with and without strake. The geometrical data of the wings are given in the left part of Fig. 11.8: the basic wing has the leading-edge sweep ϕ0 = 32◦ , the strake ϕ0 = 75◦ . The simple strake with an additional area of 10 per cent of the basic wing was in the subsonic region as effective as an increase of the basic-wing area by 50 per cent for the same maximum

Fig. 11.9 Maximum lift C L max of the basic wing with three different strake types as function of the flight Mach number M∞ [6]. Wind-tunnel data

368

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.10 Lift-to-drag ratio L/D of the basic wing with three different strake types as function of the lift coefficient C L at M∞ = 0.5 [6]

lift. Regarding buffet-onset its efficiency increased by more than 50 per cent. Note that in the transonic regime around M∞ = 0.8 the values of C L bo drop, and that by a greater amount for the strake wing. The increase of the lift due to the strake is always combined with a pitch-up tendency of the wing-body configuration. A tailless configuration is not possible in this case, but a low positioned horizontal tail allows for a correction, see the F-16 and the F-18 in Fig. 10.5 as examples. The influence of three strake types on the maximum lift coefficient C L max as function of M∞ is shown in Fig. 11.9. Strake 1 has a straight leading edge, strake 5 (F16) is of Concorde wing type and strake 9 (F-18) a “gothic” one. All strakes are effective below M∞ = 0.9, strake 9 being the most effective one. Regarding the lift-to-drag ratio L/D—the aerodynamic efficiency—of the basic wing with the three strakes, at M∞ = 0.5 strake 5 is the best with L/D = 11.8 at C L = 0.3, Fig. 11.10.

11.3 Wing Sections and Leading-Edge Flaps Besides wing-planform optimization the shaping of wing sections is an approach to improve and optimize the performance of a small aspect-ratio delta-type wings. On the other hand leading-edge flaps also allow to improve the performance of such wings. This section gives short discussions of both topics.

11.3 Wing Sections and Leading-Edge Flaps

369

11.3.1 Wing Sections The chord profile of the wing is usually defined by its nose radius r◦ , the thickness ratio t/L and the camber ratio f /L, L being the reference length of the wing, see Sect. 10.2.1. With respect to lee-side vortex systems the driving parameter from the wing-section side is the physical dimension of the leading-edge radius. The larger this radius, the larger—in terms of the angle of attack—is the domain of attached flow around it, which is fully developed and stable, if the flow is subsonic, Sect. 10.2.4. Note again the importance of the normal angle of attack α N and the normal leading-edge Mach number M N . These are the relevant criteria when we look at the flow past wings with highly swept leading edges, Sect. 10.2.6. The type of separation may be ordinary, locally separation bubbles may be present. If at a delta-type wing a lee-side vortex system is present, its development and existence domain, as well as the vortex breakdown disposition can be taken from the correlations given in Sect. 10.2.6—Fig. 10.22 for wings with sharp leading edges and Fig. 10.23 with round leading edges—and in Sect. 10.2.5. In Fig. 10.23 the correlations correspond to a leading-edge Reynolds number Re N = r N v N /ν =  2·104 . r N , respectively r◦ , is the leading-edge radius, v N is the freestream velocity component normal to the leading edge, ν the kinematic viscosity. If the magnitude of the Reynolds number is above that value, fully developed flow around the leading edge is present. The lower boundary of the leading-edge Reynolds number is Re N ≈ 7·103 . At this number “full” leading-edge suction is beginning to appear. In design aerodynamics of delta-type wings the concept of the leading-edge suction analogy of E.C. Polhamus [7, 8], was and still is used. That concept is based on an analogy between the lee-side suction pressure due to the lee-side vortices at a sharp-edged wing—see for instance Fig. 10.17—and the suction pressure at the leading edge of a blunt-edged wing.2 The concept allows to obtain the so called vortex lift as well as the induced drag due to it. Here it is used in order to quantify the Reynolds-number effect. We give only an overall picture, for details see [7], and also [2]. The lift coefficient results from the suction analogy to CL = CL p + CLv ,

(11.1)

where C L p is the—nota bene linear—potential-theory (Model 4) lift and C L v the lift due to the lee-side vortex system. These two parts are written as

2 The suction-pressure concept goes back to the application of circulation theory (Model 4). Consider

an airfoil at angle of attack. The only force is the lift force L, normal to the freestream direction. Its component in x-direction of the airfoil is Px = −L sinα = S, i.e., the suction force, which is forward directed [9]. It is attributed to the low pressure—the suction pressure—due to the flow around the leading edge of the airfoil, which is a high-speed flow. The higher the angle of attack, the larger is S. In our context it regards the flow around the swept leading edges of delta wings and the resulting non-linear lift due to the lee-side vortex system.

370

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

C L p = K p sin α cos2 α,

(11.2)

C L v = K v sin2 α cos α,

(11.3)

with K p = dC L p /dα and

with K v = (K p −

K 2p

1 ) πΛ cos ϕ0

(11.4)

as derivative of the normal-force coefficient of the by 90◦ rotated, fictitious leadingedge suction force, i.e., K v = dC L /dα2 . Remembering Eq. (8.2), the induced drag can be written in a general way as C Di = K i C L2 ,

(11.5)

with K i being the induced-drag factor. This factor varies between two limits. The first limit is K 1 = 1/π Λ, which is considered to represent full leading-edge suction. Zero leading-edge suction is given in the limit K 2 ≈ 1/(dC L /dα). K 2 is an approximation, because the “exact” induced drag coefficient in this case is C Di = C L tan α. This simple consideration directly shows that K 2 and hence C Di are reduced by the development of the non-linear lift due to the lee-side vortex system, as for a demanded constant lift a lower angle of attack would be necessary. Coming back to the influence of the above leading-edge Reynolds number Re N on the development of the vortex lift, Fig. 11.11 shows the percentage of leading-edge suction S ∗ attained at a cropped delta-wing/body configuration with rounded leading edge as function of the leading-edge Reynolds number Re N in the low-speed regime, see [10] and also [11]. The experimental data were found for dC L /dα values always corresponding to the lift condition C L = C L (L/D)opt . For the Reynolds number Re N  2·104 , the value underlying Fig. 10.23, it is demonstrated that—for this case—the maximum attainable lee-side suction pressure is about 90 per cent. Compared to the situation with sharp leading edges, the development of a stable lee-side vortex system is shifted to higher angles of attack and in consequence the amount of useable nonlinear lift is reduced. Consequently, the induced drag at a given angle of attack reduces with increasing leading-edge radius, which, for instance, may be helpful in view of the specific excess power (SEP) goals of the design process.3 In general, one has to be careful using wind-tunnel data, because Reynolds-number effects can play a large role.

3 The

specific excess power is the power surplus needed/available for the acceleration of the flight vehicle or for its climbing performance.

11.3 Wing Sections and Leading-Edge Flaps

371

Fig. 11.11 Effect of the Reynolds number Re N on the development of the leading-edge suction force [10]. The full line graph is the mean value of measured data S ∗ (hatched area)

11.3.2 Leading-Edge Flaps The effect of the wing section, in particular the leading-edge or nose radius, was the topic of the preceding sub-section. Now the matter of—movable—leading-edge flaps at small aspect-ratio delta-type wings is considered. Such flaps allow to some extent to manipulate the development, position and effects of lee-side vortex systems in the relevant angle-of-attack regime [10, 11]. The manipulation via the extension of leading-edge flaps is done in view of performance, stability around all axes and handling qualities of the considered aircraft. Magnitude (and direction) of the flap deflection should correspond to the intention of finding/producing the most effective angle of attack the leading-edge region is “feeling” for a given flight condition. In the background of the following two figures is the lift condition C L = C L (L/D)opt . Figure 11.12 shows for sharp-edged delta wings with different chordwise airfoil shapes the comparison of the development of leading-edge suction S ∗ as function of the leading-edge sweep angle ϕ0 . The lowest development of S ∗ is found for the symmetric chordwise shape, warping with the shown shape increases S ∗ , and an even stronger increase is found for the nose-flap shape. Up to a leading-edge sweep angle ϕ0 ≈ 40◦ the leading-edge suction is nearly constant, for higher sweep angles it drops markedly. We are treating mainly small aspect-ratio delta-type wings. This means high leading-edge sweep angles ϕ0 . Figure 11.13 shows the development of the leadingedge suction for a rounded leading-edge delta wing with two different chord (camber) shapes. The wing has the leading-edge sweep ϕ0 = 63◦ . Because it is a delta wing, the aspect ratio is Λ = 2 and the factor for induced drag for 100 per cent leading-edge suction hence is K 1 = 1/(π Λ) = 0.16, that for zero suction is K 2 ≈ 0.34. Therefore a loss of 10 per cent of S ∗ will increase K 1 by (K 2 − K 1 )/K 1 × 0.1 ≈ 11 per cent.

372

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.12 Effect of nose flap and wing warp for sharp-edged delta wings as function of the leading-edge sweep angle ϕ0 [10]

Fig. 11.13 Effect of conical camber of a delta wing as function of the free-stream Mach number M∞ [10]

Note that the wing in Fig. 11.13 has a five per cent thick chord section with rounded leading edge, the wings in Fig. 11.12 have sharp leading edges. Our example is found in Fig. 11.12 to the right for ϕ0 = 63◦ . The figure gives at that angle for the symmetric chord section S ∗ ≈ 40 per cent, for the warped one S ∗ ≈ 50 per cent, and for that with the nose flap S ∗ ≈ 62 per cent. The corresponding factors of lift-dependent drag, the actual K 1 -factors, then are 0.268, 0.25, 0.23 (K 1 = 0.16 for 100 per cent S ∗ , see before). We conclude that the

11.3 Wing Sections and Leading-Edge Flaps

373

leading-edge flap has produced a reduction of the lift-dependent drag—for the case of a sharp leading edge—of (0.268 − 0.23)/0.268 = 14.2 per cent. Deeper insights in our theme can be found in the papers by R.K. Nangia et al. [12], J.M. Brandon et al. [13], the latter showing flight results of the F-106B aircraft, and also by N.T. Frink et al. [14].

11.4 Fuselage Forebody Strakes Fuselage forebody strakes are designed to (re-)stabilize the—usually military—deltatype aircraft at high, up to post-stall angles of attack in lateral/directional motion (clβ , cn β ). Deflecting strakes and/or asymmetrical activated strakes allow to generate vortices on the forebody. Correct positioning of such devices on the forebody—x- and zlocation—is mandatory and in many cases a hard task for the aircraft designer. When making use of forebody-strakes, one has to be aware of two imminent dangers: 1. Forebody strakes contribute to static directional stability at high angle of attack, but their effect regarding dynamic stability can be adverse, because they can lead to auto-rotation of the aircraft around its vertical axis (spin). 2. When at sideslip condition the windward forebody-strake vortex (with the strake’s higher effective angle of attack and dynamic pressure) is overlapping the leeward one, the nose-strake effect becomes destabilizing for the aircraft in its lateral motion. Complete knowledge of the flow field around the aircraft therefore is mandatory for success. We show as example a few figures from a study of S.M. Hitzel and R. Osterhuber [15]. Investigated was a supersonic combat configuration in order to extend its maneuverability to very high angles of attack. At the configuration leading-edge vortex flow with mutual interaction and breakdown effects dominate the aircraft’s behavior. URANS simulations around the configuration with all its geometrical details exhibited that asymmetric vortex breakdown and interaction features at side slip were the cause of the limitations of maneuverability. The study finally led to the application of fuselage forebody strakes and leadingedge root extensions (LERX), see below, which considerably enhanced the lateral stability and the attainable angle of attack. All that was confirmed by wind-tunnel and flight tests. Figure 11.14 from the publication [15] shows very detailed the computed flow field over the basic—Standar d—delta-canard configuration at high angle of attack and sideslip. Note that also the deployed slats were modeled and hence the resulting slat vortices were taken into account. The large figure shows the side of the aircraft with the advancing wing, the small figure at the low left shows the aircraft with receding wing (sideslip toward to the

374

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.14 Computed surface pressure, vorticity cross-sections and vortex-core paths at the Standar d configuration at subsonic speed and high angle of attack [15]

port side of the aircraft). This is the same also in the next figure. In both figures it can be seen that the canard vortex (canard wake) wake passes high over the aircraft. The interaction with the lee-side flow field appears to be small. The realized changes on the Standar d configuration in order to improve its behavior were rather small. The narrow rectangular fuselage fence on both sides below the cockpit was replaced by a delta-shaped fuselage strake. Further a leadingedge root extension, connecting the intake-ramp side edge with the wing’s leading edge, was implemented. The combination of the fuselage strake and the LERX led to the desired improvements. The configuration is called the E F E M configuration, which stands for EF2000 Enhanced Maneuverability. A visualization of the resulting flow field is shown in Fig. 11.15. The changes of the flow field compared to that of the Standar d configuration are obvious. The vortex from the intake ramp now appears to be stronger. The vortex axis—yellow color—is not absorbed by the strake vortex, in contrast to the standard case, where it merges with that of the fence vortex. The E F E M fuselage strake vortex—axis in red color—is much stronger than the Standar d vortex and it follows the wing surface closer. The wing’s lee-side vortex system as whole is stabilized and over the wing the non-linear lift is increased. All this is stronger at the advancing wing than at the receding one and leads to the stable rolling moment of the E F E M fuselage. This is demonstrated in the next two figures. Figure 11.16 shows for the subsonic flight domain, how the rolling-moment coefficient cl (α) for the two configurations behaves.

11.4 Fuselage Forebody Strakes

375

Fig. 11.15 Computed surface pressure, vorticity cross-sections and vortex-core paths at the E F E M configuration at subsonic speed and high angle of attack [15] Fig. 11.16 Wind-tunnel results of the rolling moment cl as function of the angle of attack α and the sideslip angles β = 0◦ and ∓ 5◦ of both the Standar d and the E F E M aircraft in the subsonic domain [15]

For the sideslip angle β = 0◦ the configurations in a large angle-of-attack range behave well, but then become uncontrollable. For the Standar d configuration at β = ∓5◦ the rolling moment cl abruptly drops to zero at a lower angle of attack. In contrast to that the E F E M configuration exhibits an acceptable roll-moment evolution to a much larger angle of attack. For the transonic flight domain a similar pattern emerges, Fig. 11.17. At β = 0◦ for both configurations the favorable angle-of-attack range is larger than in the subsonic domain. For β = ∓5◦ the Standar d configuration shows a markedly asymmetric behavior, whereas the E F E M configuration apparently behaves better than in the subsonic case. With this short presentation of selected results from [15] we have shown that simple forebody strake devices can be employed in order to influence and enhance

376

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.17 Wind-tunnel results of the rolling moment cl as function of the angle of attack α and the sideslip angles β = 0◦ and ∓5◦ of both the Standar d and the E F E M aircraft in the transonic domain [15]

the lateral stability of small aspect-ratio delta-type aircraft. For more information and insight regarding forebody strakes and their function the reader is referred to [16, 17] and also the paper of the third author of this book [18].

11.5 Spanwise Blowing The basic ideas regarding the technique of spanwise (concentrated) blowing were published in the late 1960s. They can be attributed—to the knowledge of the authors—to J.J. Cornish and C.J. Dixon, both with Lockheed Corporation, USA [19, 20], and to Ph. Poisson-Quinton from ONERA, France. Therefore this technique can be regarded as the result of a “transatlantic alliance”. The topic “spanwise blowing” was booming in the 1970s. The effects and the efficiency of this technique have been well established from the experimental side, spanning from the improvement of performance—increase of lift and reduction of lift-dependent drag—to stability issues around all axes, depending on the location of the blowing jet (wing, body, tail). Also control augmentation of rudders and ailerons have been a topic. The fourth author of this book was the leader of the German (MBB, DFVLR) and French (ONERA) working group “Wings with Controlled Separation”, which from 1969 to 1978 experimentally investigated wing flows with stable leading-edge vortex systems, and from 1975 to 1982 the effects of concentrated spanwise blowing. In [21] an overview is given on the major results of the work: – The merits and and shortcomings of the technique of concentrated spanwise blowing are discussed regarding the aerodynamic performance as well as stability and control aspects of aircraft. – The limits of the aerodynamic efficiency are established on an empirical/theoretical basis and compared to the experimental results. Upper and lower application boundaries are identified.

11.5 Spanwise Blowing

377

Fig. 11.18 Low-speed modular pilot model [21]

– A view is given regarding the practical (non-) application of this technique and the reasons why the technique did not find application in operational fighter aircraft. Discussed are competitive approaches. The following short overview is on results concerning only the longitudinal motion of the aircraft. We concentrate on the vortex development and manipulation and on the major effects of spanwise blowing. The blowing coefficient cμ is defined in the following way: cμ =

m˙ v j , q∞ Ar e f

(11.6)

where m˙ is the mass flux of the jet, v j the jet velocity—their product being the jet thrust—q∞ the dynamic pressure of the free-stream, and Ar e f the wing’s reference area. The effect of spanwise blowing is demonstrated with a few results, which were found with the modular low-speed pilot model with a hybrid wing, Fig. 11.18. The slat and Fowler flap have downward deflection angles 0◦  δ  90◦ , the aileron downward and upward angles −90◦  δ  90◦ . Table 11.1 gives the main geometrical parameters of the trapezoidal wing and the strake of the modular pilot model. The main parameters of the investigation are collected in Table 11.2, with c being the mean chord length.

378

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Table 11.1 Geometrical parameters of the modular pilot model wing [21]. λt is the taper ratio of the wing ϕ0 (◦ ) Λ λt ϕ0strake (◦ ) 32

3.2

0.3

75

Table 11.2 Low-speed modular pilot model: investigation parameters [21] M∞ Re∞,c 0.2

≈2.1·106

c (m)

Angle of attack

Side-slip angle

Blowing coefficient

0.42

−5◦

−15◦

0  cμ  0.4

α

90◦

β

15◦

Spanwise blowing in this case refers to a concentrated transversal jet. That is directed over the suction side of the trapezoidal wing or the strake and blows roughly parallel to its plane and leading edge. Two applications of spanwise blowing are listed now. They can be employed separately or simultaneously depending on the kind of flow over the wing or the strake. 1. If the wing has the appropriate geometry—a delta wing or a strake planform and the adequate angle of attack—and accordingly a stable lee-side vortex system, the latter can be manipulated by spanwise blowing. Regarding the pilot model, this applies for the strake. 2. If the wing does not have the appropriate geometry, which leads to a lee-side vortex system, spanwise blowing can be employed to generate, stabilize and control a lee-side vortex system. Regarding the pilot model, this applies for the trapezoidal wing.

11.5.1 Blowing at a Pilot Model Without Strake and Forebody Fins The wing of the pilot-model, Fig. 11.18, without strake can be considered as a moderate aspect-ratio/small leading-edge sweep wing. Experiments with this wing as reference configuration were made in order to study the effect of span-wise blowing on the aerodynamic properties [22–28]. The optimum nozzle position on the fuselage side over the wing (indicated at the left wing in the figure)—including the jet-direction (sweep angle and elevation-angle over the wing surface plus the nozzle location above the wing)—was found in an experimental approach. Figure 11.19 shows the results for the longitudinal motion at M∞ = 0.2 for the near optimum jet/nozzle position, here being at 40 per cent root chord (x D /li = 0.4) at the wing/body intersection. The nozzle diameter is d = 15 mm, ϕ D = 15◦ is the nozzle angle against the 40 per cent chord line.

