Modern aerodynamic methods for direct and inverse applications 9781119580560, 1821821831, 1119580560

Table of contents :
Cover......Page 1
Modern Aerodynamic Methodsfor Direct and InverseApplications......Page 4
© 2019......Page 5
Table of Contents......Page 6
Preface......Page 13
Acknowledgements......Page 15
1 Basic Concepts, Challenges and Methods......Page 16
2 Computational Methods:Subtleties, Approaches and Algorithms......Page 48
3 Advanced Physical Modelsand Mathematical Approaches......Page 180
4 General Analysis and Inverse Methodsfor Aerodynamic Modeling......Page 275
5 Engine and Airframe Integration Methods......Page 381
Cumulative References......Page 394
Index......Page 411
About the Author......Page 423

Citation preview

Modern Aerodynamic Methods for Direct and Inverse Applications

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Modern Aerodynamic Methods for Direct and Inverse Applications

Wilson C. Chin

This edition first published 2019 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2019 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-58056-0 Cover image: Airplane Maintenance - Stoyan Yotov | Dreamstime.com Cover design by Kris Hackerott Set in size of 11pt and Times New Roman PSMT Printed in the USA 10 9 8 7 6 5 4 3 2 1

Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1. Basic Concepts, Challenges and Methods . . . . . . . . . 1 1.1 Governing Equations - An Unconventional Synopsis 1.2 Fundamental “Analysis” or “Forward Modeling” Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic “Inverse” or “Indirect Modeling” Ideas . . . . 1.4 Literature Overview and Modeling Issues . . . . . . . 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . .

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.6 . 15 . 20 . 32

2. Computational Methods: Subtleties, Approaches and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Coding Suggestions and Baseline Solutions . . . . . 2.1.1 Presentation Approach . . . . . . . . . . . . . . . 2.1.2 Programming Exercises . . . . . . . . . . . . . . 2.1.3 Model Extensions and Challenges . . . . . . . . 2.2 Finite Difference Methods for Simple Planar Flows 2.2.1 Finite Differences - Basic Concepts . . . . . . . 2.2.2 Formulating Steady Flow Problems . . . . . . . . 2.2.3 Steady Flow Problems . . . . . . . . . . . . . . . 2.2.4 Wells and Internal Boundaries . . . . . . . . . . . 2.2.5 Point Relaxation Methods . . . . . . . . . . . . . 2.2.6 Observations on Relaxation Methods . . . . . . . v

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. 33 . 33 . 35 . 36 . 39 . 39 . 45 . 46 . 55 . 62 . 64

2.3 Examples - Analysis, Direct or Forward Applications . . 75 2.3.1 Example 1 - Thickness Solution, Centered Slit in Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Example 2 - Half-Space Thickness Solution . . . . 2.3.3 Example 3 - Centered Symmetric Wedge Flow . . . 2.3.4 Example 4 - General Solution with Lift, Centered Slit . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Example 5 - Transonic Supercritical Airfoil with Type-Dependent Differencing Solution, Subsonic, Mixed Flow and Supersonic Calculations . . . . . . 2.3.6 Example 6 - Three-Dimensional, Thickness Only, Finite, Half-Space Solution. . . . . . . . . . . . . . .

. 76 . 91 . 98 . 101

. 119 . 129

2.4 Examples - Inverse or Indirect Applications . . . . . . . . 138 2.4.1 Example 1 - Constant Pressure Specification and Symmetric Thin Ellipse . . . . . . . . . . . . . . . . . . 138 2.4.2 Example 2 - Inverse Problem, Pressure Specification, Centered Sit, Trailing Edge Closed vs Opened . . . . 145 2.4.3 Example 3 - Inverse Problem, Pressure Specification, Three-Dimensional Half-Space, Closed Trailing Edge, Nonlifting Symmetric Section . . . . . . . . . . . . . . 158 3. Advanced Physical Models and Mathematical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Nonlinear Formulation for Low-Frequency Transonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Discussion and Summary . . . . . . . . . . . . . . . . 3.1.4 References . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effect of Frequency in Unsteady Transonic Flow . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.2.2 Numerical Procedure . . . . . . . . . . . . . . . 3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Concluding Remarks . . . . . . . . . . . . . . . 3.2.5 References . . . . . . . . . . . . . . . . . . . . .

vi

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. 165 . 170 . 170 . 171 . 174 . 175 . 176 . 176 . 177 . 178 . 180 . 181

3.3 Harmonic Analysis of Unsteady Transonic Flow 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.3.2 Analytical and Numerical Approach . . . . . 3.3.3 Calculated Results . . . . . . . . . . . . . . . . 3.3.4 Discussion and Closing Remarks . . . . . . . 3.3.5 References . . . . . . . . . . . . . . . . . . . .

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. 182 . 182 . 183 . 184 . 185 . 188

3.4 Supersonic Wave Drag for Nonplanar Singularity Distributions . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Summary. . . . . . . . . . . . . . . . . . . . . . . 3.4.4 References . . . . . . . . . . . . . . . . . . . . .

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. 189 . 189 . 191 . 193 . 194

3.5 Supersonic Wave Drag for Planar Singularity Distributions . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . 3.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . 3.5.3 Concluding Remarks . . . . . . . . . . . . . 3.5.4 References . . . . . . . . . . . . . . . . . . .

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. 195 . 195 . 198 . 206 . 207

3.6 Pseudo-Transonic Equation with a Diffusion Term 3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 References . . . . . . . . . . . . . . . . . . . . . . 3.7 Numerical Solution for Viscous Transonic Flow . . 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Numerical Approach . . . . . . . . . . . . . . . . 3.7.4 Sample Calculation . . . . . . . . . . . . . . . . . 3.7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . 3.7.6 References . . . . . . . . . . . . . . . . . . . . . .

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. 208 . 209 . 209 . 212 . 212 . 213 . 213 . 213 . 216 . 217 . 218 . 220

3.8 Type-Independent Solutions for Mixed Subsonic and Supersonic Compressible Flow . . . . . . . . . . . . . . 3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Numerical Approaches . . . . . . . . . . . . . . . .

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. 221 . 221 . 221 . 223

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3.8.3.1 Horizontal Line Relaxation . 3.8.3.2 Vertical Column Relaxation 3.8.4 Summary. . . . . . . . . . . . . . . . . 3.8.5 References . . . . . . . . . . . . . . .

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3.9 Algorithm for Inviscid Compressible Flow Transonic Equation . . . . . . . . . . . . . . 3.9.1 Introduction . . . . . . . . . . . . . . . . 3.9.2 Analysis . . . . . . . . . . . . . . . . . . 3.9.3 Sample Calculations . . . . . . . . . . . 3.9.4 Summary and Conclusions . . . . . . . 3.9.5 References . . . . . . . . . . . . . . . .

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. 223 . 224 . 225 . 227

Using the Viscous . . . . . . . . . 228 . . . . . . . . . 228 . . . . . . . . . 229 . . . . . . . . . 231 . . . . . . . . . 232 . . . . . . . . . 233

3.10 Inviscid Parallel Flow Stability with Nonlinear Mean Profile Distortion . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3.10.3 Discussion and Conclusion . . . . . . . . . . . . . . 3.10.4 References . . . . . . . . . . . . . . . . . . . . . . .

. 234 . 235 . 235 . 239 . 240

3.11 Aerodynamic Stability of Inviscid Shear Flow Over Flexible Membranes . . . . . . . . . . . . . . . . . . . . 3.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.11.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.11.3 Specific Examples . . . . . . . . . . . . . . . . . 3.11.4 Discussion and Concluding Remarks . . . . . . 3.11.5 References . . . . . . . . . . . . . . . . . . . . .

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. 242 . 242 . 242 . 245 . 247 . 248

3.12 Goethert’s Rule with an Improved Condition . . . . . . . . . . . . . . . . 3.12.1 Introduction . . . . . . . . . . 3.12.2 Analysis . . . . . . . . . . . . 3.12.3 Summary . . . . . . . . . . . 3.12.4 References . . . . . . . . . .

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. 249 . 249 . 250 . 253 . 253

Boundary ....... ....... ....... ....... .......

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3.13 Some Singular Aspects of Three-Dimensional Transonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3.13.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3.13.2 Discussion and Summary . . . . . . . . . . . . . . . . 257 3.13.3 References . . . . . . . . . . . . . . . . . . . . . . . . 259

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4. General Analysis and Inverse Methods for Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.1 On the Design of Thin Subsonic Airfoils 4.1.1 Introduction . . . . . . . . . . . . . . 4.1.2 Analysis . . . . . . . . . . . . . . . . 4.1.3 First-Order Problem . . . . . . . . . . 4.1.4 Second-Order Problem . . . . . . . . 4.1.5 Discussion and Conclusion . . . . . . 4.1.6 References . . . . . . . . . . . . . . .

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. 264 . 264 . 265 . 266 . 269 . 271 . 273

4.2 Airfoil Design in Subcritical and Supercritical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2.2 Streamfunction Formulation . . . . . . . . . 4.2.3 Numerical Procedure . . . . . . . . . . . . . 4.2.4 Calculated Results . . . . . . . . . . . . . . . 4.2.5 Discussion and Closing Remarks . . . . . . 4.2.6 References . . . . . . . . . . . . . . . . . . .

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. 274 . 274 . 278 . 281 . 284 . 285 . 290

4.3 Direct Approach to Aerodynamic Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.3.2 Theory and Examples . . . . . . . . . . . . . . . . . . . 295 4.3.2.1 Constant Density Planar Flows . . . . . . . . 295 4.3.2.2 Constant Density Flows Past Three-Dimensional Finite Wings . . . . . . . . . . . . . . . . . . . 299 4.3.2.3 Compressible Flows Past Finite Wings . . . 301 4.3.2.4 Flows in Fans and Cascades . . . . . . . . . . 302 4.3.2.5 Axisymmetric Compressible Flows . . . . . 303 4.3.3 Sample Calculations . . . . . . . . . . . . . . . . . . . . 304 4.3.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . 307 4.3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . 310 4.4 Superpotential Solution for Jet Engine External Potential and Internal Rotational Flow Interaction . . . . . . . . . 312 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 313 4.4.2 Rotational Flow Equations . . . . . . . . . . . . . . . . 314 4.4.3 The Linearized Problem . . . . . . . . . . . . . . . . . 316 ix

4.4.4 Application to Jet-Engine External Potential and Internal Rotational Flow Interaction. . . . . . . . . . . . . . . . 318 4.4.5 Calculated Results and Closing Discussion . . . . . . 321 4.4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 325 4.5 Thin Airfoil Theory for Planar Inviscid Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Planar Flows With Constant Vorticity . . . . . . . 4.5.2.1 Planar Flows: Inverse Problems . . . . . . 4.5.2.2 Planar Flows: Direct Formulations . . . . 4.5.2.3 Some Planar Analytical Solutions . . . . . 4.5.2.4 Analogy To Ringwing Potential Flows . . 4.5.2.5 Source and Vortex Interactions for Ringwings . . . . . . . . . . . . . . . . . . . 4.5.3 Airfoils in General Parallel Shear Flow . . . . . . 4.5.4 Numerical Results . . . . . . . . . . . . . . . . . . . 4.5.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . 4.5.6 References . . . . . . . . . . . . . . . . . . . . . . . 4.5.7 Appendix I, Three-Dimensional Constant Density Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.8 Appendix II, Planar Compressible Shear Flow of a Gas . . . . . . . . . . . . . . . . . . . . . . . . .

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. 327 . 330 . 330 . 331 . 332 . 333

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. 334 . 335 . 339 . 341 . 343

. . 344 . . 345

4.6 Class of Shock-free Airfoils Producing the Same Surface Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 4.6.1 4.6.2 4.6.3 4.6.4

Introduction . . . . . . . . . . Analysis . . . . . . . . . . . . Discussion and Conclusion . References . . . . . . . . . .

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4.7 Engine Power Simulation for Transonic Flow-Through Nacelles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 4.7.1 4.7.2 4.7.3 4.7.4

Introduction . . . . . . . . . . . . . . . . . . Analytical and Numerical Approach . . . Numerical Results and Closing Remarks References . . . . . . . . . . . . . . . . . . x

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. 355 . 356 . 357 . 360

4.8 Inviscid Steady Flow Past Turbofan Mixer Nozzles . . . 361 4.8.1 4.8.2 4.8.3 4.8.4

Introduction . . . . . . . . . . . . . . . . . . Analytical Formulation . . . . . . . . . . . Calculated Results and Closing Remarks References . . . . . . . . . . . . . . . . . .

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. 361 . 361 . 363 . 365

5. Engine and Airframe Integration Methods . . . . . . . . . 366 5.1 5.2 5.3 5.4 5.5

Big Picture Revisited . . . . . . . . . . . . . . . . Engine Component Analysis . . . . . . . . . . . . Engine Power Simulation Using Actuator Disks Mixers and Supersonic Nozzles . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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. 367 . 371 . 374 . 375 . 377

Cumulative References . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

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Preface I am privileged to write this monograph, one focused on ideas new and old, but through it all, a book aimed at conveying new approaches to students and making it simple. In doing so, I want to teach important and subtle ideas while sparing the jargon – and provide readers with usable and down-to-earth programming tools to evaluate new approaches. All of this, I am adept in. With no shortage of exposure to confusion, bewilderment and getting lost. At Caltech, where I earned my Master’s, I studied with Gerald Whitham, the aerodynamicist renown for “sonic boom” modeling, while at M.I.T., where I did my Ph.D., I worked with Marten Landahl, the leading pioneer in transonic flow. My exposure to aerodynamics was good but I was compelled to learn fast. But interestingly, by the time I met Whitham and Landahl, both had moved on to hydrodynamic stability. So I’d struggle with that too, extending Whitham’s kinematic wave theory to second-order, and learning the intricacies behind applied math and nonlinear equations. I had never programmed nor touched a computer, remarkably. Not even Fortran, the mainstay of engineering. Completely alien. All I knew were differential equations and differential equations. On my first day at Boeing, I asked my supervisor, Paul Rubbert, who also worked with Landahl for his doctorate, “What kind of answers do computers give?” Sarcastically, he replied, “Any answers you want.” Strange, I thought, but I would gradually appreciate those words of wisdom. Whereas Rubbert’s group focused on panel methods, Hideo Yoshihara (my second boss) and his team specialized in transonic flow – wow, the world sure was small. I learned a lot during the two years I worked at Boeing, publishing almost two dozen papers that probed the depths of fluid dynamics. And I would move to Pratt & Whitney Aircraft as Manager of Turbomachinery, applying new-fangled methods xii

in transonic flow analysis to the innards of turbines and compressors. Here, I’d also share a cubicle with Richard Whitcomb, the leading authority on supercritical wing design, winglets and Coke bottles, during six long months as we labored days on end to refine Pratt’s engine and airframe integration efforts. Were it not for the pungent cigars he smoked, everything would have been perfect. But I would not stay long. That defining year, I accepted new challenges with the petroleum industry – at latest count, more than two decades in oil and gas exploration, where I’d author more than twenty books with John Wiley & Sons and Elsevier Science, earn about four dozen patents, write over a hundred papers. My fascination with aerodynamics, however, had never died. The feeling goes on. I want to continue the research I once enjoyed and encourage their application in airplane design, to develop smarter models, to learn from a thriving industry as the world takes a renewed interest in subsonic, transonic and supersonic flow. It’s an exciting world and getting better by the minute. It’s great to be back. Wilson C. Chin, Ph.D., M.I.T. Houston, Texas Email: [email protected] Phone: (832) 483-6899 November 2018

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Acknowledgements The subject matter, insights and perspectives in this book were conceived years ago during my aerospace studies at Caltech and M.I.T. and subsequent affiliation with Boeing and Pratt & Whitney Aircraft. Conversations and lengthy discussions with Fritz Bark, Judson Baron, Francis Edward Ehlers, Marten Landahl, Harvard Lomax, Thomas Matoi, Donald Rizzetta, Paul Rubbert, Richard Whitcomb, Gerald Whitham, Sheila Widnall and Hideo Yoshihara were particularly helpful and molded my initial approaches to aerodynamics and fluid mechanics. Funding agencies that supported my early career work were several, including Air Force Office of Scientific Research (AFOSR), National Science Foundation (NSF), Office of Naval Research (ONR), and later, the United States Department of Energy (DOE). I also thank the Kungliga Tekniska högskolan (KTH Royal Institute of Technology) in Stockholm, Sweden, which hosted and supported my early fluid mechanics training and provided me with lasting and kind memories. As usual, many thanks to Phil Carmical, Publisher, for his interest in my varied scientific activities, first petroleum and now aerospace, and for his unwavering faith and confidence that I would not compose anything too terribly incorrect. And lastly, I express my gratitude to Jenny Zhuang, my friend and companion, for discovering these almost forgotten jewels from my academic past and encouraging me to share them with similarly curious fluid-dynamicists and aerospace engineers.

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Modern Aerodynamic Methods for Direct and Inverse Applications.Wilson C. Chin. © 2019 Scrivener Publishing LLC. Published 2019 by John Wiley & Sons, Inc.

1 Basic Concepts, Challenges and Methods The fluid dynamics world is inundated with thousands of books on the subject, volumes on theory, numerical and engineering niches to no end. Within the specialty of computational fluids, hundreds of thousands of papers have appeared within the past two decades. And in the subset dubbed “aerodynamics,” tens of thousands may be found authored by specialists from dozens of countries. This being the case, we will not offer still another “first principles” derivation of governing equations. We will cite relevant subjects and refer readers to readily available literature where excellent presentations are already available. But it will be the author’s responsibility to develop and critique significant areas of fluids research that deserve further investigation. And, just as important, introduce ambitious students to key ideas quickly and rigorously, in the least amount of time, with minimal formal course work but with objectivity and honest speculation – to prepare him to understand, contribute and write software to evaluate new ideas. To this end, we have developed a fast-paced presentation style combining “simple numerics” with modern ideas in aerodynamics. With these disclaimers said and done, we now begin discussions on many exciting subjects. 1.1 Governing Equations – An Unconventional Synopsis The equations governing fluid motions are numerous, for example, as developed in excellent books by Batchelor (1967), Schlichting (2017), Yih (1969) and others. They cover constant density and compressible fluids; liquids and gases; inviscid and viscous motions; one, two and three dimensions; steady and unsteady flows; irrotational and rotational limits; and rectangular, polar, spherical and curvilinear coordinates. For the most part, we will deal with a special subset of these properties to develop the great majority of our ideas. In two dimensions, assuming Cartesian or rectangular coordinates, the momentum and mass conservation equations governing constant density, constant viscosity flows can be written concisely in the form 1

2 Modern Aerodynamic Methods ( u/ t + u u/ x + v u/ y) = - p/ x + ( 2u/ x2 + 2u/ y2) ( v/ t + u v/ x + v v/ y) = - p/ y + ( 2v/ x2 + 2v/ y2) u/ x + v/ y = 0

(1.1.1a) (1.1.1b) (1.1.1c)

These represent a highly simplified version of the Navier-Stokes equations. Generalizations of the above have appeared for special applications. For example, in high-speed aerodynamics, the density is variable, and equations of state and energy conservation laws apply (we will describe some transonic applications in Chapters 2, 3 and 4). The viscosity shown above is constant, but in gas dynamics, it may well be a function of temperature; in meteorology and oceanography, additional dependencies of pressure on properties like humidity and salinity will appear, implying more complicated mathematical descriptions and solutions. Sometimes the stress terms on the right are replaced by an anisotropic tensor; this author has developed models of fluid flow in petroleum reservoirs in a number of books (refer to “About the Author” for further publication information). For our purposes, it suffices to note how Equations 1.1.1a,b,c and similar high-order models (with high-order derivatives) require “Navier-Stokes solvers,” which are a challenge to develop, and computationally expensive and resource-intensive to run. A simpler limit is found by eliminating at the very outset, leading to what we call “Euler’s equations,” a low-order system, namely ( u/ t + u u/ x + v u/ y) = - p/ x (1.1.2a) ( v/ t + u v/ x + v v/ y) = - p/ y (1.1.2b) u/ x + v/ y = 0 (1.1.2c) The above applies at constant density only and the great majority of applications appears in flows, for instance, with oncoming velocity shear. The reader may recall words of caution. For irrotational flows, Bernoulli’s equation “p/ + ½ (u2 + v2) = constant” applies, where the constant, fixed for the entire flowfield, is known from upstream conditions. But for rotational flows, this “constant” is only so along a streamline; in fact, it varies from streamline to streamline. What happens when the flow about a jet engine is to be modeled? The external flow, uniform upstream, is irrotational and satisfies a simple Laplace and Bernoulli equation model; however, the flow behind the actuator disk, which imparts radial position-dependent work, is sheared and requires “Euler solvers” with complicated streamline tracking. Algorithm development combining potential with Euler solvers is no small task.

Concepts, Challenges and Methods 3 Investigators have developed sophisticated Euler equation solvers requiring equally sophisticated users. And all because “potential flow solvers” for xx + yy = 0 (or “ xx + rr + 1/r r = 0,” axisymmetrically) will not apply. Every student of fluid mechanics understands how potentials only apply to flows without shear. But what if potentials did apply? What if it were possible to solve xx + rr + (1/r – 2Um’ /Um) r = 0 valid for mean background flows with strong Um(r) velocity profiles? Simple potential flow codes would, through minor modification, address new classes of important flow problems. In fact, the mathematical basis behind “superpotentials” is developed in Chapter 4 with examples. Now let’s digress and turn to “analysis problems” described by classic potential formulations, that is, solving xx + yy = 0 subject to tangency conditions for the normal derivative y along y = 0, plus a requirement on a “potential jump” [ ] related to Kutta’s condition at the trailing edge. This formulation, which determines the surface pressure due to a prescribed geometry, as old as aerodynamics itself, has been solved straightforwardly in numerous ways: Glauert’s series, panel methods, finite differences, finite elements and so on. But the complementary “inverse problem,” searching for the geometry that induces a prescribed pressure, is more subtle and also known as the “indirect” problem. And for good reason. Often, the above analysis solver is run over and over, varying all sorts of empirically defined parameters in endless ways, until some type of convergence is achieved. Is there a simple but “direct approach to indirect problems?” The answer is, “Yes.” Enter the streamfunction, the “black sheep” of modern computational fluid-dynamics. We will show that the airfoil shape is described by the ordinate y(x) = – (x,0) where xx + yy = 0 is solved, subject to normal specifications for y(x,0) = – ½ U Cp(x) along y = 0, plus a requirement on a jump [ ] related to the degree of trailing edge closure. In other words, given the surface pressure coefficient Cp(x), the shape can be directly (meaning non-iteratively) solved using any potential flow algorithm for analysis problems already available! Then again, pessimists might argue that the method is limited because it could not be extended to, say transonic supercritical problems. In developing our model, we drew upon Cauchy-Riemann conditions (from complex variables) which strictly apply to complementary equation pairs like xx + yy = 0 and xx + yy = 0. And so, “no constant density assumption, no streamfunction inversion.” Correct? Incorrect. To solve the problem, we developed a completely rigorous “engineer’s

4 Modern Aerodynamic Methods Cauchy-Riemann transform” that allowed us to create a compressible, mixed subsonic and supersonic extension of xx + yy = 0 to solve inverse formulations in a single pass. Not quite a pure partial differential equation, but one with an integral coefficient that could be just as easily solved. So that’s another success story, where we’ve solved indirect problems directly, initially with a lots of speculation and then some luck. Next consider compressible flow extensions of Equations 1.1.2a,b,c, which are interesting in a very different way. At steady cruise conditions, under the assumption of irrotationality, these equations (as we will show) lead to a potential flow model not unlike ( ) xx + yy = 0 where ( ) is somewhat tricky. At Mach zero, or flight speeds say 300 mph or less, this reduces to a purely subsonic (scaled) equation not unlike Laplace’s “ xx + yy = 0.” Dozens of classical texts, conformal maps and singular integral equation methods are and have been available for decades. Near 550 mph or so, fluid particles accelerate so rapidly around leading edges that flows become locally supersonic. Most of the time, they terminate abruptly at shockwaves – where sudden discontinuous increases in pressure lead to losses and unstable wing oscillations. Such are typical of problems suggestive of a “sonic barrier” just several decades ago. But computational methods were non-existent until the 1970s, when Murman and Cole (1971) published a pioneering “type-dependent” numerical algorithm for mixed elliptic and hyperbolic equations. Their idea was simple: use “upwind differencing” for supersonic points and central for subsonic to proper account for domains of influence and dependence. The original scheme did not conserve mass, but later researchers would introduce “conservative schemes” and curvilinear grid refinements that seemed to suggest . . . well, end of story. Just when the story was finally told, workers in the mid-1980s discovered that computed solutions could be non-unique. For a given set of flight conditions, more than a single solution existed! Was this a computational anomaly or physical reality? Was it related to buffeting and aerodynamic instability? Or was it an artifact inherent in Equations 1.1.2a,b,c, which while simpler than Equations 1.1.1a.b,c, were loworder and only partially descriptive of the physics? Non-uniqueness aside, the Murman-Cole scheme and its derivatives were not perfect. Iterations were required to “march” in the direction in which the supersonic flow evolved – which was, of course, unknown at the outset. This placed limitations on mesh generation flexibility, since coordinate lines must somehow align with the flow – but this was hopeless since

Concepts, Challenges and Methods 5 curvilinear grid definition usually bears no relationship to the physics (for now, anyway). And so, tedious local, point-by-point type-testing, often employing “rotated differencing” in more sophisticated software, would continue with only evolutionary or minor change. Early on, this author had experimented with a “viscous transonic equation” of the form “ xxx + ( ) xx + yy = 0.” where ( ) could be positive, negative or both. This work, described in Chapter 3, focused on transonic supercritical applications with embedded shockwaves. The idea was simple: this model, like Equations 1.1.1a,b,c, was high-order in the sense that the viscous shock structure of any evolving discontinuities could be modeled (here, the represents the longitudinal viscosity). We believed that, since the complete model was actually parabolic, the need for mixed subsonic and supersonic differencing was unnecessary. Moreover, “sweeping” need not proceed in the direction of the supersonic flow. A series of three papers published in the AIAA Journal documented our speculations and successes. At the time, we also speculated that since an “ xxx” term was included, then the model implicitly contained all of the requisite thermodynamic properties. In other words, the required conservation form and entropy conditions are self-contained in the viscous transonic model – thus, any nonlinear computed solution should be unique and completely determined. In fact, the role of high-order terms had been discussed in the classics Supersonic Flow and Shock Waves and Linear and Nonlinear Waves by Courant and Friedrichs (1948) and Whitham (1974). A short proof using the steady form of Burger’s equation ut + uux = uxx was given previously by the author and here in Chapter 3 – also, a uniqueness theorem for the unsteady equation was offered recently by Benea and Sadallah (2016). So much for our very brief synopsis of the governing equations. Suffice it to say, to those who believe that computational fluids has ended with modern Navier-Stokes and Euler equation solvers, we believe that greater surprises await us. In this book, the author hopes to introduce new perspectives to interpret aerodynamics by explaining ideas rigorously but simply, by injecting healthy degrees of scientific speculation, and educating the reader in problems of importance to the industry. Invariably, every new approach involves numerical solution, and recognizing the unlikelihood that new students will have studied computational methods in depth, we offer a condensed presentation in Chapter 2 with readily understood but sparse Fortran. However, mathematics is truly essential, a focus that now guides our long journey.

6 Modern Aerodynamic Methods 1.2 Fundamental “Analysis” or “Forward Modeling” Ideas In this section, we will present fundamental ideas and develop basic analytical results that will be used to broaden our understanding of both analysis and inverse problems. Our discussion is comprehensive and self-contained, and takes a unique approach to numerical analysis that is intuitive and mathematically rigorous from a formulation perspective. Fundamental equations. We start our presentation by requiring fluid “irrotationality,” this assumption thus precluding boundary layer, viscous flow and related non-ideal flow effects. If q denotes the total velocity vector, this kinematic condition can be stated precisely in the q From vector analysis, we understand that it is possible form to represent this velocity as the gradient of a “potential” , or here a . If we further express steady, “velocity potential,” that is, write q constant density, mass conservation in the form q direct substitution shows that irrotational potential flows are governed by the classical Laplace equation 2 = 0. We will restrict our attention to planar flows in this section. For rectangular or “Cartesian” x and y coordinates, this can be rewritten as xx + yy = 0, while in cylindrical polar r and coordinates, this takes the form rr + 1/r r + 1/r2 = 0. Either is transformable into the other, using the relationship “x = r cos , y = r sin ,” or equivalently, “r = (x2 + y2)½ , = tan – 1 y/x.” Both representations will find important applications in our discussions. Small-disturbance results. Although exact solutions for flows past a limited number of geometrically complicated bodies can be constructed from the theory of complex variables, often numerically, in practice, small-disturbance flows past thin airfoils approximately aligned with the rapid oncoming flow form the great majority of applications. For such problems, analytical and computational methods for the forward or analysis problem, in which pressure fields are sought when a geometry is specified, are well developed; in this section, basic developments are reviewed and studied in greater depth than is usual. This development serves multiple purposes. For one, we will later derive a direct or “forward like” inverse methodology that is discussed at much greater length in Chapter 4, drawing on our understanding of the analysis problem. Second, our constant density exposition sets the foundation for more advanced methods in mixed-type transonic flow simulation, treated in Chapter 2, and finally, the introductory work here leads us to the “viscous transonic” approaches developed in Chapter 3.

Concepts, Challenges and Methods 7 Importantly the relative simplicity behind constant density planar flow formulations allows us to explore in detail the properties of singularities related to source-like and vortex-like flows, which we will find very useful application in inverse techniques. We begin by considering small-disturbance flows in rectangular or Cartesian coordinates. Here it is customary to write the total velocity vector as q = x i + y j where x represents the horizontal speed “u” in the “x” direction having a unit vector i, while y denotes the vertical speed “v” in the “y” direction having a unit vector j. Suppose a large horizontal speed exists, e.g., the wind blowing in a wind tunnel, or the relative speed experienced by an aircraft flying at cruise. Further, suppose that this speed greatly exceeds the disturbance velocities induced by the thin airfoil. We thus write = U x + where is the so-called “disturbance potential” to the constant speed U and require U >> | x | and | y |. Substitution in 2 = 0 shows that the disturbance potential likewise satisfies 2 = 0. This equation is solved with auxiliary conditions, namely, flow tangency conditions at the airfoil surface, regularity conditions faraway at infinity and, as will be discussed in detail, a special Kutta condition not found in conventional expositions for Laplace solutions in heat transfer, electrostatics or petroleum reservoir flow. We address airfoil surface kinematic conditions first. Now, the total horizontal speed is represented by x = U + x while the vertical speed is y = y. Because the airfoil surface is solid and impenetrable to flow, steadily moving fluid particles must flow tangent to it. That is, the ratio of the vertical to horizontal speed must equal the surface slope, writing, y /(U + x) = F’(x) where y = F(x) is the airfoil ordinate and prime denotes the horizontal derivative. We emphasize that this is evaluated at the surface, so that y(x, y(x))/(U + x) = F’(x). However, since U >> | x | we consider a simpler expression along the horizontal axis itself, 0. with y(x, 0) /U F’(x). Far from the airfoil, we require that In most non-aerospace applications, this boundary value problem formulation alone would suffice. If had represented the steady-state temperature on a plate containing a portion of the slit (or thin hole) y = 0, the specification of the normal temperature gradient y together with regularity conditions would completely determine temperature to within a constant – if temperature were fixed at one additional location, the complete temperature field would be fully determined. But this is not so with inviscid aerodynamic analysis and we will see why shortly.

8 Modern Aerodynamic Methods Thickness and camber formulations. To develop the ideas suggested in above, it is convenient to understand that the airfoil ordinate y = F(x) actually consists of two functions, y = Fu(x) for the upper surface, and y = Fl(x) for the lower surface. A “camber line” function is introduced as the mean arithmetic position between upper and lower surfaces, that is, ½ (Fu + Fl) = Fc, while a “thickness function” is defined as half of the local airfoil thickness with ½ (Fu – Fl) = Ft. If we write Fu + Fl = 2Fc and Fu – Fl = 2Ft, addition and subtraction then lead to Fu = Fc + Ft and Fl = Fc – Ft. This suggests that we resolve the complete boundary value problem for y = Fu,l(x) into two simpler ones, namely, t

2

t y(x, t

=0 y = 0) 2

(1.2.1a) (1.2.1b)

t

U dF (x) /dx along chord

0 as x + y

2

(1.2.1c)

and c

2 c

=0 y(x, y = 0) c

(1.2.2a) (1.2.2b)

c

U dF (x)/dx along chord

2

0 as x + y2 plus Kutta condition (to be discussed)

(1.2.2c) (1.2.2d)

Notice that the normal derivative y reverses sign or “jumps” across the chord in Equation 1.2.1b, whereas in Equation 1.2.2b, it does not and is “continuous.” Once solutions to the foregoing problems are available, the total disturbance velocity potential is obtained by linear superposition, that is, calculated from = t + c and substituted in = U + to yield the complete potential. Differentiation yields velocities. Evaluation of pressure and lift. Under the physical assumptions stated above, Bernoulli’s equation, which follows as a specific limit to the inviscid Euler equations, applies to the calculated flowfield. If we apply the foregoing small-disturbance assumptions, we have P +½ U

2

= P + ½ | q |2 = P + ½ (U 2 + 2U

x

P + ½ (U 2 + 2U =P +½ U 2 + U or

+

x) x

x

2

+

2 y )

Concepts, Challenges and Methods 9 P

P – U

(1.2.3a)

x

that is, on combination with the definition of the pressure coefficient Cp, the well known formula Cp = (P – P ) /(½ U 2)

– 2 x /U

(1.2.3b)

Two observations will be important. First, consider P + ½ U 2 = P + ½ | q |2 or P = P + ½ U 2 – ½ | q |2. At a stagnation point where | q |2 = 0, we have P = P + ½ U 2 so that Cp in Equation 1.2.3b physically takes on an absolute maximum of 1 (however, an improperly operated small disturbance algorithm may lead to results that exceed this). This provides an excellent check point for numerical analysis methods. Second, for the purposes of our inverse formulation later, it is important to note how, for analysis problems, the normal derivative y is first specified along the chord on y = 0 while the tangential derivative x is later evaluated from to calculate pressure from the solved potential. Point singularity representations. We digress to discuss properties of “source” and “vortex” singularities which will prove useful function satisfies to developing key ideas. Earlier we noted how a Laplace’s equation, and gave both rectangular Cartesian and a cylindrical radial forms. The disturbance potential likewise satisfies these relationships. In the latter polar coordinates, we have rr

+ 1/r

r

+ 1/r2

rr

+ 1/r

r

= 0 if /

=0

(1.2.4a)

= 0, leading to

= 0 if / r = 0, leading to

= A log r + B

=C +D

(1.2.4b) (1.2.4c)

Let us study two simplifications. In Equation 1.2.4b, we had set angular dependencies to zero, so that the solution for potential is the simple logarithmic function = A log r + B, where A and B are constants. It is important to observe that is identical in all directions around the origin r = 0; thus, it cannot be associated with lift, which has a preferred vertical direction. We emphasize that is single-valued and is identical in all does not depend on . Because its velocity q = directions and also varies like 1/r er (where er is the unit vector in the radial direction), this represents that due to a point source or sink. On the other hand, if in Equation 1.2.4a we had set radial dependencies to zero, we would have the solution = C + D, where C and D are constants, and = tan – 1 y/x is an arctangent function. The

10 Modern Aerodynamic Methods potential would then depend on angle; at any point, can be represented by a given value, or that value, plus 2 . Unlike the logarithmic potential, it is double-valued and does depend on . What is this solution physically? Consider our solution = C + D with a positive value of C. Then the velocity q = = 1/r / e (where e is the unit vector in the angular direction) reduces to q = C/r e which, say, points to the right at the top and to the left at the bottom. Thus, at the top, the total velocity exceeds that of the freestream, while at the bottom, it is lower – this is just the description of vortex flow. High speeds at the top and lower ones at the bottom, via Bernoulli’s equation, imply that pressure is lower at the top and higher at the bottom. In other words, vortexes are associated with the singularities needed to model lift – again, they are multivalued potentials. Finally, note that the velocity q = varies like 1/r and decays away from the airfoil. Vortexes are associated with antisymmetric velocities and lifting effects, while sources model thickness, since they displace streamlines symmetrically, equally outwards at top and bottom. Before studying thickness and camber flows in detail, we derive a formula useful in computational applications for lift calculation. The lift L acting on an airfoil having chord C and depth D into the page is given by L = (P– – P+) D dx where P– and P+ are, respectively, pressures acting at the bottom and the top, and the integral is taken over the airfoil chord. If we now invoke Equation 1.2.3a, that is, the simplified Bernoulli equation P P – U x, we obtain L = (– U x– + U x+ )D dx or the result L = U D ( x+ – x– ) dx = U D x dx where is the “jump in potential” defined by [ ] = + – – due to vorticity effects. This result can be further simplified to give L = U D { TE – LE} where “TE” and “LE” denote trailing and leading edge values. Later, we will explain why the leading edge term LE vanishes while TE does not. For now we can write L = (½ U ) (2D TE / U ). From “ = U x + . . .” the units of potential are Length2/Time. Thus, 2D TE / U has units of area, so L is consistent with “Force = ½ U Area.” The dimensionless lift coefficient is defined by CL = L/(½ U Area) where Area = D C, so we have L= U D

(1.2.5a)

TE

CL = L/( ½ U 2 DC) = 2

TE

/ (U C)

(1.2.5b)

Concepts, Challenges and Methods 11 Equations 1.2.5a and 1.2.5b were derived for use with the Laplace potential function solvers developed in Chapter 2 which are formulated in terms of jumps in potential [ ]. We also indicate that the lift L is often expressed in the form L = U where is known as the “circulation.” For completeness, the classical Glauert (1947) solution for lift coefficient is summarized in Figure 1.1.

Figure 1.1. Glauert camber solution for CL in constant density flow. The well known Glauert (1947) solution solves an integral equation formulation for the lifting problem in constant density flow using trigonometric series – it does not apply to transonic flows with shockwaves, although for nonzero subsonic Mach numbers, scaled solutions are available using the Prandtl-Glauert transformation (e.g., see Ashley and Landahl (1965)). Furthermore, the above solution applies to two-dimensional airfoils only; for three-dimensional problems, “lifting line” and “lifting surface” approaches apply. Detailed discussions are offered in the classic book Aerodynamics of Wings and Bodies due to Ashley and Landahl (1965). We cite the above solutions because they are useful for validating numerical solutions such as those developed in Chapter 2. We next discuss properties of singularity distributions because they are essential to developing our inverse methods, which take an approach uniquely different from existing methods.

12 Modern Aerodynamic Methods Thickness formulation and properties. We now consider the thickness problem in greater detail. For Equation 1.2.1b, we had indicated how velocities above and below the axis point in opposite directions. Thus, the boundary value problem in Equations 1.2.1a,b,c represents the thickness problem. On the other hand, Equation 1.2.2b shows velocities that are identical in sign above and below the chord, so that Equations 1.2.2a,b,c,d solve the camber problem. We consider the thickness problem first using methods from singular integral equations. A closed form analytical solution can be obtained. Now the “log r” source solution derived previously, centered at the origin r = (x2 + y2) = 0, solves Laplace’s equation. It follows that log {(x- )2 + y2} centered at x = , y = 0 also satisfies Laplace’s equation, where represents only a shift in the choice of origin. Now, can be viewed as a general point source position over which the effects of numerous sources can be summed. But rather than examining multiple discrete point sources, we examine continuous line source distributions placed along a slit on y = 0 to represent the thickness distribution. This is clearly the situation physically. We therefore consider the superposition

(x,y) =

f( ) log {(x- )2 + y2} d + H

This integral also satisfies Laplace’s equation for the potential, since the governing equation is linear. Integration limits extending over the airfoil chord are understood and excluded for clarity. This represents the solution for a continuously distributed line source along y = 0 and along the chord, assumed consistently with small-disturbance theory, where H is an integration constant that we need not consider here (for example, in petroleum reservoir fracture flow in a finite circular field, Chin (2017) explicitly evaluates H when the farfield pore pressure is given at a finite distance). Here, H is zero and velocities vanish at infinity. We are next interested in developing properties of the above integral and the “source strength” f(x). Let us return to the expression for potential and differentiate it with respect to the vertical coordinate y normal to the chord. (x,y)/ y = / y { f( ) log {(x- )2 + y2} d + H} = y f( )/{(x- )2 + y2} d

Concepts, Challenges and Methods 13 Following the limit process in Yih (1969), introduce the change of coordinates = ( - x)/y so that +

(x,y)/ y = f( )/(1 + 2 ) d -

Now for small positive y’s, we find that on using x = vertical derivative satisfies

- y, that the

+

(x,0+)/ y = f( )/(1 + 2) d = f(x) -

Similarly, for small negative y’s, we obtain (x,0-)/ y = - f(x). Hence, (x,0+)/ y - (x,0-)/ y = f(x). Our results also imply (x,0+)/ y = - (x,0-)/ y, that is, the vertical velocities on either side of the slit are antisymmetric, in agreement with Equation 1.2.1b. We emphasize that, from (x,y) = f( ) log {(x- )2 + y2} d + H, the potential is an even function of y, that is, (x,y) is symmetric with respect to y = 0. Also, as anticipated from the properties of the logarithm, the potential is a continuous function in space that does not jump. These provide two key check points for numerical calculations that are frequently used later. The superposition integral itself provides still another check point for evaluating computed behavior throughout x and y space. We had proved that (x,0+)/ y = f(x). From ty(x, y = 0) U dFt(x) /dx along chord, representing the tangency condition, we find that f(x) = U dFt(x)/dx. While f(x) can be related to thickness function slope, a blunted edge or infinite slope would invalidate thin airfoil models. Camber line properties. Here we derive some properties associated with flows past camber only geometries. Previously, we showed why = C + D or = tan-1 y/x is a solution to Laplace’s equation. Following the approach used previously, we might consider a point vortex at (x = , y = 0) taken in the form = tan-1 y/(x – ), or more generally as a continuous distribution of vortexes satisfying +1

(x,y) =

g( ) tan-1 y/(x- ) d + G -1

where the arbitrary constant is set to zero in order to satisfy regularity conditions at infinity. This solution also satisfies Laplace’s equation by

14 Modern Aerodynamic Methods virtue of linear superposition and g( ) is an unknown function. Differentiation with respect to y, using standard formulas, yields +1

g( ) {(x- )/{(x- )2 + y2}} d

(x,y)/ y =

-1

If we evaluate this at y = 0 and use Equation 1.2.2b, we have

y(x,

y = 0)

U dFc(x)/dx from

+1

PV

g( ) /(x- ) d

U dFc(x)/dx = -

-1

This singular integral equation, with the Cauchy kernel 1/(x- ), governs the vortex strength g( ). The PV indicates that the integral is improper and to be evaluated using a “principal value” limit defined in calculus. Fortunately, we do not need to understand integral equation methods to solve the problem. Indeed, the general solution to the equation PV g( )/(x- )d = - h(x) is g(x) = - (1/ 2) {(1-x)/(1+x)} PV {h( ) (1+ )}/{( -x) (1- )} d + / (1-x2)

where we have omitted the integration limits for clarity. This solution is derived and discussed in classical references (Mikhlin, 1964; Muskhelishvili, 2008; Carrier, Krook, and Pearson, 1966). Note that the / (1-x2) term represents the nonuniqueness associated with solutions to our singular integral equation, with the arbitrary constant related to the so-called “circulation” of a flow. Its specific value is determined by Kutta’s condition requiring smooth flow from the trailing edge. What is the physical significance of vortex strength? If we differentiate our superposition integral with respect to x, it follows that +1

/ x= -

g( ) y/{(x- )2 + y2} d -1

This integral was studied earlier. In the limit y = 0, from earlier results, (x,0+)/ x =- g(x) and (x,0-)/ x =+ g(x). Since the velocity parallel to the camber line is proportional to / x, the camber line is responsible for a discontinuity in the tangential velocity that is proportional to g(x). The above show a net jump in the tangential derivative (i.e., velocity slip) of (x,0+)/ x - (x,0-)/ x = - 2 g(x).

Concepts, Challenges and Methods 15 1.3 Basic “Inverse” or “Indirect Modeling” Ideas In Section 1.2 we formulated and studied the “analysis,” “forward” or “direct” problem, one in which the potential field, pressure distribution, pressure coefficent and total lift were sought when an airfoil geometry was prescribed. Solutions were direct or straightforward in that the formulations of Equations 1.2.1a,b,c or Equations 1.2.2a,b,c,d could be solved in a single pass (using a relaxation solver) without further work – this is now standard given the proliferation of Laplace equation solvers and the like. Again, each of these formulations require an iterative solution, but at least, this tedious process is pursued only once (or at worst twice for general non-symmetric geometries requiring both thickness and camber solutions). In summary, the prior analysis methods provide the surface pressure coefficient Cp = – 2 x /U once the airfoil ordinates y = Fu,l(x) are given. Surface pressures are useful for calculating lift and moment, in determining viscous drag using boundary layer methods, or in assessing the likelihood of flow separation and stall. What if, however, we wanted the reverse: prescribe Cp(x) along a finite slit y = 0 and calculate the shape y = Fu,l(x) that induces the given pressure? This is the so-called “inverse” or “indirect” problem – indirect because it is not obvious how one should proceed. Many procedures based on pure guess work have been published. For example, in the numerical approach of Carlson (1975), an approximate nose shape furnishes starting slope conditions for calculations carried out in an analysis mode and, over the remainder of the chord, tangential are used in design mode calculations. Intermediate derivatives of results for geometry are monitored at different stages of the relaxation and, if the required degree of trailing edge closure is not fulfilled, the starting nose shape is modified until such is assured. There is nothing special about starting modifications at the nose – any other point might well be justified, and in fact, all other points are likely candidates. One thus observes that, while potential function formulations may be useful in engineering practice, they do require considerable experience, expertise and intuition on the part of the designer. They invariably require a human decision on the choice of a free parameter indirectly related to trailing closure so that airfoils are not opened unrealistically: the “man-in-loop” requirement arises from the fact that monotonic changes to arbitrarily defined parameters generally do not correlate with monotonic changes to the degree of closure. But is there a better or more rational approach to solving inverse problems?

16 Modern Aerodynamic Methods To be sure, is it possible to develop a direct method to solve indirect or inverse problems in a single pass as we had solved analysis problems? The answer is, “Yes.” This subject is developed in Chapter 4 for airfoils, inlets, three-dimensional wings, and so on, for irrotational and rotational flows. In this section, we motivate the method with a simple example – a elementary but powerful application for which we also derive a complementary finite difference solver in Chapter 2. To do so, we begin with our disturbance potential equation xx + yy = 0, rewritten as ( x)/ x + ( y)/ y = 0

(1.3.1)

This so-called “conservation form” suggests that we might introduce a function (x,y) such that x=

(1.3.2a)

y

and y=-

(1.3.2b)

x

which is nothing more than “0 = 0” on substitution into Equation 1.3.1. Students of complex variables will recognize these as Cauchy-Riemann conditions, which we just derived using elementary methods. What are the mathematical properties of ? Partial differentiation of the first equation with respect to y and the second with respect to x, and elimination of xy between the two, shows that (x,y) satisfies xx

+

yy

=0

(1.3.3)

Observe that if we wished to specify the pressure coefficient along a finite slip y = 0, we would have, using Equations 1.2.3b and 1.3.2a, the transformation Cp = (P – P ) /(½ U 2) – 2 x /U = – 2 y /U or y

(x, y = 0) = - ½ U Cp(x)

(1.3.4)

Further, if our airfoil is finite in extent, any disturbances to the 0 as x2 + y2 freestream that it induces must vanish faraway, with . Let us now collect our key results – + yy = 0 y (x, y = 0) = - ½ U Cp(x)

(1.3.5a) (1.3.5b)

xx

0 as x2 + y2 plus trailing edge [ ] or [

(1.3.5c) x]

jump condition (to be discussed)

(1.3.5d)

Concepts, Challenges and Methods 17 Once the solution to Equations 1.3.5a,b,c,d is available, the required airfoil geometry is easily calculated. Recall that the surface kinematic or tangency boundary condition derives from dy/dx y /U or, on applying Equation 1.3.2b, dy/dx - x /U which directly integrates to y(x) = - (x,0) /U + constant

(1.3.6)

upon evaluation along the slit y = 0. We can easily check our results for = U x + , the dimensional consistency and correctness. From velocity potentials have units of (L/T)L or L2/T (that is, length2/time). From Equations 1.3.2a,b, it is clear that also has units of L2/T. Now, the right side of Equation 1.3.6 therefore varies like (L2/T)(T/L) which correctly has dimensions of length. Some comments on mass flux or mass or volume flow rate are also worthwhile. Consider a section of space that is D deep into the page, and varying from a vertical location y to y*. The horizontal mass flux M through any line connecting y and y* is simply M = x D dy where the integral is taken over the interval y to y*. Since Equation 1.3.2a requires that x = y, we can write this integral as M = y D dy or M = D { (x,y*) - (x,y)}

(1.3.7)

In other words, the difference in the value of between any two points in space is proportional to the mass flux between those points. New direct inverse formulation. The foregoing equations completely define the boundary value problem and post-processing needed to solve the inverse problem in a direct manner! Again, Equation 1.3.5a is our familiar Laplace equation, Equation 1.3.5b specifies a convenient normal “y” derivative related to prescribed pressure, Equation 1.3.5c states the expected regularity condition, while a further prescribed jump in [ ] or [ x] furnishes a Kutta-type or circulatory statement required to render a solution to Laplace’s equation unique! The jump [ ], following Equation 1.3.7, is related to mass flow ejected from the trailing edge, that is, the degree to which it is opened. A jump in [ x] (or the slope [ y]) would measure trailing included angle. In fact, the foregoing inverse formulation is identical to the forward analysis problems defined by Equations 1.2.1a,b,c and Equations 1.2.2a,b,c,d. Thus, computer algorithms developed to solve for velocity potentials can be used to solve inverse problems with minor change and only simple reinterpretation of the dependent variable! Approaches such as those in Carlson (1975) are not necessary and inverse problems can be solved in a

18 Modern Aerodynamic Methods single step. The function is the so-called “streamfunction” well known to fluid-dynamicists, but until this author’s first publication of the above algorithm, the role of streamfunctions in inverse formulations had never been recognized in the context of inverse formulations. In Chapter 2, we will introduce the finite difference method and its application to solving elliptic partial differential equations, first concentrating on the historically important analysis problem for the velocity potential. Then, we will show how simple modifications to one algorithm can solve a well known inverse problem quickly, efficiently and simply. But our discussion on inverse methods does not stop there. In fact, a major objective of this book focuses on more general inverse problems of greater difficulty, whose solutions have actually been stymied by impediments more semantic than mathematical. Students of mathematics will recognize that potentials and streamfunctions, such as are introduced above, can be described much more elegantly than in the derivation underlying Equations 1.3.1 – 1.3.3. Usually, a background in the theory of complex variables is assumed, and authors indicate how real and imaginary parts of “analytical functions of z,” where z = x + i y is complex, are “harmonic,” that is, do. Equations 1.3.2a and they satisfy Laplace’s equation as and 1.3.2b are the classical Cauchy-Riemann conditions, named after mathematicians Cauchy and Riemann who discovered the relationships in the nineteenth century, and and are said to be “harmonic conjugates.” An important application for is streamline plotting, e.g., it can be shown that lines of constant define fluid paths in steady flow – this property provides the main practical stimulus for solving Equation 1.3.5a. Other limited applications are also available, e.g., see the classial work of Milne-Thomson (1968) provides numerous drivations for exact solutions, but we will refrain from further discussion here. When this author first discovered the practical utility in Equations 1.3.5a,b,c,d he immediately sought to extend the powerful new inverse methodology to other complicated problems in aerodynamics. However, several unfortunate “semantic traps” suggested that this was not possible. After all, the prevailing wisdom behind Cauchy-Riemann conditions and harmonic functions required governing equations of the precise form given by xx + yy = 0, a narrow subset of fluid-dynamics indeed. However, using the “engineer’s derivation” given in the development starting with conservation forms similar to Equation 1.3.1 proved to be very constructive. Soon, the axisymmetric equation rr + rr + 1/r r = 0,

Concepts, Challenges and Methods 19 important to modeling flows through engine inlets and nacelles, with and without power addition, was transformed into streamfunction format for inverse analysis. Similarly, the small-disturbance transonic flow equation {1 − M∞2 − ( + 1) M∞2 x} xx + yy = 0, by rewriting it in conservation form, led to an integro-differential equation for that could be solved by existing mixed-type partial differential equation solvers with only a single coefficient modification. The three-dimensional Laplace potential equation xx + yy + zz = 0 used in wing theory was tackled next. Streamfunctions are not widely used in three-dimensional aerodynamics because, it turns out, the streamfunction is a vector potential requiring three components. Unlike two dimensional theory, where formulations employing two velocities can be replaced by simpler ones utilizing a single potential or streamfunction, no such advantage is found three-dimensionally for the streamfunction. For such applications, the author developed a rigorously satisfying a single scalar Laplace defined “streamlike function” equation xx + yy + zz = 0. This powerful transformation suggested that existing three-dimensional analysis methods using the velocity potential, e.g., conventional lifting line and lifting surface approaches, could be used the solve three-dimension inverse problems for finite wings. Finally, several of these irrotational models were extended to rotational flows driven by the presence of strong oncoming shear. The approach started with streamfunction formulations which do apply to shear flow, recasting the governing equations to conservation form, and then introducing a “superpotential” which is governed by a simple extension of xx + yy + zz = 0 requiring only a redefinition of certain coefficients. The key contribution here allows us to extend existing potential flow codes to solve analysis problems in the presence of shear with only minor modification. The theoretical approach is shown to be completely consistent with the inviscid Euler equations. These methods are all described in Chapter 4.

20 Modern Aerodynamic Methods 1.4 Literature Overview and Modeling Issues The references listed below are recommended reading and directly support multiple chapters and sections developed in our book. With possible exceptions related to the integral equation literature, all are readily understood to audiences with some exposure to advanced math. Many of the cited works, not listed in any particular order, provide the background needed to extend and refine the ideas introduced later. Basic fluid mechanics. As our work focuses on aerodynamics, basic fluid mechanics books with an aerospace yet fundamental emphasis are recommended reading. Some of these are, for instance, Ashley, H. and Landahl, M., Aerodynamics of Wings and Bodies, Addison-Wesley, Reading, Massachusetts, 1965. Well written text covering aerodynamic theory from subsonic to supersonic flow in multiple dimensions. Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. A well known author explains issues and subtleties in fluid flow with an emphasis on physical phenomena. Katz, J. and Plotkin, A., Low Speed Aerodynamics, Second Edition, Cambridge University Press, Cambridge, 2010. An excellent aerodynamics book covering a wide range of topics rigorously – a thorough discussion of modern panel methods for low speed flow. Milne-Thomson, L.M., Theoretical Hydrodynamics, Macmillan Company, New York, 1968. Although the presentation style is somewhat out-dated, this classic presents and develops numerous key results, ideas and exact solutions in hydrodynamics and aerodynamics very rigorously. Streeter, V.L., Handbook of Fluid Dynamics, McGraw-Hill, New York, 1961. A very well written book (not your usual “handbook”) covering a wide range of fluid mechanics problems and solutions. Tsien, H.S., Collected Works of H. S. Tsien (1938-1956), Academic Press, New York, 2012. A “must read” classic. Yih, C.S., Fluid Mechanics, McGraw-Hill, New York, 1969. The author is a well known author who discusses a wide range of topics clearly and mathematically, summarizing major issues and results in an easily understood manner. Great place to learn applied math.

