Frontiers of Computational Fluid Dynamics 1998
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Frontiers of Computational Fluid Dynamics 1998
Editors
D A Caughey Cornell University
M M Hafez University of California, Davis
World Scientific Singapore • New JerseM • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
FRONTIERS OF COMPUTATIONAL FLUID DYNAMICS 1998 Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-3707-3
Printed in Singapore by Uto-Print
Dedication
This volume consists of papers presented at a symposium honoring Earll Murman and recognizing his seminal contributions to transonic aerodynamics and to computational fluid dynamics (CFD) over the past three decades. The symposium, entitled Thirty Years of CFD and Transonic Flow, was held in Everett, Washington on June 24-26, 1997. The authors were selected from among internationally known researchers working in aerodynamics and CFD, where the impact of Murman's contributions have been so important. It is the pleasure of the authors and the editors to dedicate this book to Earll in recognition of the important role he has played in our technology and in our lives. Earll Murman was born on May 12, 1942. He was raised in San Francisco, but went East to Princeton University where he received the B.S., M.A., and Ph.D. degrees in 1963, 1965, and 1967, respectively. His Ph.D. research, under the direction of Professor S. M. Bogdonoff, led to the dissertation entitled "Experimental Studies of a Laminar Hypersonic Cone Wake." He joined the Boeing Scientific Research Laboratory (BSRL) in 1967, and remained there until 1971. During this period he worked with Professor Julian Cole, who was spending a sabbatical leave from UCLA at the Boeing Laboratory. Their collaboration produced the breakthrough known as the Murman-Cole scheme, which allowed the first practical calculations of steady, transonic flow fields containing regions of supersonic flow embedded in subsonic regions. In 1971 Earll joined the NASA Ames Research Center and, in 1973, presented his fully conservative version of the Murman-Cole scheme at the first AIAA Conference on CFD, held in Palm Springs, California. His pioneering work on wind tunnel wall interference and on design and optimization also were remarkable forerunners of current attempts to develop multi-disciplinary optimization techniques. The paper describing the Murman-Cole scheme was published in the January 1971 issue of the AIAA Journal, and has been identified as a Citation Classic by Current Contents, Vol. 27, No. 45, November 9, 1987. Frontiers of Computational Fluid Dynamics - 1998 Editors: David A. Caughey k, Mohamed M. Hafez
©1998 World Scientific
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DEDICATION
Murman's collaboration with Cole continued and produced another classic, their AIAA paper on Inviscid Drag at Transonic Speeds, presented in Palo Alto in 1974. In that year, Earll joined the Flow Research Company in Kent, Washington, where he became Vice President and General Manager in 1977. He moved to MIT as Professor of Aeronautics and Astronautics in 1980, and became Department Head from 1990 to 1996. Earll has served as a consultant to Calspan, United Technologies Corporation, Pratt & Whitney, General Electric Corporation, Stellar Computer Company, Kendall Square Research, and Microcraft Technology. At MIT, Earll served as Director of the Computational Fluid Dynamics Laboratory from 1980 to 1990. He chaired the Project Athena Resource Committee and, more recently, was the motive force behind Todor, a software package designed to enhance fluid mechanics curricula using computers. Earll has taught courses on Transonic Aerodynamics, Computational Fluid Mechanics, Viscous Fluids, Heat and Mass Transfer, Fluid Dynamics of Flight and Re-entry Vehicles, and has supervised numerical and experimental projects for undergraduate students. He has advised 26 undergraduates, 20 Masters level students, and 8 Ph.D. students. His research at MIT has been directed at solution of the Euler equations, including the simulation of vortical flows. He also has worked on boundary layers and their coupling with Euler calculations, chemically reacting flows, Navier-Stokes equations, and viscous hypersonic flows, as well as flow visualization techniques. He is genuinely interested in engineering education, as is clear from his publications. He has been an active participant in many committees of the American Institute of Aeronautics and Astronautics, the National Aeronautics and Space Administration, the Department of Defense, and the aerospace industry, and has served as Director of the Lean Aircraft Initiative since 1995. He is a Fellow of the AIAA and a member of the National Academy of Engineering. In the first chapter of this book, Murman's technical contributions will be discussed in more detail, particularly their impact on transonic aerodynamics and CFD in general. Earll's contributions are not restricted to his technical ideas, his leadership, the courses he has taught, or his supervision of many talented students at MIT. Our community has been blessed to have a person like Earll, who has affected not only the people with whom he has worked directly over the years, but many others who have never had the pleasure of meeting him personally. It has been said that Murman opened the door for the flood of activity that followed his original contributions. Because of his vision his personality, and his generous nature, he is highly respected throughout the aerospace community. It is our hope that there will be a second conference in the future, dedicated to his continued contributions during the next 30 years.