11.5 Spanwise Blowing

379

Fig. 11.19 Effect of spanwise blowing (cμ = 0, 0.05, 0.1, 0.2, 0.4) on lift, pitching moment and drag coefficients [21]. Basic trapezoidal wing, strake and forebody fins off

Analyzing the effects of spanwise blowing on the lift, pitching moment and drag characteristics we find: – Maximum lift and angle of attack increase with increasing cμ , the same tendency is found for the nonlinearity of the lift and the lift curve slope dC L /dα. These effects clearly indicate that a lee-side vortex system is induced by the spanwise blowing. – For low angle of attack a quasi-camber effect is induced for the lift, but without an influence on the pitching moment. Increasing cμ extends the longitudinal stability, and hence the constant location of the neutral point up to higher angles of attack and higher lift. – The lift-dependent—induced—drag is reduced by spanwise blowing. This effect must be solely attributed to the nonlinear lift development, i.e., not to any regain of leading-edge suction.

11.5.2 Blowing at the Pilot Model with Strake and Without Forebody Fins The wing of the pilot model with strake can be considered as a high leading-edge sweep (inboard)/medium aspect-ratio hybrid wing. Also with this wing on the pilot model experimental investigations were made. The optimum position of the jet nozzle was found at 10 per cent of total root chord of the wing (indicated at the right-hand strake in Fig. 11.18). The nozzle was

380

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.20 Effects of wing planform and type of blowing on the lift increment ΔC L /cμ |opt as function of the blowing coefficient cμ [21]

housed again in the body and blowing parallel to the strake’s leading edge (sweep angle ϕ L E strake = 75◦ ) and also parallel to the wing’s surface. The effects and efficiencies of blowing on the wing and on the strake are shown in the next two figures, Figs. 11.20 and 11.21. The results are given in comparison to the results for (a) the basic wing, clean and with strake, Fig. 11.20, and (b) the wing clean and with strake in high-lift configuration—(leading-edge) slats 25◦ down and trailing-edge flaps (single slotted Fowler flaps) 30◦ down, Fig. 11.21. Figure of merit is the ratio of “induced lift increment due to jet” to “blowing coefficient” ΔC L /cμ , hence the capability to transform jet momentum into useful aerodynamic lift. Figure 11.20 gives results of blowing for the clean configuration: (a) upper curve: wing only, (b) middle curve: wing and strake, blowing over the wing only, (c) lower curve: wing and strake, blowing over the strake and the wing. Evidently spanwise blowing always is more efficient on the clean basic wing (upper curve) than on the strake-wing combination (middle curve). This was to be expected. Splitting-up the total blowing momentum on the strake and the wing in equal amounts, say combining the two optimum positions on the strake and the basic wing is unfavorable, lower curve. In each case the absolute lift increment ΔC L goes up with increasing blowing. For all cases shown, the efficiency of spanwise blowing ΔC L /cμ is reduced with increasing cμ . An analogous comparison, now for a high-lift configuration, is given in Fig. 11.21. The upper curve gives the blowing efficiency for the basic wing with slat and Fowler

11.5 Spanwise Blowing

381

Fig. 11.21 High-lift configurations: blowing efficiency ΔC L /cμ |opt as function of the blowing coefficient cμ [21]. Fowler flap δ f = 30◦ , slat δ N = −25◦ , nozzle diameter d = 15 mm

flap and spanwise blowing, the lower curve that for the same basic wing, but with a strake added and blowing over that strake. The two curves clearly demonstrate that spanwise blowing over the wing is most efficient for the basic wing in high-lift condition, which means for the configuration that is “farthest” away from spanwise concentrated vortex development. Blowing is less efficient for the wing with strake. Its absolute lift coefficient ΔC L is always only a little above 50 per cent of that of the wing without strake.

11.5.3 Summarizing Remarks Spanwise blowing is a very interesting technique, seen from a fluid mechanical and aerodynamical point of view.4 Why did it not find application in aircraft, in particular fighter aircraft? The reasons for that are manifold. We do not reproduce the discussion given in [21] regarding the aerodynamic, configurational and mission characteristics opposing spanwise blowing, we only point out some aspects of the issue. Figure 11.22 gives a comparison of the effects of wing geometry, high flap and slat settings—see legend of Fig. 11.21—and spanwise blowing with respect to the achieved maximum lift increment ΔC L max . The reference of the comparison is the pilot model with the trapezoidal wing in clean configuration. The winner in any case is the combination of trapezoidal wing and strake, not necessarily with blowing. Spanwise blowing is bound to low flight-speed applications. This becomes evident when examining Eq (11.6) in more detail. Remembering that the dynamic pressure can be written as 4 We

note that detailed investigations of the ensuing flow changes with discrete numerical methods (Model 10 and 11 of Table 1.3) would be desirable, at least from an academic point of view.

382

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.22 In reference to the clean trapezoidal wing: effects of strake, high-lift system (slat and Fowler flap) and spanwise blowing in terms of ΔC L max [21]. Left side: trapezoidal wing alone. Right side: trapezoidal wing in combination with strake. The indicated angles of attack α are those at maximum lift

q∞ =

1 1 2 ∞ u 2∞ = γ p∞ M∞ , 2 2

(11.7)

we find that the increment due to blowing decreases with increasing flight Mach number. Keeping in mind that pressure and mass-flux of the blowing devices are limited by the aircraft’s engine characteristics, it is understandable that spanwise blowing anyway is limited to low flight Mach numbers. Another aspect is that when spanwise blowing was investigated, already the combination low/medium aspect-ratio trapezoidal wing and strake was in full development. It then found its application for instance with the F-16 and the F-18, Fig. 10.5. Spanwise blowing, however, should be seen in the wider context of “jets in cross flow”. The AGARD Symposium on “Computational and Experimental Assessment of Jets in Cross Flow”, held from 19th to 22nd April 1993 in Winchester, United Kingdom, had shown the breadth of jet-in-cross-flow applications [29]. Topics like thrust vectoring, jet impingement, hypersonic cross flow, vehicle control, internal flow, and turbine cooling were presented and discussed. We would like to single out the hypersonic cross-flow jet topic. On the re-entry trajectory of the Space Shuttle Orbiter reaction control devices were active in order to

11.5 Spanwise Blowing

383

control roll, pitch and yaw partly down to angles of attack α ≈ 20 to 25◦ at M∞  5, when the aerodynamic stabilization, trim and control surfaces had left the hypersonic shadow of the fuselage and the wing, see, e.g., [5].

11.6 Design Example This section is devoted to a short presentation of a study regarding the pitch-up problem of slender wings, i.e., delta-type wings, as it was discussed very detailed in [2]. The study was based on an experimental investigation of a fighter configuration with a double-delta wing. The approach was to carefully and comprehensively determine the connection of the observed pitch-up phenomena with the occurring lee-side vortex structures. These basically multi-vortex systems were observed by means of oil-flow pictures, thread tufts and vapor-screen photographs. Six-component force measurements were made in order to determine the corresponding aerodynamic force and moment coefficients.

11.6.1 Basic Configuration The basic configuration of the supersonic light-weight fighter design (M∞ = 1.4) was a (tailless) wing-body-vertical stabilizer configuration, similar to that of the F16XL. The width of the fuselage was approximately 15 per cent of the total span of the model. Starting point for the layout of the wing geometry was a plane, 3.4 per cent thick double-delta wing with an inboard leading-edge sweep angle of ϕ0,i = 70◦ , an outboard sweep angle of ϕ0,a = 50◦ with a leading-edge kink at 69 per cent half span, see the inserted wing plan view in Fig. 11.23. The sweep angle of the trailing edge was slightly negative with ϕT E = −6.50◦ and constant along the wing’s span. Design problems were found in particular for the subsonic flight regime—say M∞ < 0.6—mainly concerning an intolerable pitch-up tendency. We show in Fig. 11.23 for the basic wing configuration the pitching moment in four different angle-of-attack regimes together with the observed multi-vortex systems in the Poincaré surfaces at x/L = 0.9.5 Four characteristic regimes of the development of the pitching moment Cm (α) can be distinguished, with the assigned structures of the velocity-field topology to a degree being arguable:

5 Note that at the lower (windward) side the primary attachment line always is assumed to be located

at the middle of the wing, see the discussion in Sects. 7.4.3 and 7.5. Note further that focus points are nodal points and therefore denoted with N , instead of F, as for instance in Sect. 7.4.3.

384

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.23 The basic wing configuration at M∞ = 0.2 [2]. The pitching-moment behavior Cm (α) and the flow-topological structures in the Poincaré surface at x/L = 0.9 (horizontal broken line in the inserted wing plan view) at four angle-of-attack positions of the wing

– (1) A linear development regime with a constant negative dCm /dα. Longitudinal stability is given for the angle-of-attack domain 0◦  α  10◦ . Above the wing we have an open lee-side flow field, Sect. 7.4.3. Regarding the topology of the velocity field we observe two primary vortices in the Poincaré surface, one due to the inner delta wing with the focus point N1 , the other due to the outer delta wing with the focus point N2 . Below these vortices we see a secondary vortex each.

11.6 Design Example

385

– (2) Pitch up develops in the angle-of-attack domain 10◦  α  20◦ . Above the wing still an open lee-side flow field exists. The vortex of the outer delta wing moves faster up than that of the inner delta wing. Both are now coupled via the free saddle point S. The outer vortex moves toward the inner and begins to break up near the wing’s trailing edge (not indicated in the figure). Both vortices then begin to rotate around the free saddle point, which in the experiment was observed as wiggly line of the vortex centers (vapor-screen pictures and thread tufts). The summary effect of the vortex development moves the non-linear lift effect forward, which leads to the pitch-up bend of the Cm curve. – (3) The maximum instability present at α = 20◦ is preserved up to the stall regime 20◦  α  30◦ . Regarding the topology of the velocity field we observe first of all that now a closed lee-side flow field exists. This means that the whole upper wing surface now is dominated by the separated flow. The vortex of the outer wing part has lost the contact to the leading edge. It almost completely has undergone breakdown. Obviously a balance is present between the increase of nonlinear lift and the forward movement of the center of pressure. The result is that pitching moment still grows, but that the gradient dCm /dα is nearly constant. – (4) Above α ≈ 28◦ the gradient dCm /dα begins to decrease and finally a high degree of stability is recovered, which leads to pitch-down beyond αmax . With the approach to the maximum angle of attack αmax only one lee-side vortex system—that from the inner delta part—is present. The secondary vortex below it is barely observable in the oil-flow and the vapor-screen picture. Vortex breakdown has moved far forward. The Poincaré surface actually does no more correspond with the location x/L = 0.9, but with a much more forward lying location. What happens, if the angle-of-attack is increased beyond αmax ? The flow pattern approaches that past a vertically located flat plate. The resulting normal force slips back into the center of the surface. The configuration becomes highly stable. Above αmax  40◦ the danger of “deep stall” arises, if the aircraft—due to insufficient control-surface function—restabilizes and a balanced flight attitude with Cm = 0 is established.

11.6.2 Modifications of the Wing Geometry To solve, respectively reduce the inherent pitch-up problems, the effects of some geometrical measures were investigated with wind-tunnel tests in the low-speed domain. The most promising variants were then checked with a high-speed model in the domain 0.5  M∞  2.0. The geometric measures are: – (a) Change of the leading-edge sweep in the apex region, the main wing, the leading-edge extension and the outboard wing.

386

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.24 Effect of the modifications of the wing planform on the pitch-up behavior [2]. Upper part: wing planforms with (1) basis, (2) and (3) apex modification, (4) apex and leading-edge modifications. Lower part: the effects of the modifications

– (b) Change of the outboard wing span. – (c) Placing of a trigger-device, a “fence”, on the wing kink. – (d) Variation of the anhedral of the outboard wing (downward) with νa = 0◦ /15◦ /30◦ /45◦ . These modifications are shown in Fig. 11.24 (upper part) together with their effects on the pitch-up behavior (lower part). To quantify the effects, the C L -Cm graph was employed. In that way the location of the neutral point x N xcog − x N dCm = lμ dC L

(11.8)

can be used to show the effect. Here xcog is the location of the center of gravity, and lμ the mean aerodynamic chord length. In order to compare the effect of

11.6 Design Example

387

the modifications, all configurations were put to the same measure of instability (dCm /dC L )|C L →0 = 0.03 or 3 per cent. The effects on the pitch-up tendency are the following: – The modification of the inner delta wing—configuration (2)—yields a reduction of the measure of the pitch-up tendency to 6 per cent, compared to 9 per cent of the original configuration (1). The effect goes together with a weak unstable break. That is the excursion from the nearly straight-line graph, marked with a fine hatching. – If the leading-edge sweep of the inner delta-wing modification of 55◦ is reduced to 48◦ —configuration (3)—the measure of the pitch-up tendency is further reduced to 4.6 per cent. – Configuration (4) is configuration (2), but with a small leading-edge extension (LEX). That is located just ahead of the outer delta wing with a final sweep of 90◦ at aft the kink-point with the outward wing. It has the lowest measure of the pitch-up tendency of the configurations with 3.8 per cent. The result is that any of these modifications is helpful in reducing the basic (strong) pitch-up tendency of 9 per cent. A trigger-device—a fence at the kink-point between the outboard and the inboard wing—had a detrimental effect. It led to a too high distortion of the inboard vortex system (not shown here).

11.6.3 The Final Configuration The final configuration was found after introducing two further variations to configuration (4), now of the outboard wing: 1. modifications of the anhedral angle νa , and/or 2. the leading-edge sweep ϕa Figure 11.25 presents the effects of the two variations. Changing the anhedral angle is very effective, left part of the figure. With νa = 30◦ the pitch-up tendency has almost disappeared, and fully with νa = 45◦ . The anhedral angles lead to a stable vortex outboard of the kink, introduced by the spanwise velocity components of the inboard leading-edge vortex system. Also helpful for the longitudinal stability is a reduction of outboard leading-edge sweep, right side of Fig. 11.25. This, however, is excluded due to supersonic design aspects. It should be noted that the anhedral reduces the effective angle of attack with its increase by a quasi twist effect of the outboard wing. Reduction of the outboard wing span/aspect ratio also reduces the proneness of pitch-up, but is not tolerable due to performance aspects. Finally—combining and extrapolating the effects from above—the wing, respectively the aircraft configuration with the desired longitudinal stability properties was found with Λ = 1.6, inner and middle leading-edge sweep 50◦ /67◦ plus leading-edge extension with 50◦ sweep, the anhedral position at 69 per cent span and the anhedral angle νa = 30◦ .

388

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

Fig. 11.25 Effect of anhedral-angle νa and leading-edge sweep ϕ0a modifications of the outer wing on the pitch-up behavior of wing No. 4 from Fig. 11.24 [2]. Upper part: wing planforms and anhedral angle modifications. Lower part: the effects of the modifications

Figure 11.26 shows the resulting pitching-moment behavior again in four different angle-of-attack regimes in combination with the observed flow-topological structures in the Poincaré surfaces at x/L = 0.9. We look again at four characteristic trends of the development of the pitching moment Cm (α) together with the structures of the velocity-field topology. The new structures are noted now with a dashed line. – (1’) A linear development regime with a constant negative dCm /dα. Above the wing again an open lee-side flow field is present. From the kink of the lowered outboard wing with the anhedral angle νa = 30◦ via flow-off separation a vortex with the focus point N3 develops. Compared to the original secondary

11.6 Design Example

389

Fig. 11.26 The final wing: flow-topological structures in the Poincaré surface and pitching-moment behavior Cm (α) at four angle-of-attack positions [2]

vortex—its focus point denoted N in (1) in Fig. 11.23—this vortex is stable and with increasing angle of attack keeps its location above the outboard wing. In the experiments it was observed that to achieve this, obviously a minimum size is necessary of the anhedral angle of about νa = 15◦ . The overall topological structures (1) and (1’) are the same. The numbers of free nodal points and saddle points are the same. – (2’) The linear development regime of the pitching moment persists. The flow topology basically is the same as that in the regime (1’). The tendency seen in (2) of the vortices N1 and N2 to rotate around each other has disappeared.

390

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

– (3’) Still linear development up to α ≈ 28◦ , but a weak pitch-up. Regarding the topology of the velocity field we observe that still an open lee-side flow field exists. The outer vortex has moved up, the inner toward the wing’s center. The gradient dCm /dα is nearly constant, but shows a 0.2 per cent pitch-up. – (4’) Above α ≈ 28◦ the gradient dCm /dα begins to drop strongly. The overall flow topology with a closed lee-side flow field is almost the same as observed in (4). The secondary vortex is not observable. Vortex breakdown again begins around α L B = 27◦ . Regarding the performance of this final configuration we note that the wing has a more or less linear characteristic of the pitching moment with only a very small pitch-up tendency. The maximum lift is slightly reduced with C L max = 1.3 at α = 32◦ , compared to C L max = 1.37 at α = 34◦ of the original wing. The modification of the aerodynamic load distribution led at the same total load to a reduction of the wing-root bending moment of about 12 per cent. Overall an improvement of the flight performance is given. The potential of flight with artificially controlled longitudinal stability is enlarged. Acceleration and steady and unsteady turning performances are enhanced as well as the flight characteristics. For the full discussion the reader is referred to [2]. The knowledge and the interpretation of the topologies of the velocity fields in Poincaré surfaces revealed the vortical flow properties as the driving influences for the configuration development. Figure 11.27 shows the photo of the high speed model of a possible aircraft configuration.

Fig. 11.27 A possible aircraft configuration with the final wing design [2]

11.7 Problems

391

11.7 Problems Problem 11.1 Consider a delta wing at angle of attack α < α L B with lee-side vortices. When α L B is reached, vortex breakdown begins to happen at the rear of the wing. Assume that it occurs symmetrically. Which aerodynamic properties of the wing are being affected and how? What is the flow-physical effect? Problem 11.2 Consider the divergence of the rolling moment Clβ and the yawing moment Cn β . What are the basic dependencies on (a) the general flight conditions, (b) the flow structure over the wing? Problem 11.3 Give the reasons for the planform geometry found for the Concorde (take-off → cruise) with respect to (a) flight performance and (b) stability and handling qualities. Problem 11.4 Compare Figs. 11.12 and 11.13 for the relative effects of (a) leadingedge radius, (b) camber, and (c) nose flap for the configuration with the aspect ratio Λ = 2, the flight Mach number M∞ = 0.3, and the leading-edge sweep angle ϕ0 = 63◦ . Problem 11.5 What are the—positive—effects of concentrated spanwise blowing? Which aerodynamic properties of an aircraft/wing are affected? What are the reasons for its “non-application”? Problem 11.6 Explain the positive effects of anhedral of the outboard wing (Sect. 11.6) via consideration of the flow structure (topological aspects). Discuss Figs. 11.23 and 11.26 .