Concepts, Challenges and Methods 21 Mathematics. Many of our models focus on finite difference analyses, which require a solid background in numerical methods. The classic volume due to Carnahan, Luther and Wilkes (1969) provides a well written introduction (with excellent examples and Fortran code) that guides beginners to productive results quickly. The integral equation references below are useful to those developing source, vortex and other singularity superposition methods and provide detailed solutions for many common formulations. They are suitable for advanced students. Carnahan, B., Luther, H.A. and Wilkes, J.O., Applied Numerical Methods, John Wiley & Sons, New York, 1969. Carrier, G.F., Krook, M. and Pearson, C.E., Functions of a Complex Variable: Theory and Technique, McGraw-Hill, New York, 1966. Mikhlin, S.G., Integral Equations, International Series of Monographs on Pure and Applied Mathematics, Vol. 4, Pergamon Press., Oxford, 1964. Muskhelishvili, N.I., Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics, translation by J.R.M. Radok, Dover Publications, New York, 2008. Tricomi, F.G., Integral Equations, Interscience Publishers, London, 1957. An exact solution to “the airfoil equation” is presented. Jameson research. The works of Jameson and his colleagues are well known to the aerospace community and are “must” reading. The references below represent a cross-section of key contributions and results. These identify main simulation issues, approaches and solutions, plus pitfalls likely to be encountered in transonic flow simulation. Jameson, A., “Iterative Solution of Transonic Flows over Airfoils and Wings, Including Flows at Mach 1,” Communications on Pure and Applied Mathematics, Vol. XXVII, 1974, pp. 283-309. Detailed discussion on transonic analysis methodology. Jameson, A., Schmidt, W. and Turkel, E., “Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” 14th Fluid and Plasma Dynamics Conference, Palo Alto, CA, 23-25 June 1981. Discussions on Euler equation method, arbitrary domains, convergence to steady state, structure of dissipation terms needed for stability.

22 Modern Aerodynamic Methods Salas, M.D., Jameson, A. and Melnik, R.E., “A Comparative Study of the Nonuniqueness Problem of the Potential Equation,” AIAA Paper 83-1888, 6th Computational Fluid Dynamics Conference, Danvers, Mass., July 13-15, 1983. Prior investigators had shown how multiple flowfield solutions are possible for given geometric and freestream input parameters, leading obviously to several disturbing questions. Are these solutions computational anomalies? If they exist, are they possibly unstable physically? Could they be related to buffeting? We will return to nonuniqueness ideas shortly. Jameson, A., “Essential Elements of Computational Algorithms for Aerodynamic Analysis and Design,” NASA/CR-97-206268, ICASE Report 97-68, Langley Research Center, National Aeronautics and Space Administration, Dec. 1997. From the Abstract – “This paper traces the development of computational fluid dynamics as a tool for aircraft design. It addresses the requirements for effective industrial use, and trade-offs between modeling accuracy and computational costs. Essential elements of algorithm design are discussed in detail, together with a unified approach to the design of shock capturing schemes. Finally, the paper discusses the use of techniques drawn from control theory to determine optimal aerodynamic shapes. In the future multidisciplinary analysis and optimization should be combined to provide an integrated design environment.” Jameson, A., “Transonic Flow Calculations,” MAE Report 1651, Princeton University, Dec. 2003. Mathematics and numerical issues. Jameson, A., “Chapter 11, Aerodynamics,” Encyclopedia of Computational Mechanics, Volume 3: Computational Fluid Dynamics, edited by Erwin Stein, Rene de Borst and Thomas J.R. Hughes, John Wiley & Sons, 2004. Write-up provides a good overall survey. Explains how boundary layer to first order increases effective thickness of the body for inviscid analysis. (Note, our own inverse modeling, where shape is sought which induces a prescribed surface pressure, incorporates displacement thickness modeling by allowing thick or opened trailing edges.) Chapter 11 emphasizes ‘simple’ checks, e.g., “Does the numerical solution of a symmetric profile at zero angle of attack preserve the symmetry, with no lift?” Paper gives brief survey of mathematical fluid flow models, conservative versus non-conservative transonic schemes, transonic type-differencing, over-relaxation and convergence acceleration.

Concepts, Challenges and Methods 23 Jameson, A., “Inverse Problems in Aerodynamics and Control Theory,” International Conference on Control, PDEs and Scientific Computing, Beijing, China, September 10-13, 2004. Jameson notes, “Because a shape does not necessarily exist for an arbitrary pressure distribution the inverse problem may be ill posed if one tried directly to enforce a specified pressure as a boundary condition” and goes on to develop his method. Our small-disturbance method always produces geometric solutions, which may include surface cross-over; uniqueness is obtained by specifying trailing edge constraints. Jameson, A. and Ou, K., “Fifty Years of Transonic Aircraft Design,” Progress in Aerospace Sciences, Elsevier, 2011. A historical overview of aircraft design, most references from 1960s and 1970s. Description of full nonlinear potential equation solver with rotated differencing scheme extends Murman-Cole method. Described Euler equation solver, unstructured meshes. In one section on optimum shape design, several methods are reviewed. Formulations of (inverse) design methods for aerodynamic problems dates to a conformal mapping of Lighthill. For transonic flow, earliest design methods were based on the hodograph method. The complex characteristics method of Garabedian and Korn is cited, along with optimization procedures of Hicks and Henne to design transonic airfoils and wings. Pironneau used optimal control techniques for the design of shapes governed by elliptic equations. Subsequently, Jameson developed the use control theory, based on the solution of adjoint problems, and applied it (together with Reuther) to the design of aerodynamic shapes in transonic and supersonic flow governed by (nonlinear) potential flow and the Euler equations, and (with Martinelli and Pierce) to Navier–Stokes equation formulations. Solution non-uniqueness. Nonunique solutions are commonplace in fluid mechanics, a classic example in the aerodynamic context illustrated by Kutta’s trailing edge constraint. However, until recently, they were unexpected in calculations of flowfields past aircraft bodies. A number of researches in this area are available, and while we do not pursue this topic in our own work, we provide for completeness an early reference (dating more than three decades ago) and one completed more recently in 2017.

24 Modern Aerodynamic Methods Salas, M.D., Jameson, A. and Melnik, R.E., “A Comparative Study of the Nonuniqueness Problem of the Potential Equation,” AIAA Paper 83-1888, 6th Computational Fluid Dynamics Conference, Danvers, Mass., July 13-15, 1983. This paper was briefly cited above under “Jameson research” topics. Here, the authors indicate, “additional evidence has been provided which supports the thesis of Ref. 1, that the nonuniqueness is a problem inherent to the conservative potential differential equation.” Further, “none of the multiple solutions obtained with the conservative potential formulation seems to be relevant to the physical problem. Rather, they seem to indicate a breakdown of the theory. It appears that to avoid the anomaly the conservative formulation must be abandoned.” Our note: This is unfortunate after years of industry effort focusing on conservative schemes. It may be that the present author’s use of the high-order “viscous transonic equation” in Chapter 3 may provide a more viable path to unique solutions since it is physically grounded. The paper just cited appeared in 1983; the next, due to Liu, Luo and Liu (2017), provides updates to a controversial subject. Liu, Y., Luo, S. and Liu, F., “Multiple Solutions and Stability of the Steady Transonic Small-Disturbance Equation,” Theoretical and Applied Mechanics Letters, Vol. 7, 292-300, 2017. From the Abstract: “Numerical solutions of the steady transonic smalldisturbance (TSD) potential equation are computed using the conservative Murman Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data. From the first paragraph of paper: “The non-uniqueness of numerical

Concepts, Challenges and Methods 25 solutions of potential equations at transonic speeds has been found for three decades. Steinhoff and Jameson first reported multiple solutions for the full potential (FP) equation. Chen first reported the existence of multiple solutions of the steady transonic smalldisturbance (TSD) equation using the nonconservative Murman Cole scheme. Nixon also found multiple solutions using the TSD equation modified with vorticity and entropy corrections. Salas et al. did extensive study on multiple solutions of the FP equation. Jameson demonstrated that non-unique solutions of Euler equations can be obtained for certain airfoils. Hafez and Guo investigated the flow over airfoils with flat and wavy surface by solving the steady potential equations, the Euler equations, and the Navier Stokes equations. They found that all of the equations can generate multiple solutions at zero angle of attack in certain Mach number ranges. Luo et al. showed that the multiple solutions of the transonic small transverse disturbance equation are independent of the difference schemes and iterative methods and found multiple solutions for a three dimensional wing.” Kutta condition. While Kutta’s trailing edge condition has been long accepted in aerodynamics, recent work, particularly the research of Rienstra (1992) points to mathematical issues that are unresolved and previously unidentified. The commonly accepted model is sumarized in our “Basic fluid mechanics” listing above, e.g., Glauert’s classic solution as explained in the books by Ashley and Landahl, Batchelor, MilneThomson and Yih. We have listed a recent papers Kutta’s model below. We emphasize that solutions to potential flows are not unique and “Kutta type” constraints are required to define circulation levels enabling flows to leave the trailing edge smoothly. In general, the “right side” streamline can leave the airfoil anywhere. But in practice, a finite angle trailing edge requires a stagnation point, while for a cusped edge, upper and lower velocities must be aligned. Many published papers, however, do not provide implementation details, and it is often unclear what is being modeled or even if the baseline computations satisfy “CL = 2 .”

Rienstra, S.W., “A Note on the Kutta Condition in Glauert’s Solution of the Thin Airfoil Problem, IWDE Report; Vol. 91-04. Eindhoven: Technische Universiteit Eindhoven, Oct. 1991. Also, Journal of Engineering Mathematics, Vol. 26, pp. 61-69, 1992, Kluwer Academic Publishers, The Netherlands. Subtle aspects related to

26 Modern Aerodynamic Methods uniqueness of Kutta’s condition in two and three-dimensional airfoils and wings and related numerical methods are discussed. Iannelli, G.,Grillo, C. and Tulumello, L., “A Kutta Condition Enforcing BEM Technology for Airfoil Aerodynamics,” Transactions on Modelling and Simulation, Vol. 15, WIT Press, 1997. Deals with correct procedures for Kutta condition enforcement and solution non-uniqueness. Mohebbi, F. and Sellier, M., “On the Kuttta Condition in Potential Flow Over Airfoil,” Volume 2014, Article ID 676912, Journal of Aerodynamics, Hindawi Publishing, April 2014. More on implementation, finite difference method and curvilinear coordinates. Radwan, S.F., “Critical Review of the Trailing Edge Condition in Steady and Unsteady Flow,” Master of Science Thesis, Lehigh University, 1981. Good discussion on real viscous fluid effects. Golberg, M.A., “The Numerical Solution of Cauchy Integral Equations With Constant Coefficients,” Vol. 9, 1985, pp. 127-151, Journal of Integral Equations, Elsevier Science Publishing. Tricomi, F.G., Integral Equations, Interscience Publishers, London, 1957. A classic reference on the “airfoil equation.” Subsonic aerodynamics. The references below cover a range of interests. For example, Drela and Youngren (2001) describe the very successful XFOIL airfoil analysis and design system developed at M.I.T., while Katz and Plotkin (2010) have authored perhaps the best low speed reference and panel method descriptions currently available. Of special importance are the classic works of van Dyke, who developed the “method of matched asymptotic expansions,” at first used to develop “inner solutions” near the leading edges of airfoils where thin airfoil, small disturbance theory breaks down. The method has proven indispensable to many areas of engineering, and will, ultimately, prove to be useful in our own small disturbance streamfunction approach to aerodynamic inverse problems. van Dyke’s leading edge corrections are, perhaps, less prominent nowadays, given the popularity of curvilinear grids (where local breakdown does not occur), but the work is useful in formulating local models that are convenient and rapid to operate. A readable description together with algorithms appears in this author’s book Managed Pressure Drilling (see “About the Author”).

Concepts, Challenges and Methods 27 Drela, M. and Youngren, H., XFOIL 6.94 User Guide, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Mass., 2001. DeJongh, J., “Computation of Incompressible Potential Flow Over and Airfoil Using a High-Order Aerodynamic Panel Method Based on Circular Arc Panels,” Doctoral Thesis, Air Force Institute of Technology, Aug. 1982. Katz, J. and Plotkin, A., Low Speed Aerodynamics, Second Edition, Cambridge University Press, Cambridge, 2010. An excellent aerodynamics book covering a wide range of topics rigorously – a thorough discussion of modern panel methods for low speed flow. van Dyke, M.D., Second-Order Subsonic Airfoil Section Theory and its Practical Application, NACA TN 3390, National Advisory Committee for Aeronautics, March 1955. van Dyke, M.D., Second Order Subsonic Airfoil Theory Including Edge Effects, Report 1274, National Advisory Committee for Aeronautics, 1956. Van Dyke, M.D., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. Transonic analysis. This section cites the original Murman-Cole publication, together with papers describing other algorithms and developments of interest in transonic flow modeling. Murman, E.M. and Cole, J.D., “Calculation of Plane Steady Transonic Flows,” AIAA Journal, Vol. 9, No. 1, 1971, pp. 114-121. Paper describes the original type-dependent differencing method used to model mixed subsonic-supersonic flows. Mostrel, M.M., “On Some Numerical Schemes for Transonic Flow Problems,” Mathematics of Computation, Vol. 52, No. 186, April 1989, 587-613. New steady and unsteady shock capturing schemes are explained. Langley, M.J., “Numerical Methods for Two-Dimensional and Axisymmetric Transonic Flows,” C.P. Number 1376, Aeronautical Research Council, Her Majesty’s Stationery Office, London, Sept. 1973. Stahara, S.S., Operational Manual for Two-Dimensional Transonic Code TSFOIL, NASA Contractor Report 3064, National Aeronautics

28 Modern Aerodynamic Methods and Space Administration, Dec. 1978. An indispensable “must have” reference on the classic Murman-Cole method. Stahara, S.S. and Spreiter, J.R., “Research on Unsteady Transonic Flow Theory,” Neilsen Engineering and Research, NR-061-215, work supported by the Office of Naval Research, Contract N0001473-C-0379, April 1977. Zheng, Z., “Survey of Transonic Flow Research in China,” Foreign Technology Division, Wright-Patterson Air Force Base, FTDID(RS)T-1077-82, Oct. 1982. Ballhaus, W.F. and Steger, J.L., “Implicit Approximate Factorization Schemes for the Low-Frequency Transonic Equation,” NASA Technical Memorandum No. 73082, National Aeronautics and Space Administration, November 1975. Supersonic flow. While supersonic flowfields are routinely obtained computationally, the following classic references describe relevant theory useful, for example, in panel method simulator design. The methods described below, and extended by Boeing research staff, in fact, motivated the exact supersonic models in Chapter 3.

Fraenkel, L.E., “The Theoretical Wave Draw of Some Bodies of Revolution,” Reports & Memoranda No. 2842, A.R.C. Technical Report, Aeronautical Research Council, Her Majesty’s Stationery Office, London, May 1951. Theoretical studies for given geometric shapes, highly mathematical. Hayes, W. D., Linearized Supersonic Flow, North American Aviation, Los Angeles, Calif., Report AL 222, 1947. A classic. Hayes, W.D., “Pseudo-transonic Similitude and First-Order Wave Structure,” Journal of the Aeronautical Sciences,Vol. 21, Nov. 1954, pp. 721-730. Heaslet, M.A. and Lomax, H., “The Use of Source-Sink and Doublet Distributions Extended to the Solution of Boundary Value Problems in Supersonic Flow,” NACA Rept. 900, 1948. Courant, R. and Friedrichs, K.O., Supersonic Flow and Shock Waves, Wiley Interscience, New York, 1948, pp. 134-138. Whitham, G.B., Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.

Concepts, Challenges and Methods 29 Inverse models. “Analysis,” “forward” or “direct models” solve for the pressure fields induced by prescribed aerodynamic shapes, while “inverse” or “indirect models” determine the geometric shapes that induce a specified pressure. Various approaches are available, and for completeness, these are listed below. Note that they all differ from the approach developed in this book. In our approach, we derive “direct methods” for aerodynamic inverse problems using a novel streamfunction formulation solved subject to a “Kutta type” trailing edge constraint. Strategies and results are outlined for various classes of problems in Chapters 2 and 4 of this book. AGARD Fluid Dynamics Panel, “Optimum Design Methods for Aerodynamics,” AGARD Report 803, Advisory Group for Aerospace Research and Development, North Atlantic Treaty Organization, November 1994. Reviewed different approaches, discusses foreign developments. Yiu, K.F.C., “Computational Methods for Aerodynamic Shape Design,” Vol. 20, No. 12, Mathematics and Computer Modelling, Elsevier Science, 1994, pp. 3-29. General review of various inverse approaches. Matsushima, K., Yamamichi, I. and Tokugawa, N., “Aerodynamic Design Method for Supersonic Slender Body Using an Inverse Problem,” 29th Congress of the International Council of the Aeronautical Sciences, St. Petersburg, Russia, September 7-12, 2014. Soemarwoto, B., “The Variational Method for Aerodynamic Optimization Using the Navier Stokes Equations,” ICASE Report No. 97-71, National Aeronautics and Space Administration, Langley Research Center, Virginia, December 1997. Jameson, A., “A Sequel to Lighthill’s Early Work – Aerodynamic Inverse Design and Shape Optimization via Control Theory,” Florida State University Lecture Series, February 15-20, 2010. Qingsheng, L., Yuxin, Z., Yiling, Z. and Hongyang, L., “The Pressure Inverse Problem of Three-Dimensional Supersonic Flow,” Acta Aerodynamic Sinica, October 2016. Soemarwoto, B., “Robust Inverse Shape Design in Aerodynamics,” Vol. 1, Issue 2, Inverse Problems in Engineering, 1995.

30 Modern Aerodynamic Methods Skinner, S.N. and Zare-Behtash, H., “State-of-the-Art in Aerodynamic Shape Optimization Methods,” Vol. 62, Applied Soft Computing, Elsevier Science, 2018, pp. 933-962. Dulikravich, G.S., Dennis, B.H., Baker, D.P., Kennon, S.R., Orlande, H.R.B. and Colaco, M.J., “Inverse Problems in Aerodynamics, Heat Transfer, Elasticity and Materials Design,” International Journal of Aeronautical and Space Sciences, 13(4), 405–420 (2012). Ren, J., “Aerodynamic Shape Optimization by Multi-Fidelity Modeling and Manifold Mapping,” Master of Science Thesis, Iowa State University, 2016. Fincham, J.H.S. and Friswell, M.I., “Aerodynamic Optimisation of a Camber Morphing Aerofoil,” Aerospace Science and Technology, Elsevier Science, Vol. 43, 2015, pp. 245-255. Mukesh, R., Lingadurai, K. and Selvakumar, U., “Airfoil Shape Optimization Using Non-Traditional Optimization Technique and Its Validation,” Vol. 26, Journal of King Saud University – Engineering Sciences , 2014, pp. 191-197. Yang, Y.L., Tan, C.S. and Hawthorne, W.R., “Aerodynamic of Turbomachinery Blading in Three-Dimensional Flow: An Application to Radial Inflow Turbines,” Journal of Turbomachinery 115(3) · July 1992 Yang, W., “Three-Dimensional Inverse Method for Aerodynamic Optimization in Compressor,” 6th International Conference on Pumps and Fans with Compressors and Wind Turbines, IOP Publishing, IOP Conf. Series: Materials Science and Engineering, Vol. 52 (2013) 012008. Mohebbi, F. and Sellier, M., “Aerodynamic Optimal Shape Design Based on Body-Fitted Grid Generation,” Mathematical Problems in Engineering, Volume 2014, Article ID 505372, Hindawi Publishing. Jian, Z., Jing, T., Ming, Q., Youqi, D. and Xiaoquan, G., “Massively Parallel Numerical Simulation Program Design of Unsteady Full Annulus Turbomachine,” Acta Aerodynamic Sinica, Dec. 2017. Dulikravich, G.S., Martin, T.J. and Dennis, B.H., “Multidisciplinary Inverse Problems,” Proceedings of Inverse Problems in Engineering: Theory and Practice, 3rd International Conference on Inverse Problems in Engineering, Port Ludlow, WA, June 13-18, 1999.

Concepts, Challenges and Methods 31 Liu, G.L. “A New Generation of Inverse Shape Design Problem in Aerodynamics and Aerothermoelasticity: Concepts, Theory and Methods,” Aircraft Engineering and Aerospace Technology, 2000, Vol. 72, Issue: 4, pp.334-344, Li, Y., Yang, D. and Feng, Z., “Inverse Problem in Aerodynamic Shape Design of Turbomachinery Blades,” ASME Turbo Expo 2006: Power for Land, Sea, and Air, Volume 6: Turbomachinery, Parts A and B, Barcelona, Spain, May 8–11, 2006, Conference Sponsors: ASME International Gas Turbine Institute, ISBN: 0-7918-4241-X, eISBN: 0-7918-3774-2 Petrucci, D. and Filho, N.M., “A Fast Algorithm for Inverse Airfoil Design Using a Transpiration Model and an Improved Vortex Panel Method,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2007, October-December 2007, Vol. XXIX, No. 4, pp. 354-365. Yang, J. and Wu, H., “The Development and Application of an Inviscid Inverse Method,” Propulsion and Power Research, May 2013, pp. 131-138. Engine and airframe integration. While entire aircraft configurations can be simulated three-dimensionally on high speed computers, often including viscous effects and simple models of turbulence, and rapidly in just minutes, engine and airframe integration represents an art in the final analysis that serves as the sole judge as to the credibility of any computed results. The following papers provide a cross-section of integration approaches take for different aircraft types across different industrial settings. These researches document our own work described in Chapter 5.

Jing, L., Zhenghong, G., Jiangtao, H. and Ke, Z, “Aerodynamic Design Optimization of Nacelle/Pylon Position on an Aircraft,” Chinese Journal of Aeronautics, April 2013, pp. 850-857. Rudnik, R., Rossow, C.C. and Geyr, H.F., “Numerical Simulation of Engine/Airframe Integration for High Bypass Engines,” Aerospace Science and Technology, Vol. 6, Issue 1, Jan. 2002, pp. 31-42. Knoth, F. and Breitsamter, C., “Aerodynamic Testing of Helicopter Side Intake Retrofit Modifications,” Vol. 4, Issue 3, Aerospace 2017.

32 Modern Aerodynamic Methods Takahashi, H., Munakata, T. and Sato, S., “Thrust Augmentation by Airframe-Integrated Linear-Spike Nozzle Concept for High-Speed Aircraft, Vol. 5, Issue 19, Aerospace, Feb. 2018. Staff, Aerodynamic Engine/Airframe Integration for High Performance Aircraft and Missiles, AGARD Conference Proceedings 498, Fluid Dynamics Panel, Advisory Group for Aerospace R & D, Fort Worth, exas, Oct. 7-10, 1991. Hoheisel, H. and Geyr, H.F., “The Influence of Engine Thrust Behaviour on the Aerodynamics of Engine Airframe Integration,” CEAS Aeronautical Journal, April 2012, Vol. 3, Issue 1, pp. 79–92. Campomanes, C.S., “External Flow Effects in the Engine/Airframe Integration Testing Technique – A New Thrust/Drag Book-Keeping Approach at the German-Dutch Wind Tunnels,” M. Sc. Thesis, Aerospace Engineering, Delft University of Technology, Oct. 2017. Sanghi, V., Kumar, S.K., Sundararajan, V. and Sane, S.K., “EngineAirframe Integration During Conceptual Design for Military Application,” AIAA Journal of Aircraft, Vol. 35, No. 3, May-June 1998, pp. 380-386. Heykena, C., Savoni, L., Friedrichs, J. and Rudnik, R., “Engine Airframe Integration Sensitivities for a STOL Commercial Aircraft Concept with Over-the-Wing Mounted UHBR-Turbofans,” GPPSNA-2018-0109, Proceedings of Montreal 2018 Global Power and Propulsion Forum, Montreal, May 7-9, 2018. Isikveren, A.T., Seitz, A, Bijewitz, J., Hornung, M., Mirzoyan, A., Isyanov, A., Godard, J.L., Stückl, S. and Van Toor, J., “Recent Advances in Airframe-Propulsion Concepts with Distributed Propulsion,” 29th Congress of the International Council of the Aeronautical Sciences (ICAS 2014), SAINT-PETERSBOURG, Russia, Sept. 2014. Oates, G.C., Aerothermodynamics of Gas Turbine and Rocket Propulsion, Fifth Printing, American Institute of Aeronautics and Astronautics (AIAA), 1998. 1.5 References

Essential references are listed above. These and all chapter references appear in “Cumulative References” at the end of this book. Other author publications are found in “About the Author.”

Modern Aerodynamic Methods for Direct and Inverse Applications.Wilson C. Chin. © 2019 Scrivener Publishing LLC. Published 2019 by John Wiley & Sons, Inc.

2 Computational Methods: Subtleties, Approaches and Algorithms We introduce the reader to finite difference methods and computational solutions to partial differential equations. We assume a minimal background aside from an exposure to partial derivatives. Only rudimentary skills in Fortran programming are required, as our listed source codes contain but the bare essentials needed to solve the physical problem at hand. Basic theory and programs are offered in Section 2.2, while Sections 2.3 and 2.4 advance quickly to modern methods (including our own) used in solving analysis and inverse problems. The discussions are self-contained, and interested readers are encouraged to consult the references in Chapter 1, and then our Chapters 3 and 4. 2.1 Coding Suggestions and Baseline Solutions 2.1.1 Presentation Approach

This book focuses not just on mathematical and physical ideas, but formulation correctness and flexibility. For example, in developing streamfunction ideas, it may be elegant to introduce complex variables methods and Cauchy-Riemann conditions. But in doing so, we would have restricted ourselves, semantically at least, to constant density flows in Cartesian coordinates. Instead, we take an “engineer’s approach” to rigorously extend these classical concepts to transonic supercritical, axisymmetric nacelle and inlet flows, and introduce “streamlike functions” satisfying an unanticipated “ xx + yy + zz = 0” equation reminiscent of Laplace’s equation for the potential. In modeling flows in the presence of very strong shears, where Euler’s equations are always invoked, along with their numerical complications and subtleties, we develop the notion of a “superpotential” equation that resembles the potential equation governing irrotational flows. In all cases, our ideas were “simple” yet not so simple; but for sure, they led to simple code structures and work efforts that allowed rapid evaluation of new ideas. 33

34 Modern Aerodynamic Methods Computational aerodynamics proliferates with source code free for the asking, e.g., public domain algorithms for XFOIL and TSFOIL among the better known. Howver, “taking apart” mature code is often more difficult than programming new algorithms from scratch. The author’s philosophy, over the years, is “Keep it simple.” Integrated threedimensional color plotting and line plot capabilities help significantly. If mistakes occur, and they generally do because of unseen typographical errors or cryptic compiler messages, problems are rapidly corrected. And candidate configurations should be simple. For instance, an analysis program should be tested with an airfoil geometry with left-right and upper-lower symmetry and centered in a rectangular computational box. Print out the numbers. If expected symmetries and antisymmetries are not found, the problem and its origin will be detected more easily. Toward these ends, we’ve kept our sample algorithms simple and their corresponding Fortran implementations readable. Section 2.1.2, “Programming Exercises,” and Section 2.1.3, “Model Extensions and Challenges,” guide the reader interested in deepening his understanding. In the former, extensions of a programming nature are suggested, which may be taken in any order, for those desiring software to serve specific purposes. In contrast, the latter suggests interesting research extensions that are publishable and useful in an industrial setting. Without apologies, we summarize all the simplifications that have been made in source code listings so that our mathematical assumptions can be readily understood – and easily extended if desired by the student. With few exceptions, our code assumes constant x, y and z meshes, and in fact, x = y = z in three-dimensional problems. Simpler grids also mean that a finite difference code, absent of arrays like X(I-1), X(I) and X(I+1), and so on, is more readable. Ghost points and similar auxiliary points are not used in Neumann boundary condition implementation. We employ an readily modified first-order method using an easily read TRIDI subroutine. Small numbers of gridblocks in the x, y and z directions are chosen so that numbers for potential and streamfunction can be printed on a 4½ inch wide page, allowing visually study of symmetries and antisymmetries, and corresponding formulation errors. These printouts are especially useful in identifying programming mistakes and how errors propagate spatially.

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Curvilinear grids are not considered in this book. Those interested in a rigorous development (without abstract mathematics) should peruse the author’s book Managed Pressure Drilling: Modeling, Strategy and Planning (Elsevier, 2012). This volume provides an exact “engineer’s derivation” of topics normally covered in differential geometry, coupled with algorithms and examples. 2.1.2 Programming Exercises

The Fortran codes in Sections 2.2–2.4 were kept deliberately simple to enhance readability and to forge rapid associations between boundary conditions and programming code. Further, a reliance on flow symmetries and antisymmetries develops one’s understanding and appreciation for streamfunctions and potentials – the ability to print out and examine numbers on a 4 ½ inch page was critical, forcing us to use sparsely gridded mesh systems. All this is useful to newcomers. Often it is easier to “build up” simple code than it is to “tear down” complex programs, since complicated software is rarely well documented. The benefits in truly understanding a problem far outweigh the perceived advantages in owning complex code. For those eager to improve upon our baseline algorithms, the following enhancements or suggestions may be added, in any order or priority. Replace our finite difference model (for x = y) with an equation assuming constant values of x y, and z, if applicable. Develop finite difference equations for a general mesh xi, xi+1, xi+2 and so on, and similarly for yj and zk. Combine with a complementary grid generator, e.g., assuming algebraic or other stretching. Add over-relaxation capabilities to our TRIDI (tridiagonal matrix) iteration algorithm for convergence acceleration, plus underrelaxation capabilities to decelerate unstable iterations. Develop a convergence test which terminates relaxation solutions when key quantities cease to change or change very slowly. Set up relaxation calculations to initialize using analytical, approximate or other saved old solutions for increased speed. Add farfield regularity options physical evaluation and code testing.

and

to facilitate

36 Modern Aerodynamic Methods Develop “row relaxation,” analogous to our “column” methods. How do point relaxation methods compare in terms of convergence speed? Develop a combined iteration solver that alternates row and column solution processes (e.g., similar to “alternating direction implicit” (ADI) and “approximate factorization” (AF) solvers) for rapid convergence. Alternate between coarse and fine meshes to evaluate “multigrid” convergence acceleration properties. Refine Kutta condition model to numerically support trailing edge stagnation points, cusped streamline and smooth flow options. How does lift vary versus angle of atack? 2.1.3 Model Extensions and Challenges

The aerodynamics projects suggested here, suitable for term papers and short research assignments, are designed to challenge the interested reader. They are nontrivial mathematically and numerically, but not listed in any particular order. All appear as outgrowths to the models developed in this book; some have been solved and appear in Chapter 5. The simple Fortran algorithms given in this chapter provide good starting points for those interested in writing software development or developing their own analysis methods. Supersonic airfoil analysis. Our Murman-Cole solver in Example 5 of Section 2.3.5 for transonic supercritical flow assumes subsonic conditions calling for vanishing disturbances faraway. Consider a supersonic freestream. Develop a farfield radiation condition relating x and y spatial derivatives (using linearized supersonic flow theory) accounting for proper domains of influence and dependence. Implement your work numerically. Inverse problem with oncoming shear flow. Our direct approach to inverse problems solves a streamfunction formulation subject to Neumann conditions using / y. In Chapter 4, we show that in the presence of weak to strong shear flow, the modified boundary condition takes the form / y+ (known as a Robin specification). Modify one of our streamfunction codes to allow general fluid rotation and oncoming shear. Include ground effects by

Computational Algorithms

37

decreasing computational box size and off-centering the airfoil slit – this is an application important to aircraft take-off and descent. Farfield updates and convergence acceleration. Our algorithms for direct (and inverse) applications assume 0 or 0 (and inverse 0 or 0) for simplicity, for rapid concept evaluation and debugging. Replace these conditions with analytical asymptotic solutions (available from cited literature). Evaluate the effects on accuracy and convergence speed. Lifting line analogy for inverse applications. In threedimensional analysis problems, the potential satisfies xx + yy + zz = 0. A variable circulation [ ] together with Kelvin’s theorem leads to classic lifting line theory. Develop an analogous “thickness line” theory for inverse problems satisfying xx + yy + zz = 0 subject to variable trailing edge constraints modeled by non-constant [ ]. Analysis problem for 3D subsonic lifting wings. In Example 6 of Section 2.3.6, we developed a three-dimensional potential equation solver for a symmetric “thickness only,” non-lifting wing at zero angle of attack. The model was formulated in a computational half-space, where we extended our column relaxation sweeping for simple airfoils to model wings of finite span. We first “swept” our columns in the root plane from upstream to downstream; then, we advanced to the next spanwise station, repeating the sweeping process for all spanwise stations before returning to root plane and carrying out this process repeatedly until convergence. This example demonstrated “plane by plane” solutions, where the same columnar and planform boundary condition logic applied everywhere. If we now refer to Example 4 of Section 2.3.4, where we dealt with lifting airfoils centered in a computational box,, we find that Kutta’s condition relating to the potential jump [ ] requires different solutions to different regions of the flow. We worked with a region upstream of the leading edge, a second beneath the airfoil, a third directly above, and a fourth, downstream of the trailing edge, where we fixed [ ] along a branch cut extending downstream. Modeling wings of finite span requires additional coding. The lifting logic of Example 4 must be extended “plane by plane” along the span, and further, in the spanwise direction off the wing but without potential jump logic. Develop a three-dimensional lifting wing solver and

38 Modern Aerodynamic Methods validate computed solutions. On completion, recode the algorithm to model wings with swept planforms, introducing a suitable grid structure that accommodates leading and trailing edge sweep, properly accounting for curvilinear coordinate transformations. Inverse problem for 3D wings. Guidelines useful to solving the analysis problem for three-dimensional wings were listed in the above bulleted item. The foregoing model solves xx + yy + zz = 0 subject to a variable circulation [ ] chosen to satisfy Kutta’s condition. Chapter 4 shows how three-dimensional inverse problems follow a similar “ xx + yy + zz = 0” equation for a “streamlike function.” Develop a general 3D inverse solver where upper and lower planform surface pressures may differ. Allow two trailing edge options, the first for [ ] = 0 assuming complete closure, and a second, supporting nonzero prescribed values to allow thick viscous wake or open-edge effects. Ringwing flows. Airfoils in rectangular x-y space are governed by “ xx + yy = 0,” plus Neumann tangency conditions at solid surfaces, regularity conditions at infinity, and Kutta’s condition at the trailing edge which defines the appropriate value of [ ]. Axisymmetric ringwing and “nacelle only” flowfields satisfy “ xx + rr + 1/r r = 0” in x-r space and auxiliary conditions that are identical in form. Modify our lifting airfoil code to solve for such flows. On the other hand, pitched ringwings require an added “1/r2 ” contribution to playing the role of the spanwise the potential equation with variable. How would a simulation algorithm be formulated? Nacelles and inlets. Idealized aircraft nacelles and engine inlets may be axisymmetric, but often the requirement for a bottom “accessory bump” to house gear boxes, or a pylon to attach the engine to a wing, requires us to optimize three-dimensional geometries for aerodynamic efficiency. Further discussion and sketches are offered in Chapter 5. In Section 2.3, we will show how flows past three-dimensional wings can be solved by iterative methods, relaxing the solution “airfoil plane” by “airfoil plane” at a time. Apply this method to solve “ xx + rr + 1/r r + 1/r2 = 0, solving one x-r plane at a time discussed above, repeatedly for different azimuthal angles. This process is duplicated over and over until convergence is achieved.

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39

2.2 Finite Difference Methods for Simple Planar Flows We will introduce finite difference methods and applications to solving partial differential equations. Readers familiar with the technique may proceed to Sections 2.3 and 2.4 without loss of continuity. Although finite element, finite volume, integral equation, panel and other methods are available for solution, finite difference methods require minimal resources or specialized training for rapid evaluation of new formulations and ideas. The methods discussed do not require extended knowledge of linear algebra, numerical analysis or advanced expertise. These have served the author well over years of engineering design and research, in both aerospace and petroleum industries, supporting rapid prototyping for production code development. Although this subject is usually offered only in advanced numerical analysis courses, there is no reason to impose artificial requirements or prerequisites. Both basic and sophisticated ideas can be developed from simple notions in elementary calculus. The intuitive how to approach taken is down-to-earth, comprehensive, and, importantly, mathematically correct. But in this section we will discuss only those ideas necessary to 2 2 accomplish our objective, that is, solving Laplace’s equation F/ x + 2 2 F/ y = 0 for steady planar flows (in this book, F will denote either the velocity potential or its corresponding streamfunction). We will develop the terminology and ideas naturally, and avoid excessive mathematical formalism. We will augment the discussions with Fortran examples and source code to make the ideas clear and the methodologies widely accessible. This presentation is no substitute for a truly rigorous and formal study of numerical methods. However, like the rest of this book, it is written to be self-contained so that the application of computational methods to engineering analysis can proceed with minimal interruption. 2.2.1 Finite Differences – Basic Concepts

Before we introduce numerical relaxation schemes and their applications to partial differential equations, we need to develop the basic ideas and working vocabulary underlying finite difference discretization methods. Finite difference approximations. Let us consider the function F(x) as shown in Figure 2.2.1 and examine several representations for its derivatives. Because F(x) will be approximated at a discrete set of points, we introduce a grid or mesh of imax points x1, x2, x3, ..., ximax. In

40 Modern Aerodynamic Methods fact, we will denote any three consecutive points by xi-1, xi and xi+1, where i is an index ordered so that it increases as x increases. When the distance between successive points in this discretization process is the same, the grid is constant. On the other hand, the grid is variable if the distances vary spatially; it is adaptive in time if it adapts locally in order to track key physical events like rapid flap oscillations.

F i+1

Fi F

i-1

x i-1

A

i

B i+1

Figure 2.2.1. Finite difference formula development.

For simplicity, let us assume constant (or slowly varying meshes) first, and refer to the points A and B in Figure 2.2.1. The first derivative of F(x) at x = xA is approximated by evaluating its slope using points to its left and right, F(x )/ x = (F - F )/(x - x ) (2.1) A i i-1 i i-1 At x = x , we likewise have B F(x )/ x = (F - F )/(x -x) (2.2) B i+1 i i+1 i Thus, the second derivative at x = x , or simply i, takes the form i 2 2 F/ x ={ F(x )/ x - F(x )/ x}/{1/2 (x - x )} (2.3) B A i+1 i-1 = {(F - F )/(x - x ) - (F - F )/(x - x )}/{1/2 (x - x )} i+1 i i+1 i i i-1 i i-1 i+1 i-1

Computational Algorithms where the length ½ (x

i+1

41

- x ) in Equation 2.3 applies if the meshes

expand or contract slowly. derivative at i is

i-1

The corresponding formula for the first

F(x )/ x = (F - F )/(x -x ) (2.4) i i+1 i-1 i+1 i-1 Equations 2.3 and 2.4 are finite difference representations for 2F/ x2 and F/ x. Our use of left and right values to define geometric slopes (for both first and second derivatives) is called central differencing. Backward and forward one-sided differencing are also possible, though less accurate for the same number of points. A simple differential equation. We discuss a simple application for finite differences. In particular, let us consider the boundary value problem defined by the second-order linear differential equation and the boundary conditions (2.5) d2y/dx2 = 0 y(0) = 0

(2.6a)

y(2) = 2

(2.6b)

This equation set has the simple straight-line solution y(x) = x

(2.7)

The idea, of course, is to replicate this function numerically. For now, we specialize the analysis to constant meshes to illustrate the basic procedure. This choice of mesh system reduces Equation 2.5 to the simple form (2.8) d2y/dx2 = (y – 2 y + y )/( x) 2 i-1 i i+1 where x is an assumed mesh length, so that Equation 2.5 becomes y - 2y + y =0 (2.9) i-1 i i+1 Equation 2.9 is the finite difference model relating different y values at different positions xi. It shows that these y’s are coupled and must be determined simultaneously. To find the equations that must be solved, write Equation 2.9 for each of the internal nodes i = 2, 3, ..., (i -1). max

This leads to

42 Modern Aerodynamic Methods i=2: y – 2 y + y 1 2 3

= 0

(2.10b)

i=3:

= 0

(2.10c)

= 0

(2.10d)

y -2y +y = 0 4 5 6

(2.10e)

y – 2y + y 2 3 4

i=4:

y –2 y + y 3 4 5

i=5: . . i=i

-1:

max

y

imax-2

–2 y

imax-1

+y

imax

= 0

(2.10f)

Observe that there are two more unknowns than there are equations. The additional required equations are obtained from Equations 2.6a,b, that is, y =0 1

(2.10a)

y

(2.10g)

imax

=2

which we might introduce at the top and bottom, respectively, of the equation block. Note that Equations 2.10a to 2.10g so written assume a tridiagonal structure; the exact form will be important to the iterative schemes we consider later. For now, in our direct single-pass solution to Equations 2.5 and 2.6, we can rewrite the foregoing equations in the matrix form | 1 0

| | y | | 0 | 1 | 1 -2 1 | | y | | 0 | 2 | 1 -2 1 | | y | | 0 | 3 | 1 -2 1 | | y | | 0 | 4 | 1 -2 1 | | y | = | 0 | 5 | . | | . | | 0 | | . | | . | | 0 | | 1 -2 1 | | y | | 0 | imax-1 | 0 1 | | y | | 2 | imax

(2.11)

Computational Algorithms | b 1 | a 2 | | | | | | |

c

1 b 2 a 3

| | v c

2 b 3 a 4

1

| | v c

3 b c 4 4 a b 5 5 . . a n-1

2

| | v

3

| | v c

| | v

5

b n-1 a n

c

| | . | | . | | v

4

5

|

| w

|

| w

|

| w

|

| w

| = | w | | |

n-1 n-1 b | | v | n n

1 2 3 4 5

| . | . | w | w

n-1 n

43

| | | | |

(2.12)

| | | |

Equation 2.11 is a special instance of the more general tridiagonal matrix in Equation 2.12. The indexed quantities ai, bi, ci, vi and wi are called column vectors of dimension n, although sometimes, they are simply vectors denoted by the boldfaced symbols a, b, c, v, and w. The matrix at the left is “tridiagonal,” a special case of a diagonal banded matrix. We will not deal with matrix inversion in this book. Suffice it to say that the last (or the first) row, which involves two unknowns only, is usually used to reduce the number of unknowns along each row, right up (or down) the matrix, resulting in a bidiagonal system. Then, repeating the process in the opposite direction yields the solution vector v. When the coefficients in Equation 2.12 are defined as in Equation 2.11, the solution vector vi is obtained by calling standard tridiagonal solvers found in numerical analysis books. If the programming language used is Fortran, as is the case in this book, the subroutine in Figure 2.2.2 can be used (note, Fortran compilers are freely available at several Internet websites). The routine as coded destroys all original input coefficients upon inversion; if it is called successively, as solutions of partial differential equations require, the relevant coefficients must be redefined prior to each call of TRIDI. Also, we emphasize that A(1) and C(imax) should be defined and set to dummy values, “99” in our examples, even though they do not play a role in the solution. Unless this is done, certain computers will initialize their registers improperly and produce incorrect solutions. The reader should verify that the solution of Equation 2.11 does agree with the exact solution in Equation 2.7. Using TRIDI is simple. The reader only needs to identify the coefficients of b1v1 + c1v2 = w1 and anvn-1 + bnvn = wn with his boundary condition data and call the subroutine in Figure 2.2.2.

44 Modern Aerodynamic Methods

100

200

SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(11), B(11), C(11), V(11), W(11) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) CONTINUE RETURN END

Figure 2.2.2. Tridiagonal matrix solver. Variable coefficients and grids. Ordinary differential equations often contain variable coefficients, for example, 2 2 d y/dx + f (x) dy/dx + f (x) y = f (x) (2.13) 1 2 3 The f1, f2, and f3 might describe spatially dependent properties in the physical problem being modeled; in this write-up, we assume they are slowly varying and non-singular. Our treatment is heuristic. If they vary rapidly, the use of constant meshes may be inappropriate. If so, Equations 2.3 and 2.4 must be used, and the discretized values of the arrays xi, f1(xi), f2(xi) and f3(xi) must be additionally stored in computer memory. Needless to say, the matrix coefficients in Equation 2.12 now become much more complicated. We warn against blindly using the method given. For example, if the f’s are singular, discontinuous or nonstandard, special treatment is required, and an understanding of fundamental ordinary differential equations is needed. For well-behaved coefficients, grid selection is straightforward and follows several rules of thumb. If a “f” coefficient varies rapidly in some region of space, it is reasonable to increase local mesh density to improve physical resolution. However, there is always the danger that, since the value of yi affects each and all of its neighbors, spurious effects can contaminate the complete solution. Otherwise, constant grids may suffice. In any event, the typical mesh size should be small compared to the length scale of the problem, and mesh-to-mesh expansion rates should not exceed 10%. Detailed testing of the solution for mesh dependence should accompany program development.

Computational Algorithms 2.2.2

45

Formulating Steady Flow Problems

Here a discussion originally developed for petroleum reservoir simulation is offered, taking advantage of similarities and analogies with single-phase steady flow. This approach is meaningful because fluid pressures (which are potentials in the oilfield context) are more easily understood or visualized than velocity potentials and streamfunctions. “Wells” represent points and “fractures” denote lines where pressures are fixed. We discuss solutions to Laplace’s equation for the pressure P(x,y), with and without wells and fractures, using both “aquifer conditions” specifying pressure, and solid wall conditions assuming zero normal flow. We consider, for simplicity, the Cartesian form 2

P/ x2 + 2P/ y2 = 0

(2.14)

solved on a rectangular grid. Again, partial differential equations fall into three categories. Equation 2.14, for example, is elliptic; transient 2 2 2 2 compressible flows satisfy parabolic equations like u/ x + u/ y = 2 2 u/ t, while seismic waves satisfy hyberbolic equations like u/ x + 2 2 2 2 u/ y = u/ t . Here, we discuss elliptic equations only. Equation 2.14 assumes the two-dimensional, constant density flow of a liquid in an isotropic homogeneous medium. First consider a singly connected region, for example, a simple pie, square or triangle. It is known that the solution is completely defined whenever pressure is prescribed over the entire boundary. For such Dirichlet problems, the solutions, in mathematical lingo, exist and are uniquely defined. Now suppose that the flux of mass or the velocity is given, that is, that the derivative of pressure in a direction normal to the boundary contour is prescribed. What can we expect for the solution? Since the normal derivative p/ n is proportional to the fluid velocity, we would expect that it cannot be arbitrarily specified. It must be given in such a way that just enough flow leaves as enters the flow domain. Moreover, the value of the P’s obtained will be indeterminate to within a constant, since we have prescribed only derivatives. This additive level will not affect flow rates, since it differentiates to zero; but the exact pressure level is unimportant and can be conveniently set to any value at a given point. Boundary value problems where the normal derivative p/ n is specified at the boundaries are known as Neumann problems. Their solutions are not unique, but only to the extent just described. If the flow rate, which is proportional to P/ n, is prescribed over part(s) of the

46 Modern Aerodynamic Methods boundary, and pressure itself is given over the remainder, the solution is again completely determined and unique. The reason is simple: we have not unreasonably created or destroyed mass. The required mass conservation will manifest itself at the boundaries where pressure was prescribed, and a net outflow or inflow will be obtained that is physically sound. Problems where both P/ n and P are specified are referred to as mixed Dirichlet-Neumann problems problems. As noted in Chapter 4, these also arise naturally in aerodynamic inverse problems where geometries and pressures are both specified along an approximating slit. So far, we have restricted our discussion to singly connected domains, that is, uninteresting reservoir flows without wells. The presence of a well effectively punctures a hole in the circular or rectangular region of flow, creating a donut-like reservoir; such shapes are said to be doubly-connected. Again, common-sense ideas related to mass conservation apply. If the velocity (via P/ n) is prescribed over the complete outer reservoir boundary, then one cannot arbitarily assign P/ n at the well; however, specifying the pressure level itself is completely legitimate and will lead to unique and reasonable solutions. Similar considerations apply to reservoirs with multiple wells; the corresponding domains of flow are said to be multiply-connected. An understanding of pressure behavior is essential to good simulator development. It turns out that the insight gained in reservoir flow analysis is useful in aerospace applications; after all, an airfoil is “well like” and multi-element flaps sometimes mimic multiple wells. However, we emphasize that in our aerodynamics work, we do not deal with point singularities, but line distributions of sources and vortexes. 2.2.3 Steady Flow Problems

Here we obtain solutions to Equation 2.14 for several different flow problems. For simplicity, we consider constant meshes, with lengths x and y in the x and y directions. The resolution achieved with such meshes systems near wells is limited. The programs are given for illustrative purposes, but we note that resolution problems can be resolved by variable mesh spacing or through the use of curvilinear meshes. Now, the second derivative in Equation 2.3 applies to a function F(x) at x = xi , but P(x,y) depends on an additional y, say indexed by j. At any point (i,j), use of Equation 2.3 in both x and y directions with Equation 2.14 leads to the simple model

Computational Algorithms

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2

(P – 2 P +P )/( x) i-1,j i,j i+1,j + (P – 2 P +P )/( y) 2 = 0 i,j-1 i,j i,j+1

(2.15)

Note that this straightforward use of Equation 2.3 is actually subtle. For our constant density fluid, the pressure at (i,j) must depend on its neighbors at i-1,j, i+1,j, i,j-1 and i,j+1. That is, the flow at any point is influenced by every other point, and each point affects all other points. As discussed in Chapters 3 and 4, the situation is different for hyperbolic problems; disturbances created by supersonic aircraft cannot propagate ahead of the plane, so that a difference approximation that violates domains of influence and dependence cannot be used. Likewise, within the supersonic bubble above a transonic airfoil, similar dependencies apply, whereas outside the bubble, the subsonic central difference model in the present discussion holds. Hence, there are areas in physics where use of central differencing throughout is inappropriate, and one-sided models must be used. And there are applications where both must be used; we will address one such application, namely that for transonic supercritical flows, in Section 2.3.5. However, for Laplace’s equation, the approximation in Equation 2.15 is perfectly valid. We now consider the rectangular reservoir domain defined by the index ranges 1 i 11 and 1 j 11, and specifically, a Dirichlet formulation where pressures of 10, 20, 30, and 40 are specified in clockwise fashion along the four edges of the box. This no-well formulation, as discussed earlier, is associated with a unique solution. If Equation 2.15 is written for each and every node (i,j) internal to the computational box, and the assigned boundary values are included into the set of linear equations, we obtain 11 11, or 121 unknowns that are fully determined by 121 linearly independent equations. Over one hundred coupled equations are required for this very coarse mesh! Direct versus iterative solutions. The mechanics of setting up the necessary system for “direct solvers,” that is, for algorithms that obtain pressures in a single pass using a full matrix solver, are discussed in basic books on mumerical methods. Even for the coarse mesh 121 matrix is large and requires considered, the resulting 121 monumental inversion efforts. Usually, the unknowns are cleverly ordered, and cleaner inversion algorithms are used; other methods take advantage of the sparseness (that is, the large number of 0’s) in the system.

48 Modern Aerodynamic Methods Iterative methods. We will consider iterative solvers that require minimal memory resources. These flexible algorithms work well in both two and three dimensions; they are robust, stable and fast. Since an objective of this book is the development of portable tools, we will not discuss direct solvers. Suffice it to say that such solvers, the most notorious being Gaussian elimination, are well documented in the literature (e.g., see Carnahan, Luther and Wilkes, 1969). For reasons that will become obvious, let us rewrite Equation 2.15 in the form 2

2

P -2{1 + ( y/ x) } P + P = - ( y/ x) (P +P ) (2.16) i,j-1 i,j i,j+1 i-1,j i+1,j

Equation 2.16 contains the tridiagonal form given in our ordinary differential equations example. On the left, the index i stands alone. When i is fixed and “j” is incremented, a sequence of tridiagonal equations is generated. We assume that i varies from 1 to imax and j varies from j = 1 to jmax. Let us assume that some suitable first guess for the pressure field P(i,j) is available. If so, the idea is to first freeze “i” at i = 2, write Equation 2.16 for each of the internal nodes j = 2, 3, ..., (jmax - 1), apply boundary conditions at both j = 1 and j = jmax , and solve for updated values of P(2, j) along the column i = 2. Then, the same process is repeated for i = 3, i = 4, and so on, until the last column i = (imax -1) is completed: one sweep of the box is said to have taken place. This sweeping, called column relaxation, is repeated for multiple sweeps until satisfactory convergence is achieved. The columns located at i = 1 and imax are not solved because pressures have been specified along them. Relaxation is the mathematical name synonymous with the method of successive approximations. Line relaxation may proceed by columns, as we have demonstrated; or, it may proceed by rows, that is, through row relaxation by freezing j and incrementing i’s. Special schemes employing combined row and columnar operations are referred to as “alternating direction implicit” or ADI schemes. In all cases, the basic idea is to disseminate boundary conditions rapidly and to approach convergence as quickly as possible. All of these are improvements on point relaxation, developed by earlier workers; we will give examples for comparison later. In Figure 2.2.3a, we list the Fortran source code required to implement those iterations, assuming a rectangular box with 10, 20, 30, 40 boundary conditions, without any wells, a formulation corresponding to an aquifer alone flow. Figure 2.2.3b gives computed pressures at various stages in the sweeping process. Note from Figure

Computational Algorithms

49

2.2.3a that the initial guess for pressure was taken as “zero” throughout, an arbitrary choice since we knew nothing about the solution. In fact, the initial guess might have been anything; by contrast, the results in Figures 2.2.4a,b assume an initial Pi,j = i2 + j2 devoid of physics, a guess having nothing to do with the solution or reality. Both calculations converge quickly to the same pressures, requiring much less than a second on modern personal computers. C

C

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100 C

150

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153 C

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154 155

157

LAPLACE EQUATION SOLVER, CASE_1. PROGRAM MAIN DIMENSION P(11,11), A(11), B(11), C(11), V(11), W(11) OPEN(UNIT=4,FILE=’CASE_1.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. RATIO2 = (DY/DX)**2 INITIALIZE P(I,J) TO ZERO EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “10-20-30-40” BOUNDARY CONDITIONS DO 150 I=1,10 P(I,1) = 10. CONTINUE DO 151 J=1,10 P(11,J) = 20. CONTINUE DO 152 I=2,11 P(I,11) = 30. CONTINUE DO 153 J=2,11 P(1,J) = 40. CONTINUE LINE RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘) FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1)

50 Modern Aerodynamic Methods 160 C 170 C

200 C C

C C

250 300 400

CONTINUE ITERATE COLUMN BY COLUMN WITHIN BOX DO 300 I=2,10 DEFINE MATRIX COEFS FOR INTERNAL POINTS DO 200 J=2,10 A(J) = 1. B(J) = -2.*(1.+RATIO2) C(J) = 1. W(J) = -RATIO2*(P(I-1,J)+P(I+1,J)) CONTINUE RESTATE UPPER/LOWER BOUNDARY CONDITIONS NOTE “99” DUMMY VALUES A(1) = 99. B(1) = 1. C(1) = 0. W(1) = P(I,1) A(11) = 0. B(11) = 1. C(11) = 99. W(11) = P(I,11) INVOKE TRIDIAGONAL MATRIX SOLVER CALL TRIDI(A,B,C,V,W,11) UPDATE AND STORE COLUMN SOLUTION DO 250 J=2,10 P(I,J) = V(J) CONTINUE CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.3a. Aquifer-alone, solved with “0” guess.