Earll M. Murman
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Dedication
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X A Review of the Contributions of Earll Murman t o Transonic
Flow and Computational Fluid Dynamics Hafez d Caughey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions to Transonic Flow . . . . . . . . . . . . . . . . . . . . 1.3 Contributions to CFD . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Impact of Murman's Early Contributions . . . . . . . . . . . . . 1.5 Murman's More Recent Contributions . . . . . . . . . . . . . . . . . 1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Theses Supervised by Earl1 Murman . . . . . . . . . . . . . . . . . . 1.8 Publications of Earl1 Murman . . . . . . . . . . . . . . . . . . . . . . 2 Optimal Hypersonic Conical Wings
Triantafillou. Schwendeman €4 Cole . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hypersonic Small Disturbance Theory; Conical Wings . . . . . . . . 2.3 Lift and Drag Coefficients; Figure of Merit: Conical Wings . . . . . . 2.4 Numerical and Optimization Methods . . . . . . . . . . . . . . . . . 2.5 Calculated Results for Flat and Caret Wings; Optimal Wings . . . . 2.6 Acknowledgement and Disclaimer . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Geometry for Theoretical. Applied. and Educational Fluid
Dynamics Sobieczky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The transonic knowledge base . . . . . . . . . . . . . . . . . . . . . . 3.3 Geometry generator . . . . . . . . . . . . . . . . . . . .... 3.4 Generic transport aircraft . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
....
CONTENTS 4 C o m p u t a t i o n o f a n A x i s y m m e t r i c Nozzle Flow Cook. Newman. Rimbey d Schleiniger . . . . . . . . . . . ....... 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57 57 57 62 62 64
5 Analysis a n d N u m e r i c a l Simulation of t h e S u p e r b o o m P r o b l e m Cheng €4 Hafez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Sonic Boom in Non-Isothermal Atmosphere Analyzed in a Galilean Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 The Superboom Problem . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6 C o m p l e x Analysis of Transonic Flow
Chen d Garabedian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The hodograph transformation . . . . . . . . . . . . . . . . . . . . . 6.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Nonlinear boundary value problem . . . . . . . . . . . . . . . . . . . 6.5 Structure of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Paths of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Fourier analysis of the coordinates . . . . . . . . . . . . . . . . . . . 6.8 Comparison of design with analysis . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107 107 108 109 111 114 116 118 120 123
7 T r a n s o n i c S m a l l Transverse P e r t u r b a t i o n E q u a t i o n a n d its Computation Luo. Shen. d Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 Transonic Small Transverse Perturbation Equation . . . . . . . . . . 126 7.3 Computation of Axisymmetric Inlet . . . . . . . . . . . . . . . . . . 127 7.4 Multiple Solutions for Airfoil and Wing . . . . . . . . . . . . . . . . 132 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8 E x c i t a t i o n o f Absolutely U n s t a b l e D i s t u r b a n c e s i n B o u n d a r y -
L a y e r Flows Ryzhov d Terent'ev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 Extended triple-deck model . . . . . . . . . . . . . . . . . . . . . . . 143 8.3 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.4 Computed results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.5 Theoretical arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 148
CONTENTS 8.6 Conclusion . Acknowledgements REFERENCES . .
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9 O n A d j o i n t E q u a t i o n s f o r Error Analysis a n d O p t i m a l G r i d
Adaptation in C F D GQes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Finite volume analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Some concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A d d e d Dissipation i n Flow C o m p u t a t i o n s MacComack . . . . . . . . . . . . . . . . . . . . . ........... 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Viscous Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Choice of Added Numerical Viscosity . . . . . . . . . . . . . . . 10.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
11 A F o u r - O p e r a t o r s Conservative S c h e m e f o r t h e E u l e r E q u a t i o n s Chattot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A Mixed-Scheme for Burgers Equation . . . . . . . . . . . . . . . . . 11.3 1-D Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 A Characteristic Box Scheme . . . . . . . . . . . . . . . . . . . . . . 11.5 Use of Discrete Eigenvalues and Eigenvectors . . . . . . . . . . . . . 11.6 Some Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A u t o b l o c k i n g f o r W i n g s w i t h Split a n d Winged F l a p s Eberhardt €4 Wibowo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Results for Flapped Wings . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 L o c a l Preconditioning: M a n i p u l a t i n g M o t h e r N a t u r e to Fool Fat h e r T i m e Damofal d Van Leer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS 13.2 Design considerations 13.3 Current status . . . . 13.4 Future developments . Acknowledgements . . . . . REFERENCES . . . . . . .