References 1. Gursul, I., Wang, Z., Vardaki, E.: Review of flow control mechanisms of leading-edge vortices. Prog. Aerosp. Sci. 43(3), 246–270 (2007) 2. Staudacher, W.: Die Beeinflussung von Vorderkantenwirbelsystemen schlanker Tragflügel (The Manipulation of Leading-Edge Vortex Systems of Slender Wings). Doctoral Thesis, University Stuttgart, Germany (1992) 3. Gottmann, Th., Groß, U., Staudacher, W.: Flügel kleiner Streckung, Teil 1: Grundsatzuntersuchungen, Band 1: Analysebericht. MBB/LKE127/S/R/1563 (1985) 4. Shortal, J.A., Maggin, B.: Effect of Sweepback and Aspect Ratio on Longitudinal Stability Characteristics of Wings at Low Speeds. NACA TN-1093 (1985) 5. Hirschel, E.H., Weiland, C.: Selected Aerothermodynamic Design Problems of Hypersonic Flight Vehicles. Progress in Aeronautics and Astronautics, AIAA, Reston, VA, vol. 229. Springer, Berlin, Heidelberg (2009) 6. Staudacher, W.: Zum Einfluss von Flugelgrundrissmodifikationen auf die aerodynamischen Leistungen von Kampflugzeugen. Jahrestagung DGLR/OGFT, Innsbruck, Austria, DGLR Nr. 73–71, 24–28 (1973)

392

11 Selected Flow Problems of Small Aspect-Ratio Delta-Type Wings

7. Polhamus, E.C.: A Concept of the Vortex Lift of Sharp Edge Delta Wings, Based on a LeadingEdge Suction Analogy. NASA TN D-3767 (1966) 8. Polhamus, E.C.: Application of the Leading-Edge Suction Analogy of Vortex Lift to the Drag Due to Lift of Sharp-Edged Delta Wings. NASA TN D-4739 (1968) 9. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, vol. 1 and 2, Springer, Berlin/Gättingen/Heidelberg, 1959, also: Aerodynamics of the Aeroplane, 2nd edn. (revised). McGraw Hill Higher Education, New York (1979) 10. Staudacher, W.: Abschätzung des Reynolds-Zahl-Einflusses auf den induzierten Widerstand schlanker Flügel. MBB-UFE122-Aero-Mt-399 (1980) 11. Henderson, W.P.: Effects of Wing Leading Egde Radius and Reynoldy Number on Longitudinal Aerodynamic Characteristics of Highly Swept Wing-Body Configurations at Subsonic Speeds. NASA TN D-8361 (1976) 12. Nangia, R.K., Miller, A.S.: Vortex flow dilemmas and control of wing planforms for high speeds. In: Proceedings RTO AVT Symposium on Vortex Flow and High Angle of Attack Aerodynamics. Loen, Norway, May 7 to 11, 2001. RTO-MP-069, Paper Nr. 9 (2002) 13. Brandon, J.M., Hallissy, J.B., Brown, P.W., Lamar, J.E.: In-flight flow visualization results of the F-106B with a vortex flap. In: Proceedings RTO AVT Symposium on Vortex Flow and High Angle of Attack Aerodynamics. Loen, Norway, May 7 to 11, 2001. RTO-MP-043, Paper Nr. 43 (2002) 14. Frink, N.T., Huffman, J.K., Johnson Jr., T.D.: Vortex Flap Flow Reattachment Line and Subsonic Longitudinal Aerodynamic Data on 50◦ to 74◦ Delta Wings on Common Fuslage. NASA TM 84618 (1983) 15. Hitzel, S.M., Osterhuber, R.: Enhanced maneuverability of a delta-canard combat aircraft by vortex flow control. J. Aircraft 55(3), 1–13 (2017) 16. Erickson, G.E., Gilbert, W.P.: Experimental Investigation of Forebody and Wing Leading-Edge Vortex Interactions at High Angles of Attack. AGARD-CP-342, 11-1–11-28 (1983) 17. Malcolm, G.N., Skow, A.M.: Enhanced Controllability Through Vortex Manipulation on Fighter Aircraft at High Angles of Attack. AIAA-Paper 86–277 (1986) 18. Breitsamter, C.: Strake effects on the turbulent fin flowfield of a high-performance fighter aircraft. In: W. Nitsche, H.-J. Heinemann, R. Hilbig (eds.), New Results in Numerical and Experimental Fluid Mechanics II. Contributions to the 11th AG STAB/DGLR Symposium Berlin, Germany 1998. Notes on Numerical Fluid Mechanics, vol. 72, pp. 69–76. ViewegVerlag, Braunchweig, Wiesbaden (1999) 19. Cornish, J.J.: High Lift Applications of Spanwise Blowing. 7th ICAS Congress, ICAS Paper 70-09 (1970) 20. Dixon, C.J.: Lift and Control Augmentation by Spanwise Blowing Over Trailing Edge Flaps and Control Surfaces. AIAA-Paper 72–781 (1972) 21. Staudacher, W.: Effects, Limits and Limitations of Spanwise Blowing. AGARD-CP-534, 261–26-10 (1993) 22. Staudacher, W.: Flügel mit kontrollierter Abläsung. DGLR Nr. 77–028, 24–28 (1977) 23. Staudacher, W., Egle, S., Boddener, W., Wulf, R.: Grundsätzliche Untersuchungen über spannweitiges Blasen und stabilisierten Wirbelauftrieb. MBB, Ottobrunn, I.B. 77-125/UFE 1320 (1977) 24. Staudacher, W.: Interference Effects of Concentrated Blowing and Vortices on a Typical Fighter Configuration. AGARD-CP-285, 19-1–19-13 (1980) 25. Staudacher, W.: Influence of Jet-Location on the Efficiency of Spanwise Blowing. 12th ICAS Congress, ICAS Paper 13-02 (1980) 26. Staudacher, W.: Verbesserung der aerodynamischen Leistungen durch konzentriertes Ausblasen. MBB, Ottobrunn, FE 122 /S/R 1499 (1980) 27. Gottmann, Th., Hünecke, K., Staudacher, W.: Einfluss des Spannweitigen Blasens auf die Aerodynamischen Leistungen. Handbuch der Luftfahrttechnik (LTH) (1986) 28. Huffman, J.K., Hahne, D.E., Johnson Jr., T.D.: Aerodynamic effect of distributed spanwise blowing on a fighter configuration. J. Aircraft 24(10), 673–679 (1987) 29. N.N.: Computational and experimental assessment of jets in cross flow. In: Proceedings of the AGARD Symmposium, Winchester, UK, April 19–22, 1993. AGARD-CP-534 (1993)

Chapter 12

Solutions of the Problems

Useful relations are given in Appendix A and and constants and atmospheric data in Appendix B. Perfect gas is assumed throughout with γ = 1.4.

12.1 Problems of Chapter 2 Problem 2.1 The speed of sound is a∞ = 299.5 m/s, the flight speed u ∞ = 149.8 m/s at M∞ = 0.5 u = 4.247·106 and u ∞ = 239,6 m/s at M∞ = 0.8. The unit Reynolds numbers are Re∞ u 6 for M∞ = 0.5 and Re∞ = 6.796·10 for M∞ = 0.8. For the viscosity we use the power-law approximation with ω = 0.65. At the location x/c = 0.5 we obtain for M∞ = 0.5 the wall shear stress τw = 0.94 N m−2 and the displacement thickness δ1 = 0.0013 m, and for M∞ = 0.8 τw = 1.89 N m−2 and δ1 = 0.0011 m. At the location x/c = 1 for M∞ = 0.5 the results are τw = 0.67 N m−2 and δ1 = 0.0019 m. and at M∞ = 0.8 τw = 1.33 N m−2 and δ1 = 0.0016 m. Problem 2.2 For turbulent flow we find at the location x/c = 0.5 for M∞ = 0.5 τw = 10.2 N m−2 and δ1 = 0.0046 m, and for M∞ = 0.8 τw = 23.8 N m−2 and δ1 = 0.0042 m. At the location x/c = 1 for M∞ = 0.5 the results are τw = 8.91 N m−2 and δ1 = 0.0079 m. and at M∞ = 0.8 τw = 20.8 N m−2 and δ1 = 0.0074 m. Problem 2.3 For laminar flow we find at the location x/c = 0.5 for M∞ = 0.5 the wall shear stress τw = 0.877 N m−2 and the displacement thickness δ1 = 0.0018 m, and for M∞ = 0.8 τw = 1.75 N m−2 and δ1 = 0.0015 m. At the location x/c = 1 for M∞ = 0.5 the results are τw = 0.62 N m−2 and δ1 = 0.0026 m. and at M∞ = 0.8 τw = 1.24 N m−2 and δ1 = 0.0022 m. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3_12

393

394

12 Solutions of the Problems

Problem 2.4 For turbulent flow we find at the location x/c = 0.5 for M∞ = 0.5 τw = 8.03 N m−2 and δ1 = 0.0052 m, and for M∞ = 0.8 τw = 18.1 N m−2 and δ1 = 0.0048 m. At the location x/c = 1 for M∞ = 0.5 the results are τw = 6.99 N m−2 and δ1 = 0.009 m. and at M∞ = 0.8 τw = 15.7 N m−2 and δ1 = 0.0084 m. Problem 2.5 The results at once show that the skin friction decreases with increasing x, whereas the displacement thickness increases. This of course can directly be seen when looking at the equations. The Mach number does not have directly an effect. The effect is due to the fact that the Reynolds number increases with increasing Mach number. Important is that with increasing wall temperature a strong decrease of the skin friction happens, but only for the turbulent case. This is a typical thermal-surface effect, which is of special importance with high supersonic and hypersonic flow [1]. Regarding the total drag of, for instance, an airfoil, the skin-friction drag can be reduced with an increase of the wall temperature. But because the displacement thickness increases, the viscous-effects induced pressure drag, i.e., the form drag, increases too. Hence one has to look at the overall effect of a temperature increase. Problem 2.6 At H = 10 km altitude we have the speed of sound a∞ = 299.5 m/s and the density ρ∞ = 0.4135 kg/m3 . The flight velocities are u ∞ = 59.9 m/s, 119,8 m/s, 239,6 m/s and the drag at these velocities D = 4.45 N, 17.8 N, 71.2 N. The drag increases quadratically with the flight speed. Problem 2.7 The Prandtl–Glauert rule reads Clc = 

Clic 2 1 − M∞

.

The incompressible lift coefficient results to Clic = 0.1908, and the compressible coefficients are Clc = 0.194, 0.208, 0.31. Problem 2.8 L = 143,9 N, 617.3 N, 3,680.3 N. Problem 2.9 To arrive at actual forces, the force coefficients have to be multiplied with the dynamic pressure q∞ and the reference area Ar e f . The forces hence depend on the square of the flight velocity and the density at the flight altitude. The problems reflect the situation in the compressible flight domain below the critical Mach number. There the drag coefficient of an airfoil or wing is more or less independent of the Mach number and the lift coefficient changes with the Mach number according to the Prandtl–Glauert rule.

12.2 Problems of Chapter 3

395

12.2 Problems of Chapter 3 Problem 3.1 (a) Model 4, circulation theory. (b) The circulation Γ0 is constant on all four legs. (c) Helmholtz’s theorems. (d) The two-dimensional airfoil is a wing of infinite span. Hence the trailing vortices are located at y = ±∞. (e) The situation is the same as at the airfoil. Problem 3.2 Steady level flight: lift = weight. For the flight altitude of H = 10 km, the atmosphere data (Table B.2) are: p∞ = 26,500 Pa, ρ∞ = 0.4135 kg/m 3 . Hence, at a Mach number of M∞ = 0.82 the flight velocity is u ∞ = 245.6 m/s (with γ = 1.4). Referring to force equilibrium, the lift coefficient can be calculated: L = m g. C L = 2mg/(ρ∞ u ∞ b2 ) = 0.45. Using Eq. (3.40), and taking into account the spanwise load factor for an elliptical circulation distribution of s = π/4, the induced downward velocity w0 is (it can be also calculated directly inserting the expression for C L ): w0 = 1.59 m/s. The vertical distance Δz of the wake vortex downward movement for a flight path length of Δx = 50 km then is (Δx = u ∞ Δt, Δz = w0 Δt): Δz = w0 /u ∞ Δx = 324 m. Problem 3.3 Applying Eq. (3.38), the root circulation Γ0 is obtained to Γ0 = 475.34 m 2 /s. The expressions for the tangential (circumferential) velocity vθ related to the Lamb-Oseen vortex model and the Burnham-Hallock vortex model are given in Eqs. (3.17) and (3.21). At the core radius r = rc we find then for the first one vθ,max = 30,1 m/s (β = 1.256), and for the second one vθ,max = 20,0 m/s. The difference reflected by the ratio v L O /v B H = 2 (1 - e−β ) = 1.43 is attributed to the approach of characterizing the vortex core structure: the Lamb-Oseen model as an analytical solution accounting for a temporal vortex core growth due to viscosity, whereas the Burnham-Hallock model relaxes the peak velocity at the viscous core by quadratic terms. Problem 3.4 Applying the Kutta–Joukowsky relation with the lift L is the same for the spanwise circulation distribution Γ (y) and a distribution with spanwise constant Γ0 we obtain via Eq. (3.34) with L = ρ∞ u ∞ Γ0 b0 : 1 L = ρ∞ u ∞ b and



+b/2 −b/2

1 Γ (y)dy = ρ∞ u ∞ Γ0 b



+b/2

−b/2

  2y 2 1 − ( ) dy b

396

12 Solutions of the Problems

Fig. 12.1 Biconvex circular arc airfoil 10 per cent thick

1 b0 = b



+b/2 −b/2

   +b/2   2y 2 4 y3 2 1 1 − ( ) dy = y − 2 = b, =b 1− b b 3 −b/2 3 3

and finally s=

b0 2 = . b 3

The lateral distance b0 of the rolled-up wake vortex system or the center of the free circulation centroid, respectively, is located at 66 per cent of the wing span b. Consequently, the distance between the trailing vortex trajectories is smaller compared, for instance, to that related to an elliptical circulation distribution (b0 = 0.785 b, Eq. (3.37)). The parabolic circulation distribution also reflects a more inboard loading. The reduced lateral distance may result in an earlier vortex trajectory contact caused by the Crow instability when progressing downstream and, therefore, an earlier wake vortex decay (Fig. 12.1). Problem 3.5 As indicated in the sketch, the radius for a circle intersecting the chord line in x = ±0.5 and the z-axis in z = 0.05 is R 2 = 0.52 + (R − 0.05)2 ;

R = 2.525.

The equations for the circular contour and its slope are x 2 + (2.525 − 0.05 + z)2 = 2.5252 ,

 z = −2.475 + 6.38 − x 2 , dz x = −√ . dx 6.38 − x 2

The vertical velocity in the first collocation point w1 consists of contributions from all three source distributions w1 = w11 + w12 + w13 . The boundary conditions

12.2 Problems of Chapter 3

397

wi = u ∞ (

dz )i dx

gives the source intensities w1 = σ1 /2; σ1 /u ∞ = 2(dz/d x)1 = 2 ∗ 0.1331 = 0.2662. σ2 /u ∞ = 0. w2 = 0; w3 = σ3 /2; σ3 /u ∞ = −0.2662. From the known source intensities the longitudinal disturbance velocities can be calculated: u 11 = 0; collocation point is in the middle of source distribution. u 12 = 0; second source distribution has zero intensity. σ3 u 13 = 2π ln| 1/6−2/3 | = 0.0216 u ∞ . 1/6−1 In small disturbance theory c p = −2 u/u ∞ , hence c p = −0.0432. Problem 3.6 The lift force may be written alternatively as L = ρu ∞ Γ 1 1 L = ρu 2∞ C L c = ρu 2∞ 2παc. 2 2 Thus Γ = u ∞ παc. Problem 3.7 The vertical velocities in the collocation points are: 1 1 + Γ2 , 2πc/2 2π(c/2 + c) 1 1 w2 = u ∞ α − Γ 1 − Γ2 , 2π(3c/2 + c) 2πc/2 w1 = u ∞ α − Γ1

w should be zero. Substitute u ∞ απc with A:

398

12 Solutions of the Problems

A = Γ1 − A=

1 Γ2 , 1 + 2

1 Γ1 + Γ2 . 3 + 2

The solution is 3 + 2 , 2(1 + ) 1 + 2 Γ2 = A . 2(1 + ) Γ1 = A

And because the lift force is L = ρu ∞ Γ , then 1 3 + 2 . 2 1+ 1 1 + 2 . 2παc) = 2 1+

L 1 /(1/2ρ∞ u 2∞ 2παc) = L 2 /(1/2ρ∞ u 2∞

The lift on a single airfoil would be L ∞ = 1/2ρu 2∞ 2πα, and one may write 1 3 + 2 , 2 1+ 1 1 + 2 . = 2 1+

L 1 /L ∞ = L 2 /L ∞

Problem 3.8 The speed of sound is a∞ = 299.5 m/s, the flight speed u ∞ = 239.6 m/s at M∞ = 0.8. The results are: lift coefficient C L = 0.413, induced drag C Di = 0.00861, total drag C D = 0.0258, lift-to-drag ratio L/D = 16.

12.3 Problems of Chapter 4 Problem 4.1 With the relations from Sect. A.5.4 we find for the laminar case: x = 2.5 m, δ/x = 0.0015, x = 5 m, δ/x = 0.0011, and for the turbulent case: x = 2.5 m, δ/x = 0.015, x = 5 m, δ/x = 0.013. √ Even if these data do not scale with the 1/ Re assumption in Sect. 4.1, the result is acceptable.

12.3 Problems of Chapter 4

399

Problem 4.2 The vorticity-content integral over the hatched area is   2π  re   2π  re 1 d(r vθ ) r dθ dr = 2πre vθe . ωz r dθ dr = r dr 0 0 0 0 Problem 4.3 The terms suction side and pressure side are colloquial. Suction side means a negative pressure coefficient c p is present, pressure side a positive one, compare Fig. 2.3. There the better (?) terms upper side and lower side are used. The lift is due to the pressure difference between the lower and the upper side. Problem 4.4 A vortex line is defined by v × ω = 0. We make the proof with the vorticity-content vector Ω. We obtain with v = [u, 0, 0] and Ω = [Ωs , 0, 0] ⎡

⎤ es et e z ⎣ u 0 0 ⎦ = es (0 + 0) + et (0 + 0) + e z (0 + 0) = 0. Ωs 0 0 Problem 4.5 Circulation theory adds to the translative free stream a potential (Rankine) vortex with such a circulation Γ that the combined flow field obeys the Kutta condition, i.e., that the flow at the trailing edge does not turn around to the upper side of the airfoil, Fig. 4.14, but smoothly leaves the trailing edge, Fig. 4.15, Problem 4.6 For y = 0 kinematically active vorticity content leaves the wing’s trailing edge in the trailing vortex layer, thus reducing Γ . After the rolling-up of the trailing vortex layer the vorticity is concentrated in the trailing vortices and Γ0 is present again. The condition is the compatibility condition, presented in Sect. 4.4. Problem 4.7 The trailing vortex layer must be sketched as an rectangle. It carries only an kinematically inactive vorticity content, coming from the boundary layers at the upper and the lower side of the wing. Problem 4.8 The finite extent shows the strength of the trailing vortex layer, which, however, at that location carries only kinematically inactive vorticity.