Computational Algorithms 10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 24.1 29.3 31.4 32.4 33.1 33.6 34.1 34.3 33.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 17.6 14.4 12.7 11.9 22.1 17.6 15.0 13.7 24.6 19.8 16.8 15.2 26.1 21.4 18.3 16.6 27.2 22.8 19.7 18.0 28.2 24.1 21.3 19.7 29.2 25.6 23.2 21.8 30.1 27.2 25.3 24.3 30.5 28.7 27.6 27.0 30.0 30.0 30.0 30.0

NSWEEP = 10 10.0 10.0 10.0 11.8 12.1 13.0 13.3 13.8 15.0 14.7 15.1 16.3 16.0 16.4 17.3 17.4 17.7 18.3 19.1 19.2 19.5 21.2 21.1 21.0 23.8 23.5 23.1 26.7 26.5 26.0 30.0 30.0 30.0

10.0 15.0 17.0 18.0 18.6 19.2 19.9 20.8 22.1 24.5 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.6 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.4 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 50 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 200 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

Figure 2.2.3b. Aquifer-alone, solved with “0” guess.

51

52 Modern Aerodynamic Methods C

C

C

100 C

150

151

152

153 C

C

154 155

157 160 C 170 C

200

LAPLACE EQUATION SOLVER, CASE_2. PROGRAM MAIN DIMENSION P(11,11), A(11), B(11), C(11), V(11), W(11) OPEN(UNIT=4,FILE=’CASE_2.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. RATIO2 = (DY/DX)**2 INITIALIZE P(I,J) TO SOMETHING ABSURD EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = I**2 + J**2 CONTINUE SET “10-20-30-40” BOUNDARY CONDITIONS DO 150 I=1,10 P(I,1) = 10. CONTINUE DO 151 J=1,10 P(11,J) = 20. CONTINUE DO 152 I=2,11 P(I,11) = 30. CONTINUE DO 153 J=2,11 P(1,J) = 40. CONTINUE LINE RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘) FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE ITERATE COLUMN BY COLUMN WITHIN BOX DO 300 I=2,10 DEFINE MATRIX COEFS FOR INTERNAL POINTS DO 200 J=2,10 A(J) = 1. B(J) = -2.*(1.+RATIO2) C(J) = 1. W(J) = -RATIO2*(P(I-1,J)+P(I+1,J)) CONTINUE

Computational Algorithms C C

C C

250 300 400

RESTATE UPPER/LOWER BOUNDARY CONDITIONS NOTE “99” DUMMY VALUES A(1) = 99. B(1) = 1. C(1) = 0. W(1) = P(I,1) A(11) = 0. B(11) = 1. C(11) = 99. W(11) = P(I,11) INVOKE TRIDIAGONAL MATRIX SOLVER CALL TRIDI(A,B,C,V,W,11) UPDATE AND STORE COLUMN SOLUTION DO 250 J=2,10 P(I,J) = V(J) CONTINUE CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END 2

2

Figure 2.2.4a. Aquifer-alone, with (absurd) P = i + j guess. i,j 10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 27.1 34.9 39.3 41.9 43.4 43.9 43.2 41.1 37.3 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 23.0 21.7 21.0 20.3 32.5 31.6 30.9 29.7 39.2 39.4 39.0 37.5 43.7 45.0 45.0 43.3 46.3 48.3 48.5 46.8 47.1 49.1 49.5 47.7 45.8 47.6 47.8 46.2 42.5 43.6 43.6 42.3 37.2 37.5 37.4 36.7 30.0 30.0 30.0 30.0

NSWEEP = 10 10.0 10.0 10.0 19.2 18.0 16.9 27.7 25.1 22.5 34.7 30.9 26.8 39.9 35.2 29.9 43.0 37.8 31.7 44.0 38.7 32.5 42.8 38.0 32.2 39.7 35.9 31.3 35.2 33.1 30.4 30.0 30.0 30.0

10.0 16.9 20.6 23.0 24.6 25.6 26.0 26.1 26.0 26.6 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.4 31.7 34.8 36.5 37.4 37.7 37.6 36.9 35.0 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.8 17.2 15.8 15.0 26.4 23.1 21.0 19.5 30.6 27.4 25.0 23.2 33.2 30.4 28.1 26.0 34.8 32.3 30.1 28.1 35.5 33.3 31.3 29.4 35.5 33.5 31.8 30.1 34.7 33.0 31.6 30.3 32.9 31.8 31.0 30.3 30.0 30.0 30.0 30.0

NSWEEP = 20 10.0 10.0 10.0 14.4 14.1 14.3 18.4 17.7 17.6 21.7 20.6 19.9 24.3 22.8 21.6 26.2 24.5 22.8 27.5 25.7 23.8 28.4 26.7 24.7 29.0 27.6 25.8 29.5 28.7 27.4 30.0 30.0 30.0

10.0 15.6 18.2 19.7 20.7 21.4 21.9 22.5 23.4 25.2 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

53

54 Modern Aerodynamic Methods 10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.4 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.4 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 50 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.6 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 150 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 200 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

2

2

Figure 2.2.4b. Aquifer-alone, with (absurd) P = i + j guess. i,j

Computational Algorithms

55

Convergence acceleration. The implication above is that an initial guess close to the final solution will converge more rapidly than one that is not. This can be used beneficially when performing a sequence of flow simulations where single (or multiple) parameter(s), such as well position, rate constraint, or pressure level, or reservoir heterogeneity size or shape, vary only incrementally from one run to the next. In the context of this book, the solution for a quickly-obtained thickness distribution at zero pitch might be stored and used to initialize slightly altered geometries at angle of attack. The results of each run can be used to intelligently initialize the next, with each using close physical information that accelerates convergence. Whereas direct methods will solve N problems using N calls of a (complicated) matrix solver, iterative methods applied in the foregoing sense solve subsequent problems much more rapidly and make minimal use of computer memory. In code development or project work, it is also conceivable to have libraries of close solutions stored on disk to initialize solutions. Such a philosophy should prove productive when large volumes of calculations must be performed. That our calculations converge to the same answer regardless of starting guess is more than fortuitous. This may surprise beginning students in numerical analysis, who are forever seeking (unstable) roots to nonlinear equations. Unlike the iterative root solvers used for such problems, where the initial closeness to different multiple roots will cause problems, the convergence of steady-state flow problems to unique solutions is assured for several reasons. For one, mathematical theory guarantees that solutions to Dirichlet and mixed flow problems – when proper boundary conditions are used – exist and are unique. And, as we will later show, the iterative process mimics the search for steady solutions to the transient heat equation. Any homemaker will explain how the equilibrium steady-state (room) temperature that a loaf of bread seeks is independent of its origins from the oven or the refrigerator! Here again, convergence to a unique solution is independent of the guess.

2.2.4 Wells and Internal Boundaries

Cases 1 and 2 above deal with uninteresting pressure distributions, corresponding to flowing reservoirs without wells. Here we consider uniform boundary pressures (say, 100 psi) specified at the edges of the computational box, plus the effect of constant pressure (say, 1 psi) prescribed at the center of the domain of flow. The next example illustrates doubly-connected well effects crudely.

56 Modern Aerodynamic Methods C

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150 151

152

153 C

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154 155

157 160 C 170 C

C

LAPLACE EQUATION SOLVER, CASE_3. PROGRAM MAIN DIMENSION P(11,11), A(11), B(11), C(11), V(11), W(11) OPEN(UNIT=4,FILE=’CASE_3.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. RATIO2 = (DY/DX)**2 ONE = 1. INITIALIZE P(I,J) TO ZERO EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “100” BOUNDARY CONDITION AT BOX EDGES DO 150 I=1,10 P(I,1) = 100. CONTINUE DO 151 J=1,10 P(11,J) = 100. CONTINUE DO 152 I=2,11 P(I,11) = 100. CONTINUE DO 153 J=2,11 P(1,J) = 100. CONTINUE LINE RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘) FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE ITERATE COLUMN BY COLUMN WITHIN BOX DO 300 I=2,10 DEFINE MATRIX COEFS FOR INTERNAL POINTS DO 200 J=2,10 A(J) = 1. B(J) = -2.*(1.+RATIO2) C(J) = 1. W(J) = -RATIO2*(P(I-1,J)+P(I+1,J)) SET INTERNAL BOUNDARY CONDITION

Computational Algorithms

200 C C

C C

250 300 400

57

IF(I.EQ.6.AND.J.EQ.6) A(J) = 0. IF(I.EQ.6.AND.J.EQ.6) B(J) = 1. IF(I.EQ.6.AND.J.EQ.6) C(J) = 0. IF(I.EQ.6.AND.J.EQ.6) W(J) = ONE CONTINUE RESTATE UPPER/LOWER BOUNDARY CONDITIONS NOTE “99” DUMMY VALUES A(1) = 99. B(1) = 1. C(1) = 0. W(1) = P(I,1) A(11) = 0. B(11) = 1. C(11) = 99. W(11) = P(I,11) INVOKE TRIDIAGONAL MATRIX SOLVER CALL TRIDI(A,B,C,V,W,11) UPDATE AND STORE COLUMN SOLUTION DO 250 J=2,10 P(I,J) = V(J) CONTINUE CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.5a. Centered well in aquifer-driven reservoir. Local corrections. Reference to the converged pressure in Figure

2.2.5b shows that results are symmetric about the x and y axes passing through the box center, where the well pressure is unity. The solutions are also symmetric with respect to 45 degree diagonals passing through this origin. This is evidence of error-free programming and code development. However, the results need not be completely correct. Note that pressure changes near the well, while rapid, are not quite logarithmic as required in an exact solution; the use of Cartesian meshes, in this sense, does not provide enough flow resolution near producers and injectors. Suffice it to say that there are ad hoc numerical procedures used to repair such solutions after-the-fact. Analogies are found in aerodynamics as well. In the “small disturbance, thin airfoil theory” developed in Section 2.2, errors near airfoil leading edge always appear because disturbances are large – after all, stagnation points are found nearby. Although our coding is correct, corrections to these errors are only found by applying extended “inner” flow models valid near the nose, emphasizing that no degree of grid refinement will help.

58 Modern Aerodynamic Methods P(I,J) SOLUTION FOR NSWEEP =

10

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

95.9

92.5

89.8

88.0

87.7

88.9

91.2

94.1

97.1

99.9

99.9

92.2

85.5

80.1

76.2

75.0

77.7

82.6

88.5

94.4

99.9

99.9

89.2

79.7

71.3

64.4

60.9

66.0

74.3

83.3

92.0

99.9

99.9

87.2

75.5

64.1

52.2

41.2

53.5

67.1

79.3

90.2

99.9

99.9

86.5

73.8

60.3

41.4

1.0

42.1

63.0

77.6

89.5

99.9

99.9

87.2

75.5

64.1

52.2

41.2

53.5

67.1

79.3

90.2

99.9

99.9

89.2

79.7

71.3

64.4

60.9

66.0

74.3

83.3

92.0

99.9

99.9

92.2

85.5

80.1

76.2

75.0

77.7

82.6

88.5

94.4

99.9

99.9

95.9

92.5

89.8

88.0

87.7

88.9

91.2

94.1

97.1

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

P(I,J) SOLUTION FOR NSWEEP =

20

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

97.9

95.8

93.8

92.3

91.6

92.3

93.9

95.9

97.9

99.9

99.9

95.7

91.5

87.3

83.7

82.1

83.8

87.5

91.7

95.9

99.9

99.9

93.7

87.2

80.2

73.4

69.1

73.5

80.5

87.5

93.9

99.9

99.9

92.2

83.6

73.3

60.5

47.8

60.7

73.6

83.9

92.4

99.9

99.9

91.5

81.9

69.0

47.8

1.0

47.9

69.3

82.2

91.7

99.9

99.9

92.2

83.6

73.3

60.5

47.8

60.7

73.6

83.9

92.4

99.9

99.9

93.7

87.2

80.2

73.4

69.1

73.5

80.5

87.5

93.9

99.9

99.9

95.7

91.5

87.3

83.7

82.1

83.8

87.5

91.7

95.9

99.9

99.9

97.9

95.8

93.8

92.3

91.6

92.3

93.9

95.9

97.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

P(I,J) SOLUTION FOR NSWEEP = 200 99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

98.0

96.0

94.0

92.5

91.9

92.5

94.0

96.0

98.0

99.9

99.9

96.0

91.8

87.7

84.1

82.4

84.1

87.7

91.8

96.0

99.9

99.9

94.0

87.7

80.8

73.9

69.6

73.9

80.8

87.7

94.0

99.9

99.9

92.5

84.1

73.9

61.0

48.2

61.0

73.9

84.1

92.5

99.9

99.9

91.9

82.4

69.6

48.2

1.0

48.2

69.6

82.4

91.9

99.9

99.9

92.5

84.1

73.9

61.0

48.2

61.0

73.9

84.1

92.5

99.9

99.9

94.0

87.7

80.8

73.9

69.6

73.9

80.8

87.7

94.0

99.9

99.9

96.0

91.8

87.7

84.1

82.4

84.1

87.7

91.8

96.0

99.9

99.9

98.0

96.0

94.0

92.5

91.9

92.5

94.0

96.0

98.0

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

Figure 2.2.5b. Centered well in aquifer-driven reservoir. Derivative discontinuities. An “oddity” with the solution in Figure 2.2.5b seems to be the behavior of derivatives at the well where P is given. To the left, pressure decreases; that is, P/ x is a nonzero negative number. And to the right, P/ x is a nonzero positive number with the same magnitude. The same behavior is seen above and below the well; P/ y takes on equal and opposite nonzero values. There is nothing wrong with these results. Since the velocities are proportional to the first derivatives of pressure, the sign change only indicates that fluid is flowing from opposite directions into the source point or well.

Computational Algorithms

59

We have seen this before: from Chapter 1, source points are always associated with opposing velocities and thus discontinuities in first derivatives. Derivative discontinuities are commonplace in aerodynamic theory. In flows past general airfoils, discontinuities in the velocity perpendicular to the chord are associated with thickness, while discontinuities in the tangential component are associated with the camber line. Another type of discontinuity found in aerodynamics is that for the potential itself. Again, never blindly differentiate across discontinuities because the derivative may not exist! The velocity potential, as we have seen in Chapter 1, is multivalued – thus, special caution must be taken in differentiating this function to obtain velocities. We only take this opportunity to emphasize that the region abutting the chord must be very carefully modeled and understood. Several interesting applications will be introduced and discussed in Section 2.3. In airfoil theory, the jump in normal velocity is related to the local surface slope – the larger the slope, the larger the jump. In Section 2.3, we will show how such flowfields with velocity antisymmetry are calculated which contain slope effects. In petroleum reservoir flow simulation, where normal velocity discontinuities are found along the fractures typical of hydraulic fracturing, the computations are much simpler. If viscous flow effects along the fracture are unimportant, the fracture flow can be modeled by using a distribution of constant pressure points. We can create such a flow by modifying the above Fortran; we change the single well logic and now allow unit pressures to extend horizontally for several gridblocks. The results are shown in Figures 2.2.6a,b (the 100 boundary pressures assumed in the source code, for the next several examples, were changed to 99.9 after the calculations, for typesetting and formatting reasons). Note how, in Figure 2.2.6b, the vertical derivatives P/ y at the fracture slit are equal and opposite; also observe that the fracture or thickness singularity indicated in Chapter 1 is captured poorly in the numerics. As in the earlier calculations, our solutions were stably and rapidly obtained, requiring only minor changes to add or delete wells, or to change wells to line fractures.

60 Modern Aerodynamic Methods C

C

C

100 C

150

151

152

153 C

C

154 155

157 160 C 170 C

LAPLACE EQUATION SOLVER, CASE_4. PROGRAM MAIN DIMENSION P(11,11), A(11), B(11), C(11), V(11), W(11) OPEN(UNIT=4,FILE=’CASE_4.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. RATIO2 = (DY/DX)**2 ONE = 1. INITIALIZE P(I,J) TO ZERO EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “100” BOUNDARY CONDITION AT BOX EDGES DO 150 I=1,10 P(I,1) = 100. CONTINUE DO 151 J=1,10 P(11,J) = 100. CONTINUE DO 152 I=2,11 P(I,11) = 100. CONTINUE DO 153 J=2,11 P(1,J) = 100. CONTINUE LINE RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘) FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE ITERATE COLUMN BY COLUMN WITHIN BOX DO 300 I=2,10 DEFINE MATRIX COEFS FOR INTERNAL POINTS DO 200 J=2,10 A(J) = 1. B(J) = -2.*(1.+RATIO2) C(J) = 1. W(J) = -RATIO2*(P(I-1,J)+P(I+1,J))

Computational Algorithms C

200 C C

C C

250 300 400

SET INTERNAL BOUNDARY CONDITION MODE = 0 IF(I.GE.4.AND.I.LE.8) MODE = 1 IF(MODE.EQ.1.AND.J.EQ.6) A(J) = 0. IF(MODE.EQ.1.AND.J.EQ.6) B(J) = 1. IF(MODE.EQ.1.AND.J.EQ.6) C(J) = 0. IF(MODE.EQ.1.AND.J.EQ.6) W(J) = ONE CONTINUE RESTATE UPPER/LOWER BOUNDARY CONDITIONS NOTE “99” DUMMY VALUES A(1) = 99. B(1) = 1. C(1) = 0. W(1) = P(I,1) A(11) = 0. B(11) = 1. C(11) = 99. W(11) = P(I,11) INVOKE TRIDIAGONAL MATRIX SOLVER CALL TRIDI(A,B,C,V,W,11) UPDATE AND STORE COLUMN SOLUTION DO 250 J=2,10 P(I,J) = V(J) CONTINUE CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.6a. Centered fracture, aquifer-driven reservoir.

61

62 Modern Aerodynamic Methods 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 94.7 89.4 84.2 79.5 76.7 79.5 84.2 89.4 94.7 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 90.0 86.4 84.3 83.7 79.8 72.2 67.8 66.7 69.2 56.2 49.5 47.9 57.9 35.5 27.8 26.0 48.4 1.0 1.0 1.0 57.9 35.5 27.8 26.0 69.2 56.2 49.5 47.9 79.8 72.2 67.8 66.7 90.0 86.4 84.3 83.7 99.9 99.9 99.9 99.9

NSWEEP = 10 99.9 99.9 99.9 84.8 87.3 91.0 68.6 73.6 81.5 50.4 57.7 70.9 28.3 36.4 59.2 1.0 1.0 49.2 28.3 36.4 59.2 50.4 57.7 70.9 68.6 73.6 81.5 84.8 87.3 91.0 99.9 99.9 99.9

99.9 95.4 90.6 85.5 80.6 77.6 80.6 85.5 90.6 95.4 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 95.6 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.6 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 91.6 88.2 86.1 85.4 82.4 75.2 70.8 69.4 71.9 59.3 52.6 50.6 60.0 37.5 29.7 27.8 49.7 1.0 1.0 1.0 60.0 37.5 29.7 27.8 71.9 59.3 52.6 50.6 82.4 75.2 70.8 69.4 91.6 88.2 86.1 85.4 99.9 99.9 99.9 99.9

NSWEEP = 20 99.9 99.9 99.9 86.1 88.2 91.6 70.8 75.2 82.4 52.6 59.3 71.9 29.7 37.5 60.1 1.0 1.0 49.8 29.7 37.5 60.1 52.6 59.3 71.9 70.8 75.2 82.4 86.1 88.2 91.6 99.9 99.9 99.9

99.9 95.6 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.6 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 91.6 88.2 86.1 85.4 82.4 75.2 70.9 69.4 72.0 59.3 52.6 50.6 60.1 37.5 29.7 27.8 49.8 1.0 1.0 1.0 60.1 37.5 29.7 27.8 72.0 59.3 52.6 50.6 82.4 75.2 70.9 69.4 91.6 88.2 86.1 85.4 99.9 99.9 99.9 99.9

NSWEEP = 150 99.9 99.9 99.9 86.1 88.2 91.6 70.9 75.2 82.4 52.6 59.3 72.0 29.7 37.5 60.1 1.0 1.0 49.8 29.7 37.5 60.1 52.6 59.3 72.0 70.9 75.2 82.4 86.1 88.2 91.6 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

Figure 2.2.6b. Centered fracture, aquifer-driven reservoir. 2.2.5 Point Relaxation Methods So far, we have discussed column and row implementations of line relaxation. These methods require matrix inversion, although inverting tridiagonal matrices is a relatively straightforward task. But this was not so when computing machines were not widely available. Historically, point relaxation methods requiring simple hand calculations only (and no matrix inversion) were the first ones developed. This solution is useful for several reasons: (1) it is easily programmed, almost trivially on a

Computational Algorithms

63

modern spreadsheet, (2) it is readily implemented in irregular domains where rows and columns of constant length are difficult to define, and (3) large-scale calculations may be divided among different machines in parallel processing. Point relaxation is best explained by assuming equal constant mesh widths x = y. Then, Equation 2.16 can be rewritten in the form P = (P i,j

i-1,j

+P

i+1,j

+P

i,j-1

+P

i,j+1

)/4

(2.17)

This states that center values equal the arithmetic average of their neighbors to the left, right, top and bottom, when the mesh widths are equal. This remarkable property holds everywhere in the flowfield; that is, it holds “in the small” locally. And from Figures 2.2.3b and 2.2.4b, we find that it holds “in the large” also, the center value of 25 being the arithmetic average of the four boundary values 10, 20, 30 and 40. This explains why elliptic operators are used to smooth numerical fields in image processing. In Figures 2.2.7a,b, we revisit the 10, 20, 30, 40, no well problem in Figure 2.2.3a. However, we solve it using a simple scheme, taking Equation 2.17 as the recursion formula, again assuming Pi,j = 0 as the initial guess. Similarly, we reconsider the fracture flow problem in Figures 2.2.6a,b and solve it with point relaxation. The results are shown in Figures 2.2.8a,b. In both cases, pressures are identical to earlier ones. For our last example, we treat the implementation of no-flow solid wall boundary conditions. We have chosen to rework Case 3 (see Figures 2.2.5a,b), and add no-flow conditions along the vertical line i = 1 as well as the horizontal line j = 1. Now, Darcy’s law guarantees zero flow in any direction provided two consecutive pressures along the tangent vector are identical (this is similar to solid wind tunnel wall boundary conditions for velocity potential). This condition is enforced along j = 1 by choosing B(1) = 1, C(1) = -1 and W(1) = 0. In other words, P(I,1) - P(I,2) = 0; P(I,1) and P(I,2) are solved simultaneously along with other columnar unknowns. Along i = 1, which falls outside the I = 2,10 range of the sweeping process, the simple update procedure P(1,J) = P(2,J) suffices. The required Fortran is shown in Figure 2.2.9a, while the corresponding results are shown in Figure 2.2.9b. Note how the top two rows and the left two columns, respectively, satisfy vanishing values of P/ y and P/ x.

64 Modern Aerodynamic Methods 2.2.6 Observations on Relaxation Methods

We summarize important observations and facts about relaxation methods. These comments are based on the author’s experience in developing aerodynamics and reservoir simulation models. Easy to program and maintain. By modifying the source code in Figure 2.2.3a to handle problems of increasing difficulty in a sequence of examples, we have shown how a finite difference model can be easily understood and extended to describe wells, fractures, aquifers and solid walls. Multiple wells and fractures, and general combinations of aquifer and no-flow boundary conditions, of course, are just as easily treated: the basic engine driving the models requires but twenty lines of Fortran. The same ease of use is found in aerodynamics. For example, once the basic code for a lifting camber line is developed, it is a simple matter to derive from it codes for wind tunnel applications with close walls, codes for ringwing analysis, and so on. Importantly, this powerful methodology requires little programming or numerical analysis experience. C

C

C

100 C

150

151

152

153 C

LAPLACE EQUATION SOLVER, CASE_5. PROGRAM MAIN DIMENSION P(11,11) OPEN(UNIT=4,FILE=’CASE_5.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. INITIALIZE P(I,J) TO ZERO EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “10-20-30-40” BOUNDARY CONDITION AT BOX EDGES DO 150 I=1,10 P(I,1) = 10. CONTINUE DO 151 J=1,10 P(11,J) = 20. CONTINUE DO 152 I=2,11 P(I,11) = 30. CONTINUE DO 153 J=2,11 P(1,J) = 40. CONTINUE POINT RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170

Computational Algorithms C

154 155

157 160 C 170

300 400

PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘) FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE ITERATE POINT BY POINT WITHIN BOX DO 300 I=2,10 DO 300 J=2,10 P(I,J) = (P(I-1,J) +P(I+1,J) +P(I,J-1) +P(I,J+1))/4. CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.7a. Aquifer alone, point relaxation. 10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 23.0 27.3 28.8 29.4 29.9 30.5 31.4 32.4 32.7 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 15.6 11.9 9.9 9.0 18.6 13.1 9.9 8.2 20.0 13.8 9.8 7.8 20.7 14.2 10.0 7.7 21.4 15.0 10.7 8.5 22.5 16.6 12.6 10.5 24.3 19.1 15.7 13.9 26.6 22.6 20.0 18.7 28.8 26.4 25.0 24.3 30.0 30.0 30.0 30.0

NSWEEP = 10 10.0 10.0 10.0 8.9 9.5 11.1 8.0 9.1 11.5 7.5 8.8 11.6 7.4 8.9 11.8 8.2 9.6 12.5 10.2 11.5 13.9 13.6 14.6 16.4 18.5 19.0 19.9 24.1 24.3 24.4 30.0 30.0 30.0

10.0 14.0 15.2 15.5 15.7 16.2 17.0 18.4 20.4 23.7 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 24.2 29.4 31.7 32.9 33.7 34.3 34.8 34.9 33.9 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 17.6 14.4 12.7 11.8 22.3 17.8 15.1 13.6 25.1 20.3 17.1 15.3 26.8 22.1 18.9 17.0 28.2 23.8 20.7 18.8 29.3 25.4 22.6 20.8 30.3 26.9 24.5 22.9 31.0 28.3 26.5 25.3 31.1 29.4 28.3 27.7 30.0 30.0 30.0 30.0

NSWEEP = 20 10.0 10.0 10.0 11.6 11.9 12.8 13.1 13.5 14.7 14.6 14.9 16.0 16.2 16.3 17.1 17.9 17.8 18.3 19.8 19.5 19.6 22.0 21.6 21.3 24.5 24.0 23.4 27.2 26.8 26.2 30.0 30.0 30.0

10.0 14.9 16.8 17.8 18.5 19.1 19.9 20.8 22.2 24.6 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

65

66 Modern Aerodynamic Methods 10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 150 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

10.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

10.0 25.0 30.9 33.7 35.1 36.0 36.4 36.5 36.1 34.6 30.0

P(I,J) SOLUTION FOR 10.0 10.0 10.0 10.0 19.1 16.3 14.9 14.0 25.0 21.3 19.1 17.7 28.7 25.0 22.4 20.7 30.9 27.6 25.0 23.1 32.3 29.3 26.9 25.0 33.2 30.5 28.2 26.5 33.5 31.1 29.2 27.6 33.2 31.2 29.7 28.5 32.2 30.8 30.0 29.3 30.0 30.0 30.0 30.0

NSWEEP = 200 10.0 10.0 10.0 13.6 13.5 13.9 16.8 16.5 16.8 19.5 18.9 18.8 21.8 20.8 20.3 23.5 22.4 21.5 25.0 23.7 22.5 26.3 25.0 23.6 27.5 26.4 25.0 28.7 28.0 27.0 30.0 30.0 30.0

10.0 15.4 17.8 19.2 20.0 20.7 21.3 22.0 23.0 25.0 30.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 30.0

Figure 2.2.7b. Aquifer alone, point relaxation. C C

100 C 150 151 152 153 C

155

LAPLACE EQUATION SOLVER, CASE_6. DIMENSION P(11,11) OPEN(UNIT=4,FILE=’CASE_6.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS AND INITIALIZE P(I,J) TO ZERO DX = 1. DY = 1. ONE = 1. DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “100” BOUNDARY CONDITION AT BOX EDGES DO 150 I=1,10 P(I,1) = 100. CONTINUE DO 151 J=1,10 P(11,J) = 100. CONTINUE DO 152 I=2,11 P(I,11) = 100. CONTINUE DO 153 J=2,11 P(1,J) = 100. CONTINUE POINT RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3)

Computational Algorithms

157 160 170

300 400

DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE DO 300 I=2,10 DO 300 J=2,10 MODE = 0 IF(I.GE.4.AND.I.LE.8) MODE = 1 IF(MODE.EQ.1.AND.J.EQ.6) MODE = 2 P(I,J) = (P(I-1,J) + P(I+1,J) + P(I,J-1) + P(I,J+1))/4. IF(MODE.EQ.2) P(I,J) = ONE CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.8a. Fracture flow, point relaxation.

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 92.1 85.5 80.1 75.6 72.8 74.9 79.5 85.7 92.8 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 85.6 80.9 78.1 77.1 73.4 64.3 58.9 57.2 62.7 48.9 41.6 39.4 52.6 31.1 23.4 21.4 43.8 1.0 1.0 1.0 50.9 28.3 20.0 17.8 61.3 46.6 38.7 36.4 73.5 64.0 58.5 56.9 86.7 82.1 79.3 78.6 99.9 99.9 99.9 99.9

NSWEEP = 10 99.9 99.9 99.9 78.2 81.4 86.6 59.1 65.0 74.9 41.8 49.6 64.1 23.5 31.4 53.5 1.0 1.0 44.4 20.2 29.0 52.5 39.3 48.0 63.7 59.3 65.6 75.8 79.9 83.1 88.1 99.9 99.9 99.9

99.9 93.2 87.1 81.8 77.1 74.2 76.8 81.8 87.7 94.0 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 95.2 90.4 85.3 80.4 77.4 80.3 85.3 90.4 95.3 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 90.9 87.4 85.2 84.6 81.4 73.9 69.5 68.2 70.9 58.1 51.4 49.5 59.2 36.8 29.0 27.1 49.1 1.0 1.0 1.0 59.0 36.6 28.8 26.9 70.8 58.0 51.3 49.3 81.5 74.1 69.7 68.3 91.1 87.6 85.5 84.8 99.9 99.9 99.9 99.9

NSWEEP = 20 99.9 99.9 99.9 85.4 87.6 91.2 69.7 74.3 81.8 51.6 58.5 71.3 29.2 37.0 59.5 1.0 1.0 49.4 28.9 36.9 59.5 51.5 58.4 71.3 69.9 74.5 81.9 85.6 87.8 91.3 99.9 99.9 99.9

99.9 95.4 90.7 85.7 80.7 77.7 80.7 85.7 90.8 95.5 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

67

68 Modern Aerodynamic Methods 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 91.6 88.2 86.1 85.4 82.4 75.2 70.9 69.4 72.0 59.3 52.6 50.6 60.1 37.5 29.7 27.8 49.8 1.0 1.0 1.0 60.1 37.5 29.7 27.8 72.0 59.3 52.6 50.6 82.4 75.2 70.9 69.4 91.6 88.2 86.1 85.4 99.9 99.9 99.9 99.9

NSWEEP = 150 99.9 99.9 99.9 86.1 88.2 91.6 70.9 75.2 82.4 52.6 59.3 72.0 29.7 37.5 60.1 1.0 1.0 49.8 29.7 37.5 60.1 52.6 59.3 72.0 70.9 75.2 82.4 86.1 88.2 91.6 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

P(I,J) SOLUTION FOR 99.9 99.9 99.9 99.9 91.6 88.2 86.1 85.4 82.4 75.2 70.9 69.4 72.0 59.3 52.6 50.6 60.1 37.5 29.7 27.8 49.8 1.0 1.0 1.0 60.1 37.5 29.7 27.8 72.0 59.3 52.6 50.6 82.4 75.2 70.9 69.4 91.6 88.2 86.1 85.4 99.9 99.9 99.9 99.9

NSWEEP = 200 99.9 99.9 99.9 86.1 88.2 91.6 70.9 75.2 82.4 52.6 59.3 72.0 29.7 37.5 60.1 1.0 1.0 49.8 29.7 37.5 60.1 52.6 59.3 72.0 70.9 75.2 82.4 86.1 88.2 91.6 99.9 99.9 99.9

99.9 95.7 91.0 86.0 81.0 77.9 81.0 86.0 91.0 95.7 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

Figure 2.2.8b. Fracture flow, point relaxation. C

C

C

100 C 151 152 C C

154

LAPLACE EQUATION SOLVER, CASE_7. PROGRAM MAIN DIMENSION P(11,11), A(11), B(11), C(11), V(11), W(11) OPEN(UNIT=4,FILE=’CASE_7.DAT’,STATUS=’NEW’) DEFINE GRID PARAMETERS DX = 1. DY = 1. RATIO2 = (DY/DX)**2 ONE = 1. INITIALIZE P(I,J) TO ZERO EVERYWHERE DO 100 I=1,11 DO 100 J=1,11 P(I,J) = 0. CONTINUE SET “100” BOUNDARY CONDITION AT BOX EDGES DO 151 J=1,10 P(11,J) = 100. CONTINUE DO 152 I=2,11 P(I,11) = 100. CONTINUE LINE RELAXATION BEGINS DO 400 NSWEEP=1,200 IF(MOD(NSWEEP,10).NE.0) GO TO 170 PRINT OUT “X-Y” RESULTS WRITE(*,154) WRITE(4,154) WRITE(*,155) NSWEEP WRITE(4,155) NSWEEP FORMAT(‘ ‘)

Computational Algorithms 155

157 160 C 170 C

C

200 C C

C C 250 300 C 350 400

FORMAT(‘ P(I,J) SOLUTION FOR NSWEEP = ‘,I3) DO 160 J=1,11 WRITE(*,157) (P(I,J),I=1,11) WRITE(4,157) (P(I,J),I=1,11) FORMAT(1X,11F6.1) CONTINUE ITERATE COLUMN BY COLUMN WITHIN BOX DO 300 I=2,10 DEFINE MATRIX COEFS FOR INTERNAL POINTS DO 200 J=2,10 A(J) = 1. B(J) = -2.*(1.+RATIO2) C(J) = 1. W(J) = -RATIO2*(P(I-1,J)+P(I+1,J)) SET INTERNAL BOUNDARY CONDITION IF(I.EQ.6.AND.J.EQ.6) A(J) = 0. IF(I.EQ.6.AND.J.EQ.6) B(J) = 1. IF(I.EQ.6.AND.J.EQ.6) C(J) = 0. IF(I.EQ.6.AND.J.EQ.6) W(J) = ONE CONTINUE RESTATE UPPER/LOWER BOUNDARY CONDITIONS NOTE “99” DUMMY VALUES A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. A(11) = 0. B(11) = 1. C(11) = 99. W(11) = P(I,11) INVOKE TRIDIAGONAL MATRIX SOLVER CALL TRIDI(A,B,C,V,W,11) UPDATE AND STORE COLUMN SOLUTION DO 250 J=1,11 P(I,J) = V(J) CONTINUE CONTINUE SET NO-FLOW CONDITION DO 350 J=1,11 P(1,J) = P(2,J) CONTINUE CONTINUE CLOSE(4,STATUS=’KEEP’) STOP END

Figure 2.2.9a. Implementing no-flow boundary conditions. 2.5 2.5 3.5 5.6 9.5 15.8 25.4 38.9 56.4 77.2 99.9

2.5 2.5 3.5 5.6 9.5 15.8 25.4 38.9 56.4 77.2 99.9

P(I,J) SOLUTION FOR 3.9 6.7 11.7 20.0 3.9 6.7 11.7 20.0 4.9 7.4 12.1 19.9 7.1 9.2 12.7 19.1 11.0 12.3 13.4 15.5 17.5 18.1 15.5 1.0 27.6 28.9 29.4 30.2 41.3 43.2 45.4 49.1 58.4 60.3 62.5 65.9 78.4 79.5 80.8 82.7 99.9 99.9 99.9 99.9

NSWEEP = 10 31.9 47.0 64.1 31.9 47.0 64.1 31.9 47.2 64.4 31.7 47.6 64.9 31.1 48.5 66.1 31.2 51.2 68.5 43.8 58.5 73.0 57.5 67.8 78.8 71.3 78.1 85.5 85.5 88.9 92.6 99.9 99.9 99.9

82.1 82.1 82.3 82.6 83.3 84.7 86.8 89.6 92.9 96.4 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

69

70 Modern Aerodynamic Methods 22.1 22.1 23.2 25.7 29.9 36.6 46.0 57.7 71.0 85.3 99.9

22.1 22.1 23.2 25.7 29.9 36.6 46.0 57.7 71.0 85.3 99.9

P(I,J) SOLUTION FOR 24.9 28.7 33.8 40.9 24.9 28.7 33.8 40.9 25.7 28.8 33.2 39.6 27.5 29.3 31.6 36.1 30.9 30.6 28.8 27.0 36.8 34.6 26.6 1.0 46.3 45.0 41.9 38.5 58.2 58.1 57.8 59.0 71.5 71.9 72.5 74.2 85.6 85.9 86.4 87.5 99.9 99.9 99.9 99.9

NSWEEP = 20 50.4 61.7 74.1 50.4 61.7 74.1 49.3 61.1 73.8 46.9 60.0 73.5 42.8 58.9 73.6 38.9 59.7 74.9 52.3 66.3 78.5 66.0 74.8 83.5 78.2 83.4 89.0 89.3 91.8 94.5 99.9 99.9 99.9

87.0 87.0 86.9 86.9 87.1 87.9 89.6 91.9 94.5 97.3 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

52.6 52.6 52.6 53.0 54.3 57.6 63.3 71.2 80.2 90.0 99.9

52.6 52.6 52.6 53.0 54.3 57.6 63.3 71.2 80.2 90.0 99.9

P(I,J) SOLUTION FOR 52.6 53.0 54.3 57.6 52.6 53.0 54.3 57.6 52.3 52.0 52.4 55.1 52.0 50.2 48.4 48.9 52.4 48.4 41.9 35.4 55.1 48.9 35.4 1.0 61.3 56.9 49.9 42.7 69.9 67.5 64.6 63.7 79.6 78.5 77.5 77.9 89.7 89.3 89.1 89.4 99.9 99.9 99.9 99.9

NSWEEP = 150 63.3 71.2 80.2 63.3 71.2 80.2 61.3 69.9 79.6 56.9 67.5 78.5 49.9 64.6 77.5 42.7 63.7 77.9 56.1 69.5 80.9 69.5 77.4 85.4 80.9 85.4 90.3 90.8 92.8 95.2 99.9 99.9 99.9

90.0 90.0 89.7 89.3 89.1 89.4 90.8 92.8 95.2 97.6 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

52.6 52.6 52.6 53.0 54.3 57.6 63.3 71.2 80.2 90.0 99.9

52.6 52.6 52.6 53.0 54.3 57.6 63.3 71.2 80.2 90.0 99.9

P(I,J) SOLUTION FOR 52.6 53.0 54.3 57.6 52.6 53.0 54.3 57.6 52.3 52.0 52.4 55.1 52.0 50.2 48.4 48.9 52.4 48.4 41.9 35.4 55.1 48.9 35.4 1.0 61.3 56.9 49.9 42.7 69.9 67.5 64.6 63.7 79.6 78.5 77.5 77.9 89.7 89.3 89.1 89.4 99.9 99.9 99.9 99.9

NSWEEP = 200 63.3 71.2 80.2 63.3 71.2 80.2 61.3 69.9 79.6 56.9 67.5 78.5 49.9 64.6 77.5 42.7 63.7 77.9 56.1 69.5 80.9 69.5 77.4 85.4 80.9 85.4 90.3 90.8 92.8 95.2 99.9 99.9 99.9

90.0 90.0 89.7 89.3 89.1 89.4 90.8 92.8 95.2 97.6 99.9

99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9

Figure 2.2.9b. Implementing no-flow boundary conditions. Minimal computing resources. A rectangular grid with IMAX and JMAX meshes in x and y will have IMAX JMAX unknowns. An unoptimized direct matrix solver that does not account for sparseness and bandedness will require numerous inversion computations. The worst case is Gaussian elimination, requiring (IMAX JMAX)3 multiplies and divides. The problem is compounded in three dimensions. In our scheme, only a single tridiagonal matrix solver is needed; inverting a JMAX operations, although this is JMAX line solution requires 3

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repeated NSWEEP IMAX times. This still represents a significant improvement over direct methods.Good numerical stability. Our programs are extremely stable numerically; that is, they do not lead to divergent pressures often. The procedures are conditionally stable on a linear von Neumann stability basis. This is so because the coefficient matrixes are diagonally dominant, becoming even more so when 3D problems are solved in a columnar fashion as in our examples. Often, a planar problem that does not converge on a 2D basis can be successfully and quickly solved as the limit of the 3D problem. An unstable 2D problem can be artificially embedded in a suitable 3D problem to facilitiate convergence. As we will see in a three-dimensional wing calculation later in Section 2.3, the matrix line “+1 -4 +1)” is replaced by an even more stable “+1 -6 +1.” Fast convergence. Relaxation schemes are known to converge rapidly, at least, initially; then, the rate of convergence slows somewhat, although computing times are still tolerable. Various methods are used to accelerate convergence – for example, over-relaxation (Jameson, 1975), Shanks extrapolation (van Dyke, 1964), or multigrid methods (Wesseling, 1992) which use alternating sequences of fine and coarse meshes to host the relaxation. Perhaps the most important advantage of relaxation methods is the ability to initialize the solution to an approximate one that is already available, ideally a solution with nearly the same heterogeneities or well configuration. This is important to the study of reservoir description. Suppose a number of geological realizations are created, for example, using a geostatistical model, with each successive model being slightly different from the preceding. Then, each pressure solution should require only minimal incremental effort, when compared to a direct solution technique that assumes nothing at the outset. Similarly, the flow past one airfoil can be initialized to the solution available for another. Thus, the incremental work is only modest. On this basis, relaxation methods exceed direct solvers in speed. As we have seen, the method always seems to converge to the same answer, regardless of the initial guess. The proper initialization, of course, reduces computation times significantly. This advantage is important in the design of software that offers instantaneous user response while requiring minimal hardware resources. Why relaxation methods converge. We conclude this section by offering some quantitative insight showing why convergence to a unique solution, regardless of the initial guess, is expected, assuming of course a

72 Modern Aerodynamic Methods stable solution. Let us multiply Equation 2.17 by 4 and rewrite it with superscripts n and n-1 to describe the recursion relation used in the iteration below. n n-1 n-1 n-1 n-1 4P =P + P +P + P i,j i-1,j i+1,j i,j-1 i,j+1 n-1

We subtract 4 P n-1

P

n-1

i-1,j

-2P

i,j

n-1

i,j

+P

from each side of Equation 2.18 to obtain n-1

i+1,j

(2.18)

+ P

i,j-1

n-1

-2P

n-1

i,j

+ P

n

i,j+1

= 4 (P

i,j

n-1

- P

i,j

)

(2.19)

If we now divide Equation 2.19 throughout by ( x)2, we have n-1

(P

n-1

i-1,j

=

-2 P n

(P

i,j

i,j n-1

-P

n-1

+P i,j

i+1,j

n-1

)/( x) 2 + (P

)/ t

i,j-1

n-1

-2P

i,j

n-1

+ P

)/( x) 2

i,j+1

(2.20)

where t = ( x)2/4. We recognize Equation 2.20 as the explicitly differenced form of the dimensionless heat equation 2

P/ x2 + 2P/ y2 = P/ t

(2.21)

which governs heat propagation in solids when P(x,y,t) is the transient temperature. As is well known (Carslaw and Jaeger, 1959), the final steady-state solution (satisfying 2P/ x2 + 2P/ y2 = 0) is independent of initial conditions. Thus, it is not surprising that our solutions for steady-state pressure can be obtained independently of the initial guess. This comment applies to Equation 2.14, for a liquid, assuming pressure or flow rate boundary conditions. The usual analogy comparing relaxation with polynomial root solvers is not strictly correct, since elliptic problems, at least the ones considered here, have unique solutions. In several commercial publications, the claim is made that modern direct matrix solvers help pressure fields converge much faster than older relaxation approaches. This may be true in blind comparisons where nothing is known about the solution; but as we have seen, iterative models can be quite flexible when used cleverly. When direct solvers are used, the selection of proper matrix conditioning parameters is crucial, which requires some knowledge of the structure of the coefficient matrix. This often takes longer than the pressure solution itself. The resulting parameters, in fact, may depend on the physical characteristics of the oilfield, and will vary from problem to problem and change as oil and water saturations evolve with time. But relaxation methods, very often dummy proof, also allow

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users to initialize their solutions with analytical solutions, approximate solutions, or those available from prior simulations obtained under slightly different assumptions. Over-relaxation. Researchers have made great strides in accelerating the convergence of relaxation methods. Note from Equation 2.20 that the point relaxation scheme of Equation 2.17 is associated with a hard-coded value of heat conductivity, namely unity, in Equation 2.21. There is nothing sacred about this value; a higher conductivity will decrease the convergence time required to achieve steady-state results, while a lower one will increase it. One way to increase conductivity is to over-relax. Previously, we updated the Fortran solution using P(I,J) = V(J), where V(J) is the latest solution obtained from the columnar matrix inversion. Instead, let us update the pressure field using P(I,J) = RELAX*V(J) + (1-RELAX)*P(I,J). The choice RELAX = 1 reduces to doing nothing. However, convergence can be accelerated by overrelaxing with RELAX > 1. At other times when numerical stability is a problem, under-relaxing with RELAX < 1 may stabilize the calculations. Other authors “embed” their Laplace operators within unsteady systems that are more rapidly convergent than that of Equation 2.21. After all, the transient phases of iterative processes are unimportant if only steady results are desired; any fast artificial time variable will do. For a discussion on modern relaxation methods, the reader is referred to the paper of Jameson (1975). Line and point relaxation. Line relaxation is used for several reasons. First, the algorithm is simple to construct and maintain. Second, the tridiagonal solver requires only 3N multiplies and divides to invert an N N system. Of course, it is called dozens of times until convergence; still, the cumulative effort needed to solve a problem is small by comparison to, say, direct solutions via Gaussian elimination. If sufficiently close solutions are available for initialization, large decreases in convergence time can be achieved. Importantly, line relaxation handles two-point boundary conditions easily. Pressure data from upper and lower boundaries are communicated instantly along columns, and left and right boundary conditions quickly propagate along rows. By contrast, point relaxation methods are sluggish; they require longer computation times to converge. However, they are easily adaptable to irregular geometries, where lines having constant program dimension or vector length are difficult to define. (Curvilinear grid methods are an exception – these are explained in a simple manner in the author’s 2012

74 Modern Aerodynamic Methods book Managed Pressure Drilling.) If irregular geometries must be simulated on rectangular meshes, point relaxation is recommended because it is easily programmed, with the logic in Equation 2.17 performed only for points inside the flow domain. This simplifies development since constant mesh number lines need not be defined. Finally, there is the issue of vectorization, also referred to as scalar vs. parallel computing. Serial computers execute instructions sequentially, in specific order; parallel machines execute multiple instructions simultaneously. Often, different flow domains are apportioned to different machines, and message passing interfaces must be designed so that these domains communicate with each other in an optimal way that minimizes computation time. Point relaxation gave way to line methods when serial computers were predominant because they were slower in converging. However, they are now used on vector machines because many points can be iterated upon in parallel. On parallel machines, it is argued that the implicit schemes associated with line methods require step-by-step matrix inversion, sequential operations that do not take advantage of computer architecture. On the other hand, researchers have vectorized line methods so that large bundles of lines are solved simultaneously. Whether the reader prefers direct or iterative methods, he is cautioned against general comments, and quick and simplifying recommendations. In either approach, the issues are not as straightforward as they seem, and there is always room for ingenuity.

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2.3 Examples – Analysis, Direct or Forward Applications

In this write-up, we apply the finite difference ideas introduced in Section 2.2 to “analysis,” “direct” or “forward” applications in aerodynamics. For example, we will show how Neumann conditions are implemented to model tangency conditions, and how jumps or discontinuities in potential are coded to invoke Kutta’s trailing edge condition to model lift. Importantly, we develop the methodologies engineers need to write software and validate computational results against physical reality. Understand that our philosophies apply to other numerical methods as well, e.g., finite element, finite volume and so on. Ours is neither a comprehensive nor completely rigorous course in numerical methods. We aim to rapidly provide students with the skills needed to solve important problems, fundamental analysis formulations in this section and inverse problems in the next, so that the advanced models in Chapters 3 and 4 are understood meaningfully – and coded, used or extended as desired. As such, our codes are “bare bones,” focusing on readability and understanding. Our objective is not final “production code” for industrial applications or general engineering use. Because computational methods are judged by how well they model the physics, printing detailed numerical data for microscopic examination is a task that we strongly emphasize. Are vanishing fluxes enforced at boundaries? Do smooth results turn oscillatory and then unstable? Are expected symmetries and antisymmetries evident? Why are certain results unacceptable and why are others good? To answer these questions, we print entire spatial arrays – to preserve the “mathematical feel” behind a physical property in the entire flow domain, we unfortunately need to keep the numbers of grid blocks in both horizontal and vertical directions small, so that all critical numbers fit on a page just 4½ inches wide. To keep code readable and implementation easily understood, equations are likewise “bare bones.” Our finite difference models typically assume constant meshes, complicated interpolations are omitted, and simple boundary conditions are used to highlight, for example, computed physical differences between thickness and camber – the connections between the numerics here and the analytics in Chapter 1 are important to developing the mindset of modern engineers capable of developing numerical tools and using them wisely with care. Specific code improvements were previously suggested as exercises to the interested reader.

76 Modern Aerodynamic Methods 2.3.1

Example 1 – Thickness Solution, Centered Slit in Box Software reference, CODE-PHI-1.FOR.

2.3.1.1

Problem Setup

In this section, we explain why we formulate the numerical problem the way we do, and describe both good and bad attributes of the approach taken. We follow the column relaxation method introduced in Section 2.2 with normal derivatives simply implemented, additionally noting that row and point iterative methods have also been used successfully. We begin by observing that xx + yy = 0 can be represented in finite differences by ( i-1,j – 2 i,j + i+1,j)/ x2 + ( i,j-1 –2 i,j + i,j+1)/ y2 = 0. For simplicity, we set x = y, so that i-1,j – 4 i,j + i+1,j + i,j-1 + i,j+1 = 0. This can be rewritten in two ways, i,j-1 – 4 i,j + i,j+1 = – { i-1,j + i+1,j} and i,j = ¼ ( i-1,j + i+1,j + i,j-1 + i,j+1). In Section 2.2, we showed how these formulas can be used in column and point relaxation methods. We will use the former below and show how the latter formula can be used in an important “spot checking” exercise in Section 2.3.4. i=1,2

imax 1, imax j=jmax j=jmax 1

ABOVE j=2

FLOW

j=1

Airfoil y=0

j=jw

BELOW

j=2 j=1

j=2 j=1

Figure 2.3.1a. Computational grid setup and definitions.

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The gray area in Figure 2.3.1a is the “computational box,” square in this case. Fluid flows from left to right, over a symmetric “thickness only” airfoil that is unpitched – no lift is generated. Boundary condtions are, consistently with thin airfoil theory, applied on the centered black slit. We choose an odd number of mesh lines horizontally and vertically, so that upper-lower and left-right symmetries are easily checked; this allows rapid detection of programming errors. Note that, for the flow under consideration, it would be simpler to consider a half-space, say just the upper (or lower) flow domain. In fact, this is addressed in Section 2.3.2, however, a “full space” code is developed here because it will be needed in the lifting problem considered later in Section 2.3.4. Laplace’s equation is solved by repeatedly “sweeping” the flowfield with columnar solutions, going from left to right. We consider four types of columns, the tall ones at the left and right, and the shorter ones at the bottom and top of the airfoil. Consider that at the far left and recall the equation i,j-1 – 4 i,j + i,j+1 = – { i-1,j + i+1,j}. Identification with the matrix coefficients in Subroutine TRIDI shows that A(J) = 1, B(J) = – 4, C(J) = 1 and W(J) = – (PHI(I-1,J) + PHI(I+1,J)). This template difference equation is written for each of the indexes J = 2 to JMAX-1, for a total of JMAX-2 equations. Two more are needed to complete the system and involve farfield conditions at top and bottom. The solutions for potential are initially solved along the second column at I = 2, and then solutions are applied to the next column, and so on, until just one grid line upstream of the leading edge. We next turn to the region beneath the airfoil and examine “half columns” starting at the leading edge location I = ILE. The template equation is used in a similar manner. Now, however, the two boundary conditions would involve bottom farfield decay and nearfield tangency conditions satisfying y(x,0) = U F’(x). Solutions along these shorter columns are obtained until the last column is reached at I = ITE. The flowfield above is similarly treated. Finally, full columns (similar to those at the far left) are treated, marching from the trailing edge to JMAX-1 the far right. This completes one “sweep” of the box, at which point, conditions at the outer boundaries are updated with farfield boundary conditions – again, repeated sweeping is required until convergence is achieved. Note that we have treated right and left sides identically, as we have upper and lower halves. Since the slit is centered in the box, perfect symmetries are expected if the code is programmed properly. On convergence, surface pressure coefficients are calculated.