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212 230 236 237 237
14 Relaxation Revisited-A Fresh Look at Multtgrid for Steady Flows Roberts, Sidilkover d Swanson . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 An Approach to Multigrid . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Extension to Compressible Flow . . . . . . . . . . . . . . . . . . . . 14.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
241 241 243 244 246 249 257 258 258 258
15 Aerospace Engineering Simulations on Parallel Computers Morgan. Weatherill. Hassan. Broolces. Manzari d Said . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Memory Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261 261 262 266 273 274 274
16 Optimizing CFD Codes and Algorithms for use on Cray Computers Wigton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 16.2 Cray Computers Including the T9O . . . . . . . . . . . . . . . . . . . 278 16.3 TRANAIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 16.4 TLNS3DMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17Recent Applications in Aerodynamics with NSMB Structured MultiBlock Solver Weber, Gacherieu. Rizzi et a1 . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Program Structure and Multi Block Implementation . . . . . . . . . 17.5 High Performance Computing Strategy . . . . . . . . . . . . . . . . . 17.6 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
293 293 295 302 308 308 311
CONTENTS 17.7 Summary and conclusions Acknowledgements . . . . . . . REFERENCES . . . . . . . . .
xiii
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325 327 328
18 I n c o m p r e s s i b l e Navier-Stokes C o m p u t a t i o n s i n A e r o s p a c e Applications a n d B e y o n d Kwak. Kiris. Dacles.Mariani. Rogers B Yoon . . . . . . . . . . . . . . . . 333 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 18.2 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 18.3 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 18.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 19 Pros & C o n s o f Airfoil O p t i m i z a t i o n Drela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 19.2 Method Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 19.3 Low Reynolds Number Airfoil Application . . . . . . . . . . . . . . . 365 19.4 Transonic Airfoil Application . . . . . . . . . . . . . . . . . . . . . . 372 19.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 20 T o w a r d s I n d u s t r i a l S t r e n g t h Navier-Stokes C o d e s . A Revisit Jou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Cruise Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 High-Lift Wing Analysis and Design . . . . . . . . . . . . . . . . . . 20.4 Prediction of Handling Quality and Control Effectiveness . . . . . . 20.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
383 383 384 389 391 392 .394 394
21 W h a t H a v e We L e a r n e d f r o m C o m p u t a t i o n a l Fluid D y n a m i c s R e s e a r c h o n T r a i n Aerodynamics? Fujii & Ogawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 21.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 21.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 400 21.4 A New Quasi-One-dimensional Prediction Method . . . . . . . . . . 409 21.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 22 O n ' t h e P u r s u i t o f Value w i t h C F D Rubbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 What Aerodynamic Design Processes Must Strive For . . . . . . . . 22.3 The Overwhelming Metric Leading to Value . . . . . . . . . . . . . .
417 417 419 420
CONTENTS 22.4 The Equation that Governs Design . . . . . . . . . . . . . . . . . . . 421 22.5 Impact or Value of CFD Prior to 1990 . . . . . . . . . . . . . . . . . 422 22.6 Era of the 1990s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 22.7 The Key Enablers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 22.8 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 22.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 23 CFD at a Crossroads: An Industry Perspective Raj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 23.2 Role of CFD in Aircraft Design . . . . . . . . . . . . . . . . . . . . . 430 23.3 CFD Effectiveness for Aircraft Design . . . . . . . . . . . . . . . . . 431 23.4 CFD Progress: The Past Thirty Years . . . . . . . . . . . . . . . . . 432 23.5 Whither CFD? The Next Twenty Years . . . . . . . . . . . . . . . . 439 23.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 24 Aerospace Engineering 2000: An Integrated. Hands-On Curriculum Seebass b Peterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 24.1 "Largest Gift Ever ..." . . . . . . . . . . . . . . . . . . . . . . . . . . 449 24.2 "Best Efforts Won't Do!" . . . . . . . . . . . . . . . . . . . . . . . . 450 24.3 What Do Our Customers and Others Say? . . . . . . . . . . . . . . . 451 24.4 Integrated Teaching and Learning Laboratory . . . . . . . . . . . . . 452 24.5 Aerospace Engineering Sciences Curriculum 2000 . . . . . . . . . . . 455 24.6 Two Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 24.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 25 Computer-based Fluid Mechanics Textbook CaugheybLiggett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 25.2 Navigation and Integration . . . . . . . . . . . . . . . . . . . . . . . 467 25.3 Illustrations and Data . . . . . . . . . . . . . . . . . . . . . . . . . . 468 25.4 Computation and Utilities . . . . . . . . . . . . . . . . . . . . . . . . 472 25.5 Teaching Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Alphabetical List of Authors
. . . . . . . . . . . . . . . . . . . . . . . 483
1 A Review of the Contributions of Earll Murman to Transonic Flow and Computational Fluid Dynamics Mohamed M. Hafez1 & David A. Caughey 2
1.1
Introduction
In this chapter, the work of Earll Murman over the past thirty years will be discussed, and some of his main ideas will be examined in detail. A complete list of his publications, up to 1996, together with titles of his students' theses are included as well. (It is of interest to note that Earll's Ph.D. thesis was on experimental studies of laminar, hypersonic wakes.) One can divide Murman's papers, after his graduation from Princeton, into two groups; the first belongs to the period from 1967 to 1980 when he was at Boeing, the NASA Ames Research Center, and Flow Research, Inc. Almost all of these papers were based on the transonic, small-disturbance equation for potential flow. The papers in the second group were written from 1980 on, after he joined MIT, and they concern many topics, including solutions of the Euler and Navier-Stokes equations. The significance of Murman's contributions to transonic flow is evident if one, for example, compares the contents of the proceedings of the Symposium Transsonicum in 1962 with those of the successor conferences in 1975 and 1