400

12 Solutions of the Problems

12.4 Problems of Chapter 5 Problem 5.1 The answer lies with the integral Eq. 4.6 in Sect. 4.1. Only the velocities at the upper and the lower bound are counting, not the shape of u(z). Problem 5.2 Model 8 approaches permit, after a suitable discretization of the computation domain, a fast computation of important aerodynamic properties of the wing or flight vehicle. Compared to Model 4 approaches (today usually panel methods), compressibility effects are included. However, viscous effects, including ordinary separation can only be described with Model 9 and 10 methods. Today these are mainly RANS or URANS methods. Massive separation can only be treated with scale-resolving methods (Model 11). Generally turbulence modeling is a major topic, laminar-transition can be a big problem. Problem 5.3 The computation process basically has the following steps: (1) parametrization of the CAD body surface, (2) discretization of the body surface, (3) discretization of the computation domain. Step 2 and 3 in particular can be very cost-driving, depending on the chosen Model approach. Model 4 approaches in the form of panel methods demand only step 1 and 2, all other approaches also step 3. Shock waves appearing in the flow field in Model 8 to 11 approaches are “captured” today. This makes necessary a sufficiently fine discretization of the computation domain in the vicinity of a shock wave. That also holds for the body surface, if a shock wave originates there. Boundary layers and strong interaction domains in Model 8 to 11 approaches in particular need a very fine discretization on the body surface and in the respective flow domain, which is not the rule for trailing vortex layers and vortices. All this basically holds for rigid bodies. If flow fields past non-rigid bodies, including movables (slats, flaps, others), are to be computed, the discretization must follow the characteristic time scales. In the end all this calls for self-adapting discretization methods, which still are not fully adopted. The computation costs rise with the Model approach level. Lowest are the costs for panel methods. Statistical turbulence models in RANS/URANS methods can be nasty cost drivers, in particular also all scale-resolving approaches. In aircraft design work all this is to be considered in comparison to wind-tunnel measurements, too. There design, manufacturing and finally the instrumentation of the wind-tunnel model are very time and cost consuming. This also is the reason why the computational approaches are progressing, which of course mainly is due to the still increasing computer power, storage capacities, and not least the progress in solution-algorithm development. Problem 5.4 Case 1 for the two-dimensional, and case 2 for the three-dimensional wake. The function u(z) contributes to the entropy rise, too.

12.4 Problems of Chapter 5

401

Problem 5.5 Only a proper grid resolution yields the necessary resolution of the characteristic flow phenomena. This holds for Model 4 and 8 to 11 approaches, where the flow field as such, boundary layers, shock waves, interaction domains, trailing vortex layers and vortices need to be described to the demanded degree of accuracy.

12.5 Problems of Chapter 6 Problem 6.1 Boundary-layer decambering at a lifting airfoil is due to the different boundary-layer properties at the suction side and the pressure side at the trailing edge of the airfoil. At the suction side due to the larger loading of the boundary layer with an adverse pressure gradient the displacement thickness is larger than at the pressure side. This effect is present in both the subcritical and the supercritical case. In the supercritical case an additional effect is present. Consider the case with a supersonic flow pocket at the suction side only. The terminating shock wave impinges vertically on the airfoil’s surface. Across the shock wave a total-pressure loss occurs. Hence at the trailing edge the flow momentum at the suction side—with the shock wave—is smaller than at the pressure side—without a shock wave. This leads at the trailing edge of the airfoil to an upward deflection of the flow—shock-wave decambering—which together with the boundary-layer decambering leads to a reduction of the airfoil’s lift. Problem 6.2 From Table B.2 we find T∞ = 281.651 K, ρ∞ = 1.112 kg/m3 , μ∞ = 1.758·10−5 N s m−2 . The speed of sound is a∞ = 336.438 m/s, and the flight speed u ∞ = 33.64 m/s. With the thickness of the trailing edge h = 0.01 m, we obtain the Re = 21,280.5. Hence the Strouhal number is Sr ≈ 0.2 and the shedding frequency f ≈ 673 Hz. The shedding frequency hence is not at the upper bound of the human hearing domain, which is f ≈ 16,000–20,000 Hz. The lower bound is f ≈ 16–21 Hz. Problem 6.3 The higher the angle of attack, the higher is the actual pressure gradient, because the relevant freestream component increases with the angle of attack. Problem 6.4 cp =

p − p∞ . q∞

Vacuum means p = 0, hence we get with the squared speed of sound a 2 = γ R T∞ cp =

− p∞ −2 p∞ −2 γ ρ∞ R T∞ −2 = = = . 2 2 2 q∞ ρ∞ u ∞ γ ρ∞ u ∞ γ M∞

402

12 Solutions of the Problems

Fig. 12.2 Equivorticity lines of a wing with (a) no-slip boundary conditions and (b) with perfectslip boundary conditions

Problem 6.5 The leading edge of this wing is aerodynamically sharp and sheds a vortex sheet in Fig. 12.2b. However in Fig. 12.2a there is an interaction between the vortex and the boundary layer, which doesn’t occur in Fig. 12.2b. Problem 6.6 At higher Reynolds number the primary vortex induces a higher velocity near the surface of the cylinder that interacts and retards the motion in the boundary layer causing separation and the creation of the pair of secondary vortices. Problem 6.7 See the discussion in Sect. 3.4.2 concerning Fig. 3.4.

12.6 Problems of Chapter 7 Problem 7.1 Such lines are found in Figs. 8.27, 8.31 and 8.32 on page 205 ff., with a detailed presentation in Fig. 8.33. In Fig. 8.27 the open-type beginning of the attachment line on a wing’s leading edge is shown, in Fig. 8.31 the open-type ending of such an attachment line, and in Fig. 8.33 the open-type beginning and also ending of separation and attachment lines. Problem 7.2  For the flow field at the left Rule 2 holds with four half saddle points S and one focus point N : 1 − 0.5 · 4 = −1. For the flow field at the right the originator of the figure proposes to consider the end of the separation bubble shifted to infinity downstream. Then again Rule 2 holds. Make a sketch for yourself. Problem 7.3 We find an half-saddle each at the lower and the upper symmetry line. Rule 2 is fulfilled. Problem 7.4 The resulting eigenvalues are

12.6 Problems of Chapter 7

λ1,3

1 = 2



∂τwz ∂τwx + ∂x ∂z

403



1 ± 2



λ2 =

∂τwz ∂τwx − ∂x ∂z

2 +4

∂τwx ∂τwz , ∂z ∂x

1 ∂p . 2 ∂y

Compare with Eqs. (7.7) and (7.8). Problem 7.5 The prerequisite for plane-of-symmetry flow is that the flow occurs along a geodesic. The geodesic is defined as a curve, whose tangent vectors always remain parallel to it, if they are conveyed along it, see, e.g., [2]. It defines the shortest path between two points on a surface. Problem 7.6 On the convex side of the attachment line the approaching streamlines change from concave to convex, hence inflection points are present. Problem 7.7 Off-surface flow portraits are the patterns of the streamlines, which leave the surface, which always happens only in singular points. Problem 7.8 Write down the summary and compare with Sect. 7.3. Problem 7.9 In two-dimensional flow the wall shear-stress changes sign. In three-dimensional flow this does not happen. Instead one observes a local convergence of skin-friction lines, the occurrence of a |τw |-minimum line and the bulging of the boundary-layer thickness and the displacement thickness.

12.7 Problems of Chapter 8 Problem 8.1 The normal angle of attack results to α N = 11.93◦ and the normal leading-edge Mach number to M N = 0.179. The correlations show that at the round leading edge the flow is attached, but if the leading edge would be sharp, a lee-side vortex system could develop. Problem 8.2 The normal angle of attack results to α N = 4.88◦ and the normal leading-edge Mach number to M N = 0.205. The correlations show that at the round leading edge the flow is attached. For the sharp leading edge the correlation lies on the boundary between “attached flow” and “development of the vortex system”.

404

12 Solutions of the Problems

Problem 8.3 The application of the rule yields for the linear theory a C L ic = 0.315015, and for M∞ = 0.3 the value C L c = 0.33. This compares well with the result of the Euler simulation with C L c = 0.342. Problem 8.4 The aspect ratio of the wing is  = 9. In the ideal case with Eq. (8.1) the induced drag coefficient results to C Di = 0.007 or 28 per cent of the total drag. With an Oswald factor of e = 0.8, the result is C Di = 0.0087 or 35 per cent of the total drag. Problem 8.5 In Sect. 3.16 we find σ=

Γ . u ∞ b/2

With the Mach number M∞ and the free-stream temperature T∞ the free-stream speed is found: u ∞ = 300.47 m/s. For the three root circulations we obtain Γ0 = 538.50 m2 /s, 615.30 m2 /s, 600.29 m2 /s. With the Sutherland equation, Appendix A.5.1, the free-stream viscosity is found to be μ∞ = 1.899 · 10−5 N s m−2 , hence ρ∞ = 0.0451 kg/m3 . With Eq. (3.46) the lift coefficients for the three root-result circulations result to C L = 0.431, 0.493, 0.481, and finally C Di = 0.0082, 0.0107, 0.010. The agreements with the lift coefficient on page 200 and the induced-drag coefficient from Problem 8.4 are reasonable. Problem 8.6 The dimensionless circulation G(x ∗ ) = Γ (x ∗ )/Γ0 for the case without the horizontal tail surface is G(x ∗ ) ≈ 1, and with it G(x ∗ ) ≈ 0.85. The latter amounts to a loss of about 15 per cent for Γ0 . The lift is affected according to Eq. (3.46) and the induced drag to Eq. (3.48). Problem 8.7 (a) With Eq. (3.34) we find with ρ∞ = 0.4135 kg/m3 and b0 = (π/4)b the root circulation Γ0 =

L m2 = 566.2 . ρ∞ u ∞ b0 s

(b) For the smaller wing the root circulation results to Γ0 = 1,132.4 m2 /s. (c) The wing loading for large transport aircraft is Ws = 3,000 to 8,000 N/m2 . Here it is in the first case Ws = 6,130 N/m2 and in the second case Ws = 12,262 N/m2 , which is well above the average values. (d) If the same mass of an aircraft is transported with a smaller wing, the strength of the trailing vortex increases.

12.7 Problems of Chapter 8

405

(e) The lift coefficient is two times, and the induced drag coefficient four times higher for the smaller wing. Compute the values for yourself. Problem 8.8 The lifting-line theory in Appendix A.4, gives the incompressible flow lift coefficient for an angle of attack of α: C L = 2πα/(1 + 2/), where  = b2 /A is the aspect ratio, b the span and A the wing area. According to the Prandtl–Glauert rule for three dimensional flow, the y and z coordinates of the real configuration are to be multiplied with β to obtain the analogue configuration for incompressible flow, 2 ) = 0.6, β = sqr t (1 − M∞

0 =

2 b02 β 2 b0.8 = caverage = β0.8 = 0.6 · 10 = 6. S0 b0.8

This scaling also applies to the wing incidence, then α0 = βα0.8 : C L0 = (dC L /dα)0 · α0 = 2π/(1 + 2/6) · βα0.8 = 2.828α0.8 In the transformation back to the real M∞ = 0.8 flow, all incompressible flow pressures should be divided by β 2 , and consequently we find C L0.8 = C L0 /β 2 = 2.828 α0.8 /0.36 = 7.856 α0.8 and the lift curve slope becomes (dC L /dα)0.8 = 7.856.

12.8 Problems of Chapter 9 Problem 9.1 The limiting flow phenomena are the drag divergence in the transonic flight regime and the associated lift drop. The drag divergence is due to boundary-layer/shockwave interaction, which in the end leads to the transonic dip of the lift and to the transonic buffet phenomenon, due to the high unsteadiness of the separated wing flow. Problem 9.2 The configurational means for the wing are the wing sweep and the supercritical airfoil. The wing’s sweep usually is backward, but can also be forward. If the wing is swept, the free-stream “is seeing” a wing with smaller thickness to chord-length ratio, than it is seeing at the unswept wing. Hence the overspeed is lesser and the

406

12 Solutions of the Problems

occurrence of the supersonic flow pocket with its terminating shock wave—usually considered to be present, but not necessarily, only at the suction side of the wing—is shifted to a higher flight Mach number. The supercritical airfoil, or chord section, has the same effect due to its shape, which mainly is a flattened upper surface, and in addition a highly cambered aft section. For the whole aircraft the means is the area rule, which concerns the overall axial cross-section distribution of fuselage, wing, engine nacelles, empennage. The distribution in axial direction should be as smooth as possible in order to minimize interference drag. While the swept wing easily is discernible, the supercritical airfoil and the area ruling in a sense are “hidden” to the observer. Problem 9.3 The vortex generator basically is transporting fluid with high momentum from the upper boundary-layer regime and the adjacent inviscid external flow into the wallnear part of the boundary layer. The effect then is an increase of the flow momentum of the wall-near boundary-layer flow, and hence a reduction of separation proneness. The vortex generator in principle also works in laminar flow. But if an adverse pressure gradient is present, which is the cause of separation proneness, the flow anyway will undergo laminar-turbulent transition. Problem 9.4 In general not. The reason is that the laminar boundary layer has a less full flow profile than the turbulent one. Hence less flow momentum can be transported into the wall-near part of the boundary layer. Problem 9.5 Without a wing-root/fuselage fairing the flow coming from the forward part of the fuselage meets the wing’s root, which is a flow obstacle, at a swept wing a swept one. Ahead of that obstacle three-dimensional boundary-layer separation happens. The result is a horse-shoe like vortex around the wing-root/fuselage location. This vortex is the cause of a drag increment and further it can lead to buffet at high angles of attack. A proper wing-root/fuselage fairing leads to a smooth flow passage over the wing’s root and hence avoids separation and hence the horse-shoe vortex with its adverse side effects. Problem 9.6 The Breguet range equation is used to calculate the relative saving in fuel mass. The formula is given by Eq. (9.1). The mass m T O W related to the take-off weight refers to the sum of the masses due to empty weight m e , fuel m F and payload m P : m T O W = m e + m F + m P . The range R is kept constant and the flight speed u ∞ , the specific fuel consumption b, the lift coefficient C L and the take-off weight m T O W remain the same:

12.8 Problems of Chapter 9

407

Fig. 12.3 The resulting spanwise circulation distribution Γ (y)

CL u∞ ln R= CD b g



mT OW mT OW − m F

 = const.

The drag coefficient is composed as C D1 = C D01 + C Di1 = 0.8 C D1 + 0.2 C D1 , the index 1 denoting the baseline case. The case with reduced drag level is (index 2) C D2 = C D0 + C Di2 = 0.8o × 0.97 C D1 + 0.2 × 0.98 C D1 = 0.972 C D1 . For constant range R it follows:     CL u∞ CL u∞ mT OW mT OW = . ln ln C D1 bg m T O W − m F1 C D2 bg m T O W − m F2  ln

mT OW m T O W − m F1





m T O W − m F1 mT OW

1 = ln 0.972

0.972

 =



mT OW m T O W − m F2

m T O W − m F2 mT OW

 .

 .

m F2 = m T O W

0.972   m F1 1− 1− = 110, 483 kg. mT OW

The relative fuel saving results in Δm F = −2.23 per cent. Problem 9.7 According to Prandtl’s lifting line model a spanwise circulation distribution Γ (y) providing a minimum induced drag is of elliptical shape (blue colored area) derived from a modeling of horseshoe vortices for a planar wing surface neglecting wake vortex roll-up. Taking this elliptical circulation distribution as an optimum with respect to induced drag, vertically orientated winglets have to be loaded in an equivalent way as it would be reflected by the virtual increase in span (orange and red areas related to the winglet case; blue and orange colored area are of same magnitude for same lift level). “Unloaded” winglets do not provide substantial effects on induced drag reduction (Fig. 12.3).

408

12 Solutions of the Problems

Problem 9.8 Referring to Fig. 9.22 and Eq. (9.3), an increase in effective span may be realized by wing-tip devices independent of upward or downward orientation. From a combination of both the elastic wing deformation due to highly loaded wing-tip devices can be adjusted in such a way that deformations are limited or adjusted in a desired way for further increase of aerodynamic efficiency or reduction of structural dynamic loads. Problem 9.9 According to Chap. 9.6 the characteristic reduced frequency attributed to the Crow type instability is kCr ow ≈ 0.08. Here, the reduced frequency is defined as k = f (b/2)/u ∞ . Thus, the full scale case results in an oscillation frequency of f = kCr ow u ∞ /(b/2) = 0.21 Hz. The oscillation frequency for the wind tunnel situation then f = kCr ow u ∞ /(b/2) = 1.67 Hz.

12.9 Problems of Chapter 10 Problem 10.1 (a) More or less rectangular wings are found in the subsonic flight domain, i.e., below the critical Mach number. (b) Swept wings are typical for transport aircraft operating in the transonic flight domain. The wing’s sweep, together with a supercritical chord section, permits to fly at Mach numbers above the critical one, shifting the drag-divergence Mach number up to high values. Usually backswept wings are employed. Forward swept wings in principle are feasible, but for several reasons did not find broad use, like the oblique wing, which is swept forward at one side of the aircraft and backward at the other side. (c) The slender, delta-like wing is typical for aircraft operating in the supersonic flight domain, like the former Concorde. The main reason is to have a subsonic leading edge in order to reduce wave-drag increments. Fighter configurations either have delta wings or hybrid wings with a strake-trapezoidal wing combination. Both—still hypothetical—hypersonic airbreathers and winged re-entry vehicles— like the former Space Shuttle Orbiter—have highly swept leading edges. The re-entry vehicle on the initial flight trajectory flies a breaking mission at very high angle of attack, and on the lower flight trajectory aims for reasonable down and cross range. The airbreather must have low overall drag on the whole flight trajectory, like any other aircraft. Problem 10.2 These phenomena are vortex breakdown and vortex overlapping. Vortex breakdown— also called vortex bursting—happens, when a vortex interacts with an axially adverse pressure field. Due to their breakdown, which as a rule is beginning at the rear of the wing, the lee-side vortices loose the capability to induce the suction force beneath them. Consequently a loss of lift occurs and at the same time a pitch-up of the wing

12.9 Problems of Chapter 10

409

Fig. 12.4 Sketch of the principle spanwise distributions of C L (y)l(y) and C L (y)

happens. If vortex breakdown occurs asymmetrically, side forces and moments are induced. The higher the leading-edge sweep angle of the wing, the higher is the angle of attack, at which vortex breakdown happens. Vortex overlapping occurs at wings with very high leading-edge sweep angle and at high angles of attack. This phenomenon is also observed at fuselages. It leads to side forces at nominally zero sideslip angle and to difficulties in aircraft control. Problem 10.3 For the original round-edged configuration at α = 30◦ and M∞ = 0.3 the normal angle of attack is α N = 49.1◦ and the normal Mach number M N = 0.2. For the manipulated sharp-edged configuration this data are α N = 52.8◦ and M N = 0.19. The correlations hence for both cases show vortices fixed in span direction, but elevating. The fact that for the original case obviously no lee-side vortices were present, must be traced back to the combination of leading-edge radius and Reynolds number, which were sub-critical. Problem 10.4 Yes, in the Poincaré surface the left and right attachment points A3 , being half-saddle points, move into the center and above the wing, there combined forming a full saddle point S. The half-saddle point S4 at the center of the upper surface is retained there, but the flow directions in it change (Fig. 12.4). Problem 10.5 The spanwise lift and lift-coefficient distributions, C L (y)l(y) and C L (y), in principle can be sketched as follows: The dominating separation scenario is due to leading-edge separation resulting in a fully developed leading-edge vortex at α ≈ 20◦ , Fig. 11.2, providing an inboard loading of C L (y)l(y). Further, the wing tip area is characterized by irregular separated flow as the corresponding sections exhibit stall, which is reflected by the local outboard maximum in the C L (y) trend. Consequently, the wing tip section does not contribute effectively to lift production. Therefore often cropped delta-wing planforms are used for practical applications.