78 Modern Aerodynamic Methods 2.3.1.2 C C C C

Fortran Source Code

CODE-PHI-1.FOR Potential equation solver, model symmetric airfoil on centered slit, zero angle of attack, always nonlifting DIMENSION DIMENSION DIMENSION DIMENSION

C C

PHI(19,19),A(19),B(19),C(19),V(19),W(19) ATOP(9),BTOP(9),CTOP(9),VTOP(9),WTOP(9) ABOT(9),BBOT(9),CBOT(9),VBOT(9),WBOT(9) SLOPE(19),POSITION(19),CPTOP(19),CPBOT(19)

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JW = 10 ITMAX = 10000 C Research code for testing, hardcode special parameters. C DX and DY are grid lengths, inches; do not appear in PDE C since DX = DY assumed, but appear in BC and CP definitions. C UINF is speed at infinity, say in/sec. Thus POT is in^2/sec. C No other output has units, only PHI and dimensionless pressure C coefficients are plotted against indexes. DX = 1. DY = 1. UINF = 1. C C GENERIC SLOPE FUNCTION C Values at I = 1-5 AND 15-19 are dummies C I = 6 to 14 are biconvex symmetric in shape C Upper surface has SLOPE(I), lower is -SLOPE C Note symmetry with respect to center I = 10 SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. SLOPE( 6) = +0.3 SLOPE( 7) = +0.2 SLOPE( 8) = +0.15 SLOPE( 9) = 0. SLOPE(10) = 0. SLOPE(11) = 0. SLOPE(12) = -0.15 SLOPE(13) = -0.2 SLOPE(14) = -0.3 SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0. C C CONVENIENT CONSTANTS

Computational Algorithms JMAX = 2*JW-1 JMAXM1 = JMAX-1 JWM1 = JW-1 JWM2 = JW-2 JWP1 = JW+1 ILEM1 = ILE-1 ITEP1 = ITE+1 IMAXM1 = IMAX-1 C C

100 C C C C

110

120 C C C C C

INITIALIZE PHI TO ZERO, OTHER FUNCTION, OLD SOLUTION DO 100 I=1,IMAX DO 100 J=1,JMAX PHI(I,J) = 0. CONTINUE DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., if Grad PHI = 0 is imposed DO 110 I=1,IMAX PHI(I,1) = 0. PHI(I,JMAX) = 0. CONTINUE DO 120 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

C C

130

140 150 C

ITER=1,ITMAX

UPSTREAM OF LEADING EDGE DO 150 I=2,ILEM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PHI(I,1) A(JMAX) = 0. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = PHI(I,JMAX) CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE

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80 Modern Aerodynamic Methods C C C

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BELOW AIRFOIL SECTION Half columns vary from J = 1 to 9, keep JW line hidden J represents actual physical location DO 180 I=ILE,ITE DO 160 J=2,JWM2 ABOT(J) = +1. BBOT(J) = -4. CBOT(J) = +1. WBOT(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE ABOT(1) = 99. BBOT(1) = 1. CBOT(1) = 0. WBOT(1) = PHI(I,1) ABOT(JW-1) = 1. BBOT(JW-1) = -1. CBOT(JW-1) = 99. WBOT(JW-1) = +DY*UINF*SLOPE(I) CALL TRIDI(ABOT,BBOT,CBOT,VBOT,WBOT,JWM1) DO 170 J=1,JWM1 PHI(I,J) = VBOT(J) CONTINUE CONTINUE ABOVE AIRFOIL SECTION Half columns vary from J = 1 to 9, does not include JW line DO 220 I=ILE,ITE Here, J is an indexing parameter, not a physical location Array indexes vary from J=1 to JW-1 DO 190 J=2,JWM2 ATOP(J) = 1. BTOP(J) = -4. CTOP(J) = 1. WTOP(J) = -PHI(I-1,J+JW) -PHI(I+1,J+JW) CONTINUE ATOP(1) = 99. BTOP(1) = 1. CTOP(1) = -1. WTOP(1) = -DY*UINF*SLOPE(I) ATOP(JW-1) = 0. BTOP(JW-1) = 1. CTOP(JW-1) = 99. WTOP(JW-1) = PHI(I,JMAX) CALL TRIDI(ATOP,BTOP,CTOP,VTOP,WTOP,JWM1) DO 200 J=1,JWM1 PHI(I,JW+J) = VTOP(J) CONTINUE CONTINUE Note, no lifting "potential jump logic" used along z = 0 wake for thickness sections without lift (see lifting codes for own programming structure). Keep coding perfectly symmetric for thickness problem. DOWNSTREAM OF TRAILING EDGE

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Code identical to upstream of leading edge DO 250 I=ITEP1,IMAXM1 DO 230 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PHI(I,1) A(JMAX) = 0. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = PHI(I,JMAX) CALL TRIDI(A,B,C,V,W,JMAX) DO 240 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD POTENTIAL Zero potential gradient (loops 260, 270) or zero potential (loops 274, 276) yield slightly different potential fields and pressure coefficients. Both options shown below, hardcode your selection. Zero POT implemented below. GO TO 272 DO 260 I=1,IMAX PHI(I,1) = PHI(I,2) PHI(I,JMAX) = PHI(I,JMAXM1) CONTINUE DO 270 J=1,JMAX PHI(1,J) = PHI(2,J) PHI(IMAX,J) = PHI(IMAXM1,J) CONTINUE GO TO 300 DO 274 I=1,IMAX PHI(I,1) = 0. PHI(I,JMAX) = 0. CONTINUE DO 276 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='PHI.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX

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82 Modern Aerodynamic Methods 320 340

WRITE(7,320) (PHI(I,J),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Velocity Potential') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (PHI(I,J),I=1,IMAX) 380 FORMAT(1X,19F5.1) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write pressure coefficients CPTOP for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 CPTOP(I) = 0. CPBOT(I) = 0. 392 CONTINUE C Replace entries with calculated values at J = JW+1 and JW-1 DO 394 I=2,18 C Use central differencing over two DX boxes PHIX = (PHI(I+1,JWP1) - PHI(I-1,JWP1))/(2.*DX) C Small disturbance pressure coefficient CPTOP(I) = -2.*PHIX/UINF 394 CONTINUE DO 395 I=2,18 PHIX = (PHI(I+1,JWM1) - PHI(I-1,JWM1))/(2.*DX) CPBOT(I) = -2.*PHIX/UINF 395 CONTINUE C Prepare CP coefficients for line plotting. C Use LPLOT3 to view top pressure coefficient CPTOP(I). C Compare MYFILE3 and MYFILE4 to ensure they are identical. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW

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1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPTOP(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE4.DAT',STATUS='UNKNOW 1N') WRITE(7,400) WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPBOT(I),I=1,19) CLOSE(7,STATUS='KEEP') C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-PHI-1.FOR C -------------------------------------------------------------

84 Modern Aerodynamic Methods 2.3.1.3

Calculated Results

Run 1. Here we discuss computed results where the potential itself vanishes with = 0 at all farfield boundaries, an option supported in our sample source code. 0.0

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Figure 2.3.1b. Potential solution (horizontal red line separates symmetric upper and lower fields). 0.0

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Figure 2.3.1c. Potential solution (vertical red line separates antisymmetric left and right fields).

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The benefit in using odd numbers of horizontal and vertical mesh lines, plus a test geometry with horizontal and vertical axes of symmetry passing through its center, is obvious and important in error-checking. For instance, in Figure 2.3.1b, the horizontal red line clearly separates symmetric upper and lower potential fields, as required in our analytical results from Chapter 1, while the vertical line in Figure 2.3.1c clearly separates left and right halves with opposing potential signs due to associations with surface slopes of opposite signs. Possessing the correct symmetries is a necessary but not sufficient condition for physical correctness. For instance, in Figures 2.3.1b and 2.3.1c, examination of the left two columns and the right two columns, for rows 7-13, clearly show that the farfield spatial derivative x representing horizontal disturbance velocity is nonzero. This may resolve itself had we chosen a larger computational box, but at least for the dimensions chosen, we do not have uniform flow at infinity. On the other hand, from Figures 2.3.1h and 2.3.1i, which also support the correct symmetries and antisymmetries, examination of the two left and two right columns and the two top and two bottom rows, shows that potential gradients do vanish. Uniform flow at left and right are ensured, but this might be incorrect if the wind tunnel box being simulated is actually small in reality. Color plots are also offered in Figures 2.3.1d,e,f.

Figure 2.3.1d. Potential solution.

86 Modern Aerodynamic Methods

Figure 2.3.1e. Potential solution.

Figure 2.3.1f. Potential solution.

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Figure 2.3.1g. Pressure coefficient symmetric about mid-chord. We now consider the pressure coefficient in Figure 2.3.1g, representing the value one horizontal grid line above y = 0. Recall that the left leading edge resides at I = 6 while the right trailing edge resides at I = 14 (the pressure fields are symmetric with respect to a vertical line passing through the airfoil center for the perfect inviscid flow studied). High values of pressure coefficient near I = 6 show how the flow is slowing down and stagnating as it approaches the leading edge, while negative values a short distance later describe lower pressures associated with higher velocities, turning and acceleration. Velocities decrease somewhat near the center causing pressures to increase, although overall the pressure is still below ambient. Figure 2.3.1g assumes that = 0 at all farfield boundaries, while Figure 2.3.1m, which assumes that =0 at the farfield, shows similar behavior with only slightly different numerical values. While this is satisfying computationally, it is apparent that many possibilities (or “games,” to professional aerodynamicists) can be found to force computations to agree with experimental data. Another common “game” played is “hiding the leading edge,” in which the instability associated with blunt edges is adjusted by moving the airfoil until computed results match measured experimental values.

88 Modern Aerodynamic Methods Run 2. Results are presented for the software option allowing = 0 at all farfield boundaries. Key features of the flow have been discussed in the prior section. -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.2 -0.1 -0.1

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Figure 2.3.1h. Potential solution (horizontal red line separates symmetric upper and lower fields). -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.2 -0.1 -0.1

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Figure 2.3.1i. Potential solution (vertical red line separates antisymmetric left and right fields).

Computational Algorithms

Figure 2.3.1j. Potential solution.

Figure 2.3.1k. Potential solution.

89

90 Modern Aerodynamic Methods

Figure 2.3.1l. Potential solution.

Figure 2.3.1m. Pressure coefficient.

Computational Algorithms 2.3.1.4

91

Comments

We note that while the farfield formulations in Runs 1 and 2 are different, computed results for Cp are qualitatively similar, as are those for potential. Very often, the use of vanishing of or is interchangeable, and this is acceptable so long as the computational box is large. In other studies, the behavior due to an analytical model itself is used to update the farfield. Of course, not all problems can be solved in this manner since analytical solutions are not always available. 2.3.2

Example 2 – Half-Space Thickness Solution Software reference, CODE-PHI-2.FOR

2.3.2.1 Problem Setup

In Example 1, we demonstrated how to calculate the velocity potential on a full rectangular box, with the chord line centered in the computational domain. It was necessary to consider four separate domains, namely, regions upstream of the leading edge, downstream of the trailing edge, and then, above and below the chord. Because we considered an unpitched “thickness only” geometry, the existence of upper and lower symmetry means that a more efficient “half-space” formulation could have been used. The top would still represent the farfield, but the bottom would coincide with the horizontal bisector of the airfoil geometry. This would allow us to consider a single box, with slope tangency conditions conveniently applied along the entire lower boundary – an actual “ y(x,0) = U F’(x)” along the chord, and y(x,0) = 0 elsewhere, since slopes along the bottom line (or symmetry plane bisecting the airfoil) are actually zero. The reader should compare the prior source code to that given on the next page to appreciate the simplicity in programming (only a single columnar region, as opposed to four, appears). Again, half-space formulations are useful when they are physically applicable – they are also useful in evaluating new ideas and methods, e.g., in developing type-differencing methods, conservative schemes, inverse methods, test molecules for transonic supercritical applications, and so on, as we will consider shortly.

92 Modern Aerodynamic Methods 2.3.2.2 C C C C

Fortran Source Code

CODE-PHI-2.FOR Potential equation solver, model symmetric airfoil on half-space, zero angle of attack, always nonlifting DIMENSION PHI(19,19),A(19),B(19),C(19),V(19),W(19) DIMENSION SLOPE(19),POSITION(19),CPTOP(19)

C C C C

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JMAX = 19 ITMAX = 10000 C Research code for testing, hardcode special parameters. C DX and DY are grid lengths, inches; do not appear in PDE C since DX = DY assumed, but appear in BC and CP definitions. C UINF is speed at infinity, say in/sec. Thus POT is in^2/sec. C No other output has units, only PHI and dimensionless pressure C coefficients are plotted against indexes. DX = 1. DY = 1. UINF = 1. C C SLOPE FUNCTION C Streamline slopes at I = 1-5 AND 15-19 are actual zeros C I = 6 to 14 are biconvex in shape, geometry is symmetric C with respect to center I = 10 SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. C Leading edge at I=6, trailing edge at I=14 SLOPE( 6) = +0.3 SLOPE( 7) = +0.2 SLOPE( 8) = +0.15 SLOPE( 9) = 0. SLOPE(10) = 0. SLOPE(11) = 0. SLOPE(12) = -0.15 SLOPE(13) = -0.2 SLOPE(14) = -0.3 SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0. C C CONVENIENT CONSTANTS IMAXM1 = IMAX-1 JMAXM1 = JMAX-1

Computational Algorithms C C

100 C C C C

110

120 C C C C C

INITIALIZE PHI TO ZERO, OTHER FUNCTION, OLD SOLUTION DO 100 I=1,IMAX DO 100 J=1,JMAX PHI(I,J) = 0. CONTINUE DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., if Grad PHI = 0 is imposed DO 110 I=1,IMAX PHI(I,1) = 0. PHI(I,JMAX) = 0. CONTINUE DO 120 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

C C

130 C

C

140 150 C C C C C C C

ITER=1,ITMAX

Code below for one sweep through computational box DO 150 I=2,IMAXM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE Invoke tangency condition along J = 1 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = -DY*UINF*SLOPE(I) Farfield vertical velocity is zero A(JMAX) = -1. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD POTENTIAL Zero potential gradient (loops 260, 270) or zero potential (loops 274, 276) yield slightly different potential fields and pressure coefficients. Both options shown below, hardcode your selection. Zero POT implemented below.

93

94 Modern Aerodynamic Methods C 260

270 272 274

276 300 C C C C C C

320 340

GO TO 272 DO 260 I=1,IMAX PHI(I,1) = PHI(I,2) PHI(I,JMAX) = PHI(I,JMAXM1) CONTINUE DO 270 J=1,JMAX PHI(1,J) = PHI(2,J) PHI(IMAX,J) = PHI(IMAXM1,J) CONTINUE GO TO 300 DO 274 I=1,IMAX PHI(I,JMAX) = 0. CONTINUE DO 276 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='PHI.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX WRITE(7,320) (PHI(I,J),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Velocity Potential') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (PHI(I,J),I=1,IMAX) 380 FORMAT(1X,19F5.1) 385 CONTINUE CLOSE(7,STATUS='KEEP')

Computational Algorithms C C

95

Write pressure coefficients CPTOP for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 CPTOP(I) = 0. 392 CONTINUE C Replace entries with calculated values at J = JW+1 and JW-1 DO 394 I=2,18 C Use central differencing over two DX boxes PHIX = (PHI(I+1,1) - PHI(I-1,1))/(2.*DX) C Small disturbance pressure coefficient CPTOP(I) = -2.*PHIX/UINF 394 CONTINUE C Prepare CP coefficients for line plotting. C Use LPLOT3 to view top pressure coefficient CPTOP(I). OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPTOP(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-PHI-2.FOR C -------------------------------------------------------------

96 Modern Aerodynamic Methods 2.3.2.3

Calculated Results

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Figure 2.3.2a. Half-space potential.

Computational Algorithms

97

Figure 2.3.2b. Half-space potential.

Figure 2.3.2c. Half-space surface pressure coefficient. 2.3.2.4

Comments

Considering that we have twice the grid density as in Example 1, the fact that our solutions in Figures 2.3.2a,b,c yield similar results is good a good indicator of accuracy for both.

98 Modern Aerodynamic Methods 2.3.3

Example 3 – Centered Symmetric Wedge Flow Software reference, CODE- PHI-1.

2.3.3.1

Problem Setup

In this example, we consider the nonlifting flow past the symmetric wedge shown, using the “full box” algorithm of Example 1. The required source code modifications are shown in Section 2.3.3.2.

EXPANSION COMPRESSION

AMBIENT

RE COMPRESSION

Figure 2.3.3a. Problem definition. What do we expect for the physics? As the flow heads toward and up the leading edge, we expect a compression or slowdown (or higher pressure), and as it abruptly turns downward, we expect an expansion (or lower pressure). Then, the fluid must turn to the right, parallel to the dashed centerline shown, re-compressing as this is ongoing. 2.3.3.2

Fortran Source Code

Comparing files CODE-PHI-1.for and CODE-PHI-1-WEDGE.FOR ***** CODE-PHI-1.for SLOPE( 5) = 0. SLOPE( 6) = +0.3 SLOPE( 7) = +0.2 SLOPE( 8) = +0.15 SLOPE( 9) = 0. SLOPE(10) = 0.

Computational Algorithms SLOPE(11) SLOPE(12) SLOPE(13) SLOPE(14) SLOPE(15)

= = = = =

99

0. -0.15 -0.2 -0.3 0.

***** CODE-PHI-1-WEDGE.FOR SLOPE( 5) = 0. SLOPE( 6) = 0.10 SLOPE( 7) = 0.15 SLOPE( 8) = 0.20 SLOPE( 9) = 0.25 SLOPE(10) = 0.30 SLOPE(11) = 0.35 SLOPE(12) = 0.40 SLOPE(13) = 0.45 SLOPE(14) = 0.50 SLOPE(15) = 0. *****

2.3.3.3 Calculated Results

The calculated potential distribution only shows the expected symmetry about the red horizontal bisector of the wedge. There are no vertical lines for which left-right flow symmetries exist, as is clear from Figure 2.3.3a. 0.0

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0.0 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.2 -0.1 -0.1

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0.0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -0.5 -0.4 -0.4 -0.3 -0.2 -0.2 -0.1

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0.0 -0.1 -0.1 -0.2 -0.3 -0.4 -0.4 -0.5 -0.6 -0.6 -0.7 -0.6 -0.6 -0.5 -0.5 -0.3 -0.2 -0.1

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0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.8 -0.9 -0.9 -0.8 -0.7 -0.6 -0.4 -0.3 -0.1

0.0

0.0 -0.1 -0.2 -0.3 -0.4 -0.6 -0.7 -0.8 -1.0 -1.1 -1.1 -1.1 -1.1 -1.0 -0.7 -0.5 -0.3 -0.2

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0.0 -0.1 -0.2 -0.3 -0.5 -0.7 -0.8 -1.0 -1.2 -1.3 -1.4 -1.4 -1.4 -1.3 -0.9 -0.6 -0.4 -0.2

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0.0 -0.1 -0.2 -0.3 -0.5 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 -1.8 -1.9 -1.8 -1.0 -0.7 -0.4 -0.2

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0.0 -0.1 -0.2 -0.3 -0.5 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 -1.8 -1.9 -1.8 -1.0 -0.7 -0.4 -0.2

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0.0 -0.1 -0.2 -0.3 -0.5 -0.7 -0.8 -1.0 -1.2 -1.3 -1.4 -1.4 -1.4 -1.3 -0.9 -0.6 -0.4 -0.2

0.0

0.0 -0.1 -0.2 -0.3 -0.4 -0.6 -0.7 -0.8 -1.0 -1.1 -1.1 -1.1 -1.1 -1.0 -0.7 -0.5 -0.3 -0.2

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0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.8 -0.9 -0.9 -0.8 -0.7 -0.6 -0.4 -0.3 -0.1

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0.0 -0.1 -0.1 -0.2 -0.3 -0.4 -0.4 -0.5 -0.6 -0.6 -0.7 -0.6 -0.6 -0.5 -0.5 -0.3 -0.2 -0.1

0.0

0.0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -0.5 -0.4 -0.4 -0.3 -0.2 -0.2 -0.1

0.0

0.0

0.0 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.2 -0.1 -0.1

0.0

0.0

0.0

0.0 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

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100 Modern Aerodynamic Methods

Figure 2.3.3c. Potential calculation.

Figure 2.3.3d. Compression-expansion-recompression sequence.

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Comments

The pressure coefficient plot in Figure 2.3.3d is consistent with the labeled flow regimes in Figure 2.3.3a. The left positive slope of the curve shows the increasing pressure associated with compression and slowing down. The fluid turns at the upper right corner of the wedge and suddenly downward, resulting in very high speeds, expansion and acceleration and a negative pressure coefficient. Then symmetry forces the flow to turn horizontally, which is accomplished by recompression and increasing pressure. To the far right, the streamlines straighten and become horizontal. At this point, fluid fluid pressure returns to freestream values and the pressure coefficient vanishes. 2.3.4

Example 4 – General Solution with Lift, Centered Slit Software reference, CODE-PHI-3.FOR.

In Example 1, we demonstrated how to calculate velocity potential solutions for a simple geometry with upper and lower symmetry, that is, an unpitched “thickness only” problem. We centered the body in a rectangular computational box and took care to set up and solve left, right, top and bottom flow domains correctly, this involving proper treatment of upper and lower tangency conditions. Of course, since the problem was symmetric about the horizontal chord line, we could have more easily solved the half-space problem, as was demonstrated in Example 2. However, we had implemented a “full box” solution in Example 1 with the objective of later (here) addressing the lifting problem, for which top and bottom symmetries were no longer possible. This deserves special attention because problems with multivalued (potential) solutions are seldom discussed. Different authors model the effects of lift differently and our treatment represents one of several. Incorrect approach illustrated. We first illustrate a plausible but completely incorrect approach to modeling flow past an inclined flat plate. This plate is shown at the top in Figure 2.3.4a where the oncoming flow emerges from the left. From experiment, it physically impinges at the bottom left and leaves the trailing edge smoothly at the right. There is no symmetry or antisymmetry about the vertical line passing through the center. These are check points to be used for solution development. In designing our (incorrect) scheme, we start with CODE-PHI-1.FOR from Example 1. We make several “obvious” changes, initially replacing the previous “biconvex like” airfoil slope function, that is

102 Modern Aerodynamic Methods SLOPE( 6) SLOPE( 7) SLOPE( 8) SLOPE( 9) SLOPE(10) SLOPE(11) SLOPE(12) SLOPE(13) SLOPE(14)

= = = = = = = = =

+0.3 +0.2 +0.15 0. 0. 0. -0.15 -0.2 -0.3

with the following straight-line geometry with constant slope (a negative ten degree flat plate pitch is used below) VALUE = -0.0174533*10. SLOPE( 6) = VALUE SLOPE( 7) = VALUE SLOPE( 8) = VALUE SLOPE( 9) = VALUE SLOPE(10) = VALUE SLOPE(11) = VALUE SLOPE(12) = VALUE SLOPE(13) = VALUE SLOPE(14) = VALUE

Next, in the “beneath airfoil” logic, we use the same slope function as in “above airfoil,” changing only a positive sign to negative to represent the geometric properties of the zero-thickness (camber) flat plate, that is, C C

WBOT(JW-1) = +DY*UINF*SLOPE(I) Use same slope function as "above airfoil" code WBOT(JW-1) = -DY*UINF*SLOPE(I)

This completes the required changes. Running the simulator with our “obviously correct” modifications leads to error, in particular, the results in Figures 2.3.4b,c,d (for example, the surface pressure coefficient in Figure 2.3.4d is clearly wrong – it should not be strictly antisymmetric about mid-chord).

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Figure 2.3.4a. Flat plate at angle of attack. Notes, (a) Top sketch shows actual physical streamlines with smooth trailing edge flow; (b) Middle drawing displays symmetric pattern with incorrect trailing edge behavior, not observed in Nature, and (c) Bottom sketch shows how distributed vorticity induces “tangential treadmill” velocities which do not affect the prescribed normal velocity at the surface – the correct value of “circulation” moves the right streamline of (b) to the trailing edge. This is deduced analytically in exact methods employing complex variable conformal mappings which set the trailing edge as a stagnation point. However, in numerical small-disturbance theory, this may be physically and mathematically inconsistent, and alternative methods are used.

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Figure 2.3.4c. Potential distribution.

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Figure 2.3.4d. Incorrect pressures, antisymmetric about mid-chord. We observe from Figures 2.3.4b,c that in the potential distribution, there is symmetry about the vertical bisector, one that is not consistent with the physical flow at the top of Figure 2.3.4a. Also, the pressure coefficient in Figure 2.3.4d is perfectly antisymmetric about mid-chord, inconsistent with the top of Figure 2.3.4a, which shows no evidence of any such symmetries. Both are inconsistent with the physics behind the actual flow in Figure 2.3.4a and therefore do not represent reality. Since the coding in CODE-PHI-1.FOR appears to be correct, and the changes to it described above appear to be “obvious” and plausible, we ask, “What happened?” The answer is clear. Solutions to Laplace’s equation are not unique. As we have discussed in Chapter 1, there exist two types of solutions, single-valued “log” and multivalued “arctangent” functions. Logarithms are associated with thickness, while arctangents are used to describe vortexes – distributions of which, following Figure 2.3.4a, yield a “treadmill” effect. We know that vortexes are associated with lift, so clearly, the nonuniquenesses inherent in Laplace’s equation must be used in such a way that flows like the top motion in Figure 2.3.4a are properly constructed. How to do this is addressed next.

106 Modern Aerodynamic Methods 2.3.4.1

Problem Setup

Recall that, in Example 1, we considered the non-lifting problem for a symmetric “thickness only” unpitched airfoil centered in the computational box. We dealt with four distinct domains of integration, upstream of the leading edge, beneath the chord, above the chord, and downstream of the trailing edge. The column relaxation scheme possessed left and right symmetry, that is, upstream and downstream domains were treated identically. We initially applied this scheme blindly in the above paragraphs. The flow was not physically correct as a right side streamline failed to emerge from the trailing edge as required physically – the calculated symmetries and antisymmetries were incorrect. To correct this, just enough vorticity or “sliding treadmill rotation” must be added at the surface to move the streamline appropriately, as shown at the bottom of Figure 2.3.4a. But how? i=1,2

imax 1, imax j=jmax j=jmax 1

ABOVE j=2

FLOW

j=1

Airfoil y=0

j=jw

BELOW

j=2 j=1

j=2 j=1

Figure 2.3.4e. Main computational grid setup and definitions. The computational domain shown in Figure 2.3.4e was previously used for the algorithm in Example 1. As indicated above, the iterative procedure must be adjusted to include the proper physics. To this end, we use the modified grid developed next page in Figure 2.3.4f and introduce it to the right of the trailing edge in Figure 2.3.4e. A typical “full column” spanning the entire vertical extent of the box is shown.

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j=jw+2

Airfoil Trailing edge

j=jw+1 j=jw+ j=jw j=jw j=jw 1 j=jw 2

Figure 2.3.4f. Expanded view at trailing edge. In Example 1, the tridiagonal approximation to our Laplace equation, namely i,j-1 – 4 i,j + i,j+1 = - ( i-1,j + i+1,j), was written for each vertical node from J = 2 to JMAX-1. Then, boundary conditions were used corresponding to indexes J = 1 and J = JMAX, thus defining JMAX equations for JMAX number of unknowns. We will modify this procedure in one important respect. The introduction of vorticity (in other words, “arc tan” solutions) requires us to have a multivalued potential or one that accommodates a nonzero jump denoted by [ ]. We had shown earlier in Equation 1.2.5a that the lift force L will satisfy L = U D TE where the originating integral could be taken over any closed circuit surrounding the chord. As such, the jump [ ] must be the same at any streamwise station away from the trailing edge. For implementation purposes, we introduce a “branch cut” (sometimes referred to as a “wake,” although this is not a real viscous wake) taken as the straight horizontal line emerging from the trailing edge. Before discussing modifications to the above column relaxation algorithm, a number of Kutta condition related properties must be derived or proven. First, we prove that [ ] must be constant at all locations along the wake. Now, we emphasize that we are dealing with the camber component of the airfoil, one which supports a differential pressure loading at all points between leading and trailing edges. At the latter edge, however, this differential physically disappears, requiring us to set upper and lower pressures to identical (but as yet unknown) values. Noting that we had Cp = (P-P )/½ U 2, it follows that Pjw+ = Pjw- implies

108 Modern Aerodynamic Methods the equality Cp,jw+ = Cp,jw-. Since Cp = – 2 x/U, it follows that we can write x,jw+ – x,jw- = 0, [ ] x = 0 or [ ] = constant across the “wake” extending horizontally from the trailing edge. During iterations, the value of this constant, as yet unknown, is set equal to that at the trailing edge. It will vary during iterations but eventually converge to a constant TE from which the lift L = U D TE and the lift coefficient CL = 2 / (U C) are calculated in Equations 1.2.5a and 1.2.5b. TE Second, we redefine the column vector for PHI(I,J) at a fixed streamwise position to accommodate the required multivaluedness in . Recall in Example 1 how we wrote i,j-1 – 4 i,j + i,j+1 = - ( i-1,j + i+1,j) for each vertical node from J = 2 to JMAX-1 and then introduced two farfield boundary conditions. For all internal box points away from J = JW, the equation used is an expression of the physics behind Laplace’s equation. For the J = 1 and J = JMAX matrix indexes, the equations used represent boundary conditions. We now need to re-think what is meant by “the equation for J = JW.” In Figure 2.3.4f, we show two dashed lines infinitesimally close at top and bottom to a black JW line. We will denote the top location by JW+ and the lower by JW-. Thus, the potential immediately above the black line is jw+ and that just below is jw- (we may omit the “i” index for clarity). The jump in potential is therefore [ ] = jw+ – jw- . We further introduce a average quantity by avg = ½ ( jw+ + jw- ) which we denote by “ jw.” We understand that this “ jw” is a mathematical convenience and not a physical potential. Some simple manipulations show that jw+ = jw + ½ [ ] and jw- = jw – ½ [ ]. We will explain how jw+ and jw- are to be used later, but for now focus on what is meant by “the equation for J = JW.” Across the wake, we expect that the normal velocity will be continuous, that is, the normal derivative of the velocity potential is continuous. Setting y|jw+ = y|jwleads to ( jw+1 – jw+)/ y = ( jw- – jw-1)/ y, that is, simply jw-1 – jw+ – jw- + jw+1 = 0 or jw-1 – 2 “ jw” + jw+1 = 0. In other words, at internal J indexes (with exceptions) in the computational box, the usual equation i,j-1 – 4 i,j + i,j+1 = - ( i-1,j + i+1,j) applies, however, at the special wake point, we use instead jw-1 – 2 “ jw” + jw+1 = 0. Our “ jw” is a mathematical device that handles the required multivaluedness, but importantly, retains tridiagonal matrix form plus the needed diagonal dominance for numerical stability. Third, we now deal with the exceptions noted in the above paragraph. When “ i,j-1 – 4 i,j + i,j+1 = - ( i-1,j + i+1,j)” is evaluated at the

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upper index j = jw+1, we really mean i,jw+ – 4 i,jw+1 + i,jw+2 = - ( i-1,jw+1 + i+1,jw+1). And when the difference equation is evaluated at the lower index jw-1, we really mean i,jw-2 – 4 i,jw-1 + i,jw- = - ( i-1,jw-1 + i+1,jw-1). Recall we previously had jw+ = jw + ½ [ ] and jw- = jw – ½ [ ]. Thus, the equations for indexes jw+1 and jw-2, when expressed in terms of the redefined column vector with “ jw,” are actually “ i,jw” – 4 i,jw+1 + i,jw+2 = - ( i-1,jw+1 + i+1,jw+1) – ½ [ ] and i,jw-2 – 4 i,jw-1 + “ i,jw” = - ( i-1,jw-1 + i+1,jw-1) + ½ [ ] Fourth, we ask “What is the value of [ ] used above? ” Recall from Example 1 that, above and beneath the chord, we isolated the horizontal line J = JW during the column setup. In other words, potentials along J = JW were not calculated. To find [ ], we now need to obtain expressions for jw+ and jw- . We do this by linear extrapolation, recognizing that we have a constant mesh. Hence, jw+2 – jw+1 = jw+1 – jw+ and jw- – jw-1 = jw-1 – jw-2 imply that jw+ = 2 jw+1 – jw+2 and jw- = 2 jw-1 – jw-2 so that [ ] = jw+ – jw- = 2 jw+1 – jw+2 – 2 jw-1 + jw-2. In summary, our difference scheme differs from that of Example 1 in that “ i,jw” here refers to a convenient mathematical quantity and not the physical potential along J = JW. The finite difference equations used in Example 1 corresponding to matrix indexes JW-1, JW and JW+1 are replaced by modified equations related to a nonzero [ ] defined iteratively from evolving trailing edge values. Its value converges as the sweeping iterations converge. On completion, the lift L = U D TE and the lift coefficient CL = 2 TE / (U C) are calculated, and the local pressure coefficient is obtained from Cp = - 2 x/U (note that, in our Fortran source code, the pressure coefficient is for simplicity evaluated along the lines J = JW+1 and JW-1 and not extrapolated to JW). Note that the lifting problem introduces an extra potential value over the nonlifting formulation because both jw+ and jw- appear. The method used here allows us to retain the same column vector length throughout the entire computational box – the alternative would be cumbersome. For the small computational box used in our examples, our numerical results will only be approximate, although certain desired physical features will be obvious. We also note that some authors use, for improved accuracy, an analytical “arctangent” formula to update farfield boundary values, noting that, of course, such an option is not always available.

110 Modern Aerodynamic Methods 2.3.4.2 Fortran Source Code Software reference, CODE-PHI-3.FOR. C C C C C C

CODE-PHI-3.FOR General airfoil with thickness and camber, centered in computational box. Potential jump fixed along downstream wake from trailing edge. Kutta condition enforced. SLOPE function below coded for pure camber and no thickness. DIMENSION DIMENSION DIMENSION DIMENSION

C C

PHI(19,19),A(19),B(19),C(19),V(19),W(19) ATOP(9),BTOP(9),CTOP(9),VTOP(9),WTOP(9) ABOT(9),BBOT(9),CBOT(9),VBOT(9),WBOT(9) SLOPE(19),POSITION(19),CPTOP(19),CPBOT(19)

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JW = 10 ITMAX = 10000

C C C C C C C C

Research code for testing, hardcoded special parameters. DX and DY are grid lengths, inches; do not appear in PDE since DX = DY assumed, but appear in BC and CP definitions. UINF is speed at infinity, say in/sec. Thus PHI is in^2/sec. No other output has units except PHI. PHI and dimensionless pressure coefficients are plotted against indexes. DX = 1. DY = 1. UINF = 1.

C C C 20 21 C C C C

Convention, flow is left to right, positive angle of attack has negative slope. Enter SLOPE with decimal point. WRITE(*,20) FORMAT( ' Enter SLOPE (with decimal point) ... ',$) READ(*,21) VALUE FORMAT(F10.3) GENERIC SLOPE FUNCTION Values at I = 1-5 AND 15-19 are dummies I = 6 to 14 represent actual slopes used SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. SLOPE( 6) = VALUE SLOPE( 7) = VALUE SLOPE( 8) = VALUE SLOPE( 9) = VALUE SLOPE(10) = VALUE SLOPE(11) = VALUE SLOPE(12) = VALUE

Computational Algorithms SLOPE(13) SLOPE(14) SLOPE(15) SLOPE(16) SLOPE(17) SLOPE(18) SLOPE(19) C C

VALUE VALUE 0. 0. 0. 0. 0.

CONVENIENT CONSTANTS JMAX = 2*JW-1 JMAXM1 = JMAX-1 JWM1 = JW-1 JWM2 = JW-2 JWP1 = JW+1 ILEM1 = ILE-1 ITEP1 = ITE+1 IMAXM1 = IMAX-1

C C C

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INITIALIZE PHI TO ZERO, OTHER FUNCTION, OLD SOLUTION Boundary PHI's all zero unless overwritten later DO 100 I=1,IMAX DO 100 J=1,JMAX PHI(I,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

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= = = = = = =

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UPSTREAM OF LEADING EDGE DO 150 I=2,ILEM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. A(JMAX) = 1. B(JMAX) = -1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE BELOW AIRFOIL SECTION Half columns vary from J = 1 to 9, keep JW line hidden

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112 Modern Aerodynamic Methods C

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J represents actual physical location DO 180 I=ILE,ITE DO 160 J=2,JWM2 ABOT(J) = +1. BBOT(J) = -4. CBOT(J) = +1. WBOT(J) = -PHI(I-1,J)-PHI(I+1,J) CONTINUE ABOT(1) = 99. BBOT(1) = 1. CBOT(1) = -1. WBOT(1) = 0. ABOT(JW-1) = 1. BBOT(JW-1) = -1. CBOT(JW-1) = 99. WBOT(JW-1) = -DY*UINF*SLOPE(I) CALL TRIDI(ABOT,BBOT,CBOT,VBOT,WBOT,JWM1) DO 170 J=1,JWM1 PHI(I,J) = VBOT(J) CONTINUE CONTINUE PHIMINUS = PHI(ITE,JW-1) test, use below extrapolation PHIMINUS = 2.*PHI(ITE,JW-1)-PHI(ITE,JW-2) ABOVE AIRFOIL SECTION Half columns vary from J = 1 to 9, does not include JW line DO 220 I=ILE,ITE Here, J is an indexing parameter, not a physical location Array indexes vary from J=1 to JW-1 DO 190 J=2,JWM2 ATOP(J) = 1. BTOP(J) = -4. CTOP(J) = 1. WTOP(J) = -PHI(I-1,J+JW) -PHI(I+1,J+JW) CONTINUE ATOP(1) = 99. BTOP(1) = 1. CTOP(1) = -1. WTOP(1) = -DY*UINF*SLOPE(I) ATOP(JW-1) = 1. BTOP(JW-1) = -1. CTOP(JW-1) = 99. WTOP(JW-1) = 0. CALL TRIDI(ATOP,BTOP,CTOP,VTOP,WTOP,JWM1) DO 200 J=1,JWM1 PHI(I,JW+J) = VTOP(J) CONTINUE CONTINUE PHIPLUS = PHI(ITE,JW+1) test, use below extrapolation PHIPLUS = 2.*PHI(ITE,JW+1)-PHI(ITE,JW+2) Enforce constant potential jump across "wake" branch cut emanating from trailing edge DELTAPHI = PHIPLUS - PHIMINUS DOWNSTREAM OF TRAILING EDGE

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Code is NOT identical to that upstream of leading edge, unlike symmetric thickness code with zero lift. Symmetry broken because circulation added to enforce Kutta BC by fixing trailing edge potential jump across wake.

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DO 250 I=ITEP1,IMAXM1 DO 230 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -PHI(I-1,J)-PHI(I+1,J) IF(J.EQ.JWP1) W(J) = W(J) - 0.5*DELTAPHI IF(J.EQ.JWM1) W(J) = W(J) + 0.5*DELTAPHI IF(J.EQ.JW ) THEN B(J) = -2. W(J) = 0. ENDIF CONTINUE A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. A(JMAX) = 1. B(JMAX) = -1. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 240 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD POTENTIAL DO 217 J=1,JMAX PHI(IMAX,J) = PHI(IMAX-1,J) PHI(1,J) = PHI(2,J) CONTINUE All boundary conditions above involve derivatives, solution determined to within a constant. Fix PHI at one point, select trailing edge PHI to zero (not really necessary, the symmetric grid leads to horizontal line of symmetry with zero values anyway). PHI(ITE,JW) = 0.

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320

CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='PHI.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX WRITE(7,320) (PHI(I,J),I=1,IMAX) FORMAT(1X,19F5.1)

114 Modern Aerodynamic Methods 340

CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\FILENAME.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Velocity Potential') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (PHI(I,J),I=1,IMAX) 380 FORMAT(1X,19F12.3) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write pressure coefficients CPTOP for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 CPTOP(I) = 0. CPBOT(I) = 0. 392 CONTINUE C Replace entries with calculated values at J = JW+1 and JW-1 DO 394 I=2,18 C Use central differencing over two DX boxes PHIX = (PHI(I+1,JWP1) - PHI(I-1,JWP1))/(2.*DX) C Small disturbance pressure coefficient CPTOP(I) = -2.*PHIX/UINF 394 CONTINUE DO 395 I=2,18 PHIX = (PHI(I+1,JWM1) - PHI(I-1,JWM1))/(2.*DX) CPBOT(I) = -2.*PHIX/UINF 395 CONTINUE C Prepare CP coefficients for line plotting. C Use LPLOT3 to view top pressure coefficient CPTOP(I). C Compare MYFILE3 and MYFILE4 to ensure they are identical. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW 1N') WRITE(7,400)

Computational Algorithms 400

115

FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPTOP(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE4.DAT',STATUS='UNKNOW 1N') WRITE(7,400) WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPBOT(I),I=1,19) CLOSE(7,STATUS='KEEP') WRITE(*,420) VALUE 420 FORMAT(1X,' Angle (rad): ',F8.3) WRITE(*,430) DELTAPHI 430 FORMAT(1X,' "Delta" PHI: ',F8.3) CL = 2.*DELTAPHI/(UINF*8.*DX) WRITE(*,440) CL 440 FORMAT(1X,' Lift coef CL: ',F8.3) C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-PHI-3.FOR C -------------------------------------------------------------

116 Modern Aerodynamic Methods 2.3.4.3

Calculated Results

In testing CODE-PHI-3, we evaluated numerous runs and showed that the algorithm always led to convergent results in a numerically stable way. For the calculated results in the section, we selected Enter SLOPE (with decimal point) ... -0.01745

which corresponds to a flat plate at 1 deg positive pitch. The potential jump and lift coefficient are listed as Angle (rad): "Delta" PHI: Lift coef CL:

-0.017 0.940 0.235

The theoretical value for (small-disturbance) lift coefficient is CL = 2 where is the angle of attack, or when = 0.017, CL = 0.107. The discrepancy is due to the very low number of constant meshes used (chosen for presentation clarity) and low number of total grid points (used to enhance numerical displays). Other calculations showed that the potential jump correctly varied linearly with angle. The pressure behavior in Figure 2.3.4g is reasonable, showing how the greatest load is created upstream, with differential loads vanishing near the trailing edge.

Figure 2.3.4g. Pressure coefficient calculation.

Computational Algorithms

-0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0.0 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2

-0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0.0 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2

-0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 -0.1 0.0 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2

-0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 -0.1 0.0 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2

-0.3 -0.3 -0.3 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 0.0 0.1 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3

-0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.2 -0.3 0.0 0.3 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3

-0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 0.0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

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-0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.4 -0.4 0.0 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3

-0.3 -0.3 -0.3 -0.3 -0.3 -0.4 -0.4 -0.4 -0.4 0.0 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3

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-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

Figure 2.3.4h. Velocity potential, calculated results.

-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

117

-0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.5 0.0 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

118 Modern Aerodynamic Methods Calculated results for the potential field are given in Figure 2.3.4h above, and these are displayed because they provide useful symmetry checks to guard against programming errors. Note from the numerical values for the color mapping scheme at the right how perfect antisymmetry is achieved. Also, from the numerical potentials on the prior page, we observe how the top two rows of numbers are identical, as are the bottom two rows, the left two columns and the right two columns, demonstrating how imposed farfield velocities correctly vanish.

Figure 2.3.4i. Velocity potential, expanded view. Another check we normally perform is connsistency with the difference model and Laplace’s eqution. Recall that, for constant grid meshes with x = y, mesh lengths drop out of the finite difference molecule. The column relaxation takes the form i,j-1 – 4 i,j + i,j+1 = - ( i-1,j + i+1,j), which can be alternatively written as i,j = ¼ ( i-1,j + i+1,j + i,j-1 + i,j+1). This states that, for Laplace’s equation, the center value is just the arithmetic average of its neighbots. To check that this is true, consider the circled “147” in Figure 2.3.4i. It is clear that “147” = (147 + 157 + 107 + 177)/4, and likewise, “238” = (219 + 261 + 226 + 247)/4 and “286” = (266 + 306 + 287 + 284)/4 (these points were randomly chosen). 2.3.4.4 Comments

We encourage readers to pursue additional calculations, modifying the source code to allow general geometries where upper and lower surfaces may be completely different. How can the analytical formulas of Chapter 1 be used to update farfield values for rapid convergence and increased accuracy?

Computational Algorithms 2.3.5

119

Example 5 – Transonic Supercritical Airfoil with Type-Dependent Differencing Solution, Subsonic, Mixed Flow and Supersonic Calculations Software reference, CODE-PHI-4.FOR.

In this example, we demonstrate how smooth subsonic flows evolve into transonic supercritical flows with shockwaves, in our simulation runs, varying the freestream Mach numbers from 0 to 0.95; and then we continue to supersonic speeds at Mach 1.5. Our presentation is an elementary one, offering only the simplest application of “mixed differencing” or “type differencing” in the solution of subsonic flows containing embedded supersonic zones with shockwaves. Advanced discussions related to conservative differencing, curvilinear meshes, and so on, are not covered in our work (see Chapter 1 for references). 2.3.5.1 Problem Setup

Whereas our governing potential equations until now followed the simple Laplace model xx + yy = 0, in small-disturbance compressible flow we have {1 – M 2 – ( +1)M 2 x} xx + yy = 0 where is the ratio of specific heats and M is the freestream Mach number (e.g., refer to Ashley and Landahl, 1965). Note how the coefficient D = 1 – M 2 – ( +1)M 2 x contains a x contribution which is unknown at the outset and determines the physical character of the flowfield. This coefficient D is a “discriminant” function, rendering the flow subsonic if D > 0 and supersonic if D < 0. Embedded supersonic zones usually terminate with shockwaves and losses – they are also associated with unstable boundary layer interactions and aircraft instability. Supersonic zones can also transition smoothly to subsonic – such are affiliated with “shockfree” airfoils or conditions (e.g., see Harris (1990) for developments). The basic idea behind “type differencing,” due to Murman and Cole (1971), is simple. The column relaxation and sweeping process used in the foregoing examples is retained and the potential is initialized as before, say, with zero values. However, each point is tediously tested during the iterations. If D > 0 locally, the flow is subsonic; it is influenced by all other points and it influences all others. The central differenced transonic equation D ( i-1,j – 2 i,j + i+1,j)/ x2 + ( i,j-1 – 2 i,j + 2 i,j+1)/ y = 0 is used. Its column relaxation adaptation takes the form i,j-1

– 2(1 + D y2/ x2)

i,j

+

i,j+1

= - D ( y2/ x2)(

i-1,j

+

i+1,j)

120 Modern Aerodynamic Methods On the other hand, if the point is locally supersonic, it can only be affected by points upstream, and it can only influence points downstream. Thus, the streamwise term xx is backward or “upwind differenced” in space, with D( i-2,j – 2 i-1,j + i,j)/ x2 + ( i,j-1 – 2 i,j + 2 i,j+1)/ y = 0, which when rewritten for column relaxation takes the form i,j-1

+ (D y2/ x2 – 2)

i,j

+

i,j+1

= D ( y2/ x2)(2

i-1,j

–

i-2,j)

To keep the ideas and implementation simple, the half-space code in Example 2 is modified so that we study the transonic behavior of a nonlifting thickness distribution. Similar changes can be undertaken for lifting applications, and modifications to the code in Example 4 are left as exercises to the interested reader. Again, our implementation does not include conservative differencing. For a thorough discussion, the reader should consult the papers authored by Jameson and his colleagues. 2.3.5.2 C C C C C

Fortran Source Code

CODE-PHI-4.FOR Mixed type potential equation solver, symmetric airfoil on half-space, zero angle of attack, always nonlifting, transonic subcritical or supercritical DIMENSION PHI(19,19),A(19),B(19),C(19),V(19),W(19) DIMENSION SLOPE(19),POSITION(19),CPTOP(19),DISCRIM(19,19)

C C C C

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JMAX = 19 ITMAX = 10000

C C Research code for testing, hardcoded special parameters. C DX and DY are grid lengths, inches; UINF is speed at infinity, C say in/sec. Thus PHI is in^2/sec. GAMA is dimensionless C ratio of specific heats, EM is dimensionless Mach number. C DX = 1. DY = 1. UINF = 1. GAMA = 1.4 EM = 0.8 C C SLOPE FUNCTION C Streamline slopes at I = 1-5 AND 15-19 are actual zeros C I = 6 to 14 are biconvex in shape, geometry is symmetric C with respect to center I = 10

Computational Algorithms SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. Leading edge at I=6, trailing edge at I=14 SLOPE( 6) = +0.3 SLOPE( 7) = +0.2 SLOPE( 8) = +0.15 SLOPE( 9) = 0. SLOPE(10) = 0. SLOPE(11) = 0. SLOPE(12) = -0.15 SLOPE(13) = -0.2 SLOPE(14) = -0.3 SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0.

C

C C

CONVENIENT CONSTANTS IMAXM1 = IMAX-1 JMAXM1 = JMAX-1

C C

100 C C C C

110

120 C C C C C

INITIALIZE PHI TO ZERO, OTHER FUNCTION, OLD SOLUTION DO 100 I=1,IMAX DO 100 J=1,JMAX PHI(I,J) = 0. CONTINUE DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., if Grad PHI = 0 is imposed DO 110 I=1,IMAX PHI(I,1) = 0. PHI(I,JMAX) = 0. CONTINUE DO 120 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

C C C C

ITER=1,ITMAX

Code below for one sweep through computational box DO 150 I=2,IMAXM1 DO 130 J=2,JMAXM1 Discriminant D, subsonic if D > 0, supersonic if D < 0 D = 1.-EM**2

121

122 Modern Aerodynamic Methods C

125

130 C

C

140 150 C C C C C C C C 260

270 272 274

276 300 C C

1-(GAMA+1.)*EM**2.*(PHI(I,J)-PHI(I-1,J))/DX Mixed or type-differencing next IF(D.LT.0.0) GO TO 125 A(J) = +1. B(J) = -2.*(1.+D*DY**2/DX**2) C(J) = +1. W(J) = -(PHI(I-1,J)+PHI(I+1,J))*D*DY**2/DX**2 GO TO 130 A(J) = +1. B(J) = D*DY**2/DX**2 -2. C(J) = +1. W(J) = (2.*PHI(I-1,J)-PHI(I-2,J))*D*DY**2/DX**2 CONTINUE Invoke tangency condition along J = 1 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = -DY*UINF*SLOPE(I) Farfield vertical velocity is zero A(JMAX) = -1. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX PHI(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD POTENTIAL Zero potential gradient (loops 260, 270) or zero potential (loops 274, 276) yield slightly different potential fields and pressure coefficients. Both options shown below, hardcode your selection. Zero PHI implemented below. GO TO 272 DO 260 I=1,IMAX PHI(I,1) = PHI(I,2) PHI(I,JMAX) = PHI(I,JMAXM1) CONTINUE DO 270 J=1,JMAX PHI(1,J) = PHI(2,J) PHI(IMAX,J) = PHI(IMAXM1,J) CONTINUE GO TO 300 DO 274 I=1,IMAX PHI(I,JMAX) = 0. CONTINUE DO 276 J=1,JMAX PHI(1,J) = 0. PHI(IMAX,J) = 0. CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required).

Computational Algorithms C C

305 C 310 C C C C

320 340

Define discriminant field, first initialize DO 305 I=1,IMAX DO 305 J=1,JMAX DISCRIM(I,J) = 1. CONTINUE DO 310 I=2,IMAXM1 DO 310 J=2,JMAXM1 PHIX = (PHI(I+1,J)-PHI(I-1,J))/(2.*DX) PHIX = (PHI(I,J)-PHI(I-1,J))/DX DISCRIM(I,J) = 1.-EM**2-(GAMA+1.)*EM**2*PHIX CONTINUE WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='PHI.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX WRITE(7,320) (PHI(I,J),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Subsonic vs Supersonic Flow Discriminant') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (DISCRIM(I,J),I=1,IMAX) 380 FORMAT(1X,19F15.5) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write pressure coefficients CPTOP for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 CPTOP(I) = 0.

123

124 Modern Aerodynamic Methods 392

CONTINUE Replace entries with calculated values at J = JW+1 and JW-1 DO 394 I=2,18 C Use central differencing over two DX boxes PHIX = (PHI(I+1,1) - PHI(I-1,1))/(2.*DX) C Small disturbance pressure coefficient CPTOP(I) = -2.*PHIX/UINF 394 CONTINUE C Prepare CP coefficients for line plotting. C Use LPLOT3 to view top pressure coefficient CPTOP(I). OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPTOP(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-PHI-4.FOR C ------------------------------------------------------------C

2.3.5.3 Calculated Results

For the thickness distribution described previously and in more detail in the source code, freestream Mach numbers ranging from 0 to 0.95 were considered. Figures 2.3.5a-f each provide plots for the discriminant D at the left and the surface pressure coefficient Cp on the right. Color codes are given in the charts shown, which will be different for different figures. They allow us to clearly distinguish between subsonic and supersonic flow. Figures 2.3.5a,b are clearly subsonic and

Computational Algorithms

125

the corresponding pressure coefficient curves are symmetric about midchord as expected.

Figure 2.3.5a. Mach 0.0, flow discriminant and pressure coefficient; note symmetric pressure distribution for airfoil with fore-aft symmetry.

Figure 2.3.5b. Mach 0.5, flow discriminant and pressure coefficient; note symmetric pressure distribution for airfoil with fore-aft symmetry. .

126 Modern Aerodynamic Methods

Figure 2.3.5c. Mach 0.7, flow discriminant and pressure coefficient; observe onset of asymmetry for this higher Mach number.

Figure 2.3.5d. Mach 0.8, flow discriminant and pressure coefficient; shock formation is clearly evident at Mach 0.8. However, starting with Figure 2.3.5d for a Mach number of 0.8, negative discriminant values are apparent and the symmetry in the Cp curve disappears. In fact, the rapid rise in pressure just aft of midchord indicates the appearance of a shock. The effects are even more pronounced in Figures 2.3.5e,f where the Mach numbers are further increased. These results only display qualitative features which need to be refined with larger computational boxes and variable meshes. However, they suffice in illustrating the features associated with transonic behavior – for example, the adverse pressure gradients associated with local shocks may induce aerodynamic instabilities that arise from shock oscillations and boundary layer separation.

Computational Algorithms

127

Figure 2.3.5e. Mach 0.9, flow discriminant and pressure coefficient.

Figure 2.3.5f. Mach 0.95, flow discriminant and pressure coefficient. In Figure 2.3.5g, we simply changed the Mach number of 0.8 in the source code to 1.5, representing a supersonic freestream. We did not alter the farfield conditions – only to see “what happens.” We clearly observe distinct downstream (as opposed to upstream) effects, which are anticipated physically. Physical properties are transported along Mach lines. Furthermore, the calculations were stably completed. To refine our supersonic calculations, nonlinear supersonic flow theory can be used to provide farfield radiation conditions connecting x to y through Mach number. The resulting calculations will show characteristics or rays that are slightly bent due to nonlinearities, whereas those of linear theory are straight. Convergence of these rays in the farfield leads to sudden pressure rises related to sonic booms.

128 Modern Aerodynamic Methods

Figure 2.3.5g. Mach 1.5, supersonic freestream. 2.3.5.4 Comments

While type-dependent differencing has revolutionized the computer modeling of transonic flows, a serious limitation is computationintensive point-by-point testing, which is taken in the direction in which the supersonic flow evolves. This limits our flexibility in curvilinear grid generation since grid coordinate lines are not produced with flow physics, which may contain discontinuities, in mind – they are developed from elliptic equation models that emphasize smoothness. An alternative transonic method developed by this author uses the parabolic “viscous transonic equation” which eliminates type-differencing and allows sweeping in arbitrary directions. This allows us to partition the flow domain into different volumes that can be solved separately and periodically matched, with this process performed repeatedly until convergence. This approach, which offers an alternative to typedependent differencing methods, is described in Chapter 3.

Computational Algorithms 2.3.6

129

Example 6 – Three-Dimensional, Thickness-Only, Finite, Half-Space Solution Software reference, CODE-PHI-5.FOR.

Here, we show how our two-dimensional column relaxation solver can be embedded within a three-dimensional solver to model wings with finite span. To keep the ideas and coding simple, we will consider an unpitched non-lifting symmetric “thickness only” wing whose mean plane is located at the bottom surface of a computational half-space. 2.3.6.1 Problem Setup

In our prior examples, we considered only two-dimensional “airfoil” problems represented by the front dark-gray plane in Figure 2.3.6a. The reader should review Examples 1 and 2 to understand the “left to right” column relaxation process repeatedly used to converge our calculations. The changes needed to extend this three-dimensionally are readily understood, now referring to the light gray background planes shown.