Department of Mechanical and Aeronautical Engineering, University of California at Davis, Davis, California 95616. 2 Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853-7501. Frontiers of Computational Fluid Dynamics - 1998 Editors: David A. Caughey h Mohamed M. Hafez ©1998 World Scientific
HAFEZ & CAUGHEY
2
1988. The impact of Murman's work is manifested through the quality and the quantity of the papers dealing with transonic flow simulations in the latter two conferences. His paper on vortical flows over delta wings in the proceedings of the third Symposium (in 1988) reflects his emphasis on both the physics and the numerics of the associated phenomena. On the other hand, Murman's contributions to CFD are even more impressive. Introducing the concept of type-dependent differencing, the relation between upwind schemes and artificial viscosity, proper linearization, and viewing relaxation as an artificial time-dependent process, followed by the importance of conservation - not only at the differential level, but at the discrete level, as well - and the construction of proper switches to capture shock waves correctly; these were some of the newly born concepts put together for the first time to produce, using computers of the early 1970s, two- and three-dimensional solutions for the transonic flow over aerodynamic configurations with reasonable predictions, not only of lift, but of inviscid drag, in the transonic regime. His work can be said to have provided the first convincing evidence that led the aerospace industry to consider CFD seriously as an analysis tool that complemented the wind tunnel.
1.2
Contributions to Transonic Flow
Theoretical work on transonic flow dates from the beginning of the twentieth century and the work by Chaplygin on gas jets. By the late 1940s the nonlinear, transonic, small-disturbance equation for planar flows
(K*x - # / 2 ) x + vy = 0 was derived independently by Oswatitsch, Guderley, Busemann, and von Karman. In this equation, <j> is the scaled perturbation velocity potential, x and y are Cartesian coordinates parallel and perpendicular to the direction of the free stream, respectively, and K is the transonic similarity parameter. 3 In 1947 the above form of the equation, identifying the transonic similarity parameter and establishing the transonic similitude, was derived by von Karman, and at the same time the Mach number freeze principle was discovered. The transonic area rule was discovered, based on slender-body theory, by Oswatitsch & Keune, and was verified experimentally by Whitcomb. Guderley wrote the first book on T h e o r y of Transonic Flow in 1957 and Ferrari & Tricomi's Transonic A e r o d y n a m i c s appeared in 1962. Landahl's 3
For planar flows the scalings are such that K oc (1 — M 2 ) / T 2 / 3 ; for the axisymmetric version2 of 2the transonic small-disturbance theory, the similarity parameter is K «■ (l-M )/r
CONTRIBUTIONS OF EARLL MURMAN
3
U n s t e a d y Transonic Flow was published in 1960, and Ashley & Landahl's A e r o d y n a m i c s of W i n g s and B o d i e s was published in 1965. Both von Mises' M a t h e m a t i c a l T h e o r y of Compressible Fluid Flow and Bers' M a t h e m a t i c a l A s p e c t s of Subsonic and Transonic Gas D y n a m i c s were published in 1958. Bitzadze's Equations of the M i x e d T y p e and Manwell's H o d o g r a p h Equations appeared in 1964 and 1971, respectively. In this regard Germain's review article in 1964 must also be mentioned. The most comprehensive treatment to date of the transonic, small-disturbance theory was published in Transonic A e r o d y n a m i c s by Cole & Cook in 1986. Most of the early work on transonic flow was based on the hodograph method, in which the dependent and independent variables are interchanged with the result that the equations of planar flow become linear, allowing solutions to be constructed by superposition. This method is restricted to isentropic, irrotationai, steady flows in two dimensions, and the treatment of shocks is complicated. The simple geometry of a finite wedge was treated by Guderley & Yoshihara and by Cole in the early 1950s. Vincenti & Waggoner calculated flow over a double-wedge profile and an inclined flat plate at a free stream Mach number of unity. The boundary condition at a solid surface for a general airfoil shape is extremely difficult to handle in the hodograph plane. Nieuland & Boerstoel designed shock-free airfoils in the 1960s by extending the earlier work of Lighthill & Cherry. In the early 1970s Garabedian & Korn successfully applied the method of complex characteristics to handle the mixed type equation, and were able to design supercritical (shock-free) wing sections, including the widely-studied Korn Airfoil. (See the related work by Garabedian and Chen in this volume.) These contributions were particularly important in light of the common misinterpretation of the earlier work of Morawetz and others to imply the nonexistence of continuous, transonic flow past profiles. 4 In fact, that work did not rule out the existence of shock-free singular solutions having no neighboring shock-free solutions. Pearcey, and later Whitcomb, succeeded in demonstrating experimentally that essentially shock-free airfoils can be realized and used. From an engineering point of view, the presence of very weak shocks is allowed, as they do not destroy the efficiency of the design below the drag-rise Mach number. Another formulation, which is restricted to steady, two dimensional (including axisymmetric) flows, is based on the use of a stream function. In the early 1940s Emmons calculated transonic flows past airfoils using this formulation. His finite-difference solutions were calculated by hand using a relaxation procedure with shock fitting. Also in the 1940s Macoll solved Recently, computer simulations have been used to study existence and uniqueness issues. SteinhofT& Jameson, and later Salas et al., demonstrated the non-uniqueness of potential flow solutions. In 1991 Jameson showed similar problems for steady solutions to the Euler equations. The problem persists even for high Reynolds number, attached flow solutions of the Navier-Stokes equations, as demonstrated by Hafez &; Guo.