410

12 Solutions of the Problems

Fig. 12.5 Sketch of the topological representation of the closed lee-side flow field in a Poincaré surface

Problem 10.6 According to Fig. 11.3 a pitch-up tendency is present at high angle of attack. Figure 11.5 provides a correlation of wing leading-edge sweep and aspect ratio based on a variety of configurations indicating “controllable” and “unsafe” pitch-up tendencies. The combination of ϕ0 = 65◦ leading-edge sweep and aspect ratio of  = 1.9 probably results in an uncontrollable pitch-up tendency. Further measures, e.g., leading-edge flap setting, have to be implemented to account for unacceptable longitudinal stability margins. Problem 10.7 (1) The topological representation of the flow field in a cross-flow plane—a Poincaré surface—can be sketched as shown in Fig. 12.5, compare Fig. 11.23. Following the topological rule, Eq. (7.25), outlined in Sect. 7.4.3, the sketch can be proven for consistency. Two interacting primary vortices and one secondary vortex are shown resulting in three nodes and one saddle point per wing side (2 x (3N − 1S)) and one saddle point for the symmetry plane. Further, there are two separation and one attachment half saddle points S  per wing side and two attachment quarter saddle points S  at the lower side. Thus, it matches the rule of Eq. (7.25): (



N+

 1  1   1   N )−( S + S ) = −1, S+ 2 2 4

with 1 1 6 − (3 + 7 + 2) = −1. 2 4 (2) Due to the increased suction on the forward (strake) wing part a pitch-up tendency for angles of attack of 20◦ and higher most probably will be present, see Figs. 11.3 and 11.24. Vortex breakdown will take place in the rearward area of the wing, thus decreasing the suction level there, while upstream the fully developed leading-edge vortex for the strake section still produces high lift, which then results in a pitch-up tendency with respect to typical locations of the center of gravity. Following the trends given in Figs. 11.4 and 11.23, for some sideslip angle a roll reversal for angles of attack beyond 20◦ is also expected, because the windward leading-edge vortex system is prone to vortex breakdown. This is caused by the

12.9 Problems of Chapter 10

411

change in the effective wing leading-edge sweep as it decreases at the windward side and increases at the leeward side. Problem 10.8 From Appendix A.4—small aspect-ratio wing theory—one finds: l(x) = 4π · αq∞ · y(x) · dy(x)/d x, bx , y= 2c b dy/d x = , 2c b bx b = ( )2 x, l(x)/(παq∞ ) = 4 2 c 2c c  c b l(x)/(παq∞ ) · d x = ( )2 · c2 /2 = b2 /2, L/(παq∞ ) = c 0 c b M/(παq∞ ) = x · l(x)/(παq∞ ) · d x = ( )2 · c3 /3 = b2 c/3, c 0 2 xcp = M/L = c. 3 Problem 10.9 In Appendix A.4—small aspect-ratio wing theory—is an expression l(x) = 4 π α q y(x) dy(x)/d x. Thus the lift reads 

 1 l(x)d x = 4παq∞ y(x) · dy(x)/d x · d x, 0 0  1 1 2 dy = 2παq∞ · 0.1252 . L = 4παq∞ 2 x=0

L=

1

The lift curve slope is L dC L =d /dα, dα q∞ A A = 0.375, dC L /dα = 0.262. Alternatively the lift contributions from the front part and the rear part may be added together. Problem 10.10 In the two collocation points, one on the wing and one on the stabilizer, sum the vertical velocity components due to the free stream and the two vortices. On the wing the sum should be zero, and on the stabilizer −u ∞ δ.

412

12 Solutions of the Problems

1 1 + Γ2 = 0, 2π (c/2) 2π (4c + c/4 + c/8) 1 1 w2 = u ∞ · 0 − Γ 1 − Γ2 = −u ∞ δ, 2π (4c + 3c/4 + 3c/8) 2π (c/4) w1 = u ∞ · 0 − Γ1

1 4 + Γ2 = 0, πc 35π c 4 2 −Γ1 − Γ2 = −u ∞ δ. 41π c πc −Γ1

Γ1 πc = 0.0568. u∞δ

12.10 Problems of Chapter 11 Problem 11.1 Lift decreases, drag rises, and, very importantly, a pitch-up happens. This behavior is due to the loss of suction force, which disappears below the vortex pair where the breakdown happens. Because that happens at the rear of the wing, the upward directed suction force there vanishes, and the consequence in particular is the pitch-up. Problem 11.2 (a) High angle of attack and asymmetries of the configuration and/or the freestream are the basic causes. Lateral/directional divergence problems introduced by side-slip effects (Cn β , Clβ ) lead to non-controllability of the aircraft. Hence these problems must be reduced by proper configurational development and/or restrictions in the flight envelope. (b) The flow structure due to sideslip basically becomes asymmetric due to a reduction of the effective leading-edge sweep on the windward side, and an increase on the leeward side. See regarding both items also the correlations in the Sects. 10.2.5 and 10.2.6. Problem 11.3 Low aspect ratio and high leading-edge sweep are the drivers in supersonic—here M∞ = 2—wing design, which results in a subsonic leading edge. A softly varying area distribution—supersonic area ruling—gives the chance to minimize the drag. This helps to increase the lift-to-drag ratio at supersonic cruise by reduction of the (zero-lift) wave drag. To keep the induced drag at a low level, the supersonic trim drag has to be reduced, too. This demands to keep the backward shift of the neutral point at supersonic flight at low level. With that high (negative) trim-flap deflections can be avoided. Consequently the lift-to-drag ratio is increased, thus either lowering the fuel consumption or extending the flight range of the aircraft.

12.10 Problems of Chapter 11

413

Stability and handling qualities of such aircraft—here the Concorde—usually are more critical at low subsonic flight speeds, say at take-off and landing and in the high angle-of-attack flight regime. Aspects of stability divergence—pitch, yaw and roll motion—are of importance. Controllability also may be a critical design driver, see Sect. 11.2. Problem 11.4 First it is to remark that these curves only show results for the special case C L = C L opt = C L (L/D)opt . This means that C L is not constant for these curves, but is varying along them, as happens with C D0 and C Di . In Fig. 11.12 our case ( = 2, ϕ0 = 63◦ ) is included for three wing-chord shapes: symmetric, warped, full-span nose-flap. The figure is valid for sharp leading-edge wings with increasing sweep angle ϕ0 . For our case, ϕ0 = 63◦ , we find the following leading-edge suction ratios S ∗ (in percent of attained to ideal—100 per cent— leading-edge suction): • symmetric profile: S ∗ = 40 per cent, • warped profile: S ∗ = 48 per cent, • full-span nose flap: S ∗ = 64 per cent. Turning to Fig. 11.13, the data for the point  = 2, ϕ0 = 63◦ now are given for the wing with rounded leading edge and a relative thickness of 5 per cent for the cases “symmetric profile” and “conical camber” of the wing. For our flight Mach number M∞ = 0.3 we obtain: • symmetric profile: S ∗ = 60 per cent, • conical camber: S ∗ = 95 per cent. These results clearly demonstrate the superiority of wings with rounded leading edge, when we look for high suction rates at subsonic Mach numbers in order to end up with a high lift-to-drag ratio. Problem 11.5 The positive Effects of concentrated spanwise blowing over the upper side of the wing are, see Figs. 11.19 to 11.21: 1. Increase of the maximum lift by stabilizing an existing leading-edge vortex system (wings with high leading-edge sweep) or by generation—and stabilization—of concentrated vortex systems over wings with low/medium aspect ratio. In both cases the additional lift increases non-linearly with increasing angle of attack. 2. This is also reflected in the development of lift-dependent drag, which is reduced for constant lift due to the lower angle of attack necessary for it. 3. Most efficient in terms of Δ C L /cμ is blowing over wings that do not develop leeside vortex systems. But the efficiency of Δ C L /cμ is decreasing with increasing blowing coefficient cμ . The reasons for its “non-application” can be found in the definition of the blowing coefficient

414

12 Solutions of the Problems

cμ =

m˙ v j . q∞ Ar e f

The efficiency of blowing decreases with increasing flight speed: cμ ∝ 1/u 2∞ . Consequently the induced extra lift will decrease with increasing speed. Hence spanwise blowing is effective—and applicable—for low flight speeds only. But there it leads to engine thrust reduction, because of the needed mass flux of the jet m. ˙ Therefore this technique did not find its way into application. Problem 11.6 Topological flow structures and the development of the pitching moment with increasing angle of attack are presented in the Figs. 11.23 and 11.26 for the planar and for the cranked wing. Comparing the figures we find: (a) The planar wing develops a strong pitch-up tendency beginning already at medium angle of attack α ≈ 10◦ , the cranked wing on the other hand shows a quasilinear stability curve Cm α (0  α < 35◦ ) and finally a—safe—pitch-down tendency. (b) These tendencies correspond to the following flow structures: (A) Low to medium angle of attack: inserts (1) and (2) in the figures. Here both wings show quite similar flow characteristics (1): • Three nodes are present due to vortices induced by (a) the highly swept leading edge of the inboard wing ⇒ leading-edge separation. (b) the leading-edge kink (high sweep of the inboard wing, reduced sweep of the outboard wing), plus the crank of the outboard wing, (c) the outboard wing side edge. • Open lee-side flow field at both wings. • An internal saddle point due to interaction of the three vortex systems, which exist over the wings. An increase of the angle of attack—(2)—induces a pitch-up tendency for the planar wing only. The flow structure is altered by the beginning of a local unsteadiness and the increasing strength of the vortex system of the outboard wing. No change of the flow structure is found for the cranked wing, (2) in Fig. 11.26. Both wings still show an open lee-side flow field. (B) Medium to high angle of attack: inserts (3) and (4) in the figures. The main change in the flow structure of the planar wing—(2) ⇒ (3)—is the appearance of a central, free-flow saddle point—representative for a closed lee-side flow field—in combination with an upward movement of the leading-edge vortex of the inboard wing. In contrast to that the cranked wing still presents an open lee-side flow field (3) and a stronger induced effect of the crank vortex, driven by the primary leading-edge vortex of the highly swept inboard wing.

12.10 Problems of Chapter 11

415

A further increase of the angle of attack—(3) ⇒ (4)—has tremendous consequences for the planar wing. The location of vortex breakdown of the primary leadingedge vortex is now far forward over the wing, hence the lift force is concentrated there, with a strong pitch-up as a result. In contrast to that the cranked wing, (4) in Fig. 11.26, is producing two stable vortex systems: see the nodal point N1 from the leading edge vortex of the inboard wing and the nodal point N2 —induced and stabilized by N1 —originating over the crank. Now we also have a closed lee-side flow field. From this comparison it is clear that the positive strong interaction of N1 and N2 is the reason for the now “healthy” development of the pitching moment of the cranked-wing configuration.

References 1. Hirschel, E.H.: Basics of Aerothermodynamics. 2nd, revised edition. Springer, Cham Heidelberg, New York (2015) 2. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin, Heidelberg

Appendix A

Useful Relations

Useful relations are given for quantitative considerations, both exact ones and approximative ones. Details can be found, for instance, in [1–4] and in relevant monographs.

A.1

Pressure Relations

The pressure coefficient c p is defined as cp =

p − p∞ , q∞

(A.1)

where q∞ = 0.5ρ∞ u 2∞ is the dynamic pressure of the free-stream. For air as perfect gas the pressure coefficient c p reads with M∞ = u ∞ /a∞   p 2 cp = −1 , (A.2) 2 γ M∞ p∞ √ a = γ RT∞ is the speed of sound, γ = c p /cv = 1.4 (perfect gas) the ratio of specific constant (air). heats, and R = 287, 06 m2 s−2 K−1 , the specific gas  The expansion limit—the maximum speed vm = 2 c p Tt , with Tt being the total temperature—is reached with p → 0. With this we obtain the vacuum pressure coefficient c pvac = −

2 . 2 γ M∞

(A.3)

The pressure coefficient can be expressed in terms of the ratio local speed u to free-stream value u ∞ , and the free-stream Mach number M∞ :

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

417

418

Appendix A: Useful Relations

⎫ ⎧ γ   γ−1 ⎬ u 2 2 ⎨ γ−1 2 cp = [1 − ] − 1 . 1 + M ∞ 2 ⎩ ⎭ γ M∞ 2 u∞

(A.4)

In subsonic compressible flow we get for the stagnation point (isentropic compression) with u = 0   γ γ − 1 2 γ−1 2 1+ M∞ c pstag = −1 . (A.5) 2 γ M∞ 2 In the case of supersonic flow of course the total-pressure loss across the shock must be taken into account. For incompressible flow the pressure coefficient is found with the help of Bernoulli’s equation, see, e.g., [1], and reads cp = 1 −

u2 . u 2∞

(A.6)

We note that at a stagnation point c p is always larger for compressible flow than that for incompressible flow with c pstag = 1.

A.2 A.2.1

Vortex-Induced Velocity Introduction

With Fig. 3.1 we introduced several vortex models for finite-span wing flow. The following useful relations, derived from these models, provide further insight into how well these models work and can be used in actual applications [4]. We begin in two dimensions with a description of the vortex bound to the airfoil in Fig 3.4, and after that proceed to the three-dimensional case.

A.2.2

The Two-Dimensional Case

• Vortex Model for Airfoil Flow A distribution of vorticity γ(x) on an airfoil is a solution to Laplace’s equation and forms the basis for the analysis of a very thin cambered airfoil (i.e. a plate): Glauert’s thin-airfoil theory. The solution satisfies the boundary conditions, if the combination of the velocity induced by the vortices cancels the component of the freestream normal to the plate:

Appendix A: Useful Relations

419

wi (x) = u ∞ (α −

dz ), dx

where small-angle approximations have been introduced. The basic approximation of thin-airfoil theory is that the velocity induced at some point x due to the vorticity at x  may be approximated by the velocity induced at the same x position on the x-axis due to a vortex on the x-axis, i.e., the vortex can be shifted down on the x-axis. The velocity induced by this bit of vorticity is computed from the basic vortex singularity, the formula known as the Biot-Savart law, Sect. 3.9. For the element of vorticity at x  , it reads in two dimensions: d w(x) =

1 γ(x  ) d x . 2π (x − x  )

So, the total induced velocity at the point x is given by:

1 γ(x  ) 1 d x . w(x) = 2π 0 (x − x  ) Combining this expression with the flow-tangency boundary condition, we have the basic integral equation to be solved for the unknown vorticity distribution:

1 1 γ(x  ) dz d x = α − .  2π u ∞ 0 (x − x ) dx The approach to solving this equation is to change variables: cos θ = 1 − 2x, with θ varying from 0 to π, and to write γ as a Fourier series. If we consider only symmetric airfoils, i.e., dz/d x = 0—no camber—we obtain the solution γ(θ) = 2u ∞ α cot (θ/2). For an arbitrary airfoil, γ is expanded in a Fourier series that then leads to a solution. • Lumped-Vortex Method Glauert’s thin-airfoil theory (see above) describes the vortex distribution γ over a cambered airfoil. The center of gravity for this distribution lies in the point x = c/4. Replacing the continuous distribution γ(θ) with the resultant point vortex Γ = u ∞ πcα at x = c/4 , lift force per unit span and moment remain the same (Fig. A.1). (Clearly, the Kutta condition says that there shall be no vortex on the trailing edge.) With only the one variable Γ (instead of a whole distribution) it is only possible to satisfy the boundary condition in just one point. Let this point lie a distance kc

420

Appendix A: Useful Relations

Fig. A.1 Substitution of the distributed vorticity with a single vortex [4]

downstream from the leading edge. The contribution from the discrete vortex has to balance the vertical component of the free stream velocity: w(x = kc) = −

Γ + u∞α = 0 ⇒ 2π(kc − c/4)

k = 3/4.

If the airfoil chord were to be divided into several parts, N panels, and on each panel i is put a vortex Γi , it can be shown, that the boundary condition is satisfied at the 3/4 local chord on each panel.

A.3

The Three-Dimensional Case—the Elementary Horse-Shoe Vortex

From Biot-Savart’s law, the induced vertical velocity w—with the notation from the left part of the figure below—is (Fig. A.2) w(x, y, 0) =

−Γ (cos β1 − cos β2 ). 4πd

The complete horse-shoe vortex as it would be used in the vortex-lattice method with coordinates for the collocation point (x, y) and the vortex definition points (xa , ya ) and (xb , yb ), is shown in the right part of the figure. The vertical velocity for the slightly simpler case with xa = xb is

Appendix A: Useful Relations

421

Fig. A.2 Influence of an elementary horse-shoe vortex [4]

  (x − xa )2 + (y − ya )2 −Γ w(x, y, 0) = + 1+ 4π (y − ya ) x − xa   (x − xa )2 + (y − yb )2 Γ + . 1+ 4π (y − yb ) x − xa

A.4 A.4.1

(A.7)

Wing-Lift Predictions Prandtl’s Theory for the Large Aspect-Ratio Wing

• Concept Rather than representing the wing with just one horse-shoe shaped vortex, the wing is represented by several of them [4]. In this way the bound circulation in the wing can vary from the root to the tip. The strength of the trailing-vortex filaments is related to the circulation on the wing by Γwake = ΔΓwing . A wake vortex is shed from the wing whenever the wing circulation changes, as illustrated in Fig. A.3. In the limit, as the number of horse-shoe vortices goes to infinity, the trailing vortex layer is a sheet of vorticity. The trailing vortex strength per unit length in the y-direction is the derivative of the total circulation on the wing at that station. From this model we can derive the basic relations for finite-span wings. The vorticity strength in the trailing vortex layer is given by: γ = dΓ /dy, and since the wing circulation changes most quickly near the wing tip, the strength of the trailing vortex layer is largest there. The roll-up of the trailing vortex layer, which can be seen as beginning there, leads to the pair of trailing vortices. The two wing-tip vortex systems, Sect. 8.4.4, are drawn into the trailing vortex layer and hence into the trailing vortices. • Lift Prediction The derivation of the method is based on the concept that lift is caused by a change in the flow direction at the wing—the downwash at the location of the wing—induced by vortices representing the wing surface. The local effective angle of attack, αe f f , is obtained through a reduction of the geometric angle α, with the induced angle αinduc (y):

422

Appendix A: Useful Relations

Fig. A.3 Horse-shoe vortex arrangement to simulate a lifting wing [4]

αe f f (y) = α − αinduc (y). The reduction is due to the wake downstream of the wing, also a distribution of vorticity. Both wing and wake deflect the air stream. As the name implies the wing-lift distribution is replaced with a lifting line. Therefore the method can only predict how the force of the lifting line varies in the spanwise direction. The variation is described through Fourier coefficients. The coefficients are determined by the geometry of the wing in a number of positions along the span. An especially simple result is obtained  for a flat elliptical wing, for which the vortex distribution becomes Γ (y) = Γ0 1 − (2y/b)2 , where b is the wing span. The method gives for the special case of a plane elliptical wing the constant vertical velocity w(x = c/4, y) = −Γ0 /(2b). The effective angle of attack at a span station y is αe f f (y) = α − w(x = c/4, y)/u ∞ = α − Γ0 /(2bu ∞ ). Defining the load at a span station y as l(y), and the local lift coefficient cl (y), one can write l(y) = c L (y) q c(y) dy = ρ u ∞ Γ (y)dy, and

Appendix A: Useful Relations

423

Fig. A.4 Lift-curve slope dC L /dα as function of the aspect ratio Λ of wings [4]

Γo = cl (y) u ∞ co /2.

(A.8)

At each station the relation between flow deflection and the lift is taken from two dimensional flow theory cl (y) = 2π (α − Γo /(2 b u ∞ )).

(A.9)

Elimination of Γ between Eqs (A.8) and (A.9), and integration over the span gives with α in radian 2πΛ dC L = , dα Λ+2

(A.10)

where Λ is the wing aspect ratio. For Λ → ∞ we obtain the two-dimensional case dC L = 2π. dα

(A.11)

A graph is given in Fig. A.4. As the wing vortex is placed at the 1/4 chord points of all sections, the resulting lift force will act in the 1/4 chord point of the reference chord. For plane wings of elliptical planform the induced-drag coefficient C Di can be shown to be C Di = see Sect. 3.16.