Figure 2.3.6a. Three-dimensional sweep algorithm. Assuming constant density flow, we now have xx + yy + zz = 0 in three dimensions. For simplicity, suppose that x = y = z. Then, the standard central difference approximation ( i-1,j,k – 2 i,j,k + i+1,j,k)/ x2 + ( i,j-1,k – 2 i,j,k + i,j+1,k)/ y2 + ( i,j,k-1 – 2 i,j,k + i,j,k+1)/ z2 = 0 can be rewritten as i,j-1,k – 6 i,j,k + ,i,j+1,k = – ( i-1,j,k + i+1,j,k + i,j,k-1 + i,j,k+1) whereas we previously had i,j-1 – 4 i,j + ,i,j+1 = – ( i-1,j + i+1,j) in our column relaxation. We indicate that replacement of the “4” by a “6”

130 Modern Aerodynamic Methods improves computational stability since diagonal dominance is enhanced. Note that “k” is the spanwise index, and further, note the additional rightside terms. We now start with the K = 2 plane (adjacent to the K = 1 root or fuselage plane), use the “ i,j-1,k – 6 i,j,k + ,i,j+1,k = – . .” equation and perform column relaxation sweeping in the direction of increasing streamwise I values. Once the K = 2 plane is completed, we repeat this process for K = 3, 4, . . . KMAX-1. Then, potential values at K = 1 and KMAX are updated using farfield boundary conditions. Finally, this three-dimensional sweeping process is carried out repeatedly from K = 1 to KMAX until the calculations converge. In our calculations, we assume that flow perpendicular to K = 1 does not exist; in practice, normal velocities consistent with fuselage shapes may be used. 2.3.6.2 Fortran Source Code C C C C

CODE-PHI-5.FOR (developed from CODE-PHI-2.FOR) Potential equation 3D solver for symmetric finite wing on half-space, zero angle of attack, always nonlifting DIMENSION PHI(19,19,11),A(19),B(19),C(19),V(19),W(19) DIMENSION SLOPE(19),POSITION(19),CPTOP(19)

C C C C

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JMAX = 19 KMAX = 11 C Wing tip at K = KTIP, varies from 3 or 4 up to 8 KTIP = 4 ITMAX = 10000 C Research code for testing, hardcode special parameters. C DX, DY and DZ are grid lengths, inches; do not appear in PDE C since DX=DY=DZ assumed, but appear in BC and CP definitions. C UINF is speed at infinity, say in/sec. Thus POT is in^2/sec. C No other output has units, only PHI and dimensionless pressure C coefficients are plotted against indexes. DX = 1. DY = 1. DZ = 1. UINF = 1. C C SLOPE FUNCTION C Defined for K = 1 to KTIP between leading and trailing edges on C wing planform, else set SLOPE = 0 on half-space symmetry plane. C Streamline slopes at I = 1-5 AND 15-19 are zeros C I = 6 to 14 are biconvex in shape, geometry is symmetric

Computational Algorithms C

with respect to center I = 10 SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. Leading edge at I=6, trailing edge at I=14 SLOPE( 6) = +0.3 SLOPE( 7) = +0.2 SLOPE( 8) = +0.15 SLOPE( 9) = 0. SLOPE(10) = 0. SLOPE(11) = 0. SLOPE(12) = -0.15 SLOPE(13) = -0.2 SLOPE(14) = -0.3 SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0.

C

C C

CONVENIENT CONSTANTS IMAXM1 = IMAX-1 JMAXM1 = JMAX-1 KMAXM1 = KMAX-1

C C

100 C C C C

110

120 C C C C C

INITIALIZE PHI TO ZERO, OTHER FUNCTION, OLD SOLUTION DO 100 K=1,KMAX DO 100 I=1,IMAX DO 100 J=1,JMAX PHI(I,J,K) = 0. CONTINUE DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., if Grad PHI = 0 is imposed DO 110 K=1,KMAX DO 110 I=1,IMAX PHI(I,1,K) = 0. PHI(I,JMAX,K) = 0. CONTINUE DO 120 K=1,KMAX DO 120 J=1,JMAX PHI(1,J,K) = 0. PHI(IMAX,J,K) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300 DO 290

C

ITER=1,ITMAX K=1,KMAXM1

131

132 Modern Aerodynamic Methods C

130 C

135

C C 138

140 150 C C C C C C C C

C 260

270 C 272 274

Code below for one sweep through computational box DO 150 I=2,IMAXM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -6. C(J) = +1. W(J) = -PHI(I-1,J,K)-PHI(I+1,J,K)-PHI(I,J,K-1)-PHI(I,J,K+1) CONTINUE IF(K.GT.KTIP) GO TO 135 Invoke tangency condition along J = 1 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = -DY*UINF*SLOPE(I) GO TO 138 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. End wing plane boundary conditions Farfield vertical velocity is zero A(JMAX) = -1. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX PHI(I,J,K) = V(J) CONTINUE CONTINUE UPDATE FARFIELD POTENTIAL Zero potential gradient (loops 260, 270) or zero potential (loops 274, 276) yield slightly different potential fields and pressure coefficients. Both options shown below, hardcode your selection. Zero POT implemented below. KK is spanwise index, cannot use K because it redefines outer loop 290 span index. GO TO 272 DO 260 KK=1,KMAX DO 260 I=1,IMAX PHI(I,1,KK) = PHI(I,2,KK) PHI(I,JMAX,KK) = PHI(I,JMAXM1,KK) CONTINUE DO 270 KK=1,KMAX DO 270 J=1,JMAX PHI(1,J,KK) = PHI(2,J,KK) PHI(IMAX,J,KK) = PHI(IMAXM1,J,KK) CONTINUE Add loop similar to 280 here, if this block is used GO TO 300 DO 274 KK=1,KMAX DO 274 I=1,IMAX PHI(I,JMAX,KK) = 0. CONTINUE

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280 290 300 C C C C C C C C C 320 340

133

DO 276 KK=1,KMAX DO 276 J=1,JMAX PHI(1,J,KK) = 0. PHI(IMAX,J,KK) = 0. CONTINUE DO 280 I=1,IMAX DO 280 J=1,JMAX PHI(I,J,1) = PHI(I,J,2) PHI(I,J,KMAX) = PHI(I,J,KMAXM1) CONTINUE CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='PHI.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX Plot root plane potential at K = 1. If other station is desired, redefine K, recompile. WRITE(7,320) (PHI(I,J,K),I=1,IMAX) WRITE(7,320) (PHI(I,J,1),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Velocity Potential (K = 1)') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX C WRITE(7,380) (PHI(I,J,K),I=1,IMAX) WRITE(7,380) (PHI(I,J,1),I=1,IMAX) 380 FORMAT(1X,19F5.1) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write pressure coefficients CPTOP for line plotting

134 Modern Aerodynamic Methods DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 CPTOP(I) = 0. 392 CONTINUE C Replace entries with calculated values at J = JW+1 and JW-1 DO 394 I=2,18 C Use central differencing over two DX boxes C PHIX = (PHI(I+1,1,1) - PHI(I-1,1,1))/(2.*DX) for root plane C Spanwise index may vary from 1 to KMAX, hardcode, recompile. PHIX = (PHI(I+1,1,1) - PHI(I-1,1,1))/(2.*DX) C Small disturbance pressure coefficient CPTOP(I) = -2.*PHIX/UINF 394 CONTINUE C Prepare CP coefficients for line plotting. C Use LPLOT3 to view top pressure coefficient CPTOP(I). OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (CPTOP(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-PHI-5.FOR C -------------------------------------------------------------

Computational Algorithms 2.3.6.3

135

Calculated Results

A simple wing configuration was chosen for illustrative purposes. For our prior examples, we assumed a 19 19 constant grid in the x-y plane. Here, we extend this spanwise in the y or K direction, taking KMAX = 11 while locating the wing tip at K = 4. Figures 2.3.6a,b,c show the computed effects in a direction away from the root or fuselage plane. At K = 1, pressure results similar to Example 2 for a twodimensional airfoil are found. Pressures fall as the index K increases, and two other displays showing rapid decreases are given below.

Figure 2.3.6a. Plot at root plane K = 1, with results similar to Example 2, Cp is O(1).

Figure 2.3.6b. Plot at spanwise plane K = 5, Cp is O(10-1).

136 Modern Aerodynamic Methods

Figure 2.3.6c. Plot at spanwise plane K = 8, Cp is O(10-2). 2.3.6.4

Comments

To model a finite lifting wing in three dimensions, we would need to consider multiple regions, as suggested in Figures 2.3.4e,f – again, upstream of the leading edge, above the planform, beneath the planform, and downstream of the trailing edge. Of course, we need to add the volume in the spanwise direction away from the wing tip. And if our flow were not symmetric with respect to the mid-plane K = 1, for example, if we had wing “flaps up” one side and “down” on the other, then the entire three-dimensional space would be modeled.

Figure 2.3.6d. Inviscid wing flow with boundary layer strip model.

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Figure 2.3.6e. Nacelle iterations, wing spanwise z marching easily “morphed” into azimuthal direction.

Figure 2.3.6f. Engine power addition using actuator disk model – general theory developed in detail in Section 4.4. While we have used only the simplest example in our source code listing, again for the purposes of readability, the practical implications of our work are significant. For instance, in Figure 2.3.6d, we show how rewriting the inviscid flow algorithm on a swept grid allows us to model wing sweep; coupling to an elementary boundary layer strip solution

138 Modern Aerodynamic Methods enables us to examine the first-order effects of viscous drag. Our threedimensional wing algorithm for xx + yy + zz = 0, with minimal effort, can be rewritten so that nacelle-pylon-accessory bump configurations can be modeled. These satisfy xx + rr + 1/r r + 1/r2 = 0, a Laplace equation in circular cylindrical coordinates that is easily differenced. Consistently with thin airfoil theory, streamwise tangency conditions at different nacelle-inlet sections can be applied on a mean cylinder having constant radius. Whereas previously, spanwise marching was introduced in the z direction, we now replace that with iterations taken in the direction – the new finite difference equations are only slightly different, with r replacing y and replacing z. The iterations are repeated until convergence is achieved, as suggested in Figure 2.3.6e. Finally, other degrees of complexity are permitted. An important consideration is found in engine plume effects when power is added. Here, energy addition can be modeled by using an “actuator disk” where work interaction imparted by the turbomachinery is prescribed (see Figure 2.3.6f). This model is developed in detail in Section 4.4 in this book.

2.4 Examples – Inverse or Indirect Applications

Section 2.3 dealt with the analysis problem, for which the velocity potential provided a convenient host formulation. We had shown that, for the inverse problem, the governing boundary value problem is almost identical to those in the previous section if the dependent variable used is the streamfunction. In the present section, we illustrate the basic approach using several easily coded and readily understood examples. More advanced examples are cosidered in Chapter 4, where detailed theory and calculations are provided. 2.4.1 Example 1 – Constant Pressure Specification and Symmetric Thin Ellipse Software reference, CODE-SF-1.FOR.

It is well known in constant density aerodynamics that, for an unpitched thin ellipse of thickness ratio , the surface pressure coefficient is Cp = -2 away from blunt leading and trailing edges (e.g., see Katz and Plotkin (2010)). To evaluate our new inverse capabilities, we naturally ask, “What is the geometric shape corresponding to a surface pressure

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coefficient of Cp = -2 ?” Clearly, the solution should be an ellipse, and we have confirmed this very accurately using higher density mesh calculations as shown in Chapter 4. Again, for display purposes here, we use very coarse meshes which do not yield the exact thickness ratio – however, the elliptic contour is clearly evident. 2.4.1.1 Problem Setup

The numerical formulation is trivial and follows that of Example 2 in Section 2.3. Because of symmetries, we perform our simulations in a half-space. There are no jumps in streamfunction since, implicitly, halfspace analysis together with identical treatment of upstream and downstream columns offers full symmetry. Thus, the streamfunction formulation in Equations 1.3.5a,b,c below is solved with [ ] set to zero identically for closure. The surface ordinate is just y(x) = - (x,0). xx y

+

yy

=0

(x, y = 0) = - ½ U Cp(x) 0 as x2 + y2

(1.3.5a) (1.3.5b) (1.3.5c)

2.4.1.2 Fortran Source Code C C C C

CODE-SF-1.FOR SF streamfunction equation solver, for symmetric airfoil with closed trailing edge, modeled in half-space domain DIMENSION SF(19,19),A(19),B(19),C(19),V(19),W(19) DIMENSION CPCOEF(19),POSITION(19),SHAPE(19)

C C C C

C C C C C C C C

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JMAX = 19 ITMAX = 10000 Research code for testing, hardcode special parameters. DX and DY are grid lengths, inches; do not appear in PDE since DX = DY assumed, but appear in BC and CP definitions. UINF is speed at infinity, say in/sec. Thus PHI is in^2/sec. SF also has in^2/sec units. Pressure coefficient is dimensionless. DX = 1. DY = 1. UINF = 1.

C C

CPCOEF FUNCTION

140 Modern Aerodynamic Methods C C C

Values at I = 1-5 AND 15-19 are dummies. I = 6 to 14 are constant, correspond to known thin ellipse. This is a check case. CPCOEF( 1) = 0. CPCOEF( 2) = 0. CPCOEF( 3) = 0. CPCOEF( 4) = 0. CPCOEF( 5) = 0. CPCOEF( 6) = 0.2 CPCOEF( 7) = 0.2 CPCOEF( 8) = 0.2 CPCOEF( 9) = 0.2 CPCOEF(10) = 0.2 CPCOEF(11) = 0.2 CPCOEF(12) = 0.2 CPCOEF(13) = 0.2 CPCOEF(14) = 0.2 CPCOEF(15) = 0. CPCOEF(16) = 0. CPCOEF(17) = 0. CPCOEF(18) = 0. CPCOEF(19) = 0.

C C

CONVENIENT CONSTANTS IMAXM1 = IMAX-1 JMAXM1 = JMAX-1

C C

100 C C C

110

120 C C C C C

INITIALIZE SF TO ZERO, OTHER FUNCTION OR OLD SOLUTION DO 100 I=1,IMAX DO 100 J=1,JMAX SF(I,J) = 0. CONTINUE DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., Grad SF = 0, DO 110 I=1,IMAX SF(I,1) = 0. SF(I,JMAX) = 0. CONTINUE DO 120 J=1,JMAX SF(1,J) = 0. SF(IMAX,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

C C

ITER=1,ITMAX

Code below for one sweep through computational box DO 150 I=2,IMAXM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -4.

Computational Algorithms 130 C

C

140 150 C C C C C C C C C C C 260 C 270 272

274

276 300 C C C C C C

C(J) = +1. W(J) = -SF(I-1,J)-SF(I+1,J) CONTINUE Specify pressure along y = 0 or J = 1 edge line A(1) = 99. B(1) = 1. C(1) = -1. W(1) = +0.5*DY*UINF*CPCOEF(I) Farfield dSF/dy or pressure coefficient is zero A(JMAX) = -1. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX SF(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD STREAMFUNCTION (Testing purposes) Zero SF gradient (loops 260, 270) or zero SF (loops 274, 276) yield slightly different SF fields and geometries. Both options shown below, hardcode your selection. Zero SF implemented below. GO TO 272 DO 260 I=1,IMAX Do not use SF(I,1) = SF(I,2) at J = 1 boundary dSF/dy proportional to pressure coefficient, generally not zero on y = 0 away from airfoil. Do not specify. Set dSF/dy to zero at top SF(I,JMAX) = SF(I,JMAXM1) CONTINUE DO 270 J=1,JMAX dSF/dx is negative of vertical velocity, set to zero SF(1,J) = SF(2,J) SF(IMAX,J) = SF(IMAXM1,J) CONTINUE GO TO 300 DO 274 I=1,IMAX IF(I.LT.6) SF(I,1) = 0. IF(I.GT.14) SF(I,1) = 0. SF(I,JMAX) = 0. CONTINUE DO 276 J=1,JMAX SF(1,J) = 0. SF(IMAX,J) = 0. CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store disturbance velocity potential file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='SF.DAT',STATUS='UNKNOWN')

141

142 Modern Aerodynamic Methods 320 340

DO 340 J=1,JMAX WRITE(7,320) (SF(I,J),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write potential for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Streamfunction') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (SF(I,J),I=1,IMAX) 380 FORMAT(1X,19F9.3) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write y-ordinate SHAPE for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C Initialize arrays everywhere to zero DO 392 I=1,19 SHAPE(I) = 0. 392 CONTINUE DO 394 I=6,14 SHAPE(I) = -SF(I,1)/UINF 394 CONTINUE C Use LPLOT3 to view y-ordinate SHAPE(I). OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE3.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (SHAPE(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END

Computational Algorithms C

100

200 C C C

143

------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) CONTINUE RETURN END ------------------------------------------------------------END OF CODE-SF-1.FOR -------------------------------------------------------------

2.4.1.3 Calculated Results

Again, streamfunction plots (from SF.DAT in our code output) are useful because they yield airfoil ordinates directly from the theoretical result y(x) = - (x,0). Several “snapshots” from our color plotter show that the required ellipse is replicated.

Figure 2.4.1a. Disturbance streamfunction.

144 Modern Aerodynamic Methods

Figure 2.4.1b. Ellipse semi-contour predicted.

Figure 2.4.1c. Ellipse semi-contour predicted – better visualizations obtained in “full-space” Example 2 run.

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2.4.1.4 Comments

We have demonstrated the ease with which inverse problems can be solved when trailing edge closure satisfying [ ] = 0 is required. This condition is automatically enforced using a half-space formulation together with the manner in which our box boundary conditions are applied. In other applications, it may be useful or necessary to model opened trailing edges. This includes instances when mass is actually ejected downstream, or in other applications, when the effects of thick viscous wakes or flow separation are to be considered. For these problems, [ ] is nonzero and it is more convenient to adopt a full-space formulation. This is considerd in the next section. 2.4.2

Example 2 – Inverse Problem, Pressure Specification, Centered Slit, Trailing Edge Closed vs Opened Software reference: CODE-SF-2.FOR

In Example 1 of this Section 2.4, we introduced our streamfunction approach to inverse problems, where pressure is specified along a slit and the surface shape is determined. To keep ideas simple, we considered a half-space formulation. This implicitly assumes that the “other halfspace solution,” not explicitly considered, is a mirror image. Thus, the trailing edge jump in streamfunction is identically zero and we in fact physically but implicitly model a closed trailing edge. When thickness effects of viscous wakes, or actual mass ejection from the trailing edge, are to be considered, nonzero [ ] jumps must be modeled. For such problems, it is more convenient to formulate fullspace as opposed to half-space models. Because streamfunction formulations can be confusing at first, particularly because they represent a new approach to inverse problems, we review our thinking process in a detailed step-by-step manner to introduce our error-checking methods. 2.4.2.1 Problem Setup In this section, we will first review our “potential function, analysis” method, explain key ideas, results and, importantly, visual spot checking. We do this so that the similarities and analogies to our “streamfunction, inverse” can be developed more naturally and guide our thinking. Review, “potential function, analysis” formulation. In Chapter 1, to explain analytical differences between logarithmic and arctan distributions, we decomposed a general airfoil shape into thickness and camber contributions, solved by the respective problems below.

146 Modern Aerodynamic Methods t

2

t y(x, t

=0 y = 0) U dFt(x) /dx along chord 2 2 0 as x + y

(1.2.1a) (1.2.1b) (1.2.1c)

and c

2

=0 (x, y = 0) U dFc(x)/dx along chord y c 0 as x2 + y2 plus Kutta condition (to be discussed) c

(1.2.2a) (1.2.2b) (1.2.2c) (1.2.2d)

For our present numerical purposes, it is fitting to re-combine the two foregoing formulations, leading to a single integrated model, 2

=0 y(x, y = 0)

u,l

U dF (x)/dx along chord

2

0 as x + y2 plus Kutta condition (to be discussed)

(2.1a) (2.1b) (2.1c) (2.1d)

This should be compared to our general, single, numerical inverse formulation in Equations 1.3.5a – 1.3.5d, renumbered here as follows, xx y

+

yy

=0

(2.2a)

(x, y = 0) = - ½ U Cp(x) 2

(2.2b)

2

0 as x + y plus trailing edge [ ] or [

x]

jump condition (to be discussed)

(2.2c) (2.2d)

Once the solution for (x,y) is available, the required airfoil geometry is easily calculated. In Chapter 1, we noted that the kinematic tangency condition dy/dx y /U should be integrated with respect to x, in the small-disturbance limit, leading to y(x) = - (x,0) /U + constant. What is remarkable is the following: the formulations in Equations 2.1a-2.1d and 2.2a-2.2d are mathematically identical. In both cases, normal “y” derivatives are specified on the slit. For analysis problems, [ ] is specified via Kutta’s condition, but for inverse problems, [ ] will be prescribed in a much simpler manner. Now, we have in mind the inverse problem for which both upper and lower normal derivative specifications for y (x, y = 0) = - ½ U Cp(x) are identical, that is, we have Cp+(x) = Cp- (x), for which the solution for the physical geometric body will obviously be a symmetric thickness distribution.

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But to understand how the streamfunction solution should be obtained, we need to run the simulator solving Equations 2.12-2.1d for a flat plate at angle of attack, satisfying a similar continuous boundary condition, here with dFu+(x)/dx = dF-l(x)/dx. That is, the analysis problem for camber is mathematically analogous to the inverse problem for thickness. The simulation is motivated by code in CODE-PHI-3.FOR. It is well known that, for a thin ellipse of thickness ratio > 0, the pressure coefficient (away from leading and trailing edges) is, in the small disturbance limit, Cp = -2 < 0, e.g., see Katz and Plotkin (2010). In what follows, we wish to replicate this geometric shape, starting with a specification of negative Cp on both surfaces plus a constraint calling for a closed trailing edge. Successful replication will validate the streamfunction method. Then, in a subsequent calculations, we relax the constraint and show how to open our trailing edge to allow mass flow. Again, so that we can develop our numerical ideas more easily, we now run our previously listed potential flow analysis simulator for CODE-PHI-3.FOR for our inclined flat plate (which has identical y specifications on either side of the slit) in order to understand what properties are relevant for possible debugging. Aside from the fact that y is continuous or identical across the slit y = 0, the exact numerical inputs and output results are not important. So, let us display the computed results and interpret them. Enter SLOPE (with decimal point) ... -0.01745 Angle (rad): -0.017 "Delta" PHI: 0.940 Lift coef CL: 0.235 Stop - Program terminated.

Figure 2.4.2a. Flat plate analysis potential results.

148 Modern Aerodynamic Methods

Figure 2.4.2b. Flat plate analysis potential results. The above computed results are for checking and possibly subsequent debugging purposes, to reassure ourselves that we running the correct code. The right side Cp curve in Figure 2.4.2a, shows several properties: (1) it does not show symmetry about midchord, (2) it is large in magnitude near the left, and (3) it is vanishing at the right, correctly corresponding to the pressure coefficient plot for a pitched flat plate. The left color plot and its numerical scale show that, for problems in which the normal derivative of the dependent variable are continuous (as for flat plates at angle of attack), the dependent variable is antisymmetric with respect to the slit y = 0. This is quantitatively apparent in the numerical tabulation, where we have highlighted to “0” line separating antisymmetric results. Streamfunction code. Consider, for example, Column 10 and numbers adjacent to the highlighted line. Note that the vertical subtractions “-0.4 – 0 = 0 – 0.4” are consistent with the fact that the velocities y on either side are continuous and identical. This check indicates that the algorithm solving Equations 2.2a – 2.2d, with analogously continuous functions Cp+(x) = Cp- (x), should employ the same numerical formulation as CODE-PHI-3.FOR. The source code is given on the following page. The reader should note the following lines, where DELTASF = 0 enforces trailing edge closure, that is – C C C C

Do not use "DELTASF = SFPLUS - SFMINUS", but hardcode a prescribed value of DELTASF jump instead DELTASF = 0 means trailing edge is closed DELTASF = 0.0 Values tested, 0.0, -0.5, -1.0, -3.0, +1.0

Computational Algorithms 2.4.2.2 C C C C

Fortran Source Code

CODE-SF-2.FOR (derived from CODE-PHI-3.FOR). Inverse method with closed or opened trailing edges, uses multivalued jumps in dependent variable. DIMENSION DIMENSION DIMENSION DIMENSION

C C

C C C C C C C

SF(19,19),A(19),B(19),C(19),V(19),W(19) ATOP(9),BTOP(9),CTOP(9),VTOP(9),WTOP(9) ABOT(9),BBOT(9),CBOT(9),VBOT(9),WBOT(9) SLOPE(19),POSITION(19),YCOORD(19)

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JW = 10 ITMAX = 10000 Research code for testing, hardcoded special parameters. DX and DY are grid lengths, inches; do not appear in PDE since DX = DY assumed, but appear in BC and CP definitions. UINF is speed at infinity, say in/sec. Thus SF is in^2/sec. No other output has units except SF. SF plotted vs indexes. DX = 1. DY = 1. UINF = 1.

C C C C C

C C

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Consider constant negative pressure coefficient CPVALUE = -0.4 GENERIC SLOPE FUNCTION Values at I = 1-5 AND 15-19 are dummies I = 6 to 14 represent actual slopes used SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. SLOPE( 6) = -0.5*CPVALUE SLOPE( 7) = -0.5*CPVALUE SLOPE( 8) = -0.5*CPVALUE SLOPE( 9) = -0.5*CPVALUE SLOPE(10) = -0.5*CPVALUE SLOPE(11) = -0.5*CPVALUE SLOPE(12) = -0.5*CPVALUE SLOPE(13) = -0.5*CPVALUE SLOPE(14) = -0.5*CPVALUE SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0. CONVENIENT CONSTANTS

150 Modern Aerodynamic Methods JMAX = 2*JW-1 JMAXM1 = JMAX-1 JWM1 = JW-1 JWM2 = JW-2 JWP1 = JW+1 ILEM1 = ILE-1 ITEP1 = ITE+1 IMAXM1 = IMAX-1 C C C

100 C C C C C

INITIALIZE SF TO ZERO, OTHER FUNCTION, OLD SOLUTION Boundary SF's all zero unless overwritten later DO 100 I=1,IMAX DO 100 J=1,JMAX SF(I,J) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300

C C

130 C C C

140 150 C C C C

ITER=1,ITMAX

UPSTREAM OF LEADING EDGE DO 150 I=2,ILEM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -SF(I-1,J)-SF(I+1,J) CONTINUE A(1) = 99. Below SF(1)-SF(2) = 0 OR DSF/DY = 0 which implies D(PHI)/DX = 0. Used in all vertical farfields. Try other regularity conditions too. B(1) = 1. C(1) = -1. W(1) = 0. A(JMAX) = 1. B(JMAX) = -1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX SF(I,J) = V(J) CONTINUE CONTINUE BELOW AIRFOIL SECTION Half columns vary from J = 1 to 9, keep JW line hidden J represents actual physical location DO 180 I=ILE,ITE DO 160 J=2,JWM2 ABOT(J) = +1. BBOT(J) = -4. CBOT(J) = +1.

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WBOT(J) = -SF(I-1,J)-SF(I+1,J) 160 CONTINUE ABOT(1) = 99. BBOT(1) = 1. CBOT(1) = -1. WBOT(1) = 0. C SF(JMAX-1)-SF(JMAX) = -DY*UINF*SLOPE(I) below, or C (SF(JMAX)-SF(JMAX-1))/DY = UINF*SLOPE(I) is correct BC C for our convenient function SLOPE(I) = -0.5*CPVALUE ABOT(JW-1) = 1. BBOT(JW-1) = -1. CBOT(JW-1) = 99. WBOT(JW-1) = -DY*UINF*SLOPE(I) CALL TRIDI(ABOT,BBOT,CBOT,VBOT,WBOT,JWM1) DO 170 J=1,JWM1 SF(I,J) = VBOT(J) 170 CONTINUE 180 CONTINUE C SFMINUS not used in iterations or post-processing C SFMINUS = SF(ITE,JW-1) test, use below extrapolation SFMINUS = 2.*SF(ITE,JW-1)-SF(ITE,JW-2) C C ABOVE AIRFOIL SECTION C Half columns vary from J = 1 to 9, does not include JW line DO 220 I=ILE,ITE C Here, J is an indexing parameter, not a physical location C Array indexes vary from J=1 to JW-1 DO 190 J=2,JWM2 ATOP(J) = 1. BTOP(J) = -4. CTOP(J) = 1. WTOP(J) = -SF(I-1,J+JW) -SF(I+1,J+JW) 190 CONTINUE ATOP(1) = 99. C SF(1)-SF(2) = -DY*UINF*SLOPE(I) below, or C (SF(2)-SF(1))/DY = UINF*SLOPE(I) is chosen identical to C above so normal derivative is continuous in formulation BTOP(1) = 1. CTOP(1) = -1. WTOP(1) = -DY*UINF*SLOPE(I) ATOP(JW-1) = 1. BTOP(JW-1) = -1. CTOP(JW-1) = 99. WTOP(JW-1) = 0. CALL TRIDI(ATOP,BTOP,CTOP,VTOP,WTOP,JWM1) DO 200 J=1,JWM1 SF(I,JW+J) = VTOP(J) 200 CONTINUE 220 CONTINUE C SFPLUS not used in iterations or post-processing C SFPLUS = SF(ITE,JW+1) test, use below extrapolation SFPLUS = 2.*SF(ITE,JW+1)-SF(ITE,JW+2) C C Enforce constant SF jump across "wake" branch cut C emanating from trailing edge

152 Modern Aerodynamic Methods C C C

Do not use "DELTASF = SFPLUS - SFMINUS", but hardcode a prescribed value of DELTASF jump instead DELTASF = 0 means trailing edge is closed DELTASF = 0.0 Values tested, 0.0, -0.5, -1.0, -3.0, +1.0

C C C C C C

DOWNSTREAM OF TRAILING EDGE Code is NOT identical to that upstream of leading edge. Jump [SF] enforced across downstream "wake."

C C

230

240 250 C C C

217 C C C C C

DO 250 I=ITEP1,IMAXM1 DO 230 J=2,JMAXM1 A(J) = +1. B(J) = -4. C(J) = +1. W(J) = -SF(I-1,J)-SF(I+1,J) IF(J.EQ.JWP1) W(J) = W(J) - 0.5*DELTASF IF(J.EQ.JWM1) W(J) = W(J) + 0.5*DELTASF Next block, continuity of d(SF)/dy|+ = d(SF)/dh- at wake is just continuity of d(PHI)/dx of pressure. IF(J.EQ.JW ) THEN B(J) = -2. W(J) = 0. ENDIF CONTINUE A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. A(JMAX) = 1. B(JMAX) = -1. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 240 J=1,JMAX SF(I,J) = V(J) CONTINUE CONTINUE UPDATE FARFIELD SF (SUBJECT TO CHANGE) Note farfield SF(I) at J=1 and JMAX also subject to change DO 217 J=1,JMAX SF(IMAX,J) = SF(IMAX-1,J) SF(1,J) = SF(2,J) CONTINUE All boundary conditions above involve derivatives, solution determined to within a constant. Fix SF at one point, select trailing edge SF to zero to fix all numbers (antisymmetry may also fix all numbers). SF(ITE,JW) = 0.

C 300 C C C

CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required).

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WRITE SOLUTION TO FILES Store disturbance streamfunction file (L^2/T units) Display to check symmetries and antisymmetries OPEN(UNIT=7,FILE='SF.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX WRITE(7,320) (SF(I,J),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write streamfunction for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\FILENAME.DAT',STATUS='UNKNO 1WN') WRITE(7,350) 350 FORMAT(' Disturbance Streamfunction') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370) 370 FORMAT('1 1') WRITE(7,375) 375 FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (SF(I,J),I=1,IMAX) 380 FORMAT(1X,19F12.3) 385 CONTINUE CLOSE(7,STATUS='KEEP') C C Write streamfunction SF at JW+1 for line plotting. View with LPLOT4. DO 390 I=1,19 POSITION(I) = I C YCOORD is negative of streamfunction per theory YCOORD(I) = -SF(I,JW+1) 390 CONTINUE OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE5.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) C WRITE(7,410) (SF(I,JW+1),I=1,19) WRITE(7,410) (YCOORD(I),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END

154 Modern Aerodynamic Methods C

100

200 C C C

------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) CONTINUE RETURN END ------------------------------------------------------------END OF CODE-SF-2.FOR -------------------------------------------------------------

2.4.2.3

Calculated Results

Again, we have chosen a coarse mesh for presentation purposes and do not expect exact numerical agreement with theory (in fact, eight meshes over the chord, five ahead of the leading edge, five behind the trailing edge, nine above and nine below the chord). We focus principally on shape results and trailing edge closure. Run No. 1, [ ] = 0, closed trailing edge.

Figure 2.4.2a. Closed trailing edge solution.

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Figure 2.4.2b. Closed trailing edge solution. The streamfunction field is antisymmetric about chord line and symmetric about perpendicular bisector to chord. The line curve to the right of Figure 2.4.2a shows the desired closed elliptical form. The numerical normal derivative from Figure 2.4.2b, from the center column, shows a continuous 1.8 – 0 = 0 – (–1.8) as required. Not shown is the complementary line curve opposite to the one in Figure 2.4.2a – we have also printed that out and it is identical in form. In the examples below, we modify only a single line of source code, namely DELTASF = 0.0 to the values for [ ] shown. Run No. 2, [ ] = - 0.5, slightly opened trailing edge.

Figure 2.4.2c. Slightly opened trailing edge.

156 Modern Aerodynamic Methods Run No. 3, [ ] = -1, opened trailing edge.

Figure 2.4.2d. Opened trailing edge. Run No. 4, [ ] = -3, widely opened trailing edge.

Figure 2.4.2e. Widely opened trailing edge.

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Run No. 5, [ ] = +1, surface cross-over at trailing edge.

Figure 2.4.2f. Surface cross-overs possible for certain pressure assumptions and streamfunction jumps. 2.4.2.4

Comments

The examples in Runs 1-4 demonstrate the flexibility in our “direct approach to indirect problems.” In particular, the work needed to alter an existing potential flow analysis code for inverse applications is minimal, and fortunately, numerous well written codes are already available in academic and industrial settings. In the last run of this section, we show how “surface cross-overs” are possible, either off the chord line or for airfoil ordinates themselves, as is evident from Figure 2.4.2f. These are possible for different surface pressure assumptions and streamfunction jumps. This motivates the need to extend our approach, allowing us to impose a “positive ordinate only” constraint on allowable solutions or assumed boundary conditions.

158 Modern Aerodynamic Methods 2.4.3

Example 3 – Inverse Problem, Pressure Specification, Three-Dimensional Half-Space, Closed Trailing Edge, Nonlifting Symmetric Section Software reference: CODE-SF-3.FOR

In this last example, we consider for simplicity a half-space formulation, but treating a finite span wing in which surface pressures are prescribed over the planform using y (x,0,z) = - ½ U Cp(x), while off the planform, the approximation y (x,0,z) 0 is used. This approximation is removed if a full-space approach is used, however, this presentation only focuses on the use of three-dimensional marching. 2.4.3.1 Problem Setup

In the above, we noted that the approximation y (x,0,z) 0 is used off the planform. Why is this only an approximation? Recall that y is proportional to x or the pressure coefficient – this is not necessarily vanishing although it is faraway from the wing. This assumption is necessary only because we used our simpler half-space model (for convenience only). If we had used a full-space formulation, which does entail much more programming, full vertical columns covering the entire vertical extent of the computational box would have been used, which do not terminate in the plane of the wing. Complete columns would offer a more accurate and less constrained solution. 2.4.3.2 Fortran Source Code C C C C C

CODE-SF-3.FOR Derived from CODE-PHI-5.FOR and CODE-SF-2.FOR Streamlike function 3D solver, symmetric finite wing on half-space, zero angle of attack, always nonlifting. DIMENSION SF(19,19,11),A(19),B(19),C(19),V(19),W(19) DIMENSION SLOPE(19),POSITION(19)

C C C C

C

C

GENERAL COMPUTATIONAL PARAMETERS ILE = 6 ITE = 14 IMAX = 19 JMAX = 19 KMAX = 11 Wing tip at K = KTIP, may vary from 3 or 4 up to 8 or 9 KTIP = 4 KPLOT = 1 ITMAX = 10000 Research code for testing, hardcode special parameters.

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DX, DY and DZ are grid lengths, inches; do not appear in PDE since DX=DY=DZ assumed, but appear in BC and CP definitions. UINF is speed at infinity, say in/sec. Thus SF is in^2/sec. DX = DY = DZ = UINF

C C C on C

C

C C

1. 1. 1. = 1.

SLOPE FUNCTION Defined for K = 1 to KTIP between leading and trailing edges wing planform, else set to zero on half-space symmetry plane. CPVALUE = -0.4 SLOPE( 1) = 0. SLOPE( 2) = 0. SLOPE( 3) = 0. SLOPE( 4) = 0. SLOPE( 5) = 0. Leading edge at I=6, trailing edge at I=14 SLOPE( 6) = -0.5*CPVALUE SLOPE( 7) = -0.5*CPVALUE SLOPE( 8) = -0.5*CPVALUE SLOPE( 9) = -0.5*CPVALUE SLOPE(10) = -0.5*CPVALUE SLOPE(11) = -0.5*CPVALUE SLOPE(12) = -0.5*CPVALUE SLOPE(13) = -0.5*CPVALUE SLOPE(14) = -0.5*CPVALUE SLOPE(15) = 0. SLOPE(16) = 0. SLOPE(17) = 0. SLOPE(18) = 0. SLOPE(19) = 0. CONVENIENT CONSTANTS IMAXM1 = IMAX-1 JMAXM1 = JMAX-1 KMAXM1 = KMAX-1

C C

100

INITIALIZE SF TO ZERO, OTHER FUNCTION, OLD SOLUTION DO 100 K=1,KMAX DO 100 I=1,IMAX DO 100 J=1,JMAX SF(I,J,K) = 0. CONTINUE

110

DEFINE BOUNDARY CONDITIONS IF REQUIRED, ZERO OR OTHER Overwrite these later if desired, e.g., if Grad SF = 0 is imposed DO 110 K=1,KMAX DO 110 I=1,IMAX SF(I,1,K) = 0. SF(I,JMAX,K) = 0. CONTINUE

C C C C

160 Modern Aerodynamic Methods

120 C C C C C

DO 120 K=1,KMAX DO 120 J=1,JMAX SF(1,J,K) = 0. SF(IMAX,J,K) = 0. CONTINUE SWEEP EQUATION SOLVER ACROSS BOX, ONE COLUMN AT A TIME Column relaxation used, implement SLOR convergence acceleration if desired (not done below for simplicity) DO 300 DO 290

C C

ITER=1,ITMAX K=1,KMAXM1

140 150

Code below for one sweep through computational box DO 150 I=2,IMAXM1 DO 130 J=2,JMAXM1 A(J) = +1. B(J) = -6. C(J) = +1. W(J) = -SF(I-1,J,K)-SF(I+1,J,K)-SF(I,J,K-1)-SF(I,J,K+1) CONTINUE IF(K.GT.KTIP) GO TO 135 Invoke pressure condition along J = 1 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = -DY*UINF*SLOPE(I) GO TO 138 A(1) = 99. B(1) = 1. C(1) = -1. W(1) = 0. End wing plane boundary conditions A(JMAX) = -1. B(JMAX) = 1. C(JMAX) = 99. W(JMAX) = 0. CALL TRIDI(A,B,C,V,W,JMAX) DO 140 J=1,JMAX SF(I,J,K) = V(J) CONTINUE CONTINUE

260

UPDATE FARFIELD SOLUTION Zero gradient (loops 260, 270) or zero function (loops 274, 276) yield slightly different fields. Both options shown below, hardcode your selection. Zero SF implemented below. KK is spanwise index, cannot use K because it redefines outer loop 290 span index. GO TO 272 DO 260 KK=1,KMAX DO 260 I=1,IMAX SF(I,1,KK) = SF(I,2,KK) SF(I,JMAX,KK) = SF(I,JMAXM1,KK) CONTINUE

130 C

135

C 138

C C C C C C C

C

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270 C 272 274

276

280 290 300 C C C C C C C C 320 340

161

DO 270 KK=1,KMAX DO 270 J=1,JMAX SF(1,J,KK) = SF(2,J,KK) SF(IMAX,J,KK) = SF(IMAXM1,J,KK) CONTINUE Add loop similar to 280 here, if this block is used GO TO 300 DO 274 KK=1,KMAX DO 274 I=1,IMAX SF(I,JMAX,KK) = 0. CONTINUE DO 276 KK=1,KMAX DO 276 J=1,JMAX SF(1,J,KK) = 0. SF(IMAX,J,KK) = 0. CONTINUE DO 280 I=1,IMAX DO 280 J=1,JMAX SF(I,J,1) = SF(I,J,2) SF(I,J,KMAX) = SF(I,J,KMAXM1) CONTINUE CONTINUE CONTINUE End of iterative sweep through computational box, repeat until ITMAX completed or converged (test required). WRITE SOLUTION TO FILES Store 3D streamlike function file (L^2/T units) Display to visually check symmetries and antisymmetries Same data as in DATA.DAT but formatted differently OPEN(UNIT=7,FILE='SF.DAT',STATUS='UNKNOWN') DO 340 J=1,JMAX Root plane solution located at KPLOT = 1. WRITE(7,320) (SF(I,J,KPLOT),I=1,IMAX) FORMAT(1X,19F5.1) CONTINUE CLOSE(7,STATUS='KEEP')

C C Write streamlike function for 2D color plotting C After simulation completed, use DSTRANGE32 or ACTIONS32 to view. OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\filename.DAT',STATUS='UNKNO 1WN') WRITE(7,350) KPLOT C350 FORMAT(' Streamlike Function (K = 1)') for root plane 350 FORMAT(' Streamlike Function (K = ',I2,')') CLOSE(7,STATUS='KEEP') OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\DATA.DAT',STATUS='UNKNOWN') WRITE(7,360) 360 FORMAT('GRID 19 19') WRITE(7,365) 365 FORMAT('100') WRITE(7,370)

162 Modern Aerodynamic Methods 370 375 380 385 C C

FORMAT('1 1') WRITE(7,375) FORMAT('1 1') DO 385 J=1,JMAX WRITE(7,380) (SF(I,J,KPLOT),I=1,IMAX) FORMAT(1X,19F9.3) CONTINUE CLOSE(7,STATUS='KEEP')

Write SF function for line plotting DO 390 I=1,19 POSITION(I) = I 390 CONTINUE C View with LPLOT4.FOR software OPEN(UNIT=7,FILE='C:\WILEY-BOOKSAERO\MYFILE5.DAT',STATUS='UNKNOW 1N') WRITE(7,400) 400 FORMAT('ARRAY 1 19') WRITE(7,410) (POSITION(I),I=1,19) WRITE(7,410) (SF(I,1,KPLOT),I=1,19) 410 FORMAT(1X,19F10.3) CLOSE(7,STATUS='KEEP') C STOP END C ------------------------------------------------------------SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(19),B(19),C(19),V(19),W(19) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN 100 CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) 200 CONTINUE RETURN END C ------------------------------------------------------------C END OF CODE-SF-3.FOR C -------------------------------------------------------------

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2.4.3.3. Calculated Results

In the figures below, we plot airfoil ordinates at planform locations successively away from the root K = 1 plane. These show a consistent reduction in airfoil ordinate values.

Figure 2.4.3a. Root plane (K = 1) wing ordinate.

Figure 2.4.3b. Spanwise plane (K = 4) wing ordinate.

164 Modern Aerodynamic Methods

Figure 2.4.3a. Off-planform (K = 8) streamfunction value. 2.4.3.4 Comments

Figures 2.4.3a,b,c show wing surface ordinates starting from the wing root plane, where they are largest, to the tip, where there are smaller, and finally, to an off-planform location where the streamfunction magnitude is very small. Again, a more accurate threedimensional inverse implementation would call for “full-space” instead of half-space models. In Chapter 4, extensions of the model for transonic supercritical effects are derived and explained in detail, and example calculations are offered.

Modern Aerodynamic Methods for Direct and Inverse Applications.Wilson C. Chin. © 2019 Scrivener Publishing LLC. Published 2019 by John Wiley & Sons, Inc.

3 Advanced Physical Models and Mathematical Approaches In Chapters 1 and 2, we provided basic introductions to conventional aerodynamic modeling methods and raised questions about their limitations and the need for practical extensions. Chapters 3 and 4 address these concerns, develop new mathematical formulations, and solve them with custom designed algorithms. The present chapter focuses on general ideas within a range of aerospace applications, while the next deals with new approaches to analysis and inverse problems. Finally, Chapter 5 summarizes ideas associated with engine and airframe integration, presented through attractive visuals that excite the imagination in a manner that encourages usage of the methods developed in this book. We now briefly describe the topics covered in this chapter. Unsteady transonic flow. When aerospace vehicles travel at high subsonic speeds, rapid accelerations about wing leading edges may lead to local, embedded, supersonic flow regimes terminated by shockwaves. As explained earlier, the location and extent of these zones are unknown a priori and are determined numerically using iterative mixed-difference schemes. The presence of surface shocks can be destabilizing in flight. For example, the sudden increase in pressure across the shock can separate boundary layer flows; further, the possibilities of shock oscillations (say, induced by wing vibration or wind gusts) and their physical consequences must be addressed. In linearized transonic flow approaches, a main steady flow is first computed, assuming that transient disturbances are completely absent. Then, a linearized formulation is solved, also using mixed-type methods, which models unsteady properties about the steady solution just obtained – a steady solution that is obtained once and for all and which is “frozen” irrespective of any 165

166 Modern Aerodynamic Methods shockwave motions found in its presence. The linearized boundary value problem is solved for a range of frequencies, and completely transient solutions can be obtained, for instance, by Fourier superposition. While this approach is well-defined algorithmically, it is physically incorrect. It is known experimentally that the method is justified at “high frequencies” where there is insufficient time for shocks to respond (e.g., to flows induced by moving flaps), but at low frequencies, shock displacements from equilibrium may be significant. Thus, we ask, “How do we model the back-interaction of a wave-like flow on the mean steady flow?” This transonic problem is addressed in Sections 3.1, 3.2 and 3.3. In linearized formulations, not necessarily those in aerodynamics, but in other areas of continuum mechanics also, transient formulations which describe disturbance motions “derived about mean static solutions” are solved, once the baseline mean solution is available. This is incorrect. To be specific, suppose that U(x,y,z,t) represents the complete physical description to any transient system and that, in the absence of unsteady effects, its solution is U0(x,y,z). When disturbances are introduced, the net field is denoted by U(x,y,z,t) = U0(x,y,z) + U1(x,y,z,t). So far, our statements are completely general. Now, because the transients are linear and may involve phase lags, the disturbance U1(x,y,t) is usually assumed in the form U1(x,y) ei t, and a derived equation for the time-independent shape function U1(x,y) is instead solved, once for each disturbance frequency . Real solutions are extracted from converged complex solutions using Re U1(x,y) or Imag U1(x,y) depending on the adopted sign convention. In any event, the mean baseline solution U0(x,y,z) never changes – it is solved once and for all and remains unaltered for all frequencies. This is the unfortunate consequence arising from “expansions in amplitude,” that is, the development of disturbance models based on the perceived magnitude of the imposed perturbation. In our approach, we abandon amplitude expansions. Instead, we assume U(x,y,z,t) = U0(x,y,z) + U1(x,y,z,t) + U2(x,y,z,t) . . . = U0(x,y,z) +{U1(x,y,z) ei t + U1*e-i t} + {U2(x,y,z) ei t + U2*e-i t} +. . . and so on. Note that asterisks denote complex conjugates. Thus, each group of terms within curly brackets is completely real – that is, our series for U(x,y,z,t) is real at the outset and “real part” Re and or “imaginary part” Imag operations are never used. When this series is substituted in the complete formulation for U(x,y,z,t), to include boundary conditions, and sequences of boundary value problems are defined by equating coefficients of like harmonics, the formulation for U1 will be similar to

Advanced Models and Ideas 167 that previously discussed but corrected for higher harmonic contributions. However, now the equation governing U0 will have additional U1, U2, U3 (and so on) terms, so that U0 cannot be solved once and for all as before. In fact, all of the problems Un are nonlinearly coupled. In order to simplify the formulation, harmonics higher than the primary are ignored; this is, we ignore the effects due to = 2, 3, 4 and so on, leaving a fully coupled problem for U0 and U1 only. One might reasonably ask, “Is it not simpler to solve the transient problem once and for all, say using a numerical ‘alternating direction implicit’ or “approximate factorization’ scheme?” The answer is, “Yes,” in a narrow sense only. However, in many engineering applications, it is the frequency dependence of the solution that is of interest. For example, below a certain critical frequency, disturbances may be well-behaved and decay with time – higher frequencies may be associated with dangerous oscillations. What is the critical frequency? Are there more than one? What are the damping and amplification rates associated with different frequency modes? A fully numerical transient solution will not yield this information – it will need to be decomposed and Fourier analyzed, an extremely tedious and inaccurate process. On the other hand, we do emphasize that solving numerous boundary value problems for different frequencies (and each with different mean flows) is not as intimidating as it appears. Once a solution set is available, it can be used to initialize the iterative solution of the following solution set, so the entire effort only requires incremental effort and limited computing resources, if the initializing solutions are cleverly selected and stored. Hydrodynamic stability of mean shear flows. In this book, inviscid aerodynamics is considered for simplicity, but in practice, inviscid and viscous flow shock interactions are all-important. Not only are such interactions critical to aircraft instability, but even viscous flow effects alone cannot be ignored – after all, viscosity is responsible for viscous drag. And viscous drag depends on the character of the underlying shear flow – is it laminar or is it turbulent? How does the shape of the mean velocity profile affect transition to turbulence? The field of “hydrodynamic stability” addresses such concerns. The simplest problem treated historically is the inviscid stability of “parallel shear flows,” meaning a boundary layer (or other) velocity profile whose horizontal speed varies with the vertical coordinate. Inviscid problems are treated by a so-called Rayleigh formulation while viscous effects are considered in the more detailed Orr-Sommerfeld problem. In either case,

168 Modern Aerodynamic Methods a mean velocity profile is prescribed or “frozen,” and the properties of disturbances riding on the main profile are studied. It is clear that the same formulation issues arise here as in unsteady transonic flow. Given the nonlinearity underlying most fluid-dynamics processes, one would expect that propagating wave-like disturbances could have a significant back-interaction effect on the main flow. Would a disturbance flow unstable on a linear basis be less so when this nonlinear effect is considered? This problem is studied in Section 3.10 and definitive conclusions are offered. A second problem of interest is the effect of a flexible membrane surface – this was motivated by field observations related to the hydrodynamic efficiency of swimming sea animals and their possible use of soft body undulations in controlling the onset (or, possibly, the structure) of turbulence. In Section 3.11, we consider the linear hydrodynamic stability problem and provide estimates for growth rates related to the mechanical and fluid-dynamical parameters describing the system – a fascinating area of study indeed. Similar approaches can be used to evaluate oscillations of plates and shells. Supersonic wave drag for general airframes. Classical aerodynamics is well known for successful approximations required to produce usable answers. Many have been identified and developed, e.g., thin airfoil theory, induced drag via lifting line theory, vortex lattice methods, among others. The commercial viability of an aircraft depends on high lift and minimal drag – for the latter, components due to surface boundary layers, induced vortex effects, and lastly, supersonic drag in terms of radiated wave energy. In Sections 3.4 and 3.5, a general theory is developed to compute the latter, which extends a well known model to aircraft of arbitrary shape – that is, to aerodynamic bodies which need not be thin nor slender but completely general. The formulas are general and intended for use with modern supersonic panel methods. Viscous transonic equation. A loss mechanism associated with supersonic flow is that associated with ray coalescence in the farfield. This convergence of wave characteristics is due to cumulative nonlinear effects. Note that subsonic aircraft, for instance, airplanes flying at 550 mph, may find local shockwaves along upper aerodynamic surfaces – these are associated with transonic surface drag losses. An airplane flying at supersonic speeds will likely be free of surface shockwaves and the methods of Sections 3.4 and 3.5 will suffice insofar as computing wave radiation away from the aircraft is concerned. The characteristics or rays emanating from the aircraft into the farfield will typically

Advanced Models and Ideas 169 coalesce to create the “sonic booms” heard beneath the flight paths of these vehicles. These large sudden pressure disturbances can be modeled using the nonlinear inviscid methods developed by Whitham, Lightlhiil, Hayes and others. In this chapter, we introduce sonic boom modeling using the “viscous transonic equation,” but only in a peripheral manner in Section 3.6, providing an analytical model for ray convergence and the detailed physical structure within the shock. However, we are more interested in exploring the application of the viscous transonic equation to simplifying mixed-type transonic analysis methods. As discussed earlier, in relaxation methods, each nodal point must be “type tested” to determine whether it is subsonic or supersonic at that point in the iterations, and then, the appropriate central or backward difference molecule must be selected. And typically, the flowfields are solved or “swept” in a direction that is consistent with domains of influence and dependence – the shapes and extents of which are unknown at the outset and which evolve during the solution process. Not only is the process labor intensive, but it places significant limitations on curvilinear grid selection and generation, since solutions are often obtained along coordinate lines in that space. We ask, “Is it possible to devise a method that does not require type or mixed differencing which does reproduces the generally accepted features behind subsonic and supersonic flows with embedded supercritical zones with shocks?” This is the problem addressed in Sections 3.7 – 3.9, inclusive. The conventional transonic partial differential equation is a second-order equation whose discriminant is elliptic or hyperbolic according to standard definitions. What if we added a high-order term, in particular, a diffusive third-order xxx term to this potential equation where is a small, positive and real longitudinal viscosity quantity rendering it parabolic? Can we eliminate type-differencing? Is it possible to obtain physically correct solutions, now “sweeping” the flowfield unrealistically from downstream to upstream? These interesting questions are answered using a variety of test algorithms and successful results are explained. Finally, theoretical issues related to the correct forms of the boundary conditions and differential equations needed for proper flow modeling are considered in Sections 3.12 and 3.13. As explained previously, Chapter 4 is devoted to clever and innovative ways to solve analysis and inverse problems, addressing topics that are in addition to those covered in this chapter, while a Chapter 5 deals with ideas behind engine and airframe integration modeling essential in aircraft design.

170 Modern Aerodynamic Methods 3.1 Nonlinear Formulation for Low-Frequency Transonic Flow 3.1.1 Introduction

We will investigate the two-dimensional unsteady transonic flow past a thin airfoil executing small-amplitude harmonic oscillations. The usual approach (Ehlers, 1974) solves a (time) Fourier-transformed problem linearized about a prescribed steady flow obtained under the assumption that transient disturbances are completely absent. This mean flow is obtained (once and for all) from a type-dependent relaxation method (Murman and Cole, 1971), and its iterative solution determines the variable coefficients of a linearized, frequency-dependent, mixedtype problem. This linearization, however, is not uniformly valid in time for one important reason: the non-harmonic part of the total flow must change for different disturbance frequencies and amplitudes because the problem is nonlinear. The mean flow must not be assumed known for all time; the “ expansion” (that is, the straightforward linearization) therefore breaks down because it does not describe the mathematically required “back-interaction” that arises from the nonlinear harmonic interplay anticipated on physical grounds. Here this defect is remedied by equating not coefficients of like powers of a small-amplitude measure (in determining a sequential ordering of problems), as one would have in standard perturbation expansions, but coefficients of like harmonics. The resulting equations describe the feedback phenomenon, and if they can be solved exactly, no physical approximation will be involved. For simplicity, however, we will consider the effect of the primary harmonic only and use the smalldisturbance formulation (there is no conceptual difficulty in writing down more terms, however). The back-interaction effect, of course, will be important for lowfrequency problems where shock excursions are generally large, and may have a significant effect on mean shock jumps and location. The effect will likely be important in aircraft stability insofar as shockwave and viscous flow interactions are concerned. For high frequencies, this nonlinear effect is probably less important, and a linearized approach should do. Here we show how flutter criteria depend on both frequency and amplitude boundaries in a physically self-consistent way.