4
HAFEZ & CAUGHEY
the detached shock problem. Shock-fitting has been studied later by Moretti and his students: e.g., Pandolfi, Salas, Grossman, and others. In his 1969 Ph.D. thesis, Steger tried to automate Emmons' calculations and reported convergence difficulties. Sells' analysis was restricted to purely subsonic flows because of the same difficulty.5 Also, a related formulation based on the streamline curvature method was introduced in the 1950s, particularly for internal flows, but its extension to transonic flows was not successful until the work of Drela and Giles in the 1980s. Even using the potential flow formulation, where the density is defined uniquely in terms of the fluid velocity (using the Bernoulli equation), many attempts to simulate mixed flows failed miserably. To put Murman's contributions in perspective, we first discuss some of these attempts. To account for compressibility effects, the Karman-Tsien (tangent-gas) approximation was introduced in 1939. This approximation was doomed to failure even for purely subsonic flows (with lift) because of its inconsistency with the basic physics, as discussed by Lighthill in his Higher Approximations section of the Princeton Series on High-Speed Aerodynamics. Another ad hoc method introduced by Latoine in 1951 was based on the Prandtl-Glauert transformation to relate the pressure coefficient of a compressible flow to that of a related incompressible flow. A better approximation was then found if the local, rather than the free-stream, Mach number was used. A similar local-linearization approach was also adopted later by Spreiter & Alksne. Meanwhile, supersonic flows enjoyed more systematic treatment, starting with Ackeret's linearized approximation for Prandtl-Meyer flow, and the second- and third-order expansions of the oblique shock relations by Busemann. Expansions in terms of free stream Mach number M^ and thickness ratio T were carried out by Rayleigh & Janzen and by Hentzsche & Wendt, respectively, and Imai tried to extend the range of validity of these expansions to handle subsonic flows at higher Mach numbers. Oswatitsch used an integral method with an elliptic kernel for subsonic flows. The nonlinear terms are lagged, and a Poisson equation is solved at each iteration. For supersonic flows a nonhomogeneous wave equation (with constant coefficients) is solved at each iteration with the nonlinear terms (the forcing function) evaluated from the previous iteration. Again, the sequences obtained via Oswatitsch's The main difficulty encountered in the stream function approach arises from the fact that the relationship between fluid properties, such as density and pressure, and the mass flux density pq is not single-valued. There is a square-root singularity at the sonic point where the mass flux density is a maximum (corresponding to the choking phenomenon) Larger values of the flux density yield complex values of the density. This problem can be circumvented, and both irrotational and rotational mixed flows with shocks can be computed, as shown by Hafez in the 1980s.