C L2 , πΛ

(A.12)

424

Appendix A: Useful Relations

Fig. A.5 Sketch of the flow pattern as a delta wing at incidence penetrates a vertical plane normal to the flight direction [5]

A.4.2

R.T. Jones’ Theory for Small Aspect-Ratio Wings

R.T. Jones devised his theory by reasoning on physical grounds in the following way [5]. Fix a vertical reference plane in space. A slender delta wing at angle of attack α runs with a speed of u ∞ straight through the plane, and a streamline pattern, like the one in Fig. A.5, is established. It is also the pattern of a flat plate mowing downward with a velocity u ∞ α. Observe, that the width of the wing and the scale of the flow pattern will vary, as the wing moves through the plane. This increase in the width needs a force from the cut of the wing corresponding to the downwash velocity u ∞ α times the local change of the mass involved m  . The reacting force on the wing cross section is noted l(x): l(x) = u ∞ α dm  /dt = u 2∞ α dm  /d x,

(A.13)

because u ∞ = d x/dt. From two dimensional flow theory it is known that (a complicated proof is given in [4]) m  = π y12 ρ∞ , where y1 is the y-coordinate of the leading edge at a distance x from the apex, and m  is the mass in a cross section with y1 as radius. Hence we obtain l(x) = u 2∞ α 2πρ∞ y1 (x) dy1 (x)/d x = 4π α (ρ∞ u 2∞ /2) y1 (x) dy1 (x)/d x. Consider a delta wing with the chord c, the aspect ratio Λ and lift L, then y1 (x) becomes Λ x/4. Or if the lift coefficient is based on the wing area: π dC L = Λ. dα 2

(A.14)

Figure A.4 plots dC L /dα according to the theories for small and for large aspect ratios, respectively. The vortex lattice method, Fig. 3.1, is one suitable approach to fill in data for the most important interval of moderate aspect ratios. Then it will be possible to judge, within how wide errors, the analytical methods (lifting-line and

Appendix A: Useful Relations

425

small aspect-ratio) are valid. Note that even for an aspect ratio of Λ = 15 there is quite a way to the two dimensional value of dC L /dα = 2π.

A.5

Estimation of Boundary-Layer Properties

For a quick estimations of boundary-layer properties we give approximate boundarylayer relations in generalized form. In order to be self-consistent, we take them over partly from [3]. The relations are based on work of G. Simeonides [6], and hold for two-dimensional and not too strongly three-dimensional attached viscous flow, as long as the flow is of boundary-layer type. The body surface is assumed to be hydraulically smooth. The relations are valid for both laminar and turbulent flow in a Reynolds number range up to 107 . It is assumed that the temperature does not exceed 1,500 K. Although derived originally for hypersonic flow problems, the relations can be used also for all lower-speed problems. Of course it is necessary, as with all approximate relations, to check the range of applicability and to establish the error range. This holds in particular, if the relations are of empirical or semi-empirical character. In the following relations we use for laminar flow the exponent n = 0.5 and for turbulent flow n = 0.2.

A.5.1

Viscosity and Thermal Conductivity

Expressions for the viscosity of air are the Sutherland equation and simple power-law approximations. The Sutherland equation reads μ Suth = 1.458 · 10−6

T 1.5 [N s/m 2 ]. T + 110.4

(A.15)

A simple power-law approximation is μ = cμ T ωμ . For the temperature range T  200 K the approximation—with the constant cμ1 computed at T = 97 K—it can be written μ1 = cμ1 T ωμ1 = 0.702 · 10−7 T,

(A.16)

and for T  200 K—with the constant cμ2 computed at T = 407.4 K— μ2 = cμ2 T ωμ2 = 0.04644 · 10−5 T 0.65 .

(A.17)

Regarding the thermal conductivity of air we note that for temperatures up to 1,500–2,000 K, an approximate relation due to C.F. Hansen—similar to Sutherland’s equation for the viscosity of air—can be used [7]

426

Appendix A: Useful Relations

k H an = 1.993 · 10−3

T 1.5 [N /(s K )]. T + 112.0

(A.18)

A simple power-law approximation can also be formulated for the thermal conductivity: k = ck T ωk . For the temperature range T  200 K the approximation reads— with the constant ck1 computed at T = 100 K— k1 = ck1 T ωk1 = 9.572 · 10−5 T,

(A.19)

and for T  200 K—with the constant ck2 computed at T = 300 K— k2 = ck2 T ωk2 = 34.957 · 10−5 T 0.75 .

(A.20)

Often the thermal conductivity is found via the Prandtl number: Pr =

μc p , k

(A.21)

with c p [m 2 /(s 2 K )] being the (mass) specific heat at constant pressure.

A.5.2

Reference Temperature and Recovery Temperature

In order to take into account Mach-number and wall-temperature effects in boundarylayer flow, the reference-temperature concept is used [8, 9]. The reference temperature T ∗ is empirically composed of the boundary layer edge temperature Te , the wall temperature Tw , and the recovery temperature Tr : T ∗ = 0.28Te + 0.5Tw + 0.22Tr [K ]. The recovery or adiabatic wall temperature Tr is defined by1   v2 γ−1 2 Me , Tr = Te + r ∗ e = Te 1 + r ∗ 2c p 2

(A.22)

(A.23)

with r ∗ being the recovery factor, which is a function of the Prandtl number Pr at the reference temperature T ∗ , Me is the Mach number at the edge of the boundary layer. The Prandtl number depends rather weakly on the temperature, see, e.g., [1, 3]. ∗ flow the recovery factor Usually it is sufficient √ √to assume r = r = const. For laminar can be taken as r = Pr , and for turbulent flow r = 3 Pr . With the Prandtl number at low temperatures, Pr ≈ 0.74, we get rlam ≈ 0.86 and rtur b ≈ 0.90. 1 The

total temperature Tt is found with r ∗ = 1.

Appendix A: Useful Relations

427

The characteristic Reynolds number for a boundary-layer like attached viscous flow at the location x is postulated to read Rex∗ =

ρ∗ u e x . μ∗

(A.24)

Density ρ∗ and viscosity μ∗ are reference data, characteristic for the boundary layer. They are determined with the local pressure p and the reference temperature T ∗ , with ve being the external inviscid flow velocity. Introducing the boundary layer edge data as reference flow data into Eq. (A.24) yields Rex∗ =

ρ∗ μe ρe u e x ρ∗ μe = Re , e,x μe ρe μ∗ ρe μ∗

(A.25)

with Ree,x = ρe u e x/μe . This relation can be simplified. If we apply it to boundary layer like flows, we can write, because p ≈ pe ≈ pw = const. ρ∗ Te = ∗. ρe T

(A.26)

If, for simplicity, we further introduce the power-law expression for the viscosity, we obtain ∗

μ∗ (T ∗ )ω = . μe (Te )ωe

(A.27)

Only if T ∗ and Te are both in the same temperature interval, ω ∗ and ωe are equal, we get:  ∗ ω T μ∗ = . (A.28) μe Te Introducing Eqs. (A.26) and (A.28) into Eq. (A.25) reduces the latter to Rex∗

A.5.3

 = Ree,x

Te T∗

1+ω .

(A.29)

Skin Friction and Heat Transfer

For the skin friction over a flat plate we get in generalized form, with C = 0.332 for laminar flow and C = 0.0296 for turbulent flow [6]

428

Appendix A: Useful Relations

τw = Cμ∞ u ∞ x

−n



T∞ T∗

1−n 

μ∗ μ∞

n u 1−n (Re∞ ) [N /m 2 ],

(A.30)

respectively τw = Cμ∞ u ∞ x −n



T∗ T∞

n(1+ω)−1 u 1−n (Re∞ ) ,

(A.31)

u with Re∞ = ρ∞ u ∞ /μ∞ [m −1 ] being the unit Reynolds number. The skin-friction coefficient c f then is

τw = c f = 2 C x −n 0.5ρ∞ u 2∞



T∞ T∗

1−n 

μ∗ μ∞

n

u −n (Re∞ ) .

(A.32)

The heat flux in the gas at the wall reads, again with C = 0.332 for laminar flow and C = 0.0296 for turbulent flow,

qgw = C x −n k∞ Pr 1/3 (Tr − Tw )



T∞ T∗

1−n 

μ∗ μ∞

n



u Re∞

1−n

[kg/s 3 ], (A.33)

respectively qgw = C x

A.5.4

−n



k∞ Pr

1/3

T∗ (Tr − Tw ) T∞

n(1+ω)−1



u Re∞

1−n

.

(A.34)

Boundary-Layer Thicknesses

The boundary-layer thicknesses of the Blasius and the 1/7-power law— incompressible—boundary layers can be written in generalized form: δi = Ci

x 1−n [m], u )n (Re∞

(A.35)

u = ρ∞ u ∞ /μ∞ , and i = 0 for the boundarywith the unit Reynolds number being Re∞ layer thickness δ (where we leave away the lower index 0), i = 1 for the displacement thickness δ1 , i = 2 for the momentum thickness δ2 . The constants Ci for the different thicknesses and the exponent n, both for laminar and turbulent flow are given in Table A.1. u u∗ with the relation Eq. (A.25) as Re∞ and introducing Redefining the original Re∞ the reference temperature, Eq. (A.35) becomes

Appendix A: Useful Relations

429

Table A.1 Constants Ci in Eq. (A.35). The exponent in that equation is n = 0.5 for laminar flow and n = 0.2 for turbulent flow. Also given is the shape factor H12 = δ1 /δ2 δ δ1 δ2 H12 Laminar BL: Ci = Turbulent BL: Ci =

5 0.37

1.721 0.046

δi = Ci

x 1−n u )n (Re∞

δi = Ci

x 1−n u )n (Re∞



0.664 0.036

ρ∗ μ∞ ρ∞ μ∗

−n

2.591 1.278

,

(A.36)

.

(A.37)

respectively 

T∗ T∞

n(1+ω)

Alternate formulations for high Mach-number flow can be found, for instance, in [1]. The characteristic thicknesses Δ of laminar and turbulent boundary layers can be used to explain quite a number of phenomena in laminar and turbulent attached viscous flow, see [1, 3]. They also must be taken into account in grid generation for numerical methods. The characteristic thicknesses govern the wall shear stress and the heat flux in the gas at the wall of attached viscous flow of boundary layer type. In the laminar flow domain, the characteristic thickness Δlam is approximately the boundary layer thickness δlam , the 99 per-cent thickness. In the turbulent domain Δtur b is the thickness of the viscous sub-layer δvs , not the thickness δtur b . δvs is much smaller than the boundary layer thickness δtur b . However, at and in the vicinity of singular points and singular lines the characteristic thickness can not be approximated by δlam or δvs . The explicit relation for the thickness of the viscous sub-layer usually is not given in the boundary-layer literature. Exceptions are for instance the book of E.R.G. Eckert and R.M. Drake [9] and the report of G. Simeonides [10]. The relation of the viscous sub-layer, and hence of the characteristic thickness Δtur b , reads for the flat plate: δvs = Δtur b

x 0.2 = 33.78 u )0.8 (Re∞



ρ∞ μ∗ ρ∗ μ∞

0.8 ,

(A.38)

.

(A.39)

respectively: δvs = Δtur b = 33.78

x 0.2 u )0.8 (Re∞



T∗ T∞

0.8(1+ω)

430

Appendix A: Useful Relations

u u 1−n Note the different dependencies of δvs and δtur b on x and Re∞ : δvs ∝ x n (Re∞ ) 1−n u −n compared to δtur b ∝ x (Re∞ ) (Eq. (A.35)), where n = 0.2. The authors of [9] give n = 0.1 for the δvs -relation. Regarding the treatment at junctions, laminar-turbulent transition locations etc. with the concept of the virtual origin see [3].

References 1. Hirschel, E.H.: Basics of Aerothermodynamics. 2nd, revised edition. Springer, Cham, Heidelberg, New York (2015) 2. Hirschel, E.H., Weiland C.: Selected Aerothermodynamic Design Problems of Hypersonic Flight Vehicles. Progress in Astronautics and Aeronautics, AIAA, Reston, Va, vol. 229. Springer, Heidelberg (2009) 3. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin, Heidelberg (2014) 4. Rizzi, A., Oppelstrup, J.: Aircraft Aerodynamic Design with Computational Software. Cambridge University Press (2020) 5. Jones, R.T.: Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound. NACA Rep. 835 (1946) 6. Simeonides, G.: Generalized reference-enthalpy formulation and simulation of viscous effects in hypersonic flow. Shock Waves 8 3, 161–172 (1998) 7. Hansen, C.F.: Approximations for the Thermodynamic and Transport Properties of HighTemperature Air. NACA TR R-50 (1959) 8. Rubesin, M.W., Johnson, H.A.: A critical review of skin friction and heat transfer solutions of the laminar boundary layer of a flat plate. Trans. ASME 71, 385–388 (1949) 9. Eckert, E.R.G., Drake, R.M.: Heat and Mass Transfer. MacGraw-Hill, New York (1950) 10. Simeonides, G.: On the Scaling of Wall Temperature Viscous Effects. ESA/ESTEC EWP 1880 (1996)

Appendix B

Constants, Atmosphere Data, Units, and Conversions

In this book, units are in general the SI units (Système International d’Unités), see [1, 2], where also the constants can be found. In the following sections we give first constants and air properties, Sect. B.1, and then a selection of atmosphere data, Sect. B.2. The basic units, the derived units, and conversions to US units are given in Sect. B.3.1.1

B.1

Constants and Air Properties

See Table B.1. Molar universal gas constant:

Standard gravitational acceleration of earth at sea level:

R0 = 8.314472 · 103 kg m2 s−2 kmol−1 K−1 = = 4.97201·104 lbm ft2 s−2 (lbm -mol)−1 ◦ R−1

g0 = 9.80665 m s−2 = 32.174 ft s−2

Table B.1 Molecular weights and gas constants of air constituents for the low temperature domain [3, 4]. ∗ is the U.S. standard atmosphere value [5], + the value from [4] Gas Molecular weight M [kg kmol−1 ] Specific gas constant R [m2 s−2 K−1 ] Air N2 O2

1 Details

28.9644∗ (28.97+ ) 28.02 32.00

287.06 296.73 259.83

can be found, for instance, at http://physics.nist.gov/cuu/Reference/contents/html

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

431

432

Appendix B: Constants, Atmosphere Data, Units, and Conversions

Table B.2 Properties of the 15 ◦ C U.S. standard atmosphere as function of the altitude [5] Altitude H Temperature Pressure p Density ρ Dynamic Thermal [km] T [K] [Pa] [kg m−3 ] viscosity μ conductivity k [N s m−2 ] [W m−1 K−1 ] 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0

B.2

288.150 281.651 275.154 268.659 262.166 255.676 249.187 242.700 236.215 229.733 223.252 216.650 216.650

1.013 · 105 8.988 · 104 7.950 · 104 7.012 · 104 6.166 · 104 5.405 · 104 4.722 · 104 4.110 · 104 3.565 · 104 3.080 · 104 2.650 · 104 1.940 · 104 1.417 · 104

1.225 · 100 1.112 · 100 1.007 · 100 9.092 · 10−1 8.193 · 10−1 7.364 · 10−1 6.601 · 10−1 5.900 · 10−1 5.258 · 10−1 4.671 · 10−1 4.135 · 10−1 3.119 · 10−1 2.279 · 10−1

1.789 · 10−5 1.758 · 10−5 1.726 · 10−5 1.694 · 10−5 1.661 · 10−5 1.628 · 10−5 1.595 · 10−5 1.561 · 10−5 1.527 · 10−5 1.493 · 10−5 1.458 · 10−5 1.421 · 10−5 1.421 · 10−5

2.536 · 10−2 2.485 · 10−2 2.433 · 10−2 2.381 · 10−2 2.329 · 10−2 2.276 · 10−2 2.224 · 10−2 2.170 · 10−2 2.117 · 10−2 2.063 · 10−2 2.009 · 10−2 1.953 · 10−2 1.953 · 10−2

Atmosphere Data

See Table B.2.

B.3

Units and Conversions

Basic and derived SI units are listed of the major flow, transport, and thermal entities. In the left column name and symbol are given and in the right column the unit (dimension), with → the symbol used in Appendix C, and in the line below its conversion.

B.3.1

SI Basic Units

length, L

[m], → [L] 1.0 m = 100.0 cm = 3.28084 ft 1,000.0 m = 1.0 km

mass, m

[kg], → [M] 1.0 kg = 2.20462 lbm

Appendix B: Constants, Atmosphere Data, Units, and Conversions

time, t

[s] (= [sec]), → [t]

temperature, T

[K], → [T] 1.0 K = 1.8 ◦ R ⇒ TKelvin = (5/9) (TFahrenheit + 459.67) ⇒ TKelvin = TCelsius + 273.15

amount of substance, mole

[kmol], → [mole] 1.0 kmol = 2.20462 lbm -mol

B.3.2

SI Derived Units

area, A

[m2 ], → [L2 ] 1.0 m2 = 10.76391 ft2

volume, V

[m3 ], → [L3 ] 1.0 m3 = 35.31467 ft3

speed, velocity, v

[m s−1 ], → [L t −1 ] 1.0 m s−1 = 3.28084 ft s−1

force, F

[N] = [kg m s−2 ], → [M L t−2 ] 1.0 N = 0.224809 lbf

pressure, p

[Pa] = [N m−2 ], → [M L−1 t −2 ] 1.0 Pa = 10−5 bar = 9.86923·10−6 atm = = 0.020885 lbf ft−2

density, ρ

[kg m−3 ], → [M L−3 ] 1.0 kg m−3 = 0.062428 lbm / ft −3 = = 1.94032·10−3 lbf s2 ft−4

(dynamic) viscosity, μ

[Pa s] = [N s m−2 ], → [M L−1 t −1 ] 1.0 Pa s = 0.020885 lbf s ft −2

kinematic viscosity, ν

[m2 s−1 ], → [L2 t −1 ] 1.0 m2 s−1 = 10.76391 ft2 s−1

shear stress, τ

[Pa] = [N m−2 ], → [M L−1 t −2 ] 1.0 Pa = 0.020885 lbf ft −2

433

434

Appendix B: Constants, Atmosphere Data, Units, and Conversions

energy, enthalpy, work, quantity of heat

[J] = [N m], → [M L2 t −2 ] 1.0 J = 9.47813·10−4 BTU = = 23.73036 lbm ft2 s−2 = 0.737562 lbf s−2

(mass specific) internal energy e, enthalpy h

[J kg−1 ] = [m2 s−2 ], → [L2 t −2 ] 1.0 m2 s−2 = 10.76391 ft2 s−2

(mass) specific heat, cv , c p specific gas constant, R

[J kg−1 K−1 ] = [m2 s−2 K−1 ], → [L2 t −2 T−1 ] 1.0 m2 s−2 K−1 = 5.97995 ft2 s−2 ◦ R−1

power, work per unit time

[W] = [J s−1 ] = [N m s−1 ], → [M L2 t −3 ] 1.0 W = 9.47813·10−4 BTU s−1 = = 23.73036 lbm ft2 s−3

thermal conductivity, k

[W m−1 K−1 ] = [N s−1 K−1 ], → [M L t−3 T−1 ] 1.0 W m−1 K−1 = = 1.60496·10−4 BTU s−1 ft −1 ◦ R−1 = = 4.018342 lbm ft s−3 ◦ R−1

heat flux, q

[W m−2 ] = [J m−2 s−1 ], → [M t−3 ] 1.0 W m−2 = 0.88055·10−4 BTU s−1 ft −2 = = 2.204623 lbm s−3

References 1. Taylor, B.N. (ed.): The International System of Units (SI). US Dept. of Commerce, National Institute of Standards and Technology, NIST Special Publication 330 (2001), US Government Printing Office, Washington, D.C. (2001) 2. Taylor, B.N.: Guide for the Use of the International System of Units (SI). US Dept. of Commerce, National Institute of Standards and Technology, NIST Special Publication 816 (1995), US Government Printing Office, Washington, D.C. (1995) 3. Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Wiley, New York (corrected printing) (1964) 4. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002) 5. N.N.: U.S. Standard Atmosphere. Government Printing Office, Washington, D.C. (1976)

Appendix C

Symbols

Only the important symbols are listed. If a symbol appears only locally or infrequent, it is not included. Dimensions are given in terms of the SI basic units: length [L], time [t], mass [M], temperature [T], and amount of substance [mole], Appendix B. For actual dimensions and their conversions see Appendix B.3.