Advanced Models and Ideas 171 3.1.2

Analysis

Here we let M be the subsonic freestream Mach number, U the freestream speed, C the chord, the thickness ratio, and the ratio of specific heats. We normalize the streamwise coordinate x0, the normal coordinate y0, and the time t0 by defining nondimensional variables using x = x0/C, y=[(1+ ) M 2] 1/3 y0/C, and t = [(1+ ) M 2] 2/3Ut0/M 2C. Next introduce a nondimensional disturbance potential by expanding the total dimensional velocity potential in the form UCx + 2/3 UC (x,y,t)/ [(1+ ) M 2] 1/3. Let us also define a reduced frequency by k = C/U, where is the frequency of the airfoil oscillation, and a nondimensional frequency = M 2k/[(1+ ) M 2] 2/3. Then substitution into the full potential equation leads to the small-perturbation equation (K* -

x) xx +

yy

=2

xt

+ (k/ )

(3.1.1)

tt

where the so-called “transonic similarity parameter” satisfies the definition K* = (1 - M 2)/ [(1+ ) M 2] 2/3. Let y = gu,l(x,t) represent upper and lower wing surfaces. Then Equation 3.1.1, which is nonlinear, is solved along with the linearized boundary conditions: y

(0 < x < l; y = 0) = ( / x + k/ x + k/

2 x

+

2 y

x + k/

t)x>l, y=0+ = 2

/ t) gu,l(x,t)

t)x>l, y=0-

(3.1.2) (3.1.3)

2

0 as x + y

(3.1.4)

We assume that the unsteady motion is a small perturbation on the steady flow, and characterize it by a small nondimensional displacement and the reduced frequency. In view of the preceding discussion, we expand both the airfoil motion and the disturbance potential in real harmonic series, g(x,t) = g0(x) + ½ {g1(x) ei t + g1*e-i t} + . . . (x,y,t) = 0(x,y) + ½ { 1(x,y) ei t +

1

* -i t

e

} +. . .

(3.1.5) (3.1.6)

asterisks denoting complex conjugates. Higher harmonics, not shown, are related to higher-order amplitude effects that can be consistently neglected. These will generally take the form “( ) ein t + ( )* e-in t ” where n = 2, 3, 4 and so on. Substitution of Equations 3.1.5 and 3.1.6 in Equations 3.1.1 - 3.1.3 leads to the mean flow formulation {K*

0x –

½

2 0x

– ¼ 2|

2 1x| }/

x+ {

0y}/

y =0

(3.1.7)

172 Modern Aerodynamic Methods 0y(0

< x < l; y = 0) = g0u,l(x)/ x

0x(x > l;

y = 0+) =

0x(x > l;

(3.1.8)

y = 0-)

(3.1.9)

for 0(x.y), on equating coefficients of the zeroth harmonic, and on examining the first harmonic, to the following 1 formulation: (K* 1y(0 1x

0x) 1xx +

1yy

–(

0xx + 2i

)

1x + k

1

= 0

(3.1.10)

< x < l; y = 0) = (g1x + ikg1)u,l

+ ik 1)(x > l; y = 0+) =

1x

(3.1.11)

+ ik 1)(x > l; y = 0-)

(3.1.12)

This process can be repeated for n = 2, 3, 4 and so on, if desired. Regularity conditions, of course, are imposed on both of the preceding formulations (the inclusion of the tt term renders the theory applicable to reduced frequencies of order unity). If the 2 term in Equation 3.1.7 is dropped, 0 decouples from 1, and the governing unsteady equations reduce to the usual linearized equations of Ehlers (1974) and Landahl (1961). These however, as noted, do not account for modifications to the mean flow (induced by the unsteady, nonlinear back-interaction that must become important over large time scales). To retain the nonlinear coupling requires a simultaneous solution for 0 and 1, which will, in practice, be mixed elliptic and hyperbolic, requiring numerical methods. The nonlinear formulation must be completed by proper specification of “shock conditions” in the event that discontinuities form. It is essential to use the physically correct conservation form of Equation 3.1.1. This is achieved by recognizing Equation 3.1.1 as an expression for mass conservation – here, the time rate of change of density, for example, contributes a (k/ ) tt term and one xt, while the remaining xt term arises from the streamwise mass flux term. When viewed in this manner, the required conservation form is (–

x

– k t/

)/ t + (K*

x–

½

2 x

–

t

)/ x + ( y )/ y = 0

(3.1.13)

We will also invoke irrotationality, that is, vx - uy = 0, where u and v are streamwise and stream-normal velocity perturbations, respectively (in three dimensions, the corresponding equations are much more complicated). Now let (x,y,t) = x - xs(y,t) = 0 describe the instantaneous shock surface x = xs(y,t). Corresponding to Equation 3.1.13, we have [–

x

– k t/ ]

t

+ [K*

x–

½

2 x

– t]

x + [ y]

y

=0

(3.1.14)

Advanced Models and Ideas 173 where [ ] denotes the jump of the enclosed quantity across x = xs(y,t), while from irrotationality, we have [ y]

x–

[ x]

y

=0

(3.1.15)

Next, expand xs(y,t) in the form xs(y,t)= xM(y) + ½ {ei t f(y) + e–i t f *(y)} . . .

(3.1.16)

where the subscript M denotes the unknown mean shock location and the asterisk denotes the complex conjugate (exponential sums such as are used in Equations 3.1.5, 3.1.6 and 3.1.16 are employed in nonlinear analyses to keep physical quantities real, in contrast to methods used in linear modeling, where real parts of complex solutions are taken upon final solution). Now, (xs) (xM) + (xs – xM) x (xM) can be rewritten by replacing xs – xM using Equation 3.1.16 and then introducing Equation 3.1.6. Substitution in [ ] = 0, which is implied by Equation 3.1.15, leads to a result that expresses jumps in 0 and 1 and their derivatives about the mean shock location xM. Equating coefficients of the zeroth harmonic then leads to [

0] xM + ¼

2

* * 1x ] xM + f [ 1x] xM}

{f [

=0

(3.1.17)

where the jump is evaluated at xM, while coefficients of the first harmonic lead to [ 1] xM + f [

0x ] xM = 0

(3.1.18)

Next let us expand Equation 3.1.14, using Equation 3.1.16. Again we express the jump conditions for both mean and unsteady problems about the mean shock location x = xM(y). Equation 3.1.16 is first substituted into Equation 3.1.14; then, we expand (xs) in a Taylor series about x = xM, and introduce Equation 3.1.6 into the resulting equation. For simplicity, the resulting equation is truncated to retain only those nonlinear effects due to the primary harmonic. Thus, the jump condition for the mean flow becomes [K*

0x –

½

2 0x

–

0y

=¼

xM’ ] xM = 2

[| + +(

2

1x|

+ |f |

0x ( 1xx f 1y

(3.1.19) 2 *

2 0xx

+

+f

* 1xx

f ’* + f ’

* 1y )

0xx ( 1x

)+

*

f +

0yx (ff

+ xM’ (f *

* 1x f

)

*

’ + f ’f *)

1yx + f

* 1yx

)

174 Modern Aerodynamic Methods * 1x

– 2i ( f –

–f*

k( 1f* –

1

*

1x )

– K* ( f *

f )]

xM

1xx –

f

* 1xx )

while for the flow related to the primary harmonic, we have [(K* – – xM’ (

0x) 1y

(

1x +

+f

f

0xx)

–i (

1

–

0x f

)–

0y

f’

0yx )] xM = 0

(3.1.20)

For convenience, the f ’s and xM’s are shown within the square brackets. Finally, the jump conditions corresponding to Equation 3.1.15 can be shown to be [

0y

–¼ [

1y

+ 2

+

0x xM’

[

]

0xx (ff 0x

xM = ’* * ’

+f f ) +(

f ’ + xM’ f

(3.1.21) 1x

*

f ’ +f

’

’ 0xx + xM 1x] xM = 0

* 1x )

+ xM’

(f

*

1xx + f

* 1xx )] xM

(3.1.22)

This completes the nonlinear formulation, noting that the preceding equation system must, in general, be solved iteratively. We observe how the formulation for the mean potential 0 contains unknown 1 effects, and that the 1 formulation similarly contains 0 influence, requiring a hsimultaneous solution for two coupled boundary value problems. It goes without saying that extension of our nonlinear interaction model to the second harmonic will require three coupled formulations, extension of the model to the third harmonic will require four formulations, and so on. 3.1.3 Discussion and Summary

We suggest a basic numerical approach that consists in solving and updating the mean and disturbance flows iteratively. Initially, all velocity potentials can be set to zero, or alternatively, to freestream conditions. For example, one relaxation cycle (taken to partial or full convergence) using Equations 3.1.7 - 3.1.9 can be followed by a similar cycle using Equations 3.1.10 - 3.1.12, taking proper account of conservation form; the resulting 1 can be used in Equations 3.1.7 - 3.1.9 to update 0, and so on, with the recursively obtained converged solution used to calculate the shock motion and unsteady surface pressures. It is not clear whether or not this type of iteration contains any implied assumptions on the weakness of the back-interaction, but algorithms developed along these lines require little more than simple

Advanced Models and Ideas 175 modifications to existing baseline software codes. In the long run, though, “alternating direction implicit” (that is, “ADI”) direct timeintegration schemes may be more economical (such an analysis is presented in Section 3.2). But if harmonic analyses are required (as in flutter and hydrodynamic stability research), the present work may be more useful insofar as indicating the extent to which purely linearized approaches, by themselves, may suffice. When the nonlinear coupling can be safely disregarded, and what the qualitative features of nonlinearity are, for example, with regard to shock jumps and location, the resolution of steady and unsteady forces, etc., are objectives of some numerical studies currently anticipated. Finally, note that some unsteady transonics algorithms solving 2kM 2 xt + {1- M 2 - ( + 1) M 2 x} xx + yy = 0 have been formulated incorrectly. One study recasts this in conservation form as (2kM 2 x)/ t + {(1- M 2) x – ½ ( +1) M 2 x2}/ x + ( y)/ y = 0, not knowing how first principles require that 2 xt partition itself into two contributions so that (kM 2 x)/ t + {kM 2 t + (1- M 2) x – ½ ( +1) M 2 x2}/ x + ( y)/ y = 0. Alternating direction implicit (ADI) and approximate factorization (AF) schemes need to be carefully reviewed in this regard. 3.1.4 References

Ehlers, F.E., “A Finite-Difference Method for the Solution of the Transonic Flow Around Harmonically Oscillating Wings,” AIAA Paper 74-543, Palo Alto, Calif., 1974. Landahl. M. T., Unsteady Transonic Flow, International Series of Monographs in Aeronautics and Astronautics, Pergamon Press, London, 1961. Murman, E.M. and Cole, J.D., “Calculation of Plane Steady Transonic Flows,” AIAA Journal, Vol. 9, No. 1, 1971, pp. 114-121.

176 Modern Aerodynamic Methods 3.2 Effect of Frequency in Unsteady Transonic Flow 3.2.1

Introduction

A characteristic of transonic unsteady flows is the potentially large phase lag between boundary motion and induced surface pressure. Moreover, net force coefficients generally exceed those in subsonic and supersonic speed regimes. These effects tend to increase the likelihood of aeroelastic instabilities, making transonic speeds most critical for aircraft flutter and airplane stability. In this Section frequency effects are systematically considered within the framework of transonic small-disturbance theory for three different dynamic configurations: airfoil pitching oscillation, trailing edge flap oscillation, and impulsive change to angle of attack. An approximate factorization method applicable to general unsteady motions is devised to investigate the net lift and moment coefficients for a NACA 64A010 airfoil section, at Mach 0.82 and for three reduced frequencies, namely, 0.05, 0.5 and 5. Computed results are then compared against those obtained in the low frequency approximation. The role of nonlinearities at supercritical speeds was well appreciated in Unsteady Transonic Flow, the classic monograph of Landahl (1961) written more than five decades ago. Section 3.1 of the present book expands upon conventional linear formulations. In particular, we explained how harmonic interactions can impose a backinteraction on the mean steady flow, which will in turn alter the behavior of the primary oscillating flow. The previous section explained how coupled dynamical fields such as these can be efficiently solved using “alternating direction implicit” (or “ADI”) or “approximate factorization” methods. This approach is pursued in the present section and physical consequences for real flows are explored computationally. We ask and answer a simple question – what really happens at transonic speeds in practical problems? Generally speaking, small surface motions can induce large changes in aerodynamic loading, as well as large increments in shock excursion. These considerations arise from the inherent nonlinearity of the mathematical problem. Thus, in the numerical sense, direct time integration, which does not bear the limiting restriction to “frozen” shock movements, typical of linearized approaches, must be used. In the low-frequency approximation, solutions to the transonic small-disturbance equation, along these lines,

Advanced Models and Ideas 177 simulate well the expected flow nonlinearity, including irregular shockwave motions. Notable among numerous are the original works of Ballhaus and Goorjian (1977, 1978) and Ballhaus and Steger (1975). Their results compare well with those obtained from the unsteady Euler equations and are in good agreement with experiment. However, their restriction to low frequencies may, in practice, be severe; more rapid oscillations, as well as unsteady gust loadings, are excluded from consideration. The essential loss is that in phase-shift information and it is remedied by retaining in the governing equations the higher frequency tt term. This forms the substance of the numerical algorithm discussed in this Section. Of course, the extent to which the low frequency approximation holds is also of fundamental interest, and comparisons are given for this purpose. This is important in assessing the effect of frequency on transonic phase shifts and bears practical significance on direct aeroelastic applications. 3.2.2

Numerical Procedure

We will consider Equation 3.1.1 in Section 3.1, or as conveniently normalized here, the nondimensional equation, A

+ 2B

=C

+

(3.2.1)

Here is a disturbance velocity potential nondimensionalized by cU 2/3, c being the chord, U the freestream speed at infinity, and the be an oscillation maximum thickness-to-chord ratio. Also let frequency and define a dimensionless reduced frequency by k = c/U . If M is the freestream Mach number, the coefficients appearing in Equation 3.2.1 are A = k2M 2/ 2/3, B = kM 2/ 2/3 and C = (1 – M 2)/ 2/3 – ( + 1) M 2 , where is the ratio of specific heats, nondimensional coordinates having been assumed with the definitions = U kt/c, = x/c and = 1/3 y/c, with x, y and t being streamwise, transverse and time variables. For small values of k, the coefficient A vanishes, and Equation 3.2.1 leads to the low-frequency approximation. In general, Equation 3.2.1 is solved together with the surface tangency condition ( ,0 , ) = f + akf along the chord 0 1. Here f is an instantaneous airfoil displacement normalized by , with a = 0 in the low-frequency limit and a = 1 in the general case. In addition, pressure continuity is enforced across the wake aft of the trailing edge, and disturbance velocities are

178 Modern Aerodynamic Methods assumed to vanish at the farfield approaching infinity. The unsteady pressure coefficient is evaluated from Cp = - 2 2/3 ( + ak ). Solutions to Equation 3.2.1 were obtained by a non-iterative alternating direction implicit (ADI) numerical scheme that advances the solution for at each mesh point from time level tn to time level tn+1. A sequential procedure consisting of a -sweep followed by a -sweep is employed. This process is invoked repetitively in time, that is, taking (2B/

)

( *-

n

n

) =D g +

{A/( )2} { n+1 – 2 n + n-1} + (2B/ ) ( n+1 – *) = ½

(

. . . . . . . . . -sweep

n+1

–

n

) ....

-sweep

(3.2.2a)

(3.2.2b)

where g = ½ {Cn * + (1 – M 2) n/ 2/3}. This decomposition is a consistent approximate factorization of Equation 3.2.1 accurate to O( ). In the low-frequency limit, it reduces to the LTRAN2 algorithm developed in Ballhaus and Goorjian (1977), where the latter method is second-order accurate in time and unconditionally stable on a linearized von Neumann basis (see Ballhaus and Steger (1975)). For general unsteady motions, the present scheme, GTRAN2, while not unconditionally stable, appears to be quite reliable and instabilities were not observed in the current run portfolio. Note that denotes the central difference operator, the backward difference streamwise operator, and D the mixed-type difference operator discussed in Ballhaus and Goorjian (1977). Details related to the implementation of tangency, wake and farfield conditions, as well as those for correct shock capture, for brevity, will not be developed here, these being straightforward modifications to LTRAN2. 3.2.3 Results

Comparisons of low and general-frequency solutions to Equation 3.2.1 for the NACA 64A010 airfoil section were made at Mach 0.82 for three types of transient motions: airfoil pitching oscillation, trailing edge flap oscillation, and impulsive change to angle of attack. All time = 0 with a potential distribution integrations were initiated at corresponding to the steady state solution. In these comparisons, all solutions were obtained using identical spatial mesh distributions and spatial differencing for physical consistency. In addition. the time step sizes in GTRAN2 and LTRAN2 were chosen so that GTRAN2

Advanced Models and Ideas 179 2 LTRAN2 ,

making the truncation errors formally of the same order. Generated solutions used 360 time steps for GTRAN2 and 48 time steps for LTRAN2 per period of oscillation at any reduced frequency. Results values indicated that these obtained using both larger and smaller choices adequately insured numerical accuracy. A nonuniform computational mesh consisting of 113 and 97 points, with 48 points = 0.0025 lying over the chord was used. Minimum grid spacings were and = 0.01. The computation box was approximately defined by the limits 200 > | | and 400 > | | Oscillation cases were time integrated over four periods for the reduced frequencies k = 0.05, 0.5 and 5.0. Figure 3.2.1 compares low and general-frequency results for a = 1 deg airfoil pitching oscillation about the mid-chord for the unsteady lift and pitching moment coefficients (the moment is taken about the pitch point). At k = 0.05, the results are similar except that peak CL’s and CM’s are slightly overpredicted by the low-frequency approximation. With k = 0.5, the results appear to agree in phase, but amplitudes differ by about 30%. At k = 5.0, a phase difference of about 45 deg is observed in addition to differences in predicted amplitude levels. As k increases, there is a reduction in lift due to a decrease in the shock excursion; the moment, however, remains high because of the shock traversing the pitch point. Physical considerations suggest that shock excursion amplitudes decrease with increasing k. This is confirmed numerically in both models; decay rates in LTRAN2, however, far exceed those observed in GTRAN2. Figure 3.2.2 compares deg trailing edge flap oscillation about = 0.75. results for a = Results for k = 0.05 are similar to those for airfoil oscillation. For k = 0.5, however, the low frequency result over-predicts the moment, unlike the airfoil oscillation case. The lift and moment are substantially lower for the flap oscillation than for the airfoil oscillation case due to the presence of a shock lying upstream of the unsteady disturbance. Finally, we considered impulsive changes in angle of attack from 01 deg. The dynamic time response to such a step change is often used in modeling gusts or for performing aeroelastic calculations by the indicial method (e.g., see Ballhaus and Goorjian (1978)). Time histories for the lift and moment taken about mid-chord are plotted in Figure 3.2.3 where k = 1.0. For small times, the general-frequency solution shows an abrupt increase in both CL and CM. This agrees qualitatively with solutions of the more complete Euler equations, for example, as described in Magnus

180 Modern Aerodynamic Methods (1977); however, this is not obtained in the low-frequency approximation. Apparently the high-frequency components of the unsteady disturbance, as retained in Equation 3.2.1, are critical during this transient period. Both solutions however tend to the same asymptote. 3.2.4

Concluding Remarks

An alternating direction implicit scheme for general unsteady motions is developed requiring only simple modifications to LTRAN2. Computed results for three different problems indicate the importance of the unsteady terms in high-frequency and gust-like motions. For the cases considered, good agreement was found for low reduced frequencies. For these problems, LTRAN2 is more cost efficient.

Figure 3.2.1 (at left). Pitch oscillations (see Figure 2 for history). Figure 3.2.2 (right). Flap oscillations, GTRANS ––, LTRAN2 - - -.

Advanced Models and Ideas 181

Figure 3.2.3. Impulsive change in angle of attack, GTRANS ––––, LTRAN2 - - - - . 3.2.5

References

Ballhaus, W.F. and Goorjian, P.M., “Implicit Finite-Difference Computations of Unsteady Transonic Flows About Airfoils,” AIAA Journal, Vol. 15, Dec. 1977, pp. 1728 - 1735. Ballhaus, W.F. and Goorjian, P.M., “Calculation of Unsteady Transonic Flows by the Indicial Method,” AIAA Journal, Vol. 16, Feb. 1978, pp. 117-124. Ballhaus. W.F. and Steger, J.L., “Implicit Approximate Factorization Schemes for the Low-Frequency Transonic Equation,” NASA TMX 73082, Nov. 1975. Landahl. M. T., Unsteady Transonic Flow, International Series of Monographs in Aeronautics and Astronautics, Pergamon Press, London, 1961. Magnus, R.J., “Calculation of Some Unsteady Transonic Flows About the NACA 64A006 and 64A010 Airfoils,” AFFDL-TR 77-46, July 1977.

182 Modern Aerodynamic Methods 3.3 Harmonic Analysis of Unsteady Transonic Flow 3.3.1 Introduction The time-averaged mean flow past an airfoil oscillating harmonically at transonic speeds will differ from the steady flow obtained at rest. This difference depends on both oscillation frequency and amplitude; it arises because both mean and disturbance flowfields interact nonlinearly. For small oscillation amplitudes, a time linearized description usually suffices: the mean flow is conveniently solved without reference to the unsteady field and the harmonic flow is solved without explicit reference to the unsteady disturbance amplitude. However, for larger amplitudes, the wave back-interaction induced by the primary disturbance on the mean flow cannot be discounted because the modified mean flow in turn affects the evolution of the harmonic flowfield. This effect may be significant because transonic flows are inherently nonlinear. Linear theory, for example, as presented in Landahl (1961, 1964), Ehlers (1974) and Traci et al. (1974), which generally assumes small unsteady disturbance amplitudes in comparison to the overall thickness, expands the unsteady solution about the known transonic flow corresponding to the stationary airfoil. For oscillatory motions the mean flowfield is therefore defined by a simplified solution rendered independent of disturbance frequency and amplitude: the model implicitly assumes high-frequency oscillations where the mean flow and mean shock position effectively freeze in space. For low-frequency motions where large shock excursions are anticipated, however, the mean flow couples more strongly with the unsteady flow and linear theory may not apply. Thus the practical need arises for a rational harmonic formulation adaptable to unsteady transonic flutter and aeroelastic analyses yielding unstable amplitude as well as frequency boundaries. In this Section, a nonlinear harmonic approach to general unsteady oscillations in transonic flow is implemented, following Section 3.1 and Chin (1978), for those engineering applications where explicit amplitude and frequency information is required. The extent to which nonlinear feedback is significant is addressed, in particular, for the symmetric NACA 64A006 airfoil, unpitched, with a quarter-chord oscillating flap executing large amplitude deflections in a subsonic freestream mildly to strongly supercritical. Details of the numerical algorithm are outlined.

Advanced Models and Ideas 183 3.3.2

Analytical and Numerical Approach

The extent to which both mean and harmonic flowfields interact nonlinearly is measured for flapping motions by a dimensionless parameter formed on dividing the maximum angular deflection by the airfoil thickness ratio. Linear theory is obtained by expanding the unsteady velocity potential in powers of the amplitude assuming a leading order nonlinear solution governed by the steady transonic small disturbance equation. To retain the effect of nonlinear feedback within the framework of harmonic analysis, we expand the unsteady potential, the shock displacement and the airfoil oscillation instead using real series where all complex Fourier wave components are accompanied by their conjugates, and, define a sequence of problems by equating coefficients of like harmonics, for example, as explained in Chin (1978). The resulting equations describe the harmonic interaction expected on physical grounds and provide an exact inviscid unsteady flow model not restricted to small amplitude oscillations. For simplicity, all harmonics higher than the primary are neglected. The resulting time transformed problem for the unsteady flow, then, superficially reduces to the formulation of Ehlers (197) and Traci et al. (1974) except that all variable coefficients related to the mean flow are now formally unknown. The mean flow, in turn, satisfies the usual steady transonic now equation modified by an added driving term whose strength is proportional to the product of 2 (which need not be small) and the streamwise derivative of an O(1) unsteady pressure magnitude squared. The complete formulation for low frequency flows so-described appears in Section 3.1 and Chin (1978). Just how important the back interaction is will depend on the particular airfoil sectional geometry, the value of , and the freestream Mach number M. The coupled mean and unsteady flowfields can be solved using type-dependent relaxation methods. For simplicity our farfield boundary conditions assumed a zero unsteady potential everywhere excepting downstream infinity where we imposed ambient pressures; also, the mean (and not the unsteady) flowfield was solved with conservative differencing implemented. With the disturbance flow initialized to zero, the computational box for the mean potential was swept once assuming zero unsteady flow. The unsteady flow equations were solved next, using variable coefficients based on latest available values of mean potential, sweeping the box once; unsteady solutions so generated were then used

184 Modern Aerodynamic Methods to evaluate the driving term in the mean flow equation, which was solved next, and so on. This “leap frog” solution for mean and transient flows is repeated as required for the duration of the unsteady simulations. The convergence speed of the present scheme appears to be slowed by the component of flow out of phase with the surface oscillation. On the other hand, computational stability is generally limited to low supercritical Mach numbers, typically M < 0.85, slow oscillations with reduced frequencies k < 0.5 based on semichord, and values of unity or less. Nevertheless, with enough trial and error, relaxation and empirical mesh size optimization, a limited number of flows showing strong nonlinear coupling have been computed. Numerical results are described next. 3.3.3 Calculated Results

Results were obtained for the unsteady transonic flow past a symmetric NACA 64A006 airfoil at zero angle of attack with an oscillating quarter-chord trailing-edge flap executing a maximum deflection angle . All calculations employed the same grid system consisting of 80 streamwise and 60 transverse constant meshes with 20 taken over the chord. Overall dimensions of the computational box were approximately 5 5 chords. Tangency conditions were enforced along a chord embedded between two horizontal meshlines while Kutta’s condition was applied, in the particular code used, slightly downstream of the trailing edge (hence, the nonzero but insignificant unsteady pressure jumps calculated at the actual trailing-edge position). This coarse mesh obviously requires refinement in high gradient regions but this was not pursued; one obvious consequence, for example, is an unrealistic smoothing of the hinge point pressure singularity. Three Mach numbers, 0.82, 0.85 and 0.90, and seven deflection angles, = 1, 2, ... , 7 deg, were considered, and k = 0.064 throughout. Calculations were terminated after 300 “cycles” where each cycle comprises of one mean flow sweep plus one unsteady flow sweep; at this stage, both mean and unsteady surface pressure solutions changed less than 5% per 50 cycles. At Mach 0.82, numerically stable results were obtained for = 1-5 deg, while at Mach 0.85 and 0.90, the calculations converged only for = 1 and 2 deg. For convenience we introduce the pressure coefficient Cp = Cp0 + (Cp1r cos kt - Cp1i sin kt) where t is a normalized time, Cp0 is the steady component, Cp1r is in phase with the boundary motion and Cp1i is out of phase. Figure 3.3.1 concisely presents all results obtained for Mach 0.82 and 0.90. In the former case the effect of nonlinear feedback

Advanced Models and Ideas 185 is unimportant even at = 3 deg, which corresponds to = 0.87; at = 5 deg, where = 1.45, the back interaction is clearly evident. Results for Mach 0.85, not shown here, indicate that the back interaction is unimportant even at = 2 deg, agreeing with results documented in earlier work in Chin (1979). This Office of Naval Research report furnishes a more complete discussion of the numerical procedure. However, at Mach 0.90 the effects of mean and harmonic flow coupling are extremely pronounced. With increasing , the mean shock location moves downstream toward the hinge point, forming a rearward region of high spatial flow gradients. The plateau region seen in the Cp1r - Mach 0.82 curve for = 1 deg, characteristic of linear theory as well as experimental small amplitude results (e.g., see Tijdeman (1977)), with increasing and M develop into highly contrasting crests and troughs. With increasing , holding M fixed, the flap oscillations influence more of the upstream flow, possibly because the increased unsteady disturbance energy enables the wave to travel farther upstream around the mean supersonic bubble. As noted, our transient solutions contained rapidly varying peaks and valleys. For the coarse mesh used, the numerical integrity of these solutions was monitored by observing their (stable) development with iteration number, by checking the smoothness of the Cp0 curve, and by comparing, with fixed, qualitative changes in the unsteady pressure obtained as a function of Mach number and iteration history. Experimental results for large deflection angles, unfortunately, are not available for comparison; however, for small angles, our computed results here and in Rizzetta and Chin (1979) for Cp1r and Cp1i agree qualitatively with the results of Tijdeman (1977). 3.3.4

Discussion and Closing Remarks

For the NACA 64A006 flapping airfoil considered here, computed results for M = 0.82, = 1, 2 and 3 deg, and M = 0.85, = 1 and 2 deg indicate that the nonlinear coupling between mean and oscillatory flowfields is not significant. For these weakly supercritical flows, linear theory still applies despite values of near unity. Hence one potential application of nonlinear harmonic analysis is clear: it can be used to assess the extent to which simpler linear approaches apply for large oscillations (this is a definite asset in aeroelastic and flutter analyses). Our results also show how the anticipated strong dependence of Cp1r and Cp1i on for large and/or M is easily obtained; more detailed analysis on a finer mesh should, however, be pursued.

186 Modern Aerodynamic Methods The linearity of the unsteady response in our Mach 0.85 calculations was somewhat unexpected. For = 1 deg our results showed a definite shockwave near midchord with Cp0 increasing 0.22 over three meshes, approximately, suggesting that the nonlinear coupling considered here should become even more important for larger values of ; for = 2 deg, calculations indicated a downstream shock movement less than 3% of chord and no noticeable change in shock strength or computed unsteady chordwise loading. Similar conclusions are noted in our prior ONR work for M = 0.80, k = 0.064 and M = 0.85, k = 0.24, for angular deflections up to 3 deg. Note that the GTRAN2 time marching scheme of Rizzetta and Chin (1979) can also be used to study mean and harmonic flow interactions except that end results must be then Fourier analyzed (this approximate factorization method applies to general oscillation amplitudes and frequencies). In light of the foregoing observations, we present a result not noted in Rizzetta and Chin (1979). GTRAN2 calculations for a NACA 64A010 section unpitched at Mach 0.82 with a 1 deg flap deflection showed shock excursions oscillating sinusoidally about the same mean position for very high to very low reduced frequencies. One might have anticipated different mean positions for different frequencies, in particular, at the lower frequencies; but, perhaps, this is not so surprising because here = 0.17 only. The linearity of the transient response for small values of is also evident from Tijdeman’s experiments from his 1977 publication and is discussed at length in the review article of Tijdeman and Seebass (1980). Finally, we observe that our general mathematical approach – that is, boundary value problem formulation and definition by equating like powers of harmonic coefficients as opposed to expansions in powers of small amplitudes – can also be used to investigate the hydrodynamic stability of inviscid parallel mean shear flows. In other words, instabilities predicted on a purely linear basis via Orr-Sommerfeld types of eigenvalue problem analyses may not actually exist or may be considerably weaker when their corrections to the main oncoming shear flow are taken into consideration. In fact, it is possible to show that nonlinear back-interaction effects smooth the inflectional mean profile and lead to stable equilibrium solutions for waves that are “self-excited” on a linear basis. Hydrodynamic stability considerations are significant to aircraft performance and economics, and these are discussed in Chin (1980) and also subsequent sections of the present chapter.

Advanced Models and Ideas 187

M = 0.82, = 1 deg M = 0.82, = 3 deg M = 0.82, = 5 deg M = 0.90, = 1 deg M = 0.90, = 2 deg

Figure 3.3.1. Calculated results for mean and unsteady surface pressure coefficients. Solution nomenclature in above chart. Unsteady results at M = 0.82 for = 1 and 3 deg are indistinguishable.

188 Modern Aerodynamic Methods 3.3.5

References

Chin, W.C., “Nonlinear Formulation for Low-Frequency Transonic Flow,” AIAA Journal, Vol. 16, June 1978, pp. 616-618. Chin, W.C., “The Transonic Oscillating Flap,” Office of Naval Research, ONR Rept. D-180-25330-1, May 1979. Chin, W.C., “Inviscid Parallel Flow Stability with Mean Profile Distortion,” Journal of Hydronautics, Vol. 14, July 1980, pp. 91-93. Ehlers, F.E., “A Finite Difference Method for the Solution of the Transonic Flow Around Harmonically Oscillating Wings,” NASA CR-2257, Jan. 1974. Landahl, M.T., Unsteady Transonic Flow, International Series of Monographs in Aeronautics and Astronautics, Pergamon Press, London, 1961. Landahl, M.T., “Linearized Theory for Unsteady Transonic Flow,” Symposium Transsonicum, edited by K. Oswatitsch, SpringerVerlag, Berlin, 1964, pp. 414-439. Rizzetta, D.P. and Chin, W.C., “Effect of Frequency in Unsteady Transonic Flow,” AIAA Journal, Vol. 17, July 1979, pp. 779-781. Tijdeman, H., “Investigations of the Transonic Flow Around Oscillating Airfoils,” National Aerospace Laboratory, The Netherlands, NLR-TR-77090-U, 1977. Tijdeman, H. and Seebass, R., “Transonic Flow Past Oscillating Airfoils,” Annual Review of Fluid Mechonics, edited by M. Van Dyke, J. V. Wehausen and J. Lumley, Vol. 12, 1980, pp. 181-222. Traci, R.M., Albano, E.D., Farr, J.L., and Cheng, H.K., “Small Disturbance Transonic Flows About Oscillating Airfoils,” AFFDLTR-74-37, June 1974.

Advanced Models and Ideas 189 3.4 Supersonic Wave Drag for Nonplanar Singularity Distributions 3.4.1

Introduction

The wave drag for a general distribution of sources and doublets on an arbitrarily curved surface in space is considered. The linear differential equation of supersonic flow is assumed, with no linearized restrictions placed on the boundary conditions. A mathematical expression is derived which extends that of Hayes (1947) for wave drag to arbitrary curved surface doublet/source representations such as are utilized by general panel-type computational methods. Wave drag comprises a significant portion of the total drag in supersonic flow, and for this reason, accurate prediction methods are desirable. Hayes’ results, e.g., Hayes (1947) and Ashley and Landahl (1965), are well known and have been accepted in general use. He examines farfield momentum considerations and, for example, arrives at a wave drag formula for a general distribution of sources without restrictions on thickness. His results showed how, for arbitrary non-lifting bodies, von Karman’s formula (see von Karman (1936)) for lineal source distributions applied locally at any azimuthal station. However, in treating the effects due to lift, certain implied near-planar assumptions were made. They arose in considering the farfield momentum flux due to a distribution of horseshoe vortices, which are restricted to lie along surfaces whose generators are aligned with the streamwise axis of the governing differential equation. The present problem is important, for example, in wave drag analysis for reentry vehicle and missile design. For such critical applications, mere seconds in travel time can determine the success or failure of a mission. It is therefore important to develop accurate drag analysis methods that accurately complement available supersonic flow models for general airframe configurations. Advances in panel method computational technology, e.g., Ehlers et al. (1976), have provided methodologies for completely non-planar representation of a configuration surface by surface source and doublet distributions. These developments also embody composite source/doublet distributions which no longer bear the classical but limiting relationships relating local source strength to local frontal area

190 Modern Aerodynamic Methods change and local vorticity strength to local lift (as an example, a fuselage can be represented entirely by a surface distribution of doublets alone, with no sources). Thus, there is a need to extend Hayes’ work to handle the types of configuration representation that are made possible with the newer panel methods. Hayes’ exact result for the general source problem is easily summarized. Essentially, consider sources of density f *(Q) where Q is the source coordinate. Define an equivalent density f such that (3.4.1)

where is an angle measured in a plane normal to the freestream; V(xi; ) is the region contained between the two Mach planes xi = x – y cos 0 – z sin 0 perpendicular to a given meridian plane = 0, and intersecting the x-axis at x = xi and x = xi + dxi. Observe that x is aligned with the undisturbed flow at infinity, and that = (M 2 – 1)½ > 1 where M is the freestream Mach number. Invariance arguments suggest local application of von Karman’s drag formula, i.e., (3.4.2)

where is the undisturbed density, U is the speed at infinity, and l is the body length. The net wave drag Dw is obtained by integration over the azimuthal coordinate, giving (3.4.3) The results mentioned previously were amended in Hayes (1947) to include the influence of elementary horseshoe vortices such as were used in classical theory to represent lateral force components (lift and side force). In particular, a function h* is defined such that (3.4.4) where U l and U S are lift and side forces per unit volume. As before, we introduce

Advanced Models and Ideas 191 (3.4.5)

Then, Equations 3.4.2 and 3.4.3 hold with f replaced by h. This constitutes the wave drag theory as developed by Hayes. Again, it is limited to the use of elementary horseshoe vortices for lift and side force representation, which is overly restrictive in terms of the surface modeling used with modern panel-type computational methods. 3.4.2

Analysis

We now consider an arbitrarily curved surface S localized in space upon which are distributed sources and doublets. In supersonic flow, the flowfield at an observation point (x,y,z) in space will depend only on the singularities upstream of the intersection defined by S and the upstream Mach cone. Let the projection of this curve on the horizontal plane be C. Then if z1 = z1(x1,y1) describes S, C will be defined by the solution x1 = C(y1) to the equation (x – x1)2 – 2 (y – y1)2 – 2 [z – z1(x1,y1)] 2 = 0, allowing us to write the source potential s symbolically in the form (3.4.6)

where f *(x1,y1) is the areal source strength, and the quantity G1(x1,y1) dx1 dy1 = (1 + z1x12 + z1y12) ½ dx1 dy1 = dS is an incremental surface area. The drag due to Equation 3.4.6 is exactly given by Equations 3.4.1–3.4.3. We next consider the effect of doublets. Thus, examine the potential D with (3.4.7)

now noting the “3/2” (versus “1/2” in Equation 3.4.6) exponent in the denominator, where

192 Modern Aerodynamic Methods (3.4.8)

Here, R, P and Q are strengths of doublets whose axes are aligned in the directions of the x, z and y axes respectively (an arbitrary doublet axis orientation is simply a linear combination of these three components). The idea is to rewrite Equation 3.4.7 in the form (3.4.9)

and integrate by parts, so that (3.4.10)

where the integrand I satisfies (3.4.11)

, y1; x,y,z) = 0, and if we discard the The first term vanishes if ( “infinite part” associated with the square root singularity, this leaves

(3.4.12)

which is to be compared with Equation 3.4.6 – note how both are now characterized with a “1/2” exponent in the denominator. It follows that an “equivalent source strength” may be defined as

Advanced Models and Ideas 193

(3.4.13)

To complete the analogy to Equation 3.4.4, we evaluate Equation 3.4.13 at large distances from the body. For this purpose, we set y = R* cos , z = R* sin , and x = x’ + R*. With x’ fixed and R* tending toward infinity, the asymptotic value of is now given by (3.4.14)

In other words, Hayes’ algorithm is modified by replacing Equation 3.4.4 by Equation 3.4.14. We can easily verify that Equation 3.4.14 reduces to Equation 3.4.4 for planar problems. This limit refers to z1 = 0, G1 = 1 and R = 0, so that (3.4.15) For planar wings, the term Px1 is proportional to the lift density l*, and Qx1 is proportional to the side-force density S*. But in general, Equation 3.4.14 must be used to calculate the drag for configurations whose surfaces are represented by nonplanar source and/or doublet distributions. 3.4.3

Summary

Hayes’ algorithm for wave drag computation has been refined to include non-planar effects. The only change consists in replacing Equation 3.4.4 by Equation 3.4.14. Modern computational techniques in supersonic aerodynamics center on the use or source/doublet panel methods, e.g., Ehlers et al. (1976). The present results, in terms of doublets, could be easily incorporated into these techniques. Implementation requires only the basic computer software codes for Equation 3.4.1 – 3.4.3, which are already in extensive use. The total

194 Modern Aerodynamic Methods inviscid drag of an airplane configuration is obtained by adding to Dw the vortex drag of the infinite wake; in addition, viscous effects must be considered to obtain the total flow resistance. 3.4.4

References

Ashley, H. and Landahl, M., Aerodynamics of Wings and Bodies, Addison-Wesley, Reading, Mass., 1965. Ehlers, F.E., Johnson, F.T. and Rubbert. P.E., “A High-Order Panel Method for Linearized Supersonic Flow,” AIAA Paper 76-381, presented at the Ninth Fluid and Plasma Dynamics Conference, San Diego, Calif., 1976. Hayes, W.D., Linearized Supersonic Flow, North American Aviation, Los Angeles, Calif., Report No. AL 222, 1947. von Karman, T., “The Problems of Resistance in Compressible Fluids,” Proceedings of the Fifth Volta Congress, R. Accad. D’Italia, Rome, 1936.

Advanced Models and Ideas 195 3.5 Supersonic Wave Drag for Planar Singularity Distributions

A closed-form expression for the supersonic wave drag of a general planar source-doublet distribution is obtained using the linearized differential equation and exact nonlinear boundary conditions. The result is useful for drag computations and analytical optimization studies, for example, in the engineering and design of modern transport aircraft, reentry vehicles and missiles. 3.5.1

Introduction

This work derives a planar counterpart to a well-known drag formula (e.g., see von Karman (1936)) for axisymmetric lineal source distributions but which includes the effect of doublets. It provides an explicit evaluation of the author’s formal results (for general nonplanar singularity distributions in Chin (1977a)) in the planar case. Current wave drag prediction methods in supersonic aerodynamics are based on the classical analyses of von Karman (1936) and Hayes (1947). The exact result due to Hayes applies only to the source problem and is summarized easily. For sources of density f *(Q), where Q is the source coordinate, define an equivalent density f such that (3.5.1)

Here is an angle measured in a plane normal to the freestream, and V(xi; ) is the region contained between two Mach planes xi = x - y cos 0 – z sin 0 perpendicular to a given meridian plane = 0 and intersecting the x axis at x = xi and x = xi + dxi. Note that x is aligned with the undisturbed flow at infinity, while = (M 2 – 1)½ > 1 where M is the supersonic freestream Mach number. Invariance arguments suggest local application of von Karman’s drag formula, that is, (3.5.2)

where is the undisturbed density, U is the speed at infinity, and l is the body length. The net wave drag Dw then is obtained by azimuthal integration, giving

196 Modern Aerodynamic Methods (3.5.3)

To amend these results for non-thickness effects, Hayes (1947) included the influence of elementary horseshoe vortices such as were used in classical theory to represent lateral force components such as lift and side force. To accomplish this, a function h* was defined such that (3.5.4)

where U 2l* and U 2S* are lift and side forces per unit volume. As suggested in Equation 3.5.1, we introduce (3.5.5) Then, Equations 3.5.2 and 3.5.3 hold with f replaced by h. However, the foregoing results contain implied near-planar body assumptions and are somewhat restrictive. They arise in considering the farfield momentum flux due to distributions of horseshoe vortices, which are restricted to lie along surfaces whose generators are aligned with the streamwise axis of the governing differential equation. The preceding equations bear limiting relationships connecting local source strength to local frontal area change and local vorticity strength to local lift. Because modern computational panel methods, e.g., see Ehlers et al. (1976), enable the modeling of completely general nonplanar aerodynamic configurations with surface singularity distributions, the need for improved wave drag prediction methods was apparent. This led to the author’s extension of Hayes’ formal algorithm to cover arbitrary source/doublet distributions on general nonplanar surfaces in Chin (1877a). In that analysis, the linearized differential equation was used, along with exact nonlinear surface boundary conditions. (Solutions to such problems are “second-order solutions,” per van Dyke (1964)). The exact wave drag for this problem, it turned out, required only simple modifications to Hayes’ original method. We assume known R, P and Q, the strengths of doublets whose axes are aligned in the directions of the x, z and y axes, respectively. (An arbitrary doublet axis orientation is simply a linear combination of these three components.) Now, the perturbation velocity potential = S + D consists of two parts, a S contribution due to a source distribution and a

Advanced Models and Ideas 197 component due to a doublet distribution given, respectively, by the equations D

(3.5.6)

and (3.5.7)

In Equations 3.5.6 and 3.5.7 note that (x1,y1;x,y,z) = R(x1,y1)(x – x1)

2

+ P(x1,y1)[z – z1(x1,y1)] + 2 Q(x1,y1)(y – y1). Also note the different “1/2” and “3/2” exponents in the denominators. In the above, the contour C is defined by the solution x1 = C(y1) to (x – x1)2 – 2 (y – y1)2 – 2 [z – z1(x1,y1)] 2 = 0. The quantity f *(x1,y1) is the areal source strength, and G1(x1,y1) dx1 dy1 = (1 + Z1x12 + Z1y12)½ dx1 dy1 = dS is an incremental surface area for Z1 = Z1(x1,y1). The wave drag due to sources alone is given exactly by Equations 3.5.1 - 3.5.3. But the total wave drag is not given by Equation 3.5.4, since it implicitly assumes a near-planar body. It is given exactly by Equation 3.5.5, though, provided that we replace Equation 3.5.4 by the more complete expression given in Equation 3.5.8. For thin, near-planar bodies, Equation 3.5.8 reduces to Equation 3.5.4. The planar limit refers to Z1 = 0, G1 = 1 and R = 0, so that Equation 3.5.4 is recovered; the term Px1 is proportional to the lift density l*, whereas Qx1 is proportional to the side force density S* – (3.5.8)

198 Modern Aerodynamic Methods 3.5.2

Analysis

The foregoing generalization to Hayes’ formal numerical procedure requires only simple modifications to existing computer codes widely in use. However, in many practical applications, especially analytical ones, it is desirable to proceed directly from a closed-form integral expression. von Karman’s drag formula for lineal source distributions has been applied to parametric engineering optimization studies. For instance, what is the ideal low-drag vehicular shape when volume is held fixed? For areal distributions, progress in terms of general results has been impeded mainly because of geometric difficulties in describing the farfield momentum flux vector, even for simple shapes. However, for planar source-doublet distributions, it turns out that the wave drag integrand can be given in closed form, and it is this evaluation that forms the subject of the present section. Our results are based on the linearized differential equation but without invoking the classical application of linearized boundary conditions on mean surfaces. The boundary value problem considered therefore corresponds to an exact solution of the zeroth-order RayleighJanzen expansion. The results, again, apply to general nonplanar configurations representable by (given) planar singularity distributions; an example might be a Rankine body with an intersecting wing. The wave drag of a planar (Z1 = 0) surface distribution of sources of areal strength f *(x1,y1) and doublets of strength P(x1,y1) oriented in the z direction can be determined by enclosing the finite body by a control surface S. The net force acting on it then can be obtained by examining the momentum flux across S. For linearized supersonic now, one therefore writes (3.5.9) where S2 is a circular control volume that encloses the singularity distribution and is aligned with the flow direction. Now enclose the body with two Mach cones (as shown in Figure 3.5.1), and define the origin (0,0,0) by the left vertex, and the right vertex by the point (l,0,0). (Note that the wave drag contributions from S1 and S3 are zero; one must add to Dw the vortex drag of the infinite wake in assessing the total inviscid drag – also, viscous drag will contribute to net aerodynamic inefficiencies.) Now if = S + D(z), and R is the radius of S2, we have

Advanced Models and Ideas 199 (3.5.10) where l3 is, as yet, an unspecified distance far downstream. The curly D’s in the integrand of Equation 3.5.10 refer to, respectively, the velocity product terms ( S / x) ( S / r), ( D / x) ( D / r), ( S / x) ( D / r) and ( D / x) ( S / R). Here, D1 and D2 are drags due to sources and doublets, respectively, and D3 and D4 are interference drags. Although Equation 3.5.10 is valid with any value of R, geometric simplifications are made possible by letting R tend to infinity. To demonstrate our analysis method, we shall carry through in detail the evaluation of the drag due to sources. We first write S in the form (3.5.11) If f (

, y1) = 0, integration by parts and differentiation lead to

(3.5.12,13) Now, in calculating D1 we retain the subscripts “1” in Equation 3.5.12 while using dummy subscripts “2” in Equation 3.5.13. Then, the integral over products, that is,

(3.5.14) can be written as the multiple integral

(3.5.15)

200 Modern Aerodynamic Methods since the outer four integrals contain constant limits. In the preceding expressions, the lower limits in the dx1 and dx2 integrations can be replaced by zero, by virtue of Figure 3.5.1. For the upper limits, we recall that the singularity distribution is finite in extent. If at an observation point P0(x,y,z) we have x - R > l, so that the singularity distribution lies wholly within the forecone emanating from P0, then events at the point P0 are influenced by all of the sources, and the upper limits can be taken over the entire body. For this, it would be sufficient to choose x1 = x2 = l whenever x’ = x – R l. We therefore can write

(3.5.16) where, in the inner integrals, the upper limits are x1,2 = x’ + R – (R2 + y1,22 – 2Ry1,2cos ) ½ or x1,2 = l depending on whether x’ is less than or greater than l. Now we evaluate the Equation 3.5.16 in the limit as R tends to infinity. By virtue of the way our rear disk S3 was chosen, l3 must be larger than R at all times. Hence, l3 approaches infinity as such that R/l3 > R, which is the case for part of the integration region, a different limiting integrand is obtained. It is convenient to split the interval of integration over x' into two parts, the first from 0 to a, and the second from a to l3 - R. In the first integral, the expression in Equation 3.5.17 can be used, whereas in the second, since a >> l, x' >> R, and x1, x2 O(l), x1 and x2 can be neglected in comparison to x'. Thus in the limit of large R, when the inner limits are also approximated to O(1/ R), we find that

(3.5.18)

Advanced Models and Ideas 201 This result can be simplified because the second term within the brackets vanishes by virtue of the requirement (3.5.19) The expression for supersonic wave drag then reduces to (3.5.20) where a >> l and (3.5.21) The domain of integration is a region in x' – x1 – x2 space whose cross-section for x' = const is a rectangle spanning 0 x1 x' + y1cos , 0 x2 x' + y2cos for x' l, and the square 0 (x1,x2) l for l < x' < a. We simplify Equation 3.5.21 by interchanging the order of integration. To do this, the proper limits are required, and these are obtained from Figure 3.5.2, which depicts the volume in x' – x1 – x2 space over which the summations are performed. Geometric considerations indicate that we can write equivalently I = I1 + I2 + I3 + I4, where (3.5.22)

refers to an integration over the lower base region in Figure 3.5.2, (3.5.23)

refers to the embedded parallelpiped in the upper pyramidal structure, and I3 and I4 below

(3.5.24,25)

202 Modern Aerodynamic Methods refer to the remainder of the pyramidal solid. Each of the preceding terms can be simplified using the integral expression (3.5.26) Equations 3.5.23 – 3.5.25 can be evaluated directly, leaving only double integrals, but Equation 3.5.22 requires special treatment. Evaluation with Equation 3.5.26 leads to (3.5.27)

where (3.5.28)

Then, making use of the fact that x1,2 – y1,2 cos O(l) and a >> l, an asymptotic expansion of the first term in Equation 3.5.28 yields (3.5.29) If this is substituted into Equation 3.5.27, the integral corresponding to ln 4a vanishes because of Equation 3.5.19, and the integral corresponding to the second term in Equation 3.5.29 vanishes as a tends to infinity, since it is O(a-1). Thus we obtain, as the final result for the supersonic wave drag due to a planar distribution of sources, (3.5.30) where “I” takes the form

Advanced Models and Ideas 203

(3.5.31) Next let us consider the drag contribution due to doublets alone, represented by the velocity potential

(3.5.32) To calculate the required derivatives, the regularity conditions P(- , y1) = Px1(- , y1) = 0 are assumed in the preliminary integration by parts, eliminating the appearance of singular integrals. We shall denote by the linear integral operator in Equation 3.5.31, that is, (3.5.33) Carrying through an analysis similar to the foregoing leads to the result that

(3.5.34) The “interaction drags” D3 and D4 can be similarly calculated, leading to the result that the total supersonic wave drag Dw is expressible in the form

(3.5.35)

204 Modern Aerodynamic Methods One can further show (e.g., refer to Chin (1976)) that the relationship { } = {[fx(x1,y1) – sin Pxx(x1,y1)] [fx(x2,y2) – sin Pxx(x2,y2)]} holds, following from a type of nonlinear reciprocity. The appearance of this product was suggested by Equation 3.5.4. However, it is not the lift density l * that belongs in the exact expression but the more general doublet derivative Px. It is interesting to see how our results reduce to conventional ones in a particular example. We calculate the wave drag due to a lineal distribution of sources by dropping the iterated integrals over y1 and y2 and setting y1 = y2 = 0 throughout. In the fourth integral of Equation 3.5.31, we exchange x1 for x2 and x2 for x1. Addition then produces the – independent expression

(3.5.36) The first multiple integral is canceled by the first term of the second, since the integrand is symmetric about the line x1 = x2. This leaves (3.5.37)

for which (3.5.38)

This reproduces von Karman’s 1936 result for lineal source distributions. The drag formula given in Equation 3.5.35 is an exact result of linearized supersonic flow and does not involve any approximations except that appearing in Equation 3.5.19. For the case where the maximum cross-dimension is small compared to l, say, it is possible to show (again, following Chin (1976)) that the interaction drags

Advanced Models and Ideas 205 are small in comparison with those due to sources and doublets, a result that is to be expected from slender-body theory. It is apparent from the foregoing discussion that drag theorems for the reversed flowfield can be derived without difficulty and without change conceptually. However, the results for the combined flowfield (that is, the sum of forward and reverse drags) are no longer as elegant as those for near-planar bodies.

Figure 3.5.1. Farfield geometry.

Figure 3.5.2. Geometry for x', x1, x2 integration.