CONTRIBUTIONS OF EARLL MURMAN
5
method do not converge, in general, for mixed (subsonic and supersonic) flows. Oswatitsch introduced a parabolic method for bodies of revolution at sonic speed. In this method, the acceleration term (the second derivative of the potential function in the flow direction) is lagged, and a parabolic equation is solved at each iteration. (An elegant asymptotic theory for sonic flow problems and the Mach number freeze principle was later developed by Ryzhov, Cole, et al.) Integral methods were applied to transonic flows by Zierep & Gullstrand, and later in the 1970s by Hancock & Nixon and by Spreiter & Ogana. A key point in these contributions is the proper treatment of the saddle point at the sonic condition. A quasi-one-dimensional approximation with a normal shock was used, and reasonable results were obtained. It is interesting to note that for a small, embedded supersonic flow region, iterations based on the Poisson equation do converge if artificial viscosity terms are added to the equation as part of the forcing function. Convergence in this case depends, however, on the amount of artificial viscosity that is added. A more general approach introduced by Morino in the 1980s is based on the time-dependent (unsteady) equation, where the integral formulation is constructed in terms of a wave equation with variable coefficients depending on the local conditions. In principle, Morino's formulation is valid for transonic (mixed) flows, but it is expensive because of the intensive computations required to evaluate the time-dependent kernels. Perhaps the most important development of the transonic small-disturbance theory is the asymptotic derivation by Cole in which the free stream Mach number M ^ —>■ 1 and the thickness ratio r —► 0, while the transonic similarity parameter K remains constant. Proper ordering, and matching with the boundary conditions, yields the celebrated nonlinear, transonic, small-disturbance equation as a first-order approximation. Transonic flow is inherently nonlinear, and all attempts based on linearization lose this important feature of the fundamental phenomenon. In Cole & Messiter and Cole's Twenty Years of Transonic Flow two derivations were made with the same final result; one starting with the full potential equation and the other with the Euler equations as the fundamental conservation laws. This exercise not only proved consistency but it shed light on the evaluation of drag, as will be discussed later. In 1965 Hayes' complete study of the higher approximations for transonic flow showed that the flow remains potential in the first approximation. It was shown also that the jump conditions, admitted by weak solutions of the derived equation in conservation form, are in agreement with the perturbation form of the Rankine-Hugoniot relations. On the other hand, the characteristic form revealed the mixed-type nature of steady, transonic flows. Cole systematically developed the far field behavior of solutions for lifting airfoils and wings in subsonic, sonic, and supersonic flows as well.
HAFEZ & CAUGHEY
6
The viscous transonic equation was derived by Cole & Sichel to describe the shock structure and the shock intersection problem. In the limit of vanishing viscosity the transonic, small-disturbance equation is recovered. The unsteady equation was studied in relation to transient phenomena. The nonlinearity remains an essential ingredient in the low frequency regime. Because one of the characteristics is aligned with the flow direction, Cole gave the equation the name parahyperbolic. The time-dependent equation can describe a well-posed initial value problem with a possible steady state solution, depending on the constraints specified. It is important to note that equations of the form t = (1 - M2)
4>XI + <j>yy
or 4>tt = ( l - M2)
4>xx + <j)yy ,
2
where 1 - M = 1 - M£, - (7 + V)M^X and M is the local Mach number, do not, in general, describe well-posed nonlinear initial value problems, while the unsteady, transonic, small-disturbance equation, given by axt = ( l — M2)
cj)xx + <j)yy,
does. In fact, this observation explains the failure of some of the standard (Jacobi and Liebman) iterative methods when applied to transonic flow problems. Following Garabedian's interpretation and analysis of successive line overrelaxation (SLOR) iterative procedures in terms of the time-dependent, partial differential equations they approximate (the so-called time-dependent analogy), Jameson explained the success of Murman's choice of line relaxation marching with the flow, since it mimics the unsteady, transonic smalldisturbance equation, particularly the appearance of a term proportional to 4>xt in the modified equation corresponding to its time-dependent analog. Hence, embedding the steady transonic equation in a higher-dimensional (time-dependent) problem of the proper type was essential to the convergence of the iterative process. Adding viscous terms, in conservation form, was essential to capture shocks having the correct strengths and locations. The combined equation is given by a4>xt = (K4>x - 4>2xl2)x + yy +
(v(f>xx)x
or the more general form 6 including the high-frequency term u + aIt = {K4>x - 4>2x/2)x + 4>yy + {v<j>xx)x This form allows vector-parallel computations of transonic flows based on three-level schemes that are explicit in time (see Keller, Jameson, South, and Hafez).