C.1 A, Ar e f a b b0 CD C D∗ i CL Cl Cl Cm Cn Cz cf cp cp c pstag c pvac D D Di Di∗ D0

Latin Letters reference area, [L2 ] speed of sound, [Lt−1 ] wing span [L] lateral distance of trailing vortex centers, [L] drag coefficient [−] drag coefficient at elliptical circulation distribution, [−] lift coefficient [−] local lift coefficient, [−] rolling-moment coefficient, [−] pitching-moment coefficient, [−] yawing-moment coefficient, [−] local normal force coefficient, [−] skin-friction coefficient, [−] (mass) specific heat at constant pressure, [L2 t −2 T−1 ] pressure coefficient, [−] stagnation pressure coefficient, [−] vacuum pressure coefficient, [−] diameter [L] drag [MLt−2 ] induced drag [MLt −2 ] induced drag at elliptic circulation distribution, [MLt−2 ] zero-lift drag, [MLt −2 ]

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

435

436

e e F G g H H k L L L∗ L/D M Me M∞ m n Pr p pb p∞ q q∞ R R0 R RN Re Reu r Sr s s T Te Tr Tt Tw T∞ T∗ t t0 t, n, z V V

Appendix C: Symbols

Oswald efficiency factor, [−] span-efficiency factor, [−] force [M Lt −2 ] normalized circulation, [−] gravitational acceleration, [Lt −2 ] altitude [L] shape factor, [−] thermal conductivity, [MLt−3 T−1 ] length [L] lift [MLt−2 ] lift at elliptical circulation distribution, [MLt−2 ] lift-to-drag ratio [−] Mach number, [−] boundary-layer edge Mach number, [−] flight Mach number [−] mass, [M] exponent in boundary-layer relations, [−] Prandtl number, [−] pressure [ML−1 t −2 ] cross-flow bluntness parameter, [−] free-stream pressure [ML−1 t −2 ] heat flux, [Mt −3 ] free-stream dynamic pressure, [ML−1 t −2 ] gas constant, [L2 t −2 T−1 ] universal gas constant, [ML2 t −2 mole−1 T−1 ] radius [L] nose radius [L] Reynolds number, [−] unit Reynolds number, [L−1 ] recovery factor, [−] Strouhal number, [−] entropy, [L 2 /t 2 T ] spanwise load factor, [−] temperature [T] boundary-layer edge temperature [T] recovery temperature, [T] total temperature, [T] wall temperature, [T] free-stream temperature [T] reference temperature, [T] time [t] time scale, [t] external inviscid streamline-oriented coordinates magnitude of velocity vector [Lt−1 ] velocity vector, [−]

Appendix C: Symbols

u, v, w u e , ve u ∞ , v∞ vn vt Ws w0 xbd x, y, z x, y, z x ∗, y∗, z∗  x i (i  = 1, 2, 3)

C.2

Cartesian velocity components [Lt−1 ] boundary-layer edge velocity, [Lt−1 ] free-stream velocity, flight speed, [Lt−1 ] cross-flow velocity component, [Lt −1 ] stream-wise velocity component, [Lt−1 ] wing loading, [M t−2 L−1 ] downward velocity, [Lt−1 ] vortex-breakdown location, [L] Cartesian coordinates, [L] body axis coordinates, [L] CRM reference coordinates, [L] Cartesian reference coordinates

Greek Letters

α αV , αV , ϕV β Γ Γ0 γ γ Δc δ δlam δtur b δvs δ1 δ2 ε ε η f lap ηslat Λ λ μ μe ρ ρe ρ∞ τw σ ϕ0

angle of attack [◦ ] vortex-axis angles, [◦ ] sideslip angle [◦ ] circulation, [L2 t −1 ] root circulation, [L2 t −1 ] flight-path angle [◦ ] ratio of specific heats, [−] characteristic boundary layer thickness, [L] flow boundary layer thickness [L] laminar boundary-layer thickness, [L] turbulent boundary-layer thickness, [L] viscous sub-layer thickness, [L] displacement thickness, [L] momentum-flow displacement thickness, [L] vortex-line angle, [◦ ] surface emissivity coefficient, [−] flap setting [◦ ] slat setting [◦ ] aspect ratio, [−] taper ratio, [−] viscosity, [ML−1 t −1 ] boundary-layer edge viscosity [ML−1 t −1 ] density [ML−3 ] boundary-layer edge density [ML−3 ] free-stream density [ML−3 ] wall shear stress, skin friction, [ML−1 t −2 ] dimensionless circulation, [−] sweep angle of leading edge [◦ ]

437

438

Appendix C: Symbols

trailing-edge flow shear angle, [◦ ] vorticity-content vector, [−] exponent in the power-law equations of viscosity and heat conductivity, [−] vorticity vector, [−]

ψe Ω ω ω

C.3

Indices

C.3.1 u ∗ ∗

unit with elliptical circulation distribution reference-temperature value

C.3.2 D e ic inv k L L LE lam ref TE t tr tur b vac vs w μ 0 ∞

Upper Indices

Lower Indices

drag boundary-layer edge, external (inviscid flow) incompressible inviscid thermal conductivity lift length leading edge laminar reference trailing edge total transition turbulent vacuum viscous sub-layer wall viscosity leading edge infinity

Appendix C: Symbols

C.4

Other Symbols

O( ) order of magnitude v, v vector < > average

439

Appendix D

Abbreviations, Acronyms

AGARD AIAA ARA AVA AVT BDW BL CFD CRM DDES DES DLR DNS DRG FFA FOI HISSS HTP ISW LE LES LTA MBB NACA NASA NATO NLR NS ONERA RAE

Advisory Group for Aerospace Research & Development of NATO American Institute of Aeronautics and Astronautics Aircraft Research Association Aerodynamische Versuchsanstalt Göttingen Applied Vehicle Technology Panel Blunt Delta Wing boundary layer computational fluid dynamics Common Research Model delayed detached-eddy simulation detached-eddy simulation German Aerospace Center direct numerical simulation Defence Research Group of NATO National Aerospace Research Center Sweden Swedish Defence Research Agency Higher-Order Subsonic-Supersonic Singularity (method) horizontal tail plane infinite swept wing leading edge large-eddy simulation large transport airplane Messerschmitt-Bölkow-Blohm GmbH National Advisory Committee for Aeronautics National Aeronautics and Space Administration North Atlantic Treaty Organization National Aerospace Laboratory of the Netherlands Navier–Stokes National Aerospace Research Center France Royal Aircraft Establishment

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

441

442

RANS RTO STO TE URANS

Appendix D: Abbreviations, Acronyms

Reynolds-averaged Navier–Stokes Research and Technology Organization of NATO Science and Technology Organization of NATO trailing edge unsteady Reynolds-averaged Navier–Stokes

Permissions

Figures reproduced with permission by Aerospace Science and Technology: Figs. 10.19, 10.39 below, 10.40, 10.42, 10.43, 10.44. Airbus: Fig. 9.6. American Institute of Aeronautics and Astronautics: Figs. 3.4, 3.11, 3.12, 9.29, 10.3, 11.14 – 11.17. Bernard & Graefe: Fig. 10.9. DGLR: Figs. 10.10, 11.8 – 11.10. DLR: Figs. 6.2, 7.2, 9.16 – 9.19. Hanser Verlag, München: Figs. 1.1 left, 1.3, 1.5 left and right, 8.23, 9.12, 9.13, 10.5, 10.14. Jefferys, D.: Fig. 1.1, right. J. Comp. Physics: Fig. 3.13, 3.15. MIT Press: Fig. 3.6. ONERA: Fig. 7.18. Progr. Aerospace Science: Figs. 3.16, 3.19, 8.1, 9.26 – 9.28. Shaker Verlag, Aachen: Figs. 9.20 – 9.24. Springer Verlag: Figs. 1.8, 2.3, 2.4, 3.2, 3.20, 3.21, 4.18, 7.5, 9.1 – 9.4, 9.7 – 9.9, 9.14, 9.15, 10.4. uhdwallpapers: Fig. 1.7. utzverlag, München: Figs. 1.2, 1.4, 3.9, 3.10, 3.22, 9.25. Verlag Dr. Hut, München: Figs. 10.49 – 10.56. The permissions to reprint all other figures have been provided directly from the authors, see Acknowledgements at the beginning of the book.

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

443

Subject Index

A Aerion AS2, 269 Air material properties, 431 transport properties, 425 Airbus A320, 254 Airbus A350, 186 Airbus A380, 259 Approach one-domain, 29, 281 three-domain, 29 two-domain, 29 Area rule, 37, 41 Aspect ratio large, 2, 12, 35f., 85, 93, 114, 119, 127, 168, 174, 181f., 187, 193, 212 small, 2, 12, 14, 40, 121, 186, 212 Attachment indicator, 165 open-type, 152, 162, 205, 208, 344 point, 148f., 162, 167 Attachment line, 15, 150f., 153, 162ff., 251 embedded, 149 inviscid, 149f., 163 primary, 149, 162, 167, 342 secondary, 149, 167, 343, 344 tertiary, 149, 167, 343, 344 AVT-183 configuration, 329

B B-58, 278 B-70, 278 Barotropic fluid, 53 Blunt Delta Wing, 339ff., 366 Boeing 707, 259 Boeing 747, 80, 259

Boeing 787, 186 Boundary-layer thickness, 167 1/7-power, 428 Blasius, 428 characteristic, 429 extremum, 164 Boundary-layer velocity profile laminar, 31 turbulent, 31 Breguet range formula, 233 Buffet, 79, 202, 366 onset boundary, 237 transonic, 131, 234, 237f. Buffeting, 80f. transonic, 237 Bypass ratio, 248

C CAST 7 airfoil, 100 Circulation bound, 84 root, 84 Circulation theory, 16 Common Research Model, 133, 187, 199ff. Concorde, 186, 270, 280, 366 Crocco’s theorem, 58, 101 Cross-flow bluntness parameter, 136, 292, 317f., 340 shock, 344 Crow instability, 68, 184, 260

D Decambering, 113, 159 boundary-layer, 22, 39, 127ff., 236 shock-wave, 22, 39, 59, 129f., 215, 236

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

445

446 Detachment line, 150f., 163, 164 point, 150f., 166 Displacement effect, 34 thickness, 34, 39, 131, 165, 403 DM-1, 277, 291, 332 Drag equivalent reduction, 254ff. form, 39, 107, 120, 185, 236, 255 induced, 16ff., 35ff., 51, 86f., 93f., 107f., 111f., 121, 185, 189, 212, 253ff., 369–371, 379 interference, 38 pressure, 37 skin-friction, 37, 107, 120, 185, 255 transonic, 39 trim, 226, 255 viscous, 38, 107, 254 wave, 38, 236, 255, 270, 366 zero-lift, 39, 87, 134 Drag divergence, 37–39, 132f., 234, 237, 270 Drag Prediction Workshop, 133, 199 Draken, 280 Droop-noose device, 239 DSF 194, 278 E EFEM, 374 Entropy, 121ff. Equivalence theorem, 37 Equivalent inviscid source distribution, 35 Euler boundary layer, 102, 121, 195 Euler wake, 120, 195 Eurofighter Typhoon, 7, 186, 274, 280 EUROLIFT II, 249 F F-102, 278 F-104, 40, 186, 280 F-106, 278 F-106B, 373 F-16, 276, 366f. F-16XL, 276 F-18, 366f. F-35A, 186 F/A-18, 81 Feeding layer, 124, 308 Flatback airfoil, 135 Flight-path angle, 5 Flow downwash, 16ff., 35, 51, 111f., 185, 199, 213, 219f., 259, 421f.

Subject Index Flow field closed lee-side, 74, 168, 174, 287, 299, 343, 344, 385, 390 open lee-side, 168, 174, 210, 287, 290, 299, 308, 309, 320, 384, 385, 388, 390 Fowler flap, 246 G Galilean invariance, 29, 47 Gas constant specific, 431 universal, 431 Geodesic, 163 Gravitational acceleration at sea level, 431 Gripen, 68, 274, 280 Ground clearance, 248 Gurney flap, 244 H HA-300, 279 Hansen equation, 425 Heat flux, 153 in the gas at the wall, 164, 428f. HF-24, 279 Horse-shoe vortex, 183, 202, 247 Hypersonic flow/flight, 56f., 79, 133, 164, 186, 269, 271f., 339, 382 Hypersonic shadow effect, 340, 383 I Interaction global, 18, 35f., 93 shock-wave/boundary-layer, 39, 57, 131, 235ff., 299 strong, 18, 29, 35f., 39, 93, 105, 128 vortex, 73f. weak, 34f., 38, 148 K Kármán vortex street, 14, 135, 150 Kolbe wing, 188ff. Krueger flap, 245 Küchemann wing tip, 255f. Kutta direction, 22, 111, 127, 131, 140 panel, 140f. panel free-floating, 140 Kutta condition, 16, 18, 23, 37, 52f., 109ff., 121, 127, 131, 150, 419 explicit, 17ff., 140 implicit, 17ff., 37, 140, 292

Subject Index Kutta-Joukowsky theorem, 82

L Lagrangian description, 47 Lambda shock, 236 Laminar wing, 238 Leading edge aerodynamically sharp, 136ff., 150, 319, 330 extension, 387 root extension, 373 subsonic, 40, 271, 273, 277, 366 suction analogy, 281, 369ff. supersonic, 271 Leading-edge flow shear angle, 310, 314 Lift enhancement surface, 213, 239 non-linear, 52, 212ff., 230, 274f., 281, 285ff., 324, 334f., 345ff., 360, 366, 370 Lift divergence, 39 Lifting-line model, 14, 16, 20, 46, 51, 113, 185, 253 Lift-to-drag ratio, 87, 185, 239, 243, 244, 272, 368 Locality principle, 36, 111 Local wake coordinate system, 105, 111 Local wing stall, 248

M Mach number critical, 11, 39 drag-divergence, 39f., 132, 234 pre-shock, 35, 39, 101, 130f., 214, 236f. subcritical, 12, 38 supercritical, 22 Me 163, 278 MiG 21, 133 Mini-TED, 244 Mirage, 280 Molecular weight, 431

N Nacelle-strake vortex, 79, 248ff. Non-parallel effects, 161 Normal angle of attack, 123, 291, 294, 306, 316, 325, 331, 339, 366, 369 Normal leading-edge Mach number, 124, 213, 292, 294, 306, 316, 325, 331, 339, 366, 369 No-slip wall condition, 9, 153

447 O Onglet vortex, 247 Onset flow, 176 Orr-Sommerfeld equation, 161 Oswald efficiency factor, 87

P Panel method, 20, 187ff. Paradox of d’Alembert, 16, 18, 108 Phase-plane analysis, 154 Phase portrait, 148, 154f., 170 Poincaré surface, 162ff., 172, 210, 213, 287, 289, 308, 320, 344, 383, 384ff. Points-of-inflection line, 164 Prandtl-Glauert rule, 42, 230 Prandtl number, 426 Pressure pmax -line, 163 Pulqui II, 279

R Rafale, 274 Re-entry flight/vehicle, 57, 270, 271, 339, 382 Reference area, 82 Airbus Gross, 82 Boeing Wimpress, 82, 199 Rolling-moment divergence, 361

S SAGITTA, 329ff. SÄNGER, 273 Separation, 36, 156, 166 control, 78 cove, 240 definition, 15 flow-off, 11ff., 16, 23, 30, 35, 109ff., 115, 127, 138, 140, 149, 159, 182, 230, 235, 248, 281, 289, 300, 347, 388 incipient, 298 indicator, 165 open-type, 30, 152, 162f., 207, 319, 343 ordinary, 11, 12, 14, 30, 35, 149, 183, 254, 281, 289, 315, 350 point, 15, 162 squeeze-off, 11 Separation line, 15, 148ff., 152f., 158, 160, 162ff., 208, 251, 342 primary, 30, 203, 309f., 332, 343f. secondary, 203, 208, 298, 311, 334, 343f. tertiary, 208, 315 Separatrix, 151 Sharklet, 254, 257

448 Shock capturing, 124 fitting, 124 Shock wake, 100 Singular line, 15, 148, 153, 162ff., 203, 320, 339, 429 Singular point, 10, 11, 15, 148f., 153ff., 156f., 159, 161, 162, 165, 170, 203, 205, 320, 343, 429 center, 155 focus, 155, 159, 165 half-nodal, 161 half-saddle, 161, 168 nodal, 155, 162, 165 quarter-saddle, 167f. saddle, 151, 155f., 162f. star node, 156 Skin friction, 153, 427 τw -min line, 164f. line, 149ff., 158, 162f., 165ff., 175, 191, 203ff., 214, 250ff., 289, 310ff., 332, 342ff., 348 Slat, 245 Slat cutout, 249 Slat-horn vortex, 247 Slip flow, 9 Solenoidality, 48 Space Shuttle Orbiter, 186, 271, 366 Span-efficiency factor, 87 Spanwise load factor, 84, 113 Specific excess power, 370 Specific impulse, 233 SR-72, 273 Stagnation point, 150f., 162, 167, 418 Stall deep, 385 Standard atmosphere, 432 Streamline, 9f., 34, 53, 57, 107, 110f., 148, 150f., 152, 156, 158, 159, 160f., 162, 189f. external inviscid, 97, 104 limiting, 150 surface, 163 Strouhal number, 135 Structural stability, 170ff. flow field, 36 Subsonic flow/flight, 36, 187 Suction peak, 12, 289, 302, 332, 340 pressure, 286, 289, 300, 334, 336, 369, 370 Supersonic flow/flight, 187 Supersonic flow pocket, 38, 39, 57, 98f., 129, 141, 234f., 239, 327

Subject Index Surface-curvature effect, 33, 161 Sutherland equation, 425 Symmetry break first, 17, 20, 56, 109, 111 second, 17, 20, 56, 111f., 115, 304

T Temperature adiabatic, 426 external inviscid flow, 426 recovery, 426f. reference, 427f. total, 426 wall, 164 Thermal state surface extrema, 164, 298 Thin wing, 40, 132, 186, 269 Topology skin-friction field, 148ff. Total pressure loss, 59, 99f., 101, 121, 122, 130f., 195, 282, 303, 308, 317, 418 Trailing edge blunt, 11, 21, 113, 132ff., 141, 150, 323 cusp, 21, 132 finite angle, 21, 132 Trailing-edge flow shear angle, 105, 111, 189, 190, 194, 215, 310 Transonic lift dip, 237 Tu-144, 270