206 Modern Aerodynamic Methods 3.5.3

Concluding Remarks

We can make several comments that explain our approach and results in perspective with those of Hayes (1947). The main point of our analysis is this: the most important restriction in Hayes’ theory is the implied near-planarity. This arises in considering the farfield momentum flux due to distributions of horseshoe vortices, which are restricted to lie along surfaces whose generators are aligned with the streamwise axis of the governing partial differential equation. The classical results therefore bear the limiting relationships connecting, for example, local vorticity strength to local lift, and fail to account properly for lift and thickness interaction. The early work implicitly assumes a planar vortex sheet behind each lifting element, whereas a more general doublet formulation allows the singularity surface to be nonplanar. A second point concerns the source drag integrand “I” given in Equation 3.5.21 and expanded in Equation 3.5.31. The operator associated with “I” enters in the more complete drag formula that includes the combined effects of lift and lift-thickness interaction (see Equations 3.5.33 and 3.5.35). In the form given by Equation 3.5.21 or Equation 3.5.31, the evaluation of “I” is straightforward, although somewhat tedious. The purpose of our evaluation is to show how the complete drag formula can be written down explicitly, noting that the availability of this result is desirable and useful in direct applications. Of course, Equation 3.5.21 could be simplified considerably (although symbolically) following Hayes’ lumping of the singularities on the x axis by integrating them along the Mach planes, and certainly, this can be done easily. The results of our very tedious asymptotic evaluation indicate that it is not the lift density l* that belongs in Hayes’ drag integral but the doublet derivative. (These two are equal only for planar bodies.) This suggests that Hayes’ method summarized by Equations 3.5.2, 3.5.3 and 3.5.5, as previously described, should be modified by replacing the classical use or Equation 3.5.4 by h* = f * - Px1 sin ; the drag integral with the operator “I” as defined in Equation 3.5.21 or 3.5.31 is therefore completely equivalent to using Hayes’ integral formula (for singularities lumped along the x axis) but using the foregoing h*, and this should afford some computational advantage. This result is part of a more general one derived in Chin (1977a) using a different but complementary approach; Equation 3.5.8, which is valid for arbitrary surfaces, reduces in the planar limit (that is, using Z1 = 0 and G1 = 1) to h* = f * – Rx1 – Px1 sin – Qx1 cos , which implicitly contains the

Advanced Models and Ideas 207 present results. [Equation 3.5.8 furthermore shows the nontrivial effect of streamwise surface slope and curvature.] The present section considers in detail the structure of the drag integral, provides an explicit evaluation of the “equivalent source,” and confirms the more general results of Chin (1977a). The use of exact, explicit formulas such as those given in Equations 3.5.8 and 3.5.35 appears in both supersonic analysis and inverse problems. Aside from their direct value to wave drag calculations, they reduce the labor necessary to compute incremental changes in Dw due to incremental changes in singularity strength or geometry, or vice-versa (for these applications, the basic linearization would be carried out about a given nonplanar geometry). Additional uses also can be found. Advanced design concepts presently pursued by the author involve the application of these exact formulas to wave drag minimization; variational calculus is used to determine optimal geometries under various physical constraints. Other uses no doubt, will appear, and various possibilities currently are being explored. 3.5.4

References

Chin. W.C., “Explicit Wave Drag Formulas for General SourceDoublet Distributions on Arbitrarily Curved Surfaces in Linearized Supersonic Flow,” Report D6-43843, The Boeing Company, Seattle, Wash., 1976. Chin, W. C., “Supersonic Wave Drag for Nonplanar Singularity Distributions,” AIAA Journal, Vol. 15, June 1977 (a), pp. 884-886. Ehlers, F.E., Johnson, F.T. and Rubbert, P.E., “A High-Order Panel Method for Linearized Supersonic Flow,” AIAA Paper 76-381, presented at the Ninth Fluid and Plasma Dynamics Conference, San Diego, Calif., 1976. Hayes, W. D., Linearized Supersonic Flow, North American Aviation, Los Angeles, Calif., Report AL 222, 1947. Van Dyke, M.D., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. von Karman, T., “The Problems of Resistance in Compressible Fluids,” Proceedings of the Fifth Volta Congress, Researche Accademia D’ltalia, Rome, 1936.

208 Modern Aerodynamic Methods 3.6 Pseudo-Transonic Equation with a Diffusion Term

When “Pseudo-transonic Equation with a Diffusion Term” first appeared in 1978, the author was strongly focused on supersonic flow, having published “Supersonic Wave Drag for Nonplanar Singularity Distributions” in 1977 and “Supersonic Wave Drag for Planar Singularity Distributions” in 1978. Calculating wave drag was important to the 1970s – the latter two papers aimed at accurate computations able to draw upon the advantages offered by the then newer “panel methods.” But the underlying linear supersonic flow models used could not predict the sonic booms that ultimately led to the demise of high speed air transportation. The author had studied with Professor Gerald Whitham at Caltech, the industry’s leading aerospace guru, who had devised methods to model ray coalescence and their disastrous effects on communities beneath supersonic flight paths. But ray computations were unwieldy – so when the author became aware of Sichel’s modification of Whitham’s nonlinear equation {1 – M 2 – ( +1) M 4 x} xx + yy = 0 to a slightly different xxx + {1 – M 2 – ( +1) M 4 x} xx + yy = 0 form, numerous interesting questions and possibilities arose. Because the new model was actually parabolic, due to the presence of a high-order diffusive “ xxx” term, the idea that ray kinematics and shock-fitting might not be necessary in calculating sonic booms took shape. This led to the general closed form analytical solutions in Sections 3.6.1 and 3.6.2 below. However, another just as significant question lurked in the background. Flight at transonic speeds was equally important to the 1970s. In fact, airplanes then and now cruise at 550 mph, approximately, a speed threshold beyond which undesirable local supersonic zones would form on otherwise subsonic wings. The locations and properties of these zones was unknown and required iterative computations using so-called “mixed type,” “mixed subsonic-supersonic” or “mixed elliptic-hyperbolic” numerical modeling schemes. These were labor intensive. Each point in three-dimensional space required “type testing” and then appropriate choices of computational molecules. Both engineers and mathematicians were frustrated. But vector (as opposed to scalar) computers would come along, this fortuitous development stimulating the development of transonic solution methods that “didn’t care” if the underlying flows were subsonic or supersonic.

Advanced Models and Ideas 209 Such was the case with the parabolic “ xxx + . . . ” equation. It didn’t care about subsonic or supersonic because it was diffusive, whereas the simpler low-order equation without the “ xxx” term could be both and therefore quite complicated. And so, the present Section introduces the “pseudo-transonic equation with a diffusion term” and its use in purely supersonic flow, while Sections 3.7, 3.8 and 3.9 describe the author’s development of mixed subsonic-supersonic flow solvers that did not require the use of Murman-Cole types of scalar algorithms. The new “type-independent” solvers for transonic would allow vector computers, which utilized “parallel processing” to solve for flows at multiple locations simultaneously and independently (synchronizing solutions at conveniently selected interfaces only periodically) and accelerate the computational process. One need no longer solve at one location before proceeding to another, this advantage significantly reducing waiting times and programming logic. Moreover, because shockwaves appeared naturally within the framework of a differential equation with a real compressive viscosity, artifacts like “conservative differencing” would not be necessary – accurate thermodynamic information is completely embedded in the “ xxx” term to describe events outside and within the shock. Thus ray coalescence and entropy changes in both supersonic and transonic flow could be modeled straightforwardly. All of these positive attributes seemed difficult to ignore – and, in the end, we would successfully demonstrate several new and important ways to model modern flows of commercial importance. 3.6.1 Introduction

In this Section, we show how a classic pseudo-transonic equation for supersonic flow, as presented in Hayes (1954), when modified by a third-order streamwise diffusion term, can be transformed into Burgers’ equation. The solution of this equation reduces to that of Whitham (1952) in the limit of vanishing viscosity. The role of physical diffusion near shocks is well known, and here, simplifying mathematical features introduced by viscosity are discussed. 3.6.2 Analysis

Ackeret’s classical solution is based on an approximate linearized equation that is a proper first approximation near the surface, but which breaks down at large distances from the airfoil. For example, classical theory fails to predict the bending of Mach lines that occurs in reality,

210 Modern Aerodynamic Methods and also, the subsequent shock formation and decay. This failure arises from the neglected cumulative effecls of locally small disturbances that grow to first-order over large distances. It is now well known that the nonuniformity in linear theory arises only from the neglect of the nonlinear “pseudo-transonic” ( +1)M 4 x xx term in the more complete model for the disturbance velocity potential given in Equation 3.6.1a. In this equation, is the ratio of specific heats, M is the freestream Mach number, x is the streamwise coordinate and y is the stream-normal coordinate. The contribution of all other nonlinear terms is uniformly of second-order. This pseudo-transonic term has a first-order cumulative effect and must be retained in addition to the usual linear ones in seeking a uniformly valid first approximation. As discussed, laborious “shock fitting” is generally required in Whitham’s approach, however, the addition of the high-order diffusion term xxx should render the fitting procedure unnecessary. It is the role played by xxx that is studied here. To the same order of accuracy, geometric tangency conditions can be imposed on the axis, with this and regularity conditions completing the formulation in Equations 3.6.1a,b,c. In this Section, we derive an analytical but approximate solution to the following diffusive system, xxx

+ {1 – M 2 – ( + 1) M

y(x,0)

= T’(x)

= 0, upstream

4

x}

xx

+

yy(x,y)

=0

(3.6.1a) (3.6.1b) (3.6.1c)

where > 0 is a small diffusion coefficient, is the thickness ratio, and T’ is a normalized slope. It is interesting that Equation 3.6.1a is just the “viscous transonic equation” as derived by Cole (1949), although in the present application we are dealing with purely supersonic applications. In the transonic case, Equation 3.6.1a is obtained through a special limiting process taking into account the effect of compressive viscosity at sonic lines and shocks, where the coefficient of xx normally would vanish in low-order theory; to the order considered, the effects of rotationality introduced by viscosity and shock curvature can be ignored. We can expect solutions of Equation 3.6.1a to reduce in the limit of 0 to solutions of the inviscid formulation; that they actually do in the transonic case was demonstrated by Sichel (1968) for planar and axisymmetric nozzles. In both cases the solution provided a viscous, shock-like transition from an inviscid, supersonic, accelerating flow to an inviscid, subsonic, decelerating flow. The addition of viscosity here

Advanced Models and Ideas 211 enables us to treat the shock structure as well, but the structure obtained is only a rough model of the real flow. In fact, Chin (2014) in his book Wave Propagation demonstrates how “conservative differencing” and the commonly accepted entropy conditions can be derived from a simplified form of Equation 3.6.1a (see Pages 322-323 in the book). For simplicity, consider a symmetric airfoil and introduce a transformation of coordinates. Two directions appear in the problem, that of the freestream and that of the outgoing freestream Mach lines. The airfoil lies nearly along the former and the wave pattern nearly along the latter. Thus the solution can be expressed more naturally in terms of the oblique coordinates = x – (M 2 – 1)½ y and = (M 2 – 1)½y as first introduced by Hayes (1954). As noted in Hayes (1954), coordinate transformations for ( , ) then produce terms like and which represent second-order effects that can be dropped. When these terms are discarded in the equation and in the surface boundary conditions, and new asterisked variables are defined by = * , x = = u = 2½(M 2 – 1)½ u*/{( + 1) M 4} and lastly = */{2(M 2 – 1)}½, the formulation as given in Equations 3.6.1a, 3.6.1b and 3.6.1.c becomes u* * + u*u* * = u*

(3.6.2a)

* *

u* = - ( +1)M 4/{2½(M 2 –1)} T’{ *(2(M 2 – 1)) -½ } = F( *) at *

u = 0, upstream

*

= 0 (3.6.2b) (3.6.2c)

This is just the initial value problem for Burger’s equation, for which the closed-form solution is known from Cole (1951). The solution is simply u*( *, *; ) =

(

*

– )/

*

exp(-G/2 ) d / exp(-G/2 ) d

(3.6.3)

where G( ; *,

*

)=

F( ) d + (

*

– )2/(2 *)

(3.6.4)

0 and can be readily re-expressed in physical coordinates. In general this requires some labor, but modern algebraic manipulation software reduces much of the tedious effort required.

212 Modern Aerodynamic Methods 3.6.3 Summary

It is clear that solutions to Equations 3.6.2a,b,c tend to those of 0 by virtue of the known properties of Burger’s inviscid theory as equation. The addition of viscosity here, in fact, actually makes the mathematical problem simpler. In the usual inviscid formulation, the solution for u is implicit, whereas that given above is explicit (this representation is useful in many applications). Furthermore, shocks must be constructed in the farfield to prevent multivalued solutions; viscous diffusion in the physical problem here automatically gives a single-value solution. In addition, the present approach furnishes a model for the shock structure and decay that, for supersonic flows, is correct in the limit M 1. As in the inviscid theory, simple surface slope discontinuities are permissible, as are slightly rounded leading edges, provided that the solution is used in the distant farfield. The present results are exactly the same as those in Whitham’s low-order theory, and the approximate structure of the shock region is that determined on the basis of Burgers equation. 3.6.4 References

Chin, W.C., Wave Propagation in Drilling, Well Logging and Reservoir Applications, John Wiley & Sons, Hoboken, New Jersey, 2014. Cole, J.D., Problems in Transonic Flow, Ph.D. Thesis, California Institute of Technology, Pasadena, Calif., 1949. Cole, J.D., “On a Quasilinear Parabolic Equation Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236. Hayes, W.D., “Pseudo-transonic Similitude and First-Order Wave Structure,” Journal of the Aeronautical Sciences,Vol. 21, Nov. 1954, pp. 721-730. Sichel, M., “Theory of Visous Transonic Flow - A Survey,” AGARD Conference Proceedings, No. 35, Transonic Aerodynamics, Sept. 1968, Paper 10, pp. 10.1 – 10.16. Whitham, G.B., “The Flow Pattern of a Supersonic Projectile,” Communications on Pure and Applied Mathematics, Vol. 5, 1952, pp. 301-348.

Advanced Models and Ideas 213 3.7 Numerical Solution for Viscous Transonic Flow 3.7.1 Introduction

The nearly inviscid transonic supercritical flow over a thin airfoil is calculated using a second-order accurate algorithm for the “viscous transonic equation.” Type-differencing, shock-point and parabolic operators are unnecessary in the present approach; good agreement is found with the results of Martin (1975) using a Murman-Cole scheme. Special advantages offered by the present formulation, which introduces longitudinal or compressive viscosity explicitly into the model equations, are discussed both mathematically and numerically. 3.7.2 Analysis

The computation of steady, inviscid, supercritica1 transonic flow over airfoils is complicated by the appearance of mixed supersonic and subsonic regions separated by unknown sonic lines and shocks. General existence and uniqueness theorems are still unavailable, but a number of successful schemes have been developed to handle mixed-type partial differential equations. A number of the methods may be considered “embedding methods” and typically involve computations in a function space other than the physical, the generalized solution of which reduces to the physical solution in some limit. One example is the “method of imaginary characteristics,” due to Garabedian (1964), where analytic continuation transforms an unstable elliptic Cauchy problem into a stable hyperbolic one in a complex space, thereby stabilizing the marching procedure. Taking a more physical approach, Magnus and Yoshihara (1970), for example, use the method of characteristics to calculate the steady asymptote of the unsteady hyperbolic Euler equations. Still another approach is the method of parametric differentiation developed by Rubbert (1965), in which the nonlinear equations are embedded in a parameter space where the governing equations are linear. The most popular approach is due to Murman and Cole (1971) employing type-differencing. Subsonic points are represented by central differences and supersonic points by backward differences, properly accounting for domains of influence and dependence. The manner in which grid points are type-tested is crucial, since divergence in the

214 Modern Aerodynamic Methods relaxation scheme is possible. Also, the inviscid finite-difference equations must be in proper conservation form, so that captured shocks satisfying the correct jump conditions appear. The procedure relies on special parabolic and shock-point operators applied at sonic lines and shocks, and high-order truncation terms must be diffusive in order to avoid unrealistic oscillatory solutions. It is clear that sophisticated program logic is required. A simpler approach is to deal directly with the high-order viscous problem. Consider, for example, Burger’s equation, uux = uxx, written in stationary coordinates, where is a positive (nonzero) number. Let us integrate this once with respect to x, where the integration is carried out across a shock from x = a to x =b. Since x =b b (½u ) | = ( ux)| x =a a 2

it follows that shock-like solutions with vanishing gradients at x = a and x = b imply straightforwardly the jump conditions u2(a) = u2(b), that is, “u2” is conserved or equal across the shock. It is also possible to derive an entropy inequality just as straightforwardly. To do this, we multiply the governing equation by u(x) throughout, so that u2 u/ x = u 2u/ x2. This can be rewritten as (1/3 u3)/ x = u 2u/ x2. If we now integrate by parts, we have (1/3 u3)+ - (1/3 u3)- = [{u u/ x - ( u/ x)2 dx }+ - {u u/ x - ( u/ x)2 dx }-] As before, the u/ x terms on either side of the shock vanish identically, but now, the positive definite integral on the right side does not. This leads to the inequality (1/3 u3)+ - (1/3 u3)- = -

( u/ x)2 dx < 0

and hence the “entropy condition” (1/3 u3)- > (1/3 u3)+ or (u3)- > (u3)+. Thus, we have shown how entropy conditions need not be derived “independently” via thermodynamic considerations, as is often the case in elementary courses. They, and indeed all of the physics, can be obtained naturally once the algebraic structure of the high-order derivative is physically known with confidence.

Advanced Models and Ideas 215 On the other hand, suppose that we had taken to be identically zero, as in a purely inviscid problem. Then, the differential equation uux = 0 can be multiplied by any power of u, with the result that (un)x = 0 for all values of n. This admits an infinity of jump conditions, so that one or more entropy conditions must be separately invoked to insure uniqueness. The jump conditions, of course, are not ambiguous; they are determined by the full, high-order problem, as we have shown. In our example, we determined that “u2” was conserved in a simple manner: we multiplied uux = uxx by the appropriate functional of u, in this case “u” itself, and integration plus setting ux = 0 on both sides of the shock gave the required conservation form. On the other hand, multiplication by u3 and subsequent integration by parts gave the desired entropy inequality. Similarly, the complicated logic involved in inviscid Murman-Coletype algorithms can be avoided by embedding the low-order equation in a high-order viscous system. For small disturbance flows, the vorticity generation within the flow can be ignored, so that the inviscid velocity potential can be used. However, it is known that inviscid theory is not completely sufficient. For example, it inadequately describes the flow near the throat of a converging-diverging nozzle during the transition from a Taylor type flow to subsonic-supersonic Meyer flow. It is also obvious that real dissipative effects must be important in narrow shock layers. To resolve the physical details near these two kinds of turning points, the usual inviscid derivation must be reconsidered to determine the circumstances under which high-order streamwise derivatives, multiplied by viscosity, are important. This special limiting process was in fact investigated by Sichel (1968) and the result is a “viscous transonic equation” that contains a third derivative term in the disturbance velocity potential with a small coefficient similar to above that accounts for the effects of compressive viscosity in shock regions. The modified equation still describes inviscid flow only, but it implicitly contains the correct jump conditions, in this more general case across the curved shocks found extending from aft airfoil surfaces. If accurate solution is possible, all of the salient physical features of flows developed within the framework of the theory would be described. This method is conceptually more attractive because the governing equation is parabolic throughout, rendering existing type-differencing unnecessary. Furthermore, special shock-point and parabolic operators used in Murman-Cole schemes near shocks and sonic lines, and conservative differencing, are no longer needed.

216 Modern Aerodynamic Methods 3.7.3 Numerical Approach

The nondimensional viscous transonic equation for the perturbation velocity potential is, as given by Sichel (1968), as follows, Kv

xxx

+ (K –

x)

xx

+

yy

=0

(3.7.1)

using a normalization slightly different from the conventional rescaling. Here Kv/K 0, Equation 3.7.1 is parabolic, and the characteristics are y = constant. This suggests a solution employing horizontal line relaxation, so that the computational box is swept upwards continually and repeated until convergence. Because of the parabolic nature, it is possible to ignore inviscid subsonic/supersonic type-differencings; the difference formulas are the same everywhere. Because horizontal line relaxation is used, consider lines of constant j, as shown in Figure 3.7.1 Along such lines, a one-dimensional “box scheme” is employed, as outlined in Keller (1974). Thus, we introduce the three-component vector ( , x, xx) = ( , U, B) at each grid point. The equation U = x, e.g., is differenced according to ( i,j – i-1,j)/hi (Ui,j + Ui-1,j)/2. B = Ux is similarly differenced, as are all x derivatives. Finally, B, U and satisfy KvBx + (K – U)Ux + yy = 0. Here, yy is approximated by central differences about the line j, so that inputs from j 1 “feed in” neighboring effects. The resulting formula is used for j = 3, 4, ... , jmax –1, since j = jmax is a boundary. Along j = 2, which lies adjacent to the airfoil, boundary conditions are incorporated using a second-order accurate formula relating at j, j+1 and j+2 to y(i, j = 1) = f ’(i). The equations so derived are second-order accurate in hi as noted in Keller (1974) and similarly for gj. Thus, a coarse mesh can be used to obtain good results, relative to the first-order accurate Murman-Cole scheme. Second-order accuracy in solving Eq. 3.7.1 is essential, since Kv 0. For inflectional instability with ci > 0, the Reynolds stress retards the flow above the inflection point and speeds it up below; it tends to smooth out the inflectional region and, as will be shown, stabilizes the primary disturbance. Throughout this process the momentum of the mean flow is constant; that is, from Equation 3.10.6, we have y2

y2

y2

U(y,t) dy = Ul (y) dy + E Ul” | N|2/|Ul – c |2 dy y1 y1 y1 y2 = Ul (y) dy = constant y1

(3.10.7)

because the shear stress given in Equation 3.10.4 vanishes y = y2. The detailed wave energy growth can be obtained from the inviscid energy equation y2

y2 / t

½ (u+2 y1

+ v+2)

dy =

(-u+v+)cyc-avg U(y,t)/ y dy y1

(3.10.8)

Here, u+ and v+ are periodic disturbance speeds and (u+v+)cyc-avg denotes a cyclic average; the left side of in the above equation represents the time rate-of-change of disturbance kinetic energy. We now assume a disturbance similar in shape to that of linear theory, but modified by a time-dependent amplitude B(t). Note that the left side of Equation 3.10.8 equals E/ t, while the -u+v+ equals the in Equation 3.10.4. Substitution in Equation 3.10.8 and integration by parts leads to

238 Modern Aerodynamic Methods y2 y E/ t = 2 iE Ul” | N|2/|Ul – c |2 dy U/ y dy (3.10.9) y1 y1 y y2 y2 2 2 = 2 iE [ U Ul” | N| /|Ul – c | dy | – UUl” | N|2/|Ul – c |2 dy] y1 y1 y1 The first bracketed term vanishes since (y1) = (y2) = 0. Use of Equation 3.10.6 in the remaining integral yields y

y2

E/ t = – 2 iE [ UlUl” | N| / |Ul – c | dy + E Ul” 2| N|4/ |Ul – c|4dy] (3.10.10) y1 y1 2

2

noting that the right-side is quadratic (or nonlinear) in E. For selfexcited waves the real and imaginary parts of the complex equation used to derive Rayleigh’s theorem imply that y2

y2 2

2

– UlUl” | N| / |Ul – c | dy = {| y1 y1

2 N’|

+ k2 | N|2} dy

(3.10.11)

Since the right side has been normalized to unity, the wave energy equation becomes y2 E/ t = + 2 iE [ 1 – E Ul” 2 | N|4/ |Ul – c|4 dy] y1

(3.10.12)

Equation 3.10.12 gives the Landau constant explicitly. Also, since the above integral is positive we conclude that, to this order, nonlinearity is stabilizing and that equilibrium solutions can be found. In fact, the equilibrium wave energy density satisfies y2

E = 1/{ Ul” 2 | N|4/ |Ul – c|4 dy

}

(3.10.13)

y1

where, again, N is a normalized eigenfunction based upon Ul and k, which are prescribed. The solution for U(y,t) is obtained by solving

Advanced Models and Ideas 239 Equation 3.10.12 for E and combining with Equation 3.10.6. The equilibrium solution for U(y,t) is just y2

U = Ul (y) + Ul” | 3.10.3

N

|

2

/{|Ul – c|

2

Ul” 2 | N|4/ |Ul – c|4 dy} y1

(3.10.14)

Discussion and Conclusion

This section examined the nonlinear stability problem for an inviscid parallel flow and, in particular, the effect of wave-induced mean flow distortion on the overall flow stability. Simple governing equations for the wave energy density and the mean velocity profile are derived which suggest that the initial effect of this distortion is stabilizing. These equations are solved using the method described in Stuart (1958). Solutions for the equilibrium wave energy density and the equilibrium mean velocity profile are given in closed form. We point out several limitations implicit in our approach. First, the neglect of higher-order harmonics was made for mathematical expediency only, so that analytical solutions are possible; thus, the method describes neither the harmonic generation expected in nonlinear processes nor the distortion of the primary wave as would be induced by the neglected harmonics. These added effects can possibly modify the Landau constant obtained here. Second, we stress that our analysis is weakly nonlinear, restricted to waves close to the neutral curve; this limitation arises because we have, in our solution method, assumed a “wave shape” similar to that of linear theory. Also, because the solution obtained here is expanded about linear theory, the well-known restriction to growing waves with ci > 0 applies; solutions to Rayleigh’s equation, when viewed within the framework of the more complete viscous Orr-Sommerfeld equation, are physically meaningful only for self-excited waves. Finally, we emphasize that we tacitly assume normal modes. For linear flows, it is known that the flow associated with the continuous spectrum decays with time, e.g., see Case (1960), thereby justifying its neglect; this may not be the case in the weakly nonlinear problem considered here and further study is needed. The study of inviscid flow stability theory for high Reynolds number flows is, in itself, a well-accepted fluid-mechanical discipline; also. some work of Landahl (1972, 1975) suggests that the breakdown mechanism leading to turbulent transition may well be inviscid. In this section, however, the study was motivated by the possibility that

240 Modern Aerodynamic Methods equilibrium solutions may exist inviscidly, without the need to assume slow temporal variations as the original viscous approach of Stuart (1958). This possibility is demonstrated here where, as in Stuart’s investigation, higher-order harmonics are neglected. In closing we point out that the mean flow Ansatz supplied in Equation 10.10.6 may be used in a Whitham-type solution for slowly varying wavetrains of wave number k(x,t) and energy density E(x,t) propagating in space and time (e.g., see Whitham (1974) or Chin (2014) for detailed analyses). In this approach, an equation for wave crest conservation using the real part of the complex frequency only, but with a nonlinear Stokes’ correction for the real frequency, is coupled with Equation 3.10.12 but modified by a spatial flux term. This “straightforward” generalization of Whitham’s conservative wave formalism is rigorously justified in the high-order extension general kinematic waves published by the author in Chin (1980). 3.10.4

References

Case, K., “Stability of Inviscid Plane Couette Flow,” Physics of Fluids, Vol. 3, 1960, pp. 143-148. Chin, W.C., “Effect of Dissipation and Dispersion on Slowly Varying Wavetrains,” AIAA Journal, Vol. 18, Feb. 1980, pp. 149158. Chin, W.C., Wave Propagation in Drilling, Well Logging and Reservoir Applications, John Wiley & Sons, Hoboken, New Jersey, 2014. Landahl, M.T., “Wave Mechanics of Breakdown,” Journal of Fluid Mechanics, Vol. 56, Dec. 1972, pp. 775-802. Landahl. M.T., “Wave Breakdown and Turbulence,” SIAM Journal of Applied Mathematics, Vol. 28, 1975, pp. 735-756. Landau, L.D. and Lifschitz, E.M., Course of Theoretical Physics Series, 1940 – Present, ten volume series now available from Amazon.com. Lin, C.C., The Theory of Hydrodynamic Stability, Cambridge Press, Cambridge, 1955.

Advanced Models and Ideas 241 Meksyn, D. and Stuart, J.T., “Stability of Viscous Motion Between Parallel Planes for Finite Disturbances,” Proceedings of the Royal Society, Series A208, 1951, p. 5l7. Schlichting, H., Boundary Layer Theory, Ninth Edition, SpringerVerlag, Berlin, 2017. Stuart. J.T., “On the Nonlinear Mechanics of Hydrodynamic Stability,” Journal of Fluid Mechanics, Vol. 4, 1958, pp. 1-21. Stuart, J.T., “On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows, Part I. The Basic Behaviour of Plane Poiseuille Flow,” Journal of Fluid Mechanics, Vol. 9, 1960, pp. 353-370. Whitham, G.B., Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974. Yih, C.S., Fluid Mechanics, McGraw-Hill, New York, 1969.

242 Modern Aerodynamic Methods 3.11 Aerodynamic Stability of Inviscid Shear Flow Over Flexible Membranes 3.11.1 Introduction

The shear flow over compliant surfaces is physically interesting and its linear stability has been studied in the viscous case. Aerodynamic applications have been motivated by observations and speculation related to possible low drag hydrodynamic performance in sea animals. Early studies contributing to our fundamental understanding include the works of Brooke Benjamin (1960) and Landahl (1962). In this section, we consider the inviscid limit and demonstrate analytical techniques which lead to exact results for self-excited disturbances. For various flow parameters, these include bounds on unstable eigenvalues, sufficient conditions for stability, and also, explicit dispersion relations. 3.11.2 Analysis

We orient along the x axis the horizontal velocity U(y) + u(x,y,t) and along the y axis the vertical velocity v(x,y,t) with t being time. Consider a flexible membrane whose deviation from equilibrium y = 0 satisfies y = n(x,t) and assume a rigid wall placed along y = - H < 0 across which v = 0. If u,v 0 so that real and imaginary parts give (3.11.4) (3.11.5) Note the expressions Gr = g + Tk2/ + s/ – mk2(cr2 – ci2)/ + mdkci / and Gi = -2mk2cr ci / – mdkcr / . Expansion of Equation 3.11.4 using Equation 3.11.5 leads to (3.11.6) assuming ci > 0. Now let A and B represent the minimum and maximum of U(y) in (-H,0). Then the identity

243

244 Modern Aerodynamic Methods

on using Equations 3.11.5 and 3.11.6 yields

(3.11.7)

Several cases are to be distinguished. For nondissipative membranes with d = 0, we note that

If g + s/ + k2(T - mAB)/ or

0, it follows that cr2 + ci2 - (A+B)cr +AB

0

(3.11.8) Thus, the complex wave velocity c for any unstable mode must lie inside the semicircle in the upper half of the complex c-plane which has the range of U for diameter. Alternatively, for massless membranes with m = 0, Equation 3.11.7 again simplifies considerably. If g + s/ + Tk2/ > 0, Equation 3.11.8 again holds with the same consequences. Results for more general membrane parameters have not been obtained and probably require the use of different inequalities. We now obtain sufficient conditions for stability for light membranes. The equation used in proving Rayleigh’s theorem, that is,

Advanced Models and Ideas 245 is evaluated using Equations 3.11.2 and 3.11.3. The resulting imaginary part is given by the equation

Setting m = 0 leads to Gi = 0 and Gr = g + Tk2/ + s/ > 0. Hence, we find that (3.11.9)

and that ci 0. Equation Now suppose that U"(y) < 0 and U'(0) 3.11.9 then requires that U(0) = max U < cr . However, if this is satisfied, ci must vanish according to the semicircle theorem. This contradicts the supposition ci 0. Hence, ci must be zero. Thus, if U" < 0 throughout and U'(0) , the flow is stable. Similarly, if U" > 0 throughout and U'(0) , the flow is also stable (these results do not apply to constant U flows since U' = U" = 0). Sufficiency conditions for more general membrane parameters have not been obtained. 3.11.3 Specific Examples

Explicit solutions for straight-line velocity profiles are easy to obtain and we thus examine more general classes of problems. However, we restrict ourselves to those yielding simple closed-form solutions. For long waves we assume | ”| >> k2 | | in Equation 3.11.1. The Rayleigh equation reduces to (U - c) ” – U” = 0. Defining = /(U – c) and using (-H) = 0 lead to an integrable equation whose solution is of the form

246 Modern Aerodynamic Methods From this, the expression for ’(y)/ (y) can be calculated. Then evaluation at y = 0 and comparison with Equation 3.11.3 yields the dispersion relation

The corresponding eigenfunction is given by

where C0 was determined by evaluating (y) at y = 0 and using the dispersion and surface kinematic conditions. The foregoing solutions hold for long waves only; the integrals are evaluated by assuming ci > 0. Closed-form solutions are also possible for a restricted class of shear flows. Assuming that U'(y) is small near the membrane, an approximate solution to Equation 3.11.1 is (y) = - a(U-c) exp(ky). Solutions of this form have been considered in various hydrodynamic stability studies. The foregoing solution satisfies the kinematic condition 0) a {U(0) - c} as well as (- ) = 0. The approximation holds whenever either of U'/k(U-c) or kU'U” is small. The first forbids critical points and the second is a restriction to long waves. The surface condition leads to {-(U(0)-c) ' (0) + (0)U' (0)} = ( g + Tk2 + s – mk2c2 - imdkc) a. Then, evaluation with the assumed solution produces the approximate dispersion relation (U0-c)2k = g + Tk2 + s – mk2c2 imdkc where U0 = U(0). Thus unstable ci’s are easily determined for all parameters. Finally, consider an interesting example for the linear velocity Couette flow U(y) = y + . In this case U” = 0 and if (U-c) is nonzero in (-H,0), the solution to ”- k2 = 0 satisfying Equation 3.11.2 is proportional to tanh kH cosh ky + sinh ky. Substitution in Equation 3.11.3 leads to the dispersion relation

Advanced Models and Ideas 247

Now this is equivalently (0) = R -1 ’(0) where R measures an effective rigidity. In the rigid wall limit we find that (0) = 0 and hence vanishes identically; the basic flow, therefore, has no eigensolutions of the discrete c-spectrum. Thus we conclude here that it is the allowance of wall flexibility that gives rise to wavelike perturbations. 3.11.4 Discussion and Concluding Remarks

The foregoing results for the inviscid shear flow stability over compliant surfaces were obtained to supplement the more detailed viscous studies of Benjamin (1960) and Landahl (1962) using the viscous Orr-Sommerfeld equation. Our work focuses essentially on explicitly obtainable eigenvalue bounds, sufficiency conditions, and dispersion relations for the Benjamin-Landahl membrane model, although, for somewhat restrictive parameter ranges. While the results reported here are not directly applicable to the analysis of finite length plates (panel flutter) per se as contrasted with infinite length membranes, the general methods may be used in conjunction with alternative structural models and with only minor modification. The basic problem, of course, is one for fluid and solid interaction, and a substantial body of more general work appears in the aeroelastic literature. Noteworthy among these are some significant theoretical models for shear flows over flexible boundaries, for example, those pursued by Dowell and his collaborators. These are typified in Dowell (1971, 1975) and Williams et al. (1977). These studies allow for both finite plate dimensions as well as fluid compressibility; their numerical results, moreover, compare favorably with the experiments of Muhlstein et al. (1968, 1971). The analytical approaches adopted here and in the cited work of Dowell and Williams are more or less equivalent. The simpler interaction model examined here, though, leads to simple closed-form results, and these may be useful in various applications.

248 Modern Aerodynamic Methods 3.11.5 References

Brooke Benjamin, T., “Effects of a Flexible Boundary on Hydrodynamic Stability,” Journal of Fluid Mechanics, Vol. 9, 1960, pp. 513-532. Dowell, E.H., “Generalized Aerodynamic Forces on a Plate Undergoing Transient Motion in a Shear Flow with an Application to Panel Flutter,” AIAA Journal, Vol. 9, 1971, pp. 834-841. Dowell, E.H., Aeroelasticity of Plates and Shells, Noordhoff International Publishing, 1975, pp. 51-64. Howard, L.N., “Note on a Paper by John W. Miles,” Journal of Fluid Mechanics, Vol. 10, 1961, pp. 509-512. Landahl, M.T., “On the Stability of a Laminar Incompressible Boundary Layer Over a Flexible Surface,” Journal of Fluid Mechanics, Vol. 13, 1962, pp. 609-632. Muhlstein, L. and Beranek, R.G., “Experimental Investigation of the Influence of the Turbulent Boundary Layer on the Pressure Distribution Over a Rigid Two-Dimensional Wavy Wall,” NASA TN D-6477, 1971. Muhlstein, L., Gaspers, P.A. and Riddle, D., “An Experimental Study of the Influence of the Turbulent Boundary layer on Panel Flutter,” NASA TN D-4486, 1968. Schlichting, H., Boundary Layer Theory, Ninth Edition, SpringerVerlag, Berlin, 2017. Williams, M.H., Chi. M.R., Ventres, C.S. and Dowell, E.H., “Aerodynamic Effects of Inviscid Parallel Shear Flows,” AIAA Journal, Vol. 15, 1977, pp. 1159-1166. Yih, C.S., Fluid Mechanics, McGraw-Hill, New York, 1969.

Advanced Models and Ideas 249 3.12 Goethert’s Rule with an Improved Boundary Condition

A Goethert rule is developed which identifies a linearized subsonic compressible flow with a family of affinely related exact incompressible flows. A modified transformation in the supersonic case casts the problem in hyperbolic canonical form, so that the results of Heaslet and Lomax (1948) are directly applicable. The boundary conditions used are obtained from approximations consistent with the linearized differential equation. These transformations extend the usability of computed compressible flow results. 3.12.1 Introduction

Goethert’s rule equating a compressible flow to an affinely related incompressible one has been extremely valuable to generations of practicing aerodynamicists. Its use has simplified the design and wind tunnel evaluation of many airfoil configurations, and forms the theoretical basis for numerous similarity rules. These rules, however, suffer from the requirement for near-planarity; they assume the linearized equation of compressible flow with a simplified tangency condition on an approximating slit. They are restrictive in view of realistic airfoils, which, for aerodynamic or structural reasons, are sufficiently thick so as to warrant a nonplanar, nonlinear theory. An improved theory can be constructed by retaining the linearized differential equation (so that superposition remains valid) but introducing a modified form of the exact tangency condition applied on the airfoil surface. In supersonic flow, such formulations have been considered in the now-classical work of Heaslet and Lomax (1948). The practical reasons justifying the need for nonplanar theories in both supersonic and subsonic regimes have stimulated much activity in aerodynamics. For example, an explicit formula for the supersonic wave drag of a general planar distribution of sources and doublets without restrictions on thickness was obtained by Chin (1976). Later, the well known formula due to Hayes (1947) for wave drag was generalized by this author to arbitrary source/doublet areal distributions on nonplanar surfaces (Chin, 1977). These results were, at the same time, complemented by rapid developments in computational panel methodology. Johnson and Rubbert (1975), for instance, have treated the subsonic nonplanar problem using distributed singularities with locally varying strengths; the extension to supersonic flow was carried out by Ehlers et al. (1976).

250 Modern Aerodynamic Methods In this Section, a modified Goethert rule is given which equates the compressible subsonic problem to an affinely related exact incompressible problem, or to the supersonic canonical form of Heaslet and Lomax (1948). This possibility was recognized during the early 1970s at Boeing, although, at the time, a self-consistent formulation was not available; slender-cone calculations using the modified boundary conditions to be derived actually showed better agreement with the exact solution than did slender-body theory. 3.12.2

Analysis

For steady, two-dimensional, irrotational, isentropic motion, the usually is differential equation for the complete velocity potential simplified by introducing small perturbation velocities u and v such that x = U + u and y = v, where the freestream speed U >> |u| and |v|. Quadratic and cubic nonlinearities in the resulting disturbance equation are then dropped, under the requirement that M 2(u/U )2 0 and z < 0, z(x, 0 ) = - ½ (1-M 2)Cp - M 2( +1) Cp2/8 specified on the chord, a vanishing jump [ ] through the downstream “wake” for trailing edge closure and ~ 0 at infinity (note here that z (x) = - (x, 0 ) represent upper and lower surfaces to within a constant), where Cp is the prescribed pressure coefficient, x and z are streamwise and transverse coordinates, M is a subsonic freestream Mach number, is the ratio of specific heats, actually follows without and approximation from the complementary analysis problem for the velocity potential (x,z) solving {1-M 2 - M 2( +1) x}

xx

+

zz

=0

subject to z(x,0 ) specified on the chord, [ ] chosen to satisfy Kutta’s at infinity (note Cp = -2 x). Shock conditions condition and vanishing for mass and vorticity apply in both cases. In Chin and Rizzetta (1979), classic solutions for ellipses and flat plates at angle of attack were obtained from known subsonic pressures. Also, known supercritical airfoils, using type-dependent numerical methods, were reproduced in one case using a shock-free Korn Cp and in a second using a Jameson Cp with a strong shock. The method was also used in Chin (1980) to design families of shock-free airfoils (as will be discussed in Section 4.6) and to examine the existence or realistic airfoils

294 Modern Aerodynamic Methods given smooth surface pressures fixed throughout a range in M . These calculations appear to have established the computational expediency of the new formulation. We stress that our approach for a given freestream always produces a closed airfoil for any prescribed pressure (unphysical surface “crossovers” are possible, however) and apparently contradicts the wellknown work of Lighthill (1945). In brief, Lighthill solves Laplace’s with the exact pressure x2 + z2 equation for the total potential prescribed on the circle to which the unknown airfoil maps. This formulation, neither Neumann nor Dirichlet, insists on three existence requirements: a consistency condition on permissible surface pressures, a condition on “x closure” and one on “z closure.” But, in their smalldisturbance model, Chin and Rizzetta (1979) enforce z closure only; their formulation, it turns out, is correct for thin airfoils nearly aligned with the freestream. This is true because, in the small-disturbance limit, the exact expression for the pressure coefficient Cp linearizes to the usual -2 x identical, of course, with the normal derivative -2 z; since z is prescribed on the slit z = 0, one has an analysis-like Neumann problem for xx + zz = 0 whose uniqueness is insured by setting = 0 at infmity and choosing a “circulation” [ ] consistently with edge requirements. Lighthill’s inverse formulation is Neumann-like for thin airfoil flows: his first pressure constraint is satisfied automatically, x closure does not apply, leaving only a requirement on z closure correctly handled by prescribing [ ]. The difficulties discussed in Lighthill (1945) apply only to the inner nose problem, not addressed here, where linearization is impossible (see Thwaites (1960) for a review of early inverse methods). Alternatively, we could have considered the small-disturbance formulation for xx + zz = 0; for any chordwise z (x,0 ), a unique analysis solution always exists given Kutta’s condition plus some statement on the leading edge singularity. Constraints on z are unnecessary even for nonzero z

(x, 0 ) dx

C if a source-like asymptote is assumed at infinity. Because our inverse is identical, the same conclusions apply without formulation using

General Analysis and Inverse Models 295 change: z(x, 0 ) can be prescribed arbitrarily provided vortex and source-like farfields are assumed for flows with nonzero z(x,

0 ) dx

C or [ ]. In the above, C denotes a complete contour integral taken about the airfoil. Prescribed trailing edge pressures should obey Kutta’s condition physically, but this is not required mathematically. The present work expands on Chin and Rizzetta (1979) by considering broader classes of aerodynamic shapes, e.g., finite wings, cascades, fans, inlets and nacelles. While the earlier study dealt only with closed edges, the present also discusses constraints on [ x] which affect included angle (in particular, a vanishing jump assures a cusped but generally opened trailing edge, modeling the inviscid displacement thickness streamline leaving the trailing edge tangent to the high Reynolds number flow far upstream). For subsonic linear flows, analytical results derived provide a number of dualities relating analysis and design formulations, as discussed in the Introduction, which may shed insight on the more difficult but numerically tractable transonic problem. That shapes with edges both cusped and closed are possible by introducing dynamically insignificant changes to prescribed Cp’s is demonstrated for linear flows, suggesting, perhaps, new variations on transonic inverse formulations. In the following discussion, basic ideas are developed from simple examples and extended to flows of increasing geometric and aerodynamic complexity. The important role played by constraints is also examined in detail. 4.3.2 Theory and Examples 4.3.2.1 Constant Density Planar Flows

Thin airfoil theory solves xx + zz = 0, with z(x, 0 ) specified ~ 0 at along -1 x 1, [ ] chosen for smooth trailing edge flow and infinity. The direct problem superposes solutions symmetric and antisymmetric with respect to chord: a “thickness” (t) satisfying an equation that implies a source strength proportional to local surface slope and a “camber” (c) (modeled by lineal vortices and) governed by an integral equation whose solution is rendered unique only by invoking Kutta’s condition plus some statement on the leading edge singularity.

296 Modern Aerodynamic Methods Design or indirect problems also require auxiliary constraints. Now, any prescribed Cp can be decomposed into symmetric and antisymmetric parts. For thickness problems, we assume sources of strength f using the integral +1 (t)

= {1/(2 )} f (s) log {(x-s)2 + z2} ds -1

and the boundary condition singular integral equation

x

(t)

(x, 0 ) = u(t)(x).

But the resulting

+1 f (s) ds /(x-s) = u(t)(x) -1 is the analysis equation for the lifting flow past a camber line if u(t)(x) is the surface slope and f is the sought vorticity! Thus various solutions are possible from the mathematical literature, e.g., referring to the classic integral equation textbooks by Miklin (1964) and Muskhelishvili (2008). For instance, the choice +1 f(x) = (1/ ) {(1-x)/(1+x)}

1/2

{(1+s)/(1-s)}1/2 u(t)(s) ds /(s-x) -1

+1 2 -1/2

– (1/ ) (1 – x )

{(1+s)/(1-s)}1/2 u(t)(s) ds -1

leads to closed airfoils satisfying +1 f (s) ds = 0 -1 but both edges are blunted unless the second integral vanishes (thus leaving a sharp trailing edge of zero radius); in contrast, +1 f(x) = (1/ ) {(1-x)/(1+x)}

1/2

{(1+s)/(1-s)}1/2 u(t)(s) ds /(s-x) -1

General Analysis and Inverse Models 297 leads to blunted leading and cusped trailing edges generally opened. This duality between the analysis problem for lift and the design problem for thickness is complemented by one for the analysis problem for thickness and the design problem for lift. The boundary condition x(c) (x, 0 ) = u(c)(x) together with +1 (c)

= - {1/(2 )} h(s) arctan z/(x-s) ds -1

(h is the vortex strength) leads to an equation with the obvious solution h(x) = 2u(c)(x), analogous to the analysis result relating source strength to local surface slope. Now let us reconsider the foregoing using as opposed to . We solve an analysis-like problem for xx + zz = 0 with +1 = (1/ ) f (s) arctan z/(x-s) ds -1 where f is determined from the Neumann condition z(x,0 ) = x(t)(x, 0 ) = u(t)(x). But interestingly, we have superposed “arctangents” just like the vortex solutions used in analysis potential formulations for camber (i.e., ~ tan-1, here, of course, “arctangent” means sources): in this sense one has “pseudo-vortices.” superposes Similarly, the design formulation for camber using vortex singularities which analytically appear as logarithms; the analogy with potential formulations for thickness (solved taking ~ log) suggests the use of “pseudo-sources.” As in “analysis potential theory,” solutions to the integral equation for “pseudo-vortex strength,” not unique, are physically fixed by Kutta-type conditions. A specified jump in measures the mass flow issuing from the trailing edge, since the streamfunction is constant along a streamline. The limit [ ] = 0, that is +1 f(s) ds = 0 -1 enforces closure; on the other hand, [ x] = [- z] = - [dz/dx] fixes the included angle. The design problem for camber, again, superposes pseudo-sources, leading to the well-known “source equation” with unique solutions. These terms appear somewhat superficial at first, but

298 Modern Aerodynamic Methods their use in three dimensions, as will be seen, broadens the duality connecting analysis and design problems. In closing this paragraph, we emphasize that streamfunction formulations for design are mathematically identical to potential ones for analysis; thus computational methods, panel, vortex-lattice, finite difference, finite element and finite volume, devised for analysis apply with minor change to design problems provided only we reinterpret inputs and outputs (again, z (x) = - (x, 0 )). The investigation in Chin and Rizzetta (1979) worked this idea for planar transonic flows; the present Section, again, extends this work to problems of increased geometric and aerodynamic complexity. An important geometric shape is the one that induces the uniform loading u(t)(x) = > 0 and possessing blunted leading and cusped trailing edges; hence, the source strength f(x) = (l - x)1/2/(l + x)1/2 is singular at x = -1 and zero at x = +1 (f, as defined, again equals the surface slope). Now, the upper surface satisfying z+(-l) = 0 is z+(x) = (1- x2)1/2 + ½ + sin-1 x); thus the thickness grows monotonically with x, producing a (contrasting with the well-known thin trailing edge gap equal to 2 ellipse). Thus, in the design problem, airfoils with blunted leading and cusped trailing edges entirely closed are possible by appending to an “original” u(t)(x) a constant loading everywhere, with just enough “2 ” to close the final shape (“starting” and appended shapes are linearly superposed)! This constant addition has no aerodynamic significance since it modifies neither the specified pressure gradient nor the lift. The cusped edge model also provides the inviscid displacement thickness streamline leaving the trailing edge tangent to the free stream; to obtain the “actual” airfoil, one subtracts a displacement thickness determined from simple boundary layer estimates, with closure ensured by modifying u(t) as discussed. Note that the amount of closure [ ] changes monotonically with , just as the lift [ ] in analysis problems changes monotonically with angle of attack. This may also be true of weakly supercritical transonic flows (in analysis problems, high angles of attack produce strong shocks and possible losses in lift). The method of Chin and Rizzetta (1979) can he reformulated to produce trailing edges both cusped and closed: drop the [ ] = 0 in favor of fixing [ x] and program changes to “intelligently” to provide the desired closure (the analysis analogy would vary angle of attack to produce a prescribed lift).

General Analysis and Inverse Models 299 4.3.2.2 Wings

Constant Density Flows Past Three-Dimensional Finite

Analysis problems solve xx + zz + yy = 0 (where “y” is the spanwise variable) with the normal derivative z(x, 0 y) specified on a flat “lifting surface” z = 0. A spanwise jump [ (y)] is chosen to satisfy Kutta’s condition and also assumed are regularity conditions at infinity. A physical wake is also required; in the simplest implementation, Prandtl’s lifting line model is used. To demonstrate similar analysis and design analogies three·dimensionally, we must show that a simple “ (x,z,y)” exists which satisfies the same boundary value problem. The more general streamfunction, however, is known to be a vector potential governed by three coupled boundary value problems, thus diminishing somewhat our hopes. But this impediment is mere semantics: any (equally rigorous) scalar “stream-like” function satisfying our requirement for a scalar function and which reduces to our planar formulation suffices. We simply invoke the obvious conservation form x (

+ yy dx)x +( z)z = 0 and define the generalized Cauch-Riemann conditions x x

z

(x,z,y) =

x

+ -

yy

dx

and x (x,z,y) = - z . This leads to xx + zz + yy = yy (- ,z,y) = 0, the required three-dimensional potential-like scalar equation! For mildly swept high aspect ratio wings, since spanwise variations are small, the approximation x z

(x,z,y) =

x

+ -

yy

dx

x

= - ½ Cp

provides a Neumann condition unchanged from planar theory, and the same “Kutta-type” trailing edge constraints and regularity conditions can be used. This three-dimensional formulatiun for (x,z,y) is identical to the analysis formulation for (x,z,y) and provides the required inverse dual to lifting surface theory: numerical methods devised for analysis problems, as in planar flows, again therefore apply to design problems.

300 Modern Aerodynamic Methods Surface streamlines are computed from the tangency condition dz (x,y)/dx = z(x,0 ,y) = - x(x,0 ,y), that is, z (x, y) = - (x,0 ,y) + g(y), where the dihedral g(y) vanishes in small-disturbance theory (potential function design methods, assuming closure is properly handled, first differentiate to obtain z and then integrate to obtain z , a procedure that may be inaccurate numerically). In planar flows, infinite line vortices with axes perpendicular to the page create lift; for finite wings, this vortex bends into the downstream flow, but because the bound circulation [ (y)] now varies with span, a wake forms that induces a local downwash that reduces the geometric angle of attack. Planar inverse formulations for thickness superpose pseudo-vortices as discussed. For finite wings, because (x,z,y) satisfies Laplace’s equation, our infinite line pseudo-vortices must likewise bend in the downstream direction. Whenever spanwise variations in [ ] exist, Stokes’ theorem requires a shed pseudo-vortex wake that arises from mass conservation as opposed to rotation: a wake of “source lines” carries “shed mass” that arises from spanwise differences in trailing edge ejected flow (these are generally nonzero for Kutta-type conditions in [ x]). Pseudo-vortex wakes induce changes in wing loading the way real wakes induce downwash; the “wake” vanishes for a trailing edge everywhere closed (or equally opened), for example, as a real wake would if all section loadings were identical. Our above explanations follow directly from analogies to classical analysis problems. Our cusped edge model is particularly relevant to high Reynolds number flows where, in fact, one has a semi-infinite chord bounded by an inviscid displacement thickness streamsurface that leaves the trailing edge tangent to the uniform flow. Since [ x] = 0, the gap will vary with span. In the direct problem, Cp is obtained, and boundary layer wing and wake flows are solved. The inverse problem is also important: the displacement stream surface (as described) is sought that induces the prescribed Cp over a given planform (the actual wing can be obtained by subtracting a boundary layer thickness as in the planar case). The resulting edge thickness [ (y)] then provides some estimate of surface viscous effects; for example, if it is large relative to the maximum thickness, the pressure is probably undesirable. This problem, easily solved computationally, also lends itself to simple analysis in the “lifting line” sense. Prandtl’s solution for flat plate wings uses planar results locally based on [ x] = 0 at the trailing edge and assumes zero tip loadings [ ] = 0; here, enforce our viscous condition at

General Analysis and Inverse Models 301 each section and assume wing tip closure through [ ] = 0. Because both formulations are identical, lifting line formulas based on the Glauert (1947) classic expansion apply without change to design. For example, the total ejected mass flow depends only on the first harmonic much like the well-known result for total lift, and the edge gap can be everywhere found (unfortunately, analogies to known results for minimum induced drag do not exist because the Trefftz plane kinetic energy y2 + z2 is unphysical). Of course, aerodynamic twist would correspond to a variable loading (y); while it can be used to produce edges both cusped and closed, as discussed, in the present context it provides a means to analyze prescribed spanwise pressure gradients to prevent unwanted boundary layer separation. Bound pseudo-vortices are responsible for closure while free pseudo-vortices induce changes in pressure. 4.3.2.3 Compressible Flows Past Finite Wings

In the above section, we considered constant density flows only. For the flow of compressible of gaseous fluids, we write for the nonlinear velocity potential equation x ((1 – M 2) but now define conditions

– ½ M 2 ( +1)

2

+ yy dx)x + ( z)z = 0 (x,z,y) through the generalized Cauchy-Riemann x

x

x z

2

= (1 – M )

x

2

– ½ M ( +1)

x

2

+ -

yy

dx

and x = – z. Thus, (1 – M 2 – M 2( +1) x) xx + zz + yy = 0 assuming that yy(- ,z,y) vanishes faraway. For mildly swept high aspect ratio wings, as before, the spanwise term in z can be neglected, so that |(1-M 2)2 – 2M 2( +1) z|1/2 xx + zz + yy = 0 holds (here the refers to pure subsonic and supersonic flows). For transonic flows with M < 1, x decays at infinity, so that z x

=–

xx(x,s,y)

ds

302 Modern Aerodynamic Methods (if M > 1, surface Cp’s provide the required reference): substitution produces the integro-differential equation of Chin and Rizzetta (1979) with an added yy term. Assuming weak spanwise effects, the definition for z and the expression Cp = -2 x lead to the Neumann boundary condition z = - ½ (1-M 2) Cp – M 2( +1)Cp2/8. Kutta-type trailing edge conditions again apply, along with the usual regularity conditions at infinity for M < 1 or radiation conditions for M > 1. This analysis-like problem is amenable to existing transonic analysis methods; again, only simple code changes are required, and wherever the local type is known beforehand, the simpler “ ” equation can be used (again, see Chin and Rizzetta (1979) for details). For linear compressible flows, (1-M 2) xx + zz + yy = 0 holds; the dual analogy to analysis results satisfying the Prandtl-Glauert equation (1-M 2) xx + zz + yy = 0 is exact! Thus the vast literature on subsonic and supersonic analysis methods provides an important new source of aerodynamic design information. 4.3.2.4 Flows in Fans and Cascades

An inverse approach to rectilinear cascades is easily formulated. The method discussed here is relevant to the design of turbine and compressor blade rows. For transonic small-disturbance flow, we will use (1 - M 2 - M 2( + l) x ) xx + zz = 0 with z x

(x,z) = - ½ Cp(x,0 ) –

xx

(x,s) ds

0 taking references along the chord; if M < 1, for example, | (1-M 2)2 – 2M 2( +1) z |1/2 xx + zz = 0 applies upstream of the leading edge and downstream of the trailing edge. This is solved with periodicity conditions, trailing edge shape constraints, and nonzero ’s specified upstream and downstream of the airfoil to enforce prescribed levels of streamline deflection consistent with momentum considerations. The deflection is determined iteratively during the relaxation using the standard relationship connecting total normal force to angular deviation. The effects of finite span and aerodynamic twist are modeled by a yy term and taking y = 0 along symmetry planes. For linear subsonic or supesonic flow, code development can be simplified using the following

General Analysis and Inverse Models 303 approximation. Simply apply existing panel methods (which satisfy regularity and not periodicity conditions) to a stacked array of flat planforms with the required aspect and gap-to-chord ratios; the “middle flow” approximately satisfies the periodicity requirement. 4.3.2.5 Axisymmetric Compressible Flows

Flows past axisymmetric nacelles without powered actuator disks satisfy (1 – M 2 – M 2( + 1) x ) xx + rr + r/r = 0 (r is the radial coordinate), with r(x,R ) = F ’(x), where r = R + F (x) are outer and inner surfaces and R is a mean radius, r(x,0) = 0, ~ 0 at infinity (for M < 1), and the usual shock relations and Kutta’s condition through the wake r = R apply. “Isolated inlets” are semi-infinite and tend to r = R far downstream; here, the level of circulation is determined by the amount of mass flow (note Cp = -2 x). These simple formulations extend planar theory; however, for axisymmetric nacelles, the effects of source and vortex rings are strongly coupled through the r/r term. For inverse problems, auxiliary conditions again appear: finite length nacelles require trailing edge constraints, as expected, but isolated inlets will enforce requirements on compressor face flow distortion. Axisymmetry introduces a new “ r/r” contribution to the planar small-disturbance potential equation – one which initially hints at difficulties in developing generalized Cauchy-Riemann conditions needed to derive streamfunction formulations. But this is not true. Let us write {r(A x – ½ B x2)}x + (r r)r = 0 and define r = r (A x – ½ B x2) and x = - r r (here A = 1 - M 2 and B = M 2( + 1) for convenience). Thus we find (A - B x) xx + rr - r /r = 0 or |A2 - 2B r /r |1/2 xx + rr r /r = 0 for pure subsonic or supersonic flow. For transonic flow, r = x /r is integrated with respect to r from “r” to “R” and differentiated with respect to x. Thus R x (x,r) = - ½ Cp (x,R) + xx(x,s)/s ds r where C p (x, R) refers to r > R and r < R, uniquely defining the A - B x coefficient (the simpler subsonic form applies off the edges, of course, if M < 1). This is then solved with the usual Neumann condition r(x,R ) = -R(½ACp + BCp2/8) with Cp satisfying Kutta’s condition for physical consistency; also, = 0 at infinity and along r = 0, and the usual jump conditions hold.