CONTRIBUTIONS OF EARLL MURMAN
7
One can think of the artificial viscosity as a regularization tool that excludes expansion shocks and enforces smooth acceleration through sonic conditions. Shocks will be smeared, and numerical solutions can be obtained using Newton's method and direct solvers based on Gaussian elimination at each iteration since the matrix including the contributions of the discrete viscous terms will be invertible even when the unsteady term is absent. Direct solution is feasible, at least for modest, two-dimensional problems. For large problems, however, iterative methods are necessary (on all but the most powerful computers), and it is essential that the time-dependent term be included, either implicitly (as in line relaxation marching with the flow) or explicitly. The importance of nonlinearity and the local character of the governing equation, the role of viscosity, and understanding the transient process associated with the unsteady equation were the necessary ingredients for a successful simulation of steady, transonic flows. Equipped with this body of knowledge, Murman started working with Cole in 1968. Murman was the right person at the right time. He was responsible for obtaining a numerical solution for steady transonic flow problems using available computers at Boeing. Many of the concepts described above were not clear in the late 1960s, and problems of implementation could open the door to more obstacles. Murman's deep understanding of the physics and the numerics led to the first practical computer simulation of steady, transonic flow with shock waves. Details of the numerics will be discussed in the following section, particularly the numerical stability of the Murman-Cole scheme, the design of the switches from one region to another, and Murman's fully-conservative operators. In the remainder of this section, Murman's papers on transonic aerodynamics are discussed. Once a tool for transonic flow simulation was available, there was obviously a demand for the accurate determination of aerodynamic loads. The lift can be calculated from the surface pressure distribution or from balancing the momentum, normal to the flight direction, over a control volume surrounding the airfoil. As a result of the use of linearized boundary conditions and the singular behavior of the solution at the leading edge, the control volume approach may be preferred for solutions based on the small-disturbance equation. If the control surface is chosen in the far field it can be shown, assuming only normal shock waves, that L = — />oos = v(G(Z),Z) etc. The general set-up in the conical projection is shown in Fig. 1. The equations of motion in (Y, Z) take the form , , r t da (v-Y)W
,
„ . da - dZ+a{oY
fdV
+ {w Z)
dw\ dz)=°
+
((.-v)£+ (.-*>£)(P +
F/7
O-T
= 0
The last of (2.8) shows that the entropy is constant on the conical projection of the stream lines: P + Hh dY v-Y — = const o n — = 1 for sufficiently large Y, Z so that the system is hyperbolic. Thus in general we are dealing with a problem of mixed type.
OPTIMAL HYPERSONIC WINGS
33
Several interesting results can be obtained from the shock relations. The question of the possibility of an attached shock at a conical edge can be answered by a local study. Near the edge Z = A the shock can be represented as straight Y, = G{Z)=W(A)+Q(A-Z)
+ ---
(2.11)
At the point just behind the shock and at the surface a regular solution must satisfy both the shock relations (2.7) and the tangent flow boundary condition (2.6). This leads to
d " «W(A» ( * g g ±
~ W
^
)
- ^
(-(A) - - ' < A » - 0
(2.12) This is a cubic equations for 6 with parameters HA, W(A), W'(A). One solution decreases the entropy and is ruled out. From the other two we assume that the weaker shock (less entropy increase) is the one that occurs. In this case the local flow is hyperbolic just aft of the shock. When the local flow becomes elliptic the shock is detached. A special class of wings for which such shocks occur are caret wings, composed of flat planes intersecting at the ridge line. From (2.12) the region of attached and detached waves at the edge can be plotted as in Fig. 2. Similar considerations apply to conical wings of arbitrary shape. We are mainly interested in calculating the flow on the high pressure side of a lifting wing and optimizing the shape to produce a minimum pres sure drag for a given lift. Formulas for lift and drag coefficients and a suitable measure of the optimum are given in the next section.
2.3
Lift a n d D r a g C o e f f i c i e n t s ; F i g u r e o f M e r i t : C o n i c a l Wings
Lift and Drag coefficients are defined conventionally except that the forces considered are only those on the lower surface
^ = -4*1' ^ = T 4 I Poo 2
A
(2-13>
"°° 2
where L = lift = f f{pi-Poo)dxdz, A = wing planform area,
D = drag = / / (pi - px)5—dxdz pi = pressure on the lower surface.
(2.14)
TRIANTAFILLOU, SCHWENDEMAN & COLE
34
Figure 2 Regions of detached shock (below solid curve) and attached shocks for caret wings containing the (Y,Z) pairs (1,0) and (W(A),±A) for H = 2.0. The regions of attached shocks contains constant 0 contours.
In hypersonic coordinates this gives CL
JL
o 2 4 / / Pidx,dz,
CD = 53j
I f
P,^dx,dz
(2.15)
where A = A/5 =scaled planform area. In general the wing will be described by a finite number iV of design variables or a design variable vector D(d1,d2,-- ■ , ^ N ) . CD = 53CD{B; H),
CL = 52CL(D;
H)
(2.16)
We seek an optimum which minimizes CD for a given lift coefficient CL by using a Lagrange multiplied A. That is minimize K = J 3 C D ( D ; H) - XS2CL(D;
H)
(2.17)
The conditions that K be stationary are ^-=352CD-2\5CL oS
=0
(2.18)
OPTIMAL HYPERSONIC WINGS dK
_3dCD
35 ,c2dCL
Eliminating A shows that d
( CD
ddj
\c3L/2
= 0,
j =
l,2,---,N.