U Unmanned Aerial System, 8, 276 Upstream effect, 35, 36

V Vacuum pressure coefficient, 417 VHBR engine, 248 Viggen, 280 Viscosity, 425 Viscous sub-layer, 429f. Vortex bound/lifting, 16, 51 control, 78 layer, 13f. lee-side, 6, 14, 36, 41, 69, 74, 77, 93, 119, 124, 161ff., 168, 172ff., 208, 213, 249, 269ff., 359ff. secondary, 36, 124, 174, 183, 206, 208f., 210, 230, 281, 282, 284, 289, 290, 298, 300f., 310, 320, 323, 346, 384, 385, 388 shedding, 12, 23 starting, 16, 46, 51ff., 99

Subject Index tertiary, 124, 183, 210, 230, 289, 290, 300f., 349 trailing, 4f., 11, 14, 16, 20, 35, 46, 51f., 63f., 66, 68f., 73, 75, 93, 113, 114, 124, 181, 183, 183f., 184, 200, 210, 219, 221f., 223, 225, 228, 230, 253, 259f., 263, 264 Vortex age, 86, 222 Vortex breakdown, 14, 75f., 80, 281, 285, 288, 293, 296, 301, 306, 315, 323ff., 331ff., 345, 349, 359f., 364, 373 bubble type, 75 spiral type, 75, 323 Vortex core jet-type flow, 75 wake-type flow, 75 Vortex filament, 49, 62f., 73, 159, 206, 251 Vortex-Flow Experiment first, 193, 283, 300, 306 second, 284 Vortex generator, 78, 243 sub-boundary-layer, 79, 243 Vortex layer trailing, 1, 11, 12, 16, 35, 51, 60, 64, 69f., 75, 84, 93, 105, 111ff., 119, 123f., 181, 185, 187, 194, 199f., 213, 216, 221f., 223f., 253f., 259f., 281f., 305 Vortex line, 46, 49, 52, 61, 73f., 107, 114, 131, 152, 189, 190, 281 Vortex-line angle, 106, 111, 131, 182, 187, 189, 190, 194, 216, 229 Vortex model Batchelor, 66 Burger, 65 Burnham-Hallock, 66 Lamb-Oseen, 65, 222 point, 49 potential, 49, 62, 64 Rankine, 49, 64f., 96, 222 ring, 51 Vortex overlapping, 285, 293, 323, 345, 361 Vortex pairing, 73, 223 Vortex re-configuration, 76, 323, 329 Vortex re-connection, 78 Vortex stretching, 59, 73 Vortex tilting, 59 Vorticity content, 30, 93f., 95, 190, 210, 229

449 kinematically active, 2, 14, 17, 22, 35f., 48f., 56, 70, 96ff., 112ff., 120f., 124, 187, 188, 198, 208, 216ff., 283, 291, 312f., 321, 345f. kinematically inactive, 3, 14, 17, 22, 35f., 48f., 56, 96ff., 219f., 240, 283, 291, 321

W Wake flow, 12, 35f., 37, 65, 68, 70, 99, 112 Wake-vortex hazard, 67, 259 Wind turbine, 135 Wing forward swept, 40, 106, 162, 186, 187, 190f., 274 hybrid, 6 infinite swept, 163f. ogive, 270, 366 ONERA M6, 55 pitch-up, 79, 293, 324, 336, 338, 361, 362, 368, 383, 385 root bending moment, 255, 390 root-fuselage fairing, 79, 199, 202 tip device, 253ff. Wing flow-fields shear, 17, 105, 110, 112, 135, 182, 245, 253, 305, 310 Winglet, 254 Wing stall, 237 Wing-tip vortex, 4, 7f., 14, 52, 75, 168, 182, 195, 208f. vortex system, 52, 113, 183, 187, 210f., 212f., 216, 230, 248, 254ff. Wing-tip reference point, 199

X X-15, 133 X-31A, 280 XF-92A, 278

Y Yawing-moment divergence, 361 Yehudi break, 133, 182, 199, 247

Author Index

A Abbott, I.H., 144 Abell, C.J., 178 Ackeret, J., 144 Allen, A., 268 Allmaras, S.R., 232 Anderson, Jr., J.D., 2, 16, 25, 37, 42, 43, 45, 89, 117, 144, 231 Andronov, A.A., 178 Atkinson, S.A., 75, 90

B Babinsky, H., 42, 266 Badcock, K.J., 356 Baker, G.R., 27 Bakker, P.G., 178 Bangga, G.S.T.A., 144 Bannink, W.J., 355, 356 Baron, J.R., 125 Batchelor, G., 66 Bate, E.J., 264, 268 Batemann, T.E.B., 117 Becker, J., 91 Behrbohm, H., 280 Behrends, K., 27, 267 Benjamin, T.B., 281, 354 Berg, D.E., 144 Betz, A., 37, 55, 89 Bier, N., 27 Bilanin, A.J., 268 Bippes, H., 178 Bird, R.B., 42, 434 Bloor, D., 16, 26, 52, 89 Boddener, W., 392 Boelens, O.J., 27, 297, 356 Bollay, W., 277, 353

Boltz, F.W., 232 Borst, H.V., 144 Brandon, J.M., 373, 392 Braza, M., 266 Breitsamter, C., 26, 89–91, 231, 267, 268, 356, 392 Brodersen, O.P., 232 Brown, C.E., 281, 354 Brown, P.W., 392 Burggraf, U., 356 Burg, J.W. van der, 267 Büscher, A., 254, 268 Busemann, A., 277, 353 Bush, R.H., 27 Bütefisch, K.A., 144, 355

C Calvo, J.B., 42 Campbell, J.F., 90, 354 Carter, E.C., 178 Cayley, G., 272 Cebeci, T., 2, 25, 42 Chambers, J.R., 90 Champion, M., 353 Charbonnier, D., 90 Chu, J., 355 Chyczewski, T.S., 27 Ciobaca, V., 267 Clareus, U., 276, 353 Cornish, J.J., 376, 392 Cousteix, J., 2, 25, 42, 90, 117, 144, 177, 232, 267, 268, 356, 392, 415, 430 Coustols, E., 268 Crippa, S., 232, 325, 348, 356, 357 Croom, D.R., 268 Crouch, J.D., 264, 268

© Springer-Verlag GmbH Germany, part of Springer Nature 2021 E. H. Hirschel et al., Separated and Vortical Flow in Aircraft Wing Aerodynamics, https://doi.org/10.1007/978-3-662-61328-3

451

452 Crow, S.C., 69, 90, 264, 268 Cummings, R.M., 316, 356 Curtiss, C.F., 434

D D’Alembert, J.-B. le Rond, 16, 18, 108 Dallmann, U., 43, 148, 177, 178 Darracq, D., 268, 355 Davey, A., 165, 178 Deck, S., 266 DeHaan, M.A., 232 Deister, F., 145 Délery, J., 1, 25, 42, 148, 152, 177, 232, 266 Dirmeier, S., 267 Dixon, C.J., 376, 392 Donaldson, C.duP., 268 Donohoe, S.R., 356 Dovgal, A.V., 25 Drake, R.M., 429, 430 Drela, M., 2, 25, 45, 59, 89 Drougge, G., 355 Duraisamy, H., 27 Durbin, P.A., 27

E Eberle, A., 26, 117, 125, 145, 283, 355, 392 Eckert, E.R.G., 429, 430 Eckert, M., 26 Edwards, C.L.W., 353 Egle, S., 392 Ehlers, T., 356 Eichelbrenner, E.A., 15, 26 Eisfeld, B., 27, 42, 232 Eliasson, P., 267 Elle, B.J., 75, 90 Elsenaar, A., 11, 26, 117, 125, 232, 355, 356 Erickson, G.E., 392 Eriksson, G., 117, 125, 232, 355 Eriksson, L.-E., 232, 282, 355 Esquieu, S., 267

F Farokhi, S., 27, 43, 144, 186, 231, 266 Fassbender, J., 232 Fellows, K.A., 178 Fischer, J., 100, 117 Flaig, A., 267 Flügge, S., 89 Fornasier, L., 117, 178, 187, 232 Foughner, J.T., 354 Frenzl, O., 37

Author Index Fricke, S., 267 Friedrich, O., 354 Frink, N.T., 373, 392 Fritz, W., 301, 316, 323, 352 Fulker, J., 266 Furman, A., 321, 356

G Gagnepain, J.-J., 353 Ganzer, W., 294, 356 Gebing, H., 43 Gehri, A., 90 Geissler, W., 266 Gerhold, T., 232 Gersdorff, K. von, 354 Gersten, K., 2, 25, 42, 277, 353 Gerz, T., 268 Geyr, H. von, 267 Gilbert, W.P., 392 Goldstein, S., 10, 26 Gordon, I.I., 178 Görtler, H., 26, 178 Görtz, S., 77, 90 Göthert, B., 37, 43, 213, 232 Gottmann, Th., 391, 392 Grasjo, I., 90 Grossi, F., 266 Groß, U., 391 Guillaume, M., 90 Gursul, I., 391

H Haase, W., 27, 117, 267 Hahne, D.E., 392 Haines A.B., 267 Hallissy, J.B., 392 Hall, M.G., 281, 354 Hansen, C.F., 425, 430 Harvey, J.K., 42, 266 Heinemann, H.-J., 392 Heinzerling, W., 354 Heller, G., 267 Helmholtz, H. von, 16, 60, 76 Henderson, W.P., 392 Henne, P.A., 144 Hentschel, R., 283, 305, 355 Herberg, H., 43 Herbst, W., 354 Herr, M., 144 Hess, J.L., 27 Hewitt, B.L., 232

Author Index Hilbig, R., 267, 392 Hilgenstock, A., 178 Hirschel, E.H., 2, , 25–27, 42, 43, 89, 90, 117, 125, 144, 145, 177–179, 232, 266, 267, 353–357, 391, 392, 415, 430 Hirschfelder, J.O., 434 Hitzel, S.M., 26, 283, 353, 355, 373, 392 Hjelmberg, L., 355 Hoarau, Y., 266 Hoder, H., 294, 356 Hoeijmakers, H.W.M., 71, 90, 125, 281, 354–356 Hoerner, S.F., 144 Holzäpfel, F., 268 Hornung, H., 178 Horst, P., 2, 25, 26, 45, 89, 231, 267, 353 Hövelmann, A., 329, 356 Hünecke, K., 268 Huffman, J.K., 392 Hummel, D., 74, 90, 280, 284, 289, 315, 316, 354–357 Hünecke, K., 392 Hunt, J.C.R., 166, 178

I Iovnovich, M., 266

J Jacob, D., 268 Jacquin, L., 266, 268 Jefferys, D., 26 Jiràsek, A., 79, 90, 267 Johnson, H.A., 430 Johnson, Jr., T.D., 392 Jones, R.T., 277, 287, 354, 355, 424, 430 Joukowski, N., 16, 26, 45

K Kaplan, W., 178 Kármán, Th. von, 26, 202, 232 Kawalki, K.H., 37 Keune, F., 37 Keye, S., 27 Klein, Ch., 356 Klein, F., 55, 89 Klopfer, G.H., 144 Knopp, T., 42 Kohlmann, D.L., 356 Kolbe, D.C., 232 Kompenhans, J., 356 Konrath, R., 356

453 Kordulla, W., 2, 25, 42, 90, 117, 144, 177, 232, 267, 356, 392, 415, 430 Körner, H., 26 Kozlov, V.V., 25 Krämer, E., 26, 144, 266, 274, 353 Krause, E., 26, 76, 90, 355 Kreuzer, P., 267 Kroll, N., 27, 232, 267 Küchemann, D., 1, 2, 25, 47, 132, 144, 186, 231, 267, 280, 353, 354, 392 Kuczera, H., 353 Kullberg, E., 90 Kutta, M.W., 16, 26, 45, 109 L Laflin, K.R., 232 Lamar, J.E., 392 Lanchester, F.W., 16, 26, 46 Laporte, F., 231 Larsson, R., 90 Lawaczeck, O., 144 Legendre, R., 280, 354 Le Moigne, Y., 73, 90 Lengers, M., 267 Leontovich, E.A., 178 Leschziner, M., 27 Levy, D.W., 232 Lightfoot, E.N., 42, 434 Lighthill, M.J., 1, 25, 27, 53, 89, 103, 114, 117, 128, 144, 151, 165, 178 Linde, M., 357 Lippisch, A., 37, 277 Lorenz-Meyer, W., 144 Lovell, J.C., 278, 291, 332, 354 Luber, W., 91 Lucchi, C.W., 144 Luckring, J.M., 27, 285, 291, 297, 299, 316, 355, 356 Lüdecke, H., 356 Ludwieg, H., 37, 277, 281, 354 Ludwig, T., 90 Lugt, H.J., 1, 25, 45, 54, 61, 89, 152, 178 Lutz, Th., 144, 266 M Mabey, D.G., 90 Madelung, G., 43, 89, 354 Maggin, B., 362, 391 Ma, H.-Y., 1, 25, 45, 89, 144 Maier, A.G., 178 Malcolm, G.N., 392 Mandanis, G., 90

454 Mangler, K.W., 117, 281, 354 Mani, M., 232 Marsden, D.J., 280, 354 Maruhn, K., 117 Masters, D., 354 Matteis, P. de, 266 Maury, B., 266 Mavriplis, D.J., 232 Mayer, R., 266 Meier, H.U., 43, 266, 353 Menter, F.R., 232 Merazzi, S., 90 Messerschmitt, W., 279 Michael, W.H., 281, 354 Miller, A.S., 392 Miller, D.S., 356 Miller, G., 268 Misegades, K., 355 Mockett, C., 27, 267 Modin, K.E., 276, 353 Moens, F., 268 Moffat, H.K., 177, 178 Molton, P., 266 Morrison, J.H., 232 Mosinskis, G.J., 42 Müller, R., 117 Murayama, M., 232 Murman, E.M., 125

N Nangia, R.K., 232, 373, 392 Newsome, R.W., 145 Nicolai, L.M., 267 Niehuis, R., 27, 267 Nitsche, W., 392 Nixon, D., 144 Norman, D., 267 Nowarra, H.J., 354

O Oberkampf, W.L, 27 Obermeier, E., 354 Obert, E., 144, 186, 231 Oppelstrup, J., 2, 26, 91, 145, 186, 231, 357, 415, 430 Örnberg, T., 280, 290, 354 Osborne, R.F., 354 Osterhuber, R., 373, 392 Oswatitsch, K., 10, 26, 37, 156, 159, 178 Özger, E., 268

Author Index P Pandolfi, M., 27 Payen, N.R., 277 Peake, D.J., 1, 25, 148, 152, 175, 177, 178 Peckham, D.H., 75, 90 Pelletier, A.J., 353 Perez, E.S., 125 Periaux, J., 353 Perry, A.E., 178 Peterka, J.A., 178 Pfnür, S., 199, 205, 231 Pironneau, O., 353 Platzer, M., 26, 178 Poisson-Quinton, Ph., 376 Polhamus, E.C., 281, 354, 369, 392 Powell, K.G., 125 Prandtl, L., 9, 10, 15, 16, 25, 26, 46, 113, 117, 185, 212, 231 Prem, H., 43, 89, 354 Probst, A., 267 Probst, S., 267 Puffert, W., 117 Pulliam, T.H., 27 R Radespiel, R., 27, 267 Rainbird, W.J., 280, 354 Raveh, D.E., 266 Reckzeh, D., 267 Redeker, G., 280, 284, 315, 354, 355 Rider, B., 232 Riedelbauch, S., 178, 339, 356, 357 Rivers, S.M., 232 Rizzi, A., 2, 26, 27, 55, 89–91, 117, 125, 145, 186, 231, 232, 282, 348, 355, 357, 392, 415, 430 Rohlmann, D., 27 Rom, J., 1, 25, 125, 355 Rosenhead, L., 27, 89, 178 Rossow, C.-C., 2, 25, 26, 45, 89, 231, 267, 353 Rossow, V.J., 268 Rubbert, P.E., 232 Rubesin, M.W., 430 Rudnik, R., 186, 231, 244, 267 Rudolph, P.K.C., 267 Rumsey, C.L., 27, 232 S Sacher, P., 232, 353 Sachs, G., 26 Sacks, A.H., 64, 70, 90, 117 Saffmann, P.G., 27, 45, 55, 89

Author Index Samuel, W.W.E., 43 Schade, N., 267 Schell, I., 268 Schiavetta, L.A., 356 Schlichting, H., 2, 25, 42, 43, 45, 89, 117, 144, 186, 231, 392 Schmatz, M.A., 232 Schmid, A., 91 Schmidt, W., 353 Schneider, W., 26, 178 Schrauf, G., 231 Schröder, A., 356 Schulte-Werning, B., 171, 178 Schütte, A., 356 Schwamborn, D., 27, 117, 178, 267 Schwarz, W., 355 Sears, W.R., 150, 178 Sedin, Y.C.J., 68, 90 Sensburg, O., 90 Serrin, J.B., 89 Shapiro, A.H., 42 Shortal, J.A., 362, 391 Simeonides, G., 425, 430 Simpson, R.W., 280, 354 Skow, A.M., 392 Small, W.J., 353 Smith, A.M.O., 42 Smith, B.R., 27 Smith, J.H.B., 117, 281, 354 Soulevant, D., 266 Spalart, P.R., 27, 232, 268 Spreiter, J.R., 64, 70, 90, 117 Squire, L.C., 294, 356 Stanbrook, A., 294, 356 Stanewsky, E., 117, 266 Staudacher, W., 353, 391, 392 Stephani, P., 90 Stewart, W.E., 42, 434 Stivers, Jr., L.S., 144 Stouflet, B., 353 Stratford, B.S., 34, 42, 139 Streeter, V.L., 42 Streit, T., 268 Strüber, H., 267 Stumpf, E., 268 Su, W.-H., 43 Sytsma, H.S., 232 Szodruch, J., 294, 356

455 T Tank, K., 279 Taylor, B.N., 434 Thiede, P., 266 Thomas, F., 237, 266 Thomas, P., 353 Thomson, W., 50 Thwaites, B., 33, 42, 139 Tinoco, E.N., 232 Tobak, M., 1, 25, 148, 152, 175, 177, 178 Tomac, M., 27 Trucano, T.G., 27 Truckenbrodt, E., 2, 25, 43, 45, 89, 117, 144, 186, 231, 392 Tsinober, A., 177, 178 Turk, M., 178

V Van Dyke, M., 36, 43, 178 Vardaki, E., 391 Vassberg, J.C., 232 Viviand, H., 178, 355 Vollmers, H., 178, 268 Von Doenhoff, A.E., 144 Vos, J.B., 90, 355 Vos, R., 27, 43, 144, 186, 231, 266

W Wagner, W., 353 Wahls, R.A., 232 Wang, K.C., 152, 178 Wang, Z., 391 Wanie, K.M., 232 Weidner, J.P., 353 Weiland, C., 27, 145, 178, 179, 353, 391, 430 Weis-Fogh, T., 56, 89 Wentz, W.H.Jr., 356 Werlé, H., 75, 90, 178, 280, 354 White, F.M., 2, 25 Wichmann, G., 267 Widnall, S.E., 64, 89 Wild, J., 267 Wilson, H.A., 278, 291, 332, 354 Winkel, M.E.M. de, 178 Winter, H., 277, 353 Wolf, K., 2, 25, 26, 45, 89, 231, 267, 353 Wood, R.M., 356 Woo, H., 178

456 Wu, J.-Z., 1, 25, 45, 89, 144 Wulf, R., 392 Z Zahm, A.F., 117

Author Index Zayas, J.R., 144 Zhang, H.-Q., 43 Zhou, M.-D., 1, 25, 45, 89, 144 Zickuhr, T., 232 Zierep, J., 117, 144