304 Modern Aerodynamic Methods Surface coordinates are then obtained from the tangency condition dr (x)/dx = r(x,R ) = - x(x,R )/R or r (x) = - (x,R )/R; for closed edges, differences between trailing edge and centerline ’s provide the disturbance mass flow. This new formulation allows inlet and afterbody interaction. In some problems afterbody effects on inlet flow may be less important than a desire for distortion-free flow vital to engine flow stability and surge suppression. Such isolated inlet problems require semi-infinite nacelles with specified lip Cp’s or r(x,R ). Edge closure no longer applies; instead, compressor face constraints are imposed (e.g., x = - r r = 0 assures a flow without radial velocity, while xx = - r( x)r = 0 guarantees a uniform streamwise flow. Improved boundary conditions are computationally possible: substitute x = - ½ (Cp + r2 ) - ½ (Cp + 2 2 x /R ) in the definition for r and evaluate x using latest relaxation values (this formulation may be ill-posed, since xx + rr - r /r = 0 is not Laplace’s equation; however, numerical results have appeared to be stable). These methods apply to diffusers, nozzles, and wind tunnel test sections with slowly varying contours; the boundary conditions are 2 (x,0), and r(x,R) = - R(½ACp + BCp /8), vanishing r(x,0) or prescribed exit and entry plane streamline deflections specifying nonzero (again, r(x) = - (x,R)/R). Slender bodies of revolution are more difficult; the formula Cp = -2 x - r2 and the definitions for imply the “Neumann” condition r = - ½ Ar(Cp + x2/r2) - Br(Cp + x2/r2)2/8 where x is evaluated using latest values. The weak dependence of r /r on x may be useful numerically; streamlines would be obtained from dr(x)/dx = r(x, r) = - x(x, r)/r. Similar ideas apply to planar flows. In concluding this section, we emphasize that the axisymmetric streamfunction equation contains a “- r /r” first-order term while the corresponding potential equation contains a “+ r /r” term. Programming care is required in adapting existing analysis codes to inverse applications. 4.3.3 Sample Calculations The ideas outlined in the foregoing sections provide a rational basis for inverse design codes that produce geometric shapes satisfying pressure and auxiliary constraints automatically. Of course, the choice of Cp that yields sections free of unwanted wiggles and surface crossover remains a perennial problem. Because the success of any inverse method ultimately depends on “good” input pressure, this, in practice, would come from analysis results for shapes undergoing minor redesign.

General Analysis and Inverse Models 305 Chin and Rizzetta (1979) addressed transonic design for Cp’s with and without shocks and while Chin (1980) examined shock-free airfoil design applications; here, we emphasize instead alternative geometric constraints for both planar and three-dimensional flows. To highlight the basic effects we keep the pressures simple, preferring, for the present, to survey quickly various results implied within the broad scope of streamfunction formulations. The airfoil thickness corresponding to a constant negative Cp = - 2 prescribed chordwise at Mach zero, assuming leading and trailing edge closure, is a thin ellipse of thickness ratio , correctly calculated by the difference method of Chin and Rizzetta; added “analysis wake logic” easily produced a monotonically growing shape of thickness ratio with an opened cusp. These calculations agreed well with the analytic solutions z = z+/ = (1-x2)1/2 and z = z+/ = ½ + (1-x2)1/2 + sin-1 x (multiples of the latter are appended to shapes whose Cp’s are modified by constant ’s to provide sharp cusped edges, as discussed). We also indicated how approximate displacement thicknesses could be subtracted from “opened cusp” solutions (based on “ modified” Cp’s) to provide “actual” shapes. Such a shape, for example, is shown in Figure 4.3.1, along with the opened cusp, the Blasius boundary layer having the same trailing edge gap and the classic ellipse. Finite span effects were considered taking Cp(x,0 ,y) = - 0.2 on a rectangular unswept planform of aspect ratio 8 at Mach zero; as before, cusped edges were obtained using wake logic, while closed edges assume single-valued ’s without “wakes” (for pure subsonic or supersonic flow, panel methods are very efficient, but we stress that type-dependent relaxation methods must be used transonically). Calculated midspan sections agreed with the planar shape just given, with three-dimensional effects confined to the wing tips, as displayed in Figure 4.3.2; both panel and finite-difference methods gave nearly identical results. Fully three-dimensional transonic design calculations were also pursued using nearly converged analysis pressures; however, shape recovery was hampered by the effects of blunt leading edges, wing sweep and, especially, truncation errors through shocks, present in both analysis and design modes. However, further computational experimentation would likely have led to improved results. For airfoils and finite wings in unbounded flow, thickness and lift decouple in small-disturbance theory, and analytical and numerical simplifications are possible: in contrast, flows induced by ring sources

306 Modern Aerodynamic Methods and vortices in axisymmetric problems are highly coupled, due to centerline effects. Thus inlet, nacelle and ringwing flowfields, which have not yet yielded to closed-form analytical solution, must be solved numerically. But this is trivial since the difference methods just described require only r /r or r /r changes. Figure 4.3.3 shows a baseline ringwing section used to generate the Cp shown; it, in turn, reproduced the baseline shape. This same Cp was next used, assuming a smaller mean radius, to produce the modified shape, which correctly shows a flatter internal (lower) surface. These constant density runs were repeated at Mach 0.8; computed shapes, not shown here, however, showed weak surface oscillations near the shock foot and slight crossover at the trailing edge. For isolated inlets, Cp = - 0.6 at Mach zero was assumed on both internal and external lips (for the latter, this slowly vanished far downstream) with an inlet diameter to lip length ratio of 6. Results for two flow constraints, zero r and uniform x, as discussed, are shown in Figure 4.3.4; the former, as expected, produces a flatter internal surface more aligned with the free stream. Finally, consider mixed analysis and inverse problems, e.g., a wing is redesigned for improved tip flow, specifying stable pressures, but also inboard geometric shape to minimize root bending moment. In general, both closure and Kutta conditions would apply, but existence also depends on compatible boundary conditions. No complete mathematical theory exists, in particular for small-disturbance flows, but special classic solutions are available in isolated exact examples are available in Thwaited (1960). Here we sketch an ad hoc approach valid threedimensionally and transonically, but, for simplicity, consider constant density airfoil flows. At the outset, z (geometry) and z (pressure) are specified arbitrarily on a slit. The first iteration solves an analysis potential problem satisfying Kutla’s condition, using z where it is prescribed and zero elsewhere; in the second, solve a design streamfunction problem satisfying closure, using z where it is prescribed, but z = x from latest estimates otherwise. Analysis calculations then follow using z = - x as needed, followed by a design streamfunction iteration, and so on. Here, pressures for a 10 percent thick unpitched biconvex section were obtained by analysis; mixed calculations specifying geometry on the upper chord and pressure on the lower, were initiated, noting that each recursive iteration in the difference solution need not individually converge. This symmetric example is nontrivial because the circulation initially present, due to numerical

General Analysis and Inverse Models 307 asymmetry, must disappear in the converged solution; Figure 4.3.5 shows encouraging results typical of others not reported here. The method probably applies to minor redesigns about known flows provided any slight incompatibility in boundary data is no worse than the discretization error. 4.3.4 Closing Remarks Inverse problems, which are analysis-like under streamfunction transformations, are easily solved by existing potential flow codes with minor change. Chin and Rizzetta (1979) and Chin (1980) pursued this idea for transonic planar flows with closed edges; in this Section, we have explored the role of constraints in flows of increased geometric and aerodynamic complexity, developed the fundamental ideas through simple examples, and illustrated their application using subcritical pressures. Essentially, prescribing pressure fixes the lift or vorticity strength [ ], so that the nonuniqueness, which must be related to source strength, is “directly” and naturally handled using [ ]. If nonzero, the thickness farfield is source-like, but doublet-like for completely closed trailing edges. Of special interest is the “stream-like” function (x,z,y): like the more conventional vector potential, it also arises from mass conservation, but it is scalar and tractable mathematically unlike its vector streamfunction counterpart (it is, in a sense, “harmonically conjugate” to (x,z,y) in three dimensions). Unfortunately, it is not possible to furnish additional calculated examples covering the broad scope of streamfunction formulations, because of time, funding and length constraints; the author, through the work here and in the cited references, hopes to stimulate further research in this promising area. Excepting the simplest cases, we have not established mathematical existence requirements, but the formulations suggested are physically plausible; other applications, e.g., wings in ground effect, separation modeling for displacement streamlines with [ x] zero or nonzero, and so on, easily lend themselves to our methodology. Finally, several research areas remain, among them, (1) the need for special “inner” design solutions near noses and wing tips (there, “Lighthill’s constraints” apply, but they are known only for constant density planar flows); (2) methods to adjust Cp, to obtain “satisfactory” shapes, if both [ ] and [ x] are imposed; (3) heat transfer analogies for design, namely, Txx + Tzz + Tyy = 0 specifying Tz(x, 0 ,y) on the planform and T = 0 at infinity (closure is automatic since the temperature T is single-valued in the absence of

308 Modern Aerodynamic Methods insulators); (4) studies of any three-dimensional results inferred from the duality between our (1 - M 2 ) xx + zz + yy = 0 and (1- M 2 ) xx + zz + yy = 0 using classical analysis literature; and possibly, (5) the evaluation of possible exact methods, which may be implicit in ss + nn = 0 and ss + nn = 0, where s and n are streamline coordinates. For alternative “potential function” perspectives on inverse methods for planar and wing-like transonic flows, the additional references cited below provide a representative survey. The book by van Dyke (1964) provides some interesting approaches in modeling flows about blunt leading edges using localized “inner coordinates” – these flows are “asymptotically matched” to “outer” thin airfoil solutions using rigorous mathematical methods discussed in the monograph. Again, the classic integral equation monographs by Miklin (1964) and Muskhelishvili (2008) are “must reading” for any researcher new to our field of inverse aerodynamics. Z Opened cusp Blasius solution

Ellipse

“Actual” X

Figure 4.3.1. Thin airfoil results.

General Analysis and Inverse Models 309 CLOSED ELLIPSOID SECTIONS

OPENED TRAILING EDGE CUSPS

Figure 4.3.2. High aspect ratio thin wing results.

Cp 0.5

Internal External

(Input) +0.5 Baseline Modified

Figure 4.3.3. Inlet/afterbody design calculation.

Constant Cp < 0 assumed r zero x uniform

Figure 4.3.4. Isolated inlet design results.

310 Modern Aerodynamic Methods Cp 0.3

(Exact)

0.0 0.3 0.0 Calculated Cp+ and lower geometry

Figure 4.3.5. Mixed analysis and design problems. 4.3.5 References

Carlson, L.A., “Transonic Airroil Coordinates,” NASA CR-2578, 1976.

Design Using

Cartesian

Chin, W.C., “On the Design of Thin Subsonic Airfuils,” ASME Journal of Applied Mechanics, Vol. 46, No. 1, Mar. 1979, pp. 6-8. Chin. W.C., “Class of Shock~Free Airfoils Producing the Same Surface Pressure,” Journal of Aircraft, Apr. 1980, pp. 286-288. Chin, W.C. and Rizzetta, D.P., “Airfoil Design in Subcritical and Supercritica1 Flows,” ASME Journal of Applied Mechanics, Vol. 46, Dec. 1979, pp. 761-766. Henne, P.A., “Inverse Transonic Wing Design Method,” Journal of Aircraft, Vol. 18, Feb. 1981, pp. 121-127. Hicks. R.M., and Henne, P.A., “Wing Design by Numerical Optimization,” AlAA Paper 77-1247, 1977. Lighthill, M.J., “A New Method of Two-Dimensional Aerodynamic Design,” Rep. Memor. Aero. Res. Coun., London, No. 2112, 1945. Miklin, S.G., Integral Equations, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York, 1964. Muskhelishvili, N.I., Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical

General Analysis and Inverse Models 311 Physics, translation by J.R.M. Radok, Dover Publications, New York, 2008. Shankar, V., and Malmuth, N.D., “Computational Transonic Inverse Procedure for Wing Design With Automatic Trailing Edge Closure,” AIAA Paper 80-1390, 1980. Shankar, V., Malmuth. N.D., and Cole, J.D., “Computational Transonic Airfoil Design in Free Air and a Wind Tunnel,” AlAA Paper 78-103, 1978. Sobieczky, H., et al., “New Method for Designing Shock-Free Transonic Configurations,” AlAA Journal, Vol. 17, July 1979, pp. 722-729. Thwaites, B., Incompressible Aerodynamics, Oxford University Press, Oxford. 1960. Yu, N.J., “Efficient Transonic Shock-Free Wing Redesign Procedure Using a Fictitious Gas Method,” AlAA Journal, Vol. 18, Feb. 1980, pp. 143-148. van Dyke, M.D., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.

312 Modern Aerodynamic Methods 4.4 Superpotential Solution for Jet Engine External Potential and Internal Rotational Flow Interaction

Flows characterized by strong velocity shear are not irrotational and cannot be studied using conventional potential methods. As a result, inviscid shear flows are often modeled using complicated Euler equation approaches, these requiring mathematical techniques vastly different from those developed for potential flows. In this Section, we develop rigorous “potential like” methods for flows with axisymmetry from first principles, which apply to applications with significant levels of shear or vorticity. In Section 4.5, the results obtained here are specialized to planar flowfields and aerodynamic applications are also given. We emphasize that potential-like approaches, formulated through our socalled “superpotentials,” provide the accuracy and rigor offered by Euler methods while taking cost-effective advantage of potential flow software codes in need of only minor programming modification. Our approach is easily described conceptually (for elementary discussions on jet engine fluid mechanics, e.g., see Oates (1998). For flows with prescribed parallel shear far upstream, the vorticity generation term in our axisymmetric disturbance streamfunction ( ) equation and the Bernoulli “constant,” both of which vary from streamsurface to with streamsurface, are explicitly evaluated as power series in curvature-dependent coefficients, using some invariant properties of vorticity. By casting the linearized streamfunction equation in conservation form, as illustrated in Sections 4.1 – 4.3, extended CauchyReimann conditions can be obtained, which then imply the existence of a “superpotential” * satisfying a “Laplace-like” equation useful for solving flows past prescribed shapes. The corresponding tangency and Kutta conditions, interestingly, also take a “potential form,” so that simple changes to existing potential flow algorithms extend their applicability to strong oncoming shears with arbitrary velocity profile curvature. One important application is discussed in detail. The theory, which is relevant to duct flows behind jet engine actuator disks imparting velocity shear, is sketched for both “analysis” and “design” formulations. In this Section, we will address the interaction between external potential and internal rotational jet-engine flows occurring through both an assumed actuator disk and a trailing edge slipstream, and provide representative numerical calculations.

General Analysis and Inverse Models 313 4.4.1 Introduction In many aerospace problems, the important effects of oncoming shear usually require a direct attack on Euler’s equations or on a very complicated streamfunction equation useful in only the simplest applications. The shear, for example, might result from radially varying work imparted by turbomachinery blade rows, or, from the effects of nonuniform winds. This section, using the above “superpotential” approach, shows how small-disturbances to strong shears with arbitrary velocity curvature can be solved, making only simple changes to existing potential flow algorithms. Both the “analysis” problem, solving for pressures induced by prescribed geometries, and the “design” or “inverse” problem, solving for the geometruc shapes inducing specified pressures, are easily handled within the theoretical framework. A particularly challenging problem not yet tackled by existing computational methods is the jet-engine external potential and internal rotational flow interaction occurring through an actuator disk and a trailing edge slipstream. The power, flexibility and simplicity of the superpotential approach is applied to this very important engineering problem and numerical results are obtained; the results obtained here may be useful in guiding the design and evaluation of experiments aimed at understanding jet engine flow downstream effects in detail. The basic ideas derive from an invariant property of vorticity: for axisymmetric flow, the vorticity convected along a streamsurface changes only in proportion to its length. If a parallel shear flow is prescribed in some part of the flow domain (not necessarily upstream infinity), then the nonlinear vorticity generation term in the disturbance streamfunction ( ) equation governing the remaining perturbed flow and the Bernoulli “constant,” both of which vary from streamsurface to streamsurface, in principle can be evaluated explicitly as functionals of the base flow. In our analysis approach, they are expanded in power series of with variable shear-dependent coefficients. The linearized streamfunction equation, put in conservation form, implies the existence of a “superpotential” * satisfying extended Cauchy-Riemann conditions: the * equation is potential-like, and the corresponding tangency and Kutta conditions, interestingly, take the usual potential form, indicating only minor required changes to existing potential flow algorithms! For our jet engine problem, the only difference with irrotational formulations turns out to be an additional but discontinuous coefficient in the

314 Modern Aerodynamic Methods governing equation; this interfacial discontinuity, handled carefully, is stably implemented. The approach taken is quite general and obvious applications to other aerodynamic processes will be seen by the reader. Our superpotential results, actually, appeared quite fortuitously in the course of extending the streamfunction method of Chin and Rizzetta (1979) and Chin (1981) for aerodynamic “design” or “inverse” problems to flows with oncoming shear. These formulations solve for geometric shapes inducing prescribed pressures subject to trailing edge shape constraints. The design or inverse formulation for rotational flow is presented here, but calculations and engineering applications are deferred to future work; here, we concentrate instead on “analysis” problems solving for pressures induced by prescribed shapes satisfying Kutta-type conditions that are more conveniently handled by potential-like formulations. Note that analysis problems prefer auxiliary conditions described by potential jumps that control vortex strength and lift, whereas design problems prefer shape constraints described by streamfunction discontinuities that control source strength or mass efflux. In particular, we have selected a very difficult analysis problem involving the interaction of an external potential flow and an internal rotational flow, a problem of significant interest in jet-engine nacelle design not previously considered. Although our numerical results are preliminary and, of course, subject to improvement, what is demonstrated is the power, simplicity and flexibility of the superpotential approach. In the following, the general theory, the required linearizations, application and numerical results are presented. 4.4.2 Rotational Flow Equations Let U and V be the full velocities in the streamwise (x) and radial (r) directions, P be the static pressure, and be the constant fluid density (extensions for compressibility effects will be discussed in Section 4.5). Now the governing momentum and continuity equations for a steady axisymmetric flow without swirl are simply UUx + VUr = - Px / , UVx + VVr = - Pr / and Ux + Vr + V/r = 0. If the latter equation is recast in conservation form, a streamfunction can be defined satisfying r = rU, = Ur – Vx as x = - rV and xx + rr – r /r = r , where we introduce the vorticity. To obtain more specific information, we note the dependence of /r on alone, that is /r = f ( ) where the function f, fixed for the entire streamsurface, can be explicitly written once the flow is known in any particular region of space. Thus,

General Analysis and Inverse Models 315 xx

+

rr

– (1/r)

r

2

=r f( )

(4.4.1)

which is not “Laplace-like” because of the negative sign. For simplicity we assume that a mean parallel shear flow far upstream exists with prescribed U = Um(r) and V = 0. The streamfunction for this mean flow 2 m(r), which is independent of x, satisfies r f ( m) = mrr mr /r. From this, the derivatives dnf( m)/d mn are easily written in terms of r and Um(r). Now if our mean flow is slightly perturbed, a disturbance satisfying = m + can be introduced which streamfunction satisfies xx

+

rr

–

r /r

= r2 { f ( 2

r

+ 1/6

{d f ( 2

m+

)-f(

m)/d

d3 f (

m

m)/d

m)}/

+½ m

3

d2 f (

m)/d

m

2

+. . . }

or, more explicitly, xx

+

rr

–

r /r

= {Um”/Um – Um’ /(Umr)}

(4.4.2)

+ 1/2{ Um’’’ /(Um2r) – Um”Um’ /(Um3r) – 3Um”/(Um2r2) + Um’2/(Um3r2) + 3Um’ /(Um2r3)} + – + –

2

1/6{ Um’’’’ /(Um3r2) – 3Um”’ Um’ /(Um4r2) – 6Um”’ /(Um3r3) Um”2/(Um4r2) + 3Um”Um’2/(Um5r2) + 11Um”Um’ /(Um4r3) 15Um”/(Um3r4) – 3Um’3/(Um5r3) – 10Um’2/(Um4r4) 15Um’ /(Um3r5)} 3 + . . .

Equation 4.4.2 explicitly evaluates the vorticity generation term by expanding df( )/d in powers of with curvature-dependent coefficients. Small-disturbances are implicit in the use of Taylor series. Linear theory is described by the first line shown of Equation 4.4.2. A similar analysis applies to the Bernoulli “constant” in the usual pressure integral, which varies from streamsurface to streamsurface. It is convenient to define a function J within a “true constant” by the relation J’( ) = f ( ). Then, the streamwise and radial momentum equations and the definition of vorticity lead to the Bernoulli equation P/ + ½ (U2 + V2) – J( ) = C

(4.4.3)

316 Modern Aerodynamic Methods where the constant C, fixed throughout space, does not change from streamsurface to streamsurface (the “variable constant” appears through J( )). J is can be determined in any particular example. Suppose that Um(r), and hence m, are given analytically upstream (this is not required in our later work). This functional relation can be inverted to yield r = g( m), which is substituted in f( m) = Um’ (r)/r to provide the required integrand in f( )d .

J=

If we now write Equation 4.4.3 using conditions at infinity, we have P/ + ½ (

2 r

+

x

2

)/r2 – J( ) = Pm(r)/ + ½

2 mr

/r2 – J(

m)

or Cp = {P – Pm(r)}/(½ Uref2) = – (2

mr

r

+

2 r

+

2 2 x )/(Uref r ) 2

(4.4.4) +2 {

/(Uref2}{J(

m

+ ) – J(

m)}/

where Cp is a nondimensional pressure normalized by a dynamic head based on the reference velocity Uref. Then, replacing the derivative-like J term in Equation 4.4.4 by its Taylor series approximation leads to Cp = {P – Pm(r)}/(½ Uref2) = – (2 +2

Um’

2

/(Uref r)

+ {Um”/(Umr2)

–

mr

r

+

2 r

+

Um’ /(Umr3)}

x 2

2

)/(Uref2r2)

(4.4.5)

2

/Uref + . . .

Summarizing, Equations 4.4.2 and 4.4.5 provide explicit working equations with coefficients expressed in terms of known upstream conditions; they were obtained using invariant properties of the streamfunction and the vorticity in axisymmetric flow. The linearized equations and their consequences are discussed next. 4.4.3 The Linearized Problem In many aerodynamics applications, simplifications are introduced by applying boundary conditions along slits and constant radius surfaces. For these small-disturbance flows, the above equations can be linearized. It turns out that the notion of the velocity potential can be extended without difficulty; we will show how existing potential flow algorithms can be simply modified to handle strong shears. Let us now drop the nonlinear terms in Equation 4.4.2. It is almost remarkable that the linearized equation can be recast in the conservation form

General Analysis and Inverse Models 317 {Um

x

/(rUref)}/ x + {Um

r

/(rUref) – Um’ /(rUref)}/ r = 0

(4.4.6)

*

= (Um/Uref)

r

/r – (Um’ /Uref) /r

(4.4.7)

* r

= (Um/Uref)

x

/r

(4.4.8)

x

where *(x,r) is our “superpotential” ( * derivatives are not velocities unless Um = 0). Equations 4.4.7 and 4.4.8 in turn show that * satisfies the “potential-like” equation xx

*

+

* rr

+ (1/r – 2Um’ /Um)

* r

=0

(4.4.9)

Thus only minor changes to existing potential flow finite-difference or finite-element alogorithms are needed to solve the analysis problem. The r*/r term in irrotational flow, incidentally, couples the effects of thickness and camber; here, we see that mean vorticity introduces additional coupling dependent on the shear gradient. The corresponding tangency conditions obtained from Equation 4.4.8 are * r (x,

{Um2(x, R )/Uref }F ’ (x)

R )

(4.4.10) ’

where r = R is a suitable mean radius and F (x) are streamwise surface slopes. For bodies with trailing edges, Kutta’s condition is required and a formula for Cp must be obtained. The linearized expression for Cp simplifies as follows, Cp

-2

x

*

/Uref

(4.4.11)

In summary, the Laplace-like Equation 4.4.9 is solved together with the potential-like tangency condition in Equation 4.4.10 and a trailing edge “potential jump” specified through a branch cut as inferred from Equation 4.4.11. The strong resemblance to potential flow formulations allows us to “think irrotationally” in algorithm development. Flows in simple ducts, for example, can be trivially solved. In contrast, flows through engine nacelles have constant Um externally and nonzero Um’ (r) behind assumed actuator disks. Here, a specific Um’ (r) is inferred from the radial blade loading, and Kutta’s condition is handled “irrotationally,” with the “potential jump” [ *] through the trailing edge slipstream specified but as a functional of x and * (the discontinuous coefficients in the differential equation will be addressed more completely later). The present “analysis” formulation solves for the surface pressure induced by a prescribed shape subject to Kutta’s condition. In “design” or “inverse” problems, the geometric shape that

318 Modern Aerodynamic Methods induces a prescribed chordwise pressure subject to auxiliary shape constraints is required. For nacelle flows, one might specify trailing edge closure or edges with opened cusps to model displacement thickness effects. A convenient dependent variable is the streamfunction, because the jump [ ] controls gap and mass efflux directly, while the jump [ xl automatically controls the included angle. These “Kutta-like” edge constraints would be solved with the linearized streamfunction equation derived earlier and the mixed Dirichlet and Newmann boundary condition -2Um(R±)

r

(x, R±)/Uref2R + 2Um’ (R±) (x, R±)/ Uref2R = Cp± (x) (4.4.12)

The required surface coordinates are then obtained from dF±/dx = F±(x) = -

x

/

r

-

x

(x, R±)/{RUm(R±)}

(x, R±)/{RUm(R±)} + constant

(4.4.13a) (4.4.13b)

Similar remarks apply to design problems in annular or co-annular ducts and pipes (for a complete discussion on streamfunction methods for aerodynamic inverse problems, the reader is referred to other sections of the present chapter). Applications to these geometries are the subject of current research. 4.4.4 Application to Jet-Engine External Potential and Internal Rotational Flow Interaction We examine the flow through a finite-length axisymmetric nacelle immersed in a uniform freestream. Without power addition, the flowfield can be modeled by potential methods; here, the internal flow is irrotational up to an assumed actuator disk location, beyond which the flow is rotational due to radially varying work imparted by the turbomachinery as shown in Figure 4.4.1. Thus, the flowfield is potential externally and “superpotential internally” and in the downstream plume; matching conditions at the disk and plume interfaces connect both dependent variables. For the external irrotational now, subscripted “e,” the superpotential is a true potential, and the oncoming freestream is constant with Um(r) = U . We introduce the nondimensional italicized variables x = x/c, r = r/c and e (x,r) = *e (x,r)/(U c) where c is the semichord and take Uref(r) = U . Then the governing equations become

General Analysis and Inverse Models 319 exx (x,r)

+

er(x,R+)

err

+ 1/r

er

=0

(4.4.14a)

= F’ e(x)

Cpe = (Pe – P )/(½ U 2) = - 2

(4.4.14b) ex (x,r)

(4.4.14c)

Next consider the internal flow. So that our previous linearizations remain physically meaningful, Uref is chosen as a suitable “maximum speed.” Here we assume a prescribed internal parallel shear flow originating at the actuator disk (that is, “upstream infinity” in the context of Sections 4.4.2 and 4.4.3) with a horizontal speed that increases monotonically outward, resulting in a maximum speed Umax found at the blade tips. With Uref = Um(R-) = Umax and a different normalization, namely, i (x,r) = *i (x,r)/(Umaxc) and Um(r) = UmaxU(r), the internal elliptic equations (subscripted “i”) become ixx (x,r)

+

ir(x,R-)

= F’ i(x)

irr

+ {1/r - 2Ur(r)/U}

ir

Cpi = (Pi – Pm(r))/(½ Umax2) = - 2

=0

(4.4.15a) (4.4.15b)

ix (x,r)

(4.4.15c)

where F’ e and F’ i are actual geometric slopes. Note that other normalizations are possible, for example, U internally, but that quadratic terms in Cp would then be retained. Next, pressure continuity through the trailing edge plume or slipstream is numerically applied along a mean radius R for simplicity. Because Cpe and Cpi are normalized differently, pressure continuity does not imply Cp continuity. Setting Pe = Pi along r = R and using Equations 4.4.14c and 4.4.15c lead to (

e

- i)x = {P – Pm(R)}/( U 2) +

(4.4.16)

ix

where we introduce the notation (Umax/U )2 = 1 + . If we now denote [ ] = we obtain [ ] x = {P – Pm(R)}/{ U 2 (1 + ½ )} +

e

-

i

avg,x /(1

and

avg

+½ )

=½(

e

+

i

),

(4.4.17)

Integrating with respect to x from the trailing edge xte to x, the wake matching condition

320 Modern Aerodynamic Methods [ ] (x) = [ ] (xte) + {P – Pm(R)} (x – xte)/{ U 2 (1 + ½ )} + [

avg(x,R)

–

avg(xTE,R)]

(4.4.18)

/(1 + ½ )

is obtained for a “potential jump” with two “power corrections” to the usual irrotational [ ] (xte) term. This is not a true jump, of course, since the superpotentials are normalized differently, Equation 4.4.18 merely describing a difference between two variables to be enforced numerically; our normalizations were in part motivated by the desire to keep the computer algorithm as “irrotational” and original as possible. For reference, specialized forms of Equation 4.4.7 are given; in the external flow, with Um(r) = Uref = U , we set e = c2 Umax e(x,r) so that er(x,r)/r

=

(4.4.19)

ex

Internally, we reference U(r)

ir

/r =

ix + Ur

i

to Umax; then

/r

i

2

= c Umax

i(x,r)

leads to (4.4.20)

Finally, we discuss actuator disk matching conditions. Actuator disks mathematically idealize changes to flow properties imparted by turbomachinery. Thus, turbomachinery performance properties must be known or approximated. For our purposes, the particular model is arbitrary and unimportant, because we are interested more in the effects of power addition arising from slipstream interaction. First, continuity of disturbance streamfunction through the actuator disk requires that we have i = e /(1 + )1/2; then continuity of the horizontal speed r /r (see Equation 4.4.7) leads to the matching condition (U

ex -

Ur

e

/r)/ (1 + )1/2 =

ix

(4.4.21)

used numerically, where a i was rewritten in terms of e. Equation 4.4.21 was motivated by a finite-difference column relaxation solution method where lines of constant x are swept from upstream to downstream. Knowing the external flow left side, and hence the gradient ix, allows us to march into the rotational flow ( e is obtained by integrating Equation 4.4.19 with respect to r, from the centerline r = 0 where e = 0). This completes the analytical formulation; next, we review some computational issues connected with the discontinuity of actual physical quantities through contact surfaces. Because the *r coefficient in the governing field equation changes discontinuously through the disk and the slipstream, the usual potentialflow difference formulas must be re-examined since certain smoothness

General Analysis and Inverse Models 321 properties are implicitly assumed. A continuous function f(r) with continuous first and second derivatives at r = rj can be described using fj ’ = (fj+1 – fj-1)/(rj+1 – rj-1) and fj ” = 4(fj+1 – 2fj + fj-1)/(rj+1 – rj-1)2. If f is discontinuous at rj, we cannot define a first derivative unless it is continuous; then, fj’ = (fj+1 – fj-1 – [f])/(rj+1 – rj-1) where the discontinuity [f] is subtracted out. If, in addition, fj ” is continuous. we can write fj ” = {(fj+1 – fj+)/(rj+1 – rj) – (fj- - fj-1)/(rj – rj-1)}/{½ (rj+1 – rj-1)}. These extended formulas allow discontinuities in f, provided f ’ and f ” are continuous; they are commonly used to difference through aerodynamic wakes. In wing flows the velocity normal to the wake is always continuous, but “ z” and “ zz” through an assumed flat wake is not – these formulas are used only with the restriction to the weak discontinuities allowed in small-disturbance theory. For our jet-engine problem we must be certain that the usual difference formulas used through the slipstream are not physically unrealistic; f should be precisely defined and the assumed continuities in f ’ and f ” checked. Let us define f = ( e, externally; i, internally) noting again that e and i, are normalized differently. In the slipstream e*r/U and i*r/Umax both equal the streamwise plume slope; thus the nondimensional variables so defined allow for f ’ continuity, as required (f ” continuity, related to smooth curvature, is also assumed). When the customary “mean difference equation” is used in the wake for fm = ½ ( e + i ), the difference approximations for the differential equation appear exactly as they would in potential flow formulations. except that the jump in f satisfies Equation 4.4.18, more complicated, but nonetheless easily implemented. 4.4.5 Calculated Results and Closing Discussion Our approach to inviscid rotational flow allows a simple “potentiallike” solution to those problems where velocity shear can be strong. The ideas were developed for cylindrical axisymmetric flows without swirl; but the same approach, with similar results, extends to “mathematically axisymmetric” flows for arbitrary, say body-fitted, three-dimensional coordinate systems, using properties of vorticity specialized to these systems. Our particular jet-engine problem, because of obvious complications, has not been examined in the literature; thus, we insisted on a simple streamwise nacelle section, so that physical intuition can check anticipated and unanticipated results.

322 Modern Aerodynamic Methods We will use an external profile corresponding to the upper half of a symmetric unpitched 10 percent thick parabolic arc airfoil and assume a perfect circular cylinder internally, with a chord-to-diameter ratio of two (boundary conditions are applied on this circular cylinder). Thus we qualitatively expect an “airfoil-like” external surface Cp with stagnation peaks near both edges, and internal flow which, being energized by power addition, continues straight more or less. The modified wake condition in Equation 4.4.18 shows that two nondimensional parameters are needed to characterize the shear, namely, = (Umax/U )2 – 1 and = (Pm(R) - P )/(½ U 2). The second is related to a total pressure increase. For simplicity we assume that Um is proportional to 1 + fr (only the ratio U’ m/U appears in the governing equation), so that f, the strength, equivalently measures ; computationally, we take f = 0, 1, 2, and 3 with = + 2 (that is, the static pressure Pm (R) equals the external total pressure plus one dynamic head). Our first set of results, shown in Figures 4.4.2a - 4.4.2d, imply the streamline pattern in Figure 4.4.3. As f increases, the external flow expands more, with Cp becoming more negative near midchord. At the same time, the internal leading edge lip flow expands less, indicating a movement of the external lip stagnation point toward the left as illustrated. Since the internal surface is flat, is constant; in this case only, from Equation 4.4.11, the Cp is proportional to the streamwise disturbance speed as in potential theory. As f increases, this speed decreases consistently, since the total pressure along the body streamline is fixed and the transverse velocity is zero. The same software code was run irrotationally using f = 0 and = 0, with and without actuator disk logic. Calculated internal results showed minor discrepancies since disk matching conditions occupied two coarse meshes. Figure 4.4.4 shows two solutions for f = 0, the first with = 2, the second with = 0 (in the former case Cp uses a different normalization downstream of the disk). A velocity slip is clearly seen in the powered flow. These finite difference relaxation was implemented using a modified potential flow code on a coarse 60 60 mesh, with 20 over the chord, and carried to convergence. In closing we emphasize that the superpotential approach requires only simple modifications to available potential flow algorithms; however, it is rigorous and founded on the exact fluid-dynamic equations. The superpotential is a consequence of linearization but without restriction to shear or vorticity strength; but it is not the Clebsch potential often used to represent rotationality, nor is it related to

General Analysis and Inverse Models 323 Lighthill’s “similar” pressure function (e.g., see Lighthill (1949)), which is restricted to weak shears. In fact, its existence was motivated by some mathematical constructions used in inviscid hydrodynamic stability theory. The particular application to jet-engine external potential and internal rotational flow interaction is not final; many code refinements are due before the prototype software becomes a working tool. Direct and inverse applications to pipes and coannular ducts are currently in progress and the general coordinate approach mentioned earlier is nearing completion. Finally, general theoretical consequences to the planar limit of our shear flow equations have been obtained for thin airfoils, in both analysis and design problems, and appear in next in Section 4.5. Uniform oncoming freestream Axisymmetric nacelle

Actuator disc

Trailing edge plume Shear flow “superpotential” formulation Centerline

Figure 4.4.1. Jet-engine external potential and internal flow interaction.

324 Modern Aerodynamic Methods A: f = 0

B: f = 1 0.7 Cp Internal External

LE D: f = 3

Figure 4.4.2.

TE

C: f = 2

= 2 rotational flow solutions.

Note, f increasing

Abrupt turn

Actuator disc Figure 4.4.3. Implied streamline pattern.

General Analysis and Inverse Models 325

Cp =0

Cp =2

Figure 4.4.4. Irrotational flow solutions with f = 0. Top, “flow-through nacelle without power.” Bottom, constant radial energy addition. 4.4.6 References

Chin. W.C., “Direct Approach to Aerodynamic Design Problems,” ASME Journal of Applied Mechanics, Vol. 48., Dec. 1981. pp. 721726. Chin, W.C., “Thin Airfoil Theory for Planar Inviscid Shear Flow,” ASME Journal of Applied Mechanics, Vol. 51, March 1984, pp. 1926. Chin. W.C., and Rizzetta, D.P. “Airfoil Design in Subcritical and Supercritical Flows,” ASME Journal of Applied Mechanics, Vol. 46, Dec. 1979, pp. 761-766. Lighthill, M.J., “The Flow Behind a Statonary Shock,” Phil. Mag., Vol. 40, 1949, pp. 214-220. Oates, G.C., Aerothermodynamics of Gas Turbine and Rocket Propulsion, Fifth Printing, American Institute of Aeronautics and Astronautics (AIAA), 1998.

326 Modern Aerodynamic Methods 4.5 Thin Airfoil Theory for Planar Inviscid Shear Flow

Steady, inviscid, planar and constant density shear flows past thin airfoils have strongly coupled thickness and camber flowfields which can be simply analyzed using some invariant properties of Euler’s equations. In this section, both the nonlinear vorticity generation term in Poisson’s disturbance streamfunction equation and the Bernoulli “constant” (which varies from streamline to streamline) are explicitly evaluated in terms of known plane parallel flow properties far upstream, recognizing that vorticity convects unstretched and unchanged along streamlines. This produces a governing partial differential equation for the streamfunction with an explicitly available right side plus a Bernoulli integral with a true constant fixed throughout the entire flowfield. Extensions to threedimensional constant density flows and planar compressible flows are also considered. Using these results for the streamfunction partial differential equation, the “inverse problem,” which solves for the geometric shape that induces a prescribed surface pressure, is formulated as a nonlinear boundary value problem with mixed Dirichlet and Neumann surface conditions. Here, trailing edge shape constraints are directly enforced by controlling jumps in the streamfunction or its streamwise derivative through the downstream wake (analogies with potential flows past axisymmetric ringwings showing similar source and vortex interactions are also described). Jumps in streamfunction prescribe the degree of closure while jumps in its streamwise derivative define included angle. For flows with arbitrary profile curvature, we show that the linearized streamfunction equation can be put in conservation form, thus leading to new and more general Cauchy-Riemann conditions which extend the notion of the velocity potential to flows with shear. The ”analysis problem,” which determines the flow past a known shape subject to Kutta’s condition, is shown to satisfy a simple boundary value problem for a “superpotential” identical to the irrotational planar formulation except for an axisymmetric-like change to the Cartesian form of Laplace’s equation (a superpotential for axisymmetric flows was previously introduced in Section 4.4). For flows with uniform vorticity, closed-form solutions are given which clearly illustrate the interaction between thickness and camber; also, for arbitrary thin airfoils, a particularly simple expression shows that the lift due to thickness varies directly as the product of vorticity and enclosed area.

General Analysis and Inverse Models 327 With more general velocity shear profiles, numerical approaches are especially convenient; these require only simple changes to existing algorithms, for example, established potential function methods readily available for analysis problems, or other “direct streamfunction methods” for inverse problems developed by this author. Numerical methods and results that cross-check and extend derived analytical solutions are given for both analysis and design problems with vorticity. Also, extensions of the basic theory to three-dimensional constant density and planar compressible shear flows are given. 4.5.1 Introduction The analysis and design of airfoil sections in flows with vorticity is important in practical engineering problems, for example, cascade design in turbomachinery flows, detailed analyses for turbine and compressor blade rows, aerodynamic stability in wind shear, and flight in the boundary layers of large obstacles or in ground effect. Unlike irrotational motions where the basic features, essentially linear, satisfy Laplace’s potential equation, flows with vorticity are inherently nonlinear: in the steady, planar, constant density and inviscid limit considered, the vorticity generation term in the governing streamfunction equation, unknown apriori, is a functional of the dependent variable, the same applying to Bernoulli’s “constant” in the pressure integral, which generally varies from streamline to streamline. Thus, well established potential function analysis methods, which prescribe Neumann boundary conditions related to normal derivative along an approximating slit, and which enforce Kutta’s condition by zeroing the jump in streamwise disturbance potential derivative through the wake, do not apply. The obvious recourse appears to be a direct attack on Euler’s equations; this, unfortunately, requires new and computationally expensive numerical methods, not to mention more precise ways to calculate pressure, since an explicit integral is unavailable. However, simplifying features appear when the upstream velocity profile is a parallel shear flow: since the vorticity and Bernoulli’s “constant” are invariant functionals of the streamfunction, they should be expressible in terms of known conditions far upstream everywhere in the flowfield. This simple idea is pursued here. The vorticity term in the and Bernoulli’s equation governing the disturbance streamfunction “constant” are written as power series in with variable coefficients depending on the mean profile curvature; these results are used to

328 Modern Aerodynamic Methods formulate a “direct” nonlinear boundary value problem for “inverse” applications, where the shape is sought that induces a prescribed pressure, with the jump in or its streamwise derivative used to specify trailing edge shape constraints. For flows with constant shear the curvature vanishes and analytical solutions can be obtained by expansion in powers of a nondimensional vorticity. Solutions illustrating the interaction between thickness and camber are presented; also given are closed-form results for the lift induced by thickness and exact analogies with potential flows past ringwings. For linearized flows with arbitrary curvature, the streamfunction equation can be put in conservation form, leading to new and more general Cauchy-Riemann conditions that extend the notion of the velocity potential; it turns out the usual planar potential formulation again applies, provided an axisymmetric-like change to the planar potential equation is made to allow source and vortex coupling, and certain redefinitions are made. These results are important because existing potential function methods for analysis problems can be trivially modified to handle the effect of weak or strong oncoming vorticity; “direct” streamfunction approaches to inverse problems, developed by this author, are also easily modified, thus avoiding direct integration of Euler’s equations. The streamfunction approach to aerodynamic inverse problems, first proposed by this author in 1979, is based on a particularly simple idea. For constant density, steady, planar, inviscid and irrotational flows, “analysis” problems solve xx + yy = 0 with the surface slope y/U specified on a mean chord and the jump [ ] chosen to satisfy Kutta’s condition, being the disturbance potential; surface pressures are then evaluated from Cp = -2 x/U . The harmonic conjugate to , namely the disturbance streamfunction , likewise satisfies Laplace’s equation. But what does a mathematically similar boundary value problem solve? Interestingly, y = x = - ½ U Cp is a pressure specification, while [ ,] and [ x] control trailing edge closure and included angle, respectively (the required surface shape is evaluated from dy (x)/dx = - x (x,0 )/U or y (x) = - (x,0 )/U + constant). Thus the streamfunction formulation solves the inverse problem – and, as a practical matter, it uses exactly the same potential formulation “analysis” algorithm! Chin (1979, 1980, 1981, 1983) and Chin and Rizzetta (1979) developed this idea extensively, at times introducing

General Analysis and Inverse Models 329 “stream-like” functions, generalizing the basic approach to handle transonic supercritical flows, axisymmetric nacelles, three-dimensional wing design, inlet flows, cascades, and so on. These extensions were restricted to irrotational problems, however. Thus, the present research aimed at inverse formulations that included the effects of oncoming shear, the results of which are described earlier. An important by-product of this work arose when, perhaps fortuitously, the linearized streamfunction equation fell into conservation form (this possibility was motivated by some mathematical constructions in hydrodynamic stability). Obviously, extended Cauchy-Riemann conditions could then be introduced, from which the notion of the “superpotential,” discussed here, would naturally appear. Here, a number of general analysis results describing airfoil aerodynamics in the presence of possibly strong shear are presented, and numerical results and applications are given. When this work was first completed, the obvious extension of the method to axisymmetric problems was immediately sought; equivalent analysis and design formulations could be directed, for example, toward duct design in turbomachinery applications. Extensions of our planar results were made to address a very challenging problem: the jet-engine external potential and internal rotational flow interaction obtained through an assumed actuator disk and through the trailing edge slipstream. In this problem, a radial shear now is prescribed at the downstream side of the actuator disk, simulating the radially varying work imparted by the turbomachinery, while the entire nacelle is immersed in a uniform irrotational flow. The complete flow pattern was numerically solved by a column relaxation method, using special derived matching operators applied at the actuator disk and crucial renormalizations of the governing dependent variables so that radial differencing through the slipstream, which separates regions characterized by distinctly different flow properties, was possible. The new approach, described in Chin (1983), expands on the author’s earlier work on jet engine power simulation methods, these being essentially flow models for problems characterized by more than one total pressure. These applications are discussed in Golden, Barber and Chin (1982), Golden, Barber and Chin (1983) and Chin (1983). At any rate, the theory and results described here and in Chin (1983) point to a new, simple and promising approach to solving “analysis” problems in flows with strong shear: the “superpotential” formulation. (This is not to be confused with the Clebsch potential, sometimes used to

330 Modern Aerodynamic Methods represent rotational flows, or Lighthill’s well-known pressure function approach, which is restricted to weak shears.) For both planar and axisymmetric problems, the linearized disturbance streamfunction equations derived here, of course, are not new; but now, they are complemented by exact Bernoulli pressure integrals with “true constants” fixed for the entire flowfield, thus permitting the construction of general “inverse” formulations designed along the lines of the author’s cited work on inverse problems for irrotational flows. In Section 4.5.2, flows with constant vorticity or linearly varying oncoming velocity will be considered, and some general results are given; Section 4.5.3 discusses general shears, introduces the notion of the “superpotential” and, finally, numerical analysis and design results are reported in Section 4.5.4. Extensions of the planar model to handle three-dimensionality and compressibility are separately given in the Appendices. 4.5.2 Planar Flows With Constant Vorticity 4.5.2.1 Planar Flows: Inverse Problems

Let U and V be x and y velocities, P be the static pressure, and be a constant liquid or gas density. The continuity equation Ux + Vy = 0 leads to a streamfunction satisfying U = y, and V = - x; together with UUx + VUy = - Px/ and UVx + VVy = -Py/ , the constancy of = Uy - Vx implies that xx + yy = with P/ + ½ (U2 + V2) constant throughout. Thus, disturbances to a uniform shear flow satisfy xx

+

yy

=0

(4.5.1)

Cp ={P – P (y)}/( ½ U 2) = - 2(1 + y/U ) +2 /U 2 – ( x2 + y2)/U 2

y/U

(4.5.2)

where = U y + ½ y2 + , U is constant, and P (y) is the pressure at infinity. For thin airfoils, the inverse problem solves Equation 4.5.1 with y

(x,

)/U -

(x,

)/U

2

= - ½ Cp(x,

)

(4.5.3)

specified on y = 0; the mixed boundary conditions strongly couple the effects of thickness and camber. Trailing edge constraints also apply; for example, the jump [ ] prescribed through the downstream “wake” controls ejected mass flow or closure while [ x] controls the included angle (both jumps can be prescribed if Cp is suilably modified). In

General Analysis and Inverse Models 331 vanishes at infinity. The required airfoil is obtained by addition, integrating the tangency condition dy (x)/dx = V/U = –

x

/(U + y +

y

)

–

x(x,

)/U

(4.5.4)

to obtain the vertical ordinate y (x) = – (x,

)/U + constant

(4.5.5)

4.5.2.2 Planar Flows: Direct Formulations

Here ( x) x + ( y - y)y = 0 suggests a “potential” N satisfying Nx = ) = U dy (x)/dx; y - y and Ny = - x . Equation 4.5.4 becomes Ny(x, is single-valued, and Kutta’s condition for closed trailing edges, requires that [Nx] vanish from Equation 4.5.3. In terms of (x,y) = N U x, the complete streamline pattern is obtained by solving the irrotational problem xx

+

y(x,

yy

=0

(4.5.6)

) = U dy (x)/dx on the chord

[ x] = 0 through the “wake” at infinity and then setting y = U + y + x and - x = y. Surface pressures are easily obtained. If y (x) = 0 at the leading edge x = xLE, then (x, ) = (xLE,0) - U y (x) from Equation 4.5.5; substitution into Equation 4.5.3 leads to Cp(x,

) = -2 x(x, )/U + 2

(xLE,0)/U

2

- 2 y (x)/U

(4.5.7)

The level (xLE,0) affects neither the pressure gradient nor the total lift; it can be determined by solving Equation 4.5.1 with y(x, ) = x(x, ), which is known, along with vanishing at infinity and appropriate edge constraints. Note that a symmetric airfoil, which has no lift in a uniform stream, has a shear induced lift Ls = 2U

y+(x) dx

(4.5.8)

chord

which varies linearly with the vorticity and the enclosed area. For a pure camber line there is no additional induced lift; in this case, the net lift can be obtained irrotationally. Thus, Equation 4.5.8 represents the

332 Modern Aerodynamic Methods only “additional” lift for a general airfoil with both thickness and camber: the effect of velocity shear is, effectively, a change in the angle of zero lift relative to irrotational solutions. 4.5.2.3 Some Planar Analytical Solutions

The foregoing problems can be solved with slight modification to existing numerical methods, e.g., those outlined in the Introduction, but it is more instructive to study representative analytical solutions. To do this, we first introduce the nondimensional variables x* = x/c, y* = y/c and *(x*,y*) = /U c where 2c is the chord. If = c/U is small, we * expand * = *0 + 1 + . . . to obtain the sequence of problems 0 xx 0y

(x,

1 xx 1y

+

+

(x,

0 yy

=0

(4.5.9)

) = - ½ Cp(x, 1 yy

)

=0

)=

0(x,

(4.5.10) )

using y (x, ) – (x, ) = - ½ Cp (x), where asterisks have been dropped. Two general observations are apparent. If Cp+ = Cp , then 0+ = - 0 ; then, the effect of weak shear, because the pressure 1 y is antisymmetric, is an induced camber line. Alternatively, if Cp+ = - Cp , then 0+ = 0 , which in turn induces a thickness distribution. Now we examine some specific solutions. First assume Cp = –2 with being a small positive constant. Then the solution to Equation 4.5.9 assuming closed edges is the classic ellipse y0 (x) = ± (1 – x2)1/2 where x = –1 and +1 are leading and trailing edges. Then, from Equation 4.5.5, 0(x,

) = – {± (1 – x2)1/2}

where the additive constant vanishes by the asymmetry in assumption +1 1=¼

h(s) log {(x-s)2 + y2} ds -1

0.

The

General Analysis and Inverse Models 333 2 1/2

leads to h(x) = -2 (1 – x ) ; thus, dy1/dx = - 1x(x,0) = x and y1(x) = ½ (x2 –1) is an upright parabola. The composite airfoil y = y0 + y1 = (1 – x2)1/2 + ½ (x2 –1) has negative camber if > 0. This is expected: since a symmetrical airfoil experiences lift, the shape due to Cp+ = Cp , with zero lift, must remove lift. Next consider Cp = - ( ) 2 (1-x)1/2/(1+x)1/2 which corresponds to the flat plate y(x) = - (x+1) in irrotational flow. The assumption +1 2 (1-s)1/2/(1+s)1/2 log{(x-s)2 + y2} ds -1 leads to 0(x, ) = (x – log 2). Since 1y(x, ) is symmetric, the required thickness distribution having closed edges can be obtained by integrating the expression +1 0(x,y)

= {1/(4 )}

dy+(x)/dx = 1/ {(1-x)/(1+x)}1/2 {(1+s)/(1-s)}1/2 -1

1y

(s,0 ) ds/(s-x)

+1 2 -1/2

– (1/ )(1-x )

{(1+s)/(1-s)} 1/2 -1

1y

(s,0 ) ds

If y+(-1) = 0, y+(x) = (½ x – log 2)(1-x2)1/2 contains both blunted leading and trailing edges. For positive , y+(x) is negative, thus reducing the “main” thickness; for negative , y+(x) is a “whale” with a large nose and an “almost sharp” tail swimming in the minus x direction, with a maximum thickness at x – 0.4. 4.5.2.4 Analogy To Ringwing Potentlal Flows

The planar inverse problem with shear is related to the potential flow past a given ringwing: in both cases the effects of sources and vorticity are coupled. For axisymmetric flows, the disturbance potential satisfies xx + rr + r /r = 0 where x and r are streamwise and radial coordinates. We assume the shape r (x) = R + cF ( ) where 2c and R are chord and mean radius and = x/c; also, define = (r - R)/c, (x,r) = U c ( , ) and = c/R. In the ringwing limit taking