Thus the optimum is found by choosing dj, di, • • •, d^ so that si
fit = figure of merit = — ^
(2.20)
is minimized. When conical symmetry is used the formula for the figure of merit becomes
fM
A /0A Pi{W{Z) = \l2 / A-
ZW'{Z))dZ x3/2
(2-21)
Caret wing wave riders or on-design caret wings are those for which the flow is uniform and hyperbolic on the lower side. The bow shock is straight and attached at both leading edges. The shape of such a wing is thus Y = W{Z) = l + -*j^-\Z\,
\Z\ v^^V* -1)
The figure of merit for such a wing is easily calculated and can be written IM =■ - ^ = =
i
j
—
(2.23)
^?+\7FP For a given H the figure of merit of an on-design caret wing is independent of A. For flat wings (W = 1) similar calculations give the regions of attached and detached shocks in Fig. 3 and the general flow structure in Fig. 4.
2.4
Numerical and Optimization Methods
In this section a brief outline is given of the numerical methods and optimization procedures used to obtain the results presented in the next
TRIANTAFILLOU, SCHWENDEMAN & COLE
36
Figure 3 Regions of detached shocks(above solid curve) and attached shocks for flat wings B = Y — 1 with various values of A and h. The region of attached shocks contains constant 0 contours.
section. For more details see[4]. The conical equations (2.8) are cast in the form u< + fy + g z = c (2.24) where u = {a, a(v - Y), a{w - Z), oE - (P + Hh)) f = (cr(v - Y),a{v g = (a(w - Z),a(v
- Y)2 + P + H/f, a{v - Y){w - Z), aE(v - Y))
- Y)(w - Y)(w - g), a{w - Z)2 + P + H/-y, aE{w - Z))
w = {a, a{v - Y), a(w - Z), aE - (P + Hh)) c = (-2 0 for all points on the lower surface. This becomes a condition on the conical slope W(Z) - ZW'(Z)
>0
(2.28)
In addition an average deflection constraint was applied so that the angle of attack over the lower surface in physical coordinates was equal to 8. In conical variables this reads
kl> i
/-A
ZW')dZ
= 1
(2.29)
A summary of the smooth optima starting from flat wings for H= 1.0, 1.5 and 2.0 and several A appears in Fig. 9. Several thousand steps in design space were required to deform an initially flat wing into an optimum. For H = 2.0, A = 2.7 the optimized wing had a figure of merit 1.8% better than the original. Both initial and final wings had attached shocks. The pressure P is close to that of the on-design caret wing P = 2.14, H = 2.0 (see Fig. 8b). Another case started with H = 2.0, A = 2.4 which is an initially optimal flat wing. The final result appears in Fig. 8c. The wing with detached shock was deformed so that an attached shock resulted.
2.6
Acknowledgement and Disclaimer
Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number F49620-97-1-0141. The US Government is authorized to reproduce and distribute reprints for govern mental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Government.
REFERENCES 1. Baysal, O. and Eleshaky, M.E., Aerodynamic Sensitivity Analysis Methods for the Compressible Euler Equation, J. Fluids Eng. 113, 1991, pp 681-688. 2. Hayes, W. and Probstein, R., Hypersonic flow theory, Academic Press, 1959.
OPTIMAL HYPERSONIC WINGS
43
3. Hillier, R., Computation of flow Past Conical Hypersonic Wings Using a Second-order Accurate Godunov Method, AGARD C P P 428, 1987. 4. Triantafillou, Susan A., Optimization of Hypersonic Wings, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, N.Y., June 1997. 5. Vanderplaats, G. N. and Moses, F., Structural Optimization by Methods of Feasible Directions, Computers and Structures, 3, 1973, pp. 739-755. 6. Van Dyke, M. D., A study of Hypersonic Small Disturbance Theory, NACA Tech. Rept. 1194, 1954, pp 885-905.
Figure 9
Summary of smooth optima and the related flat wings.
3
Geometry for Theoretical, Applied, and Educational Fluid Dynamics Helmut Sobieczky1
3.1
Introduction
Motivated by the topic of this book and by the event leading to its edition, in this contribution the attempt is made to illustrate how some of our accumulated knowledge in the field of compressible fluid dynamics may be kept alive and useful for the development of modern tools in aerospace engineering. This may be needed in a time when computational fluid dynamics (CFD) solves complex aerodynamic problems and a detailed understanding of the underlying phenomena seems to become unnecessary. Numerical modelling in transonic aerodynamics has had decisive impact from the work of Ear 11 M. Murman, who modelled the key phenomenon of transonics, mathematically described by a type change from elliptic to hyperbolic model equations, in a simple change in a numerical difference scheme [5]. In a more recent activity, Murman guided a project to provide an educational software system helping a new generation of students to understand fluid mechanics and gasdynamics by solving a collection of fluid mechanic and aerodynamic problems on the computer. We try to learn from such activities, using past work results to stay familiar in the future through the tools of information, communication and software technology. Here the background of some aerodynamic design tools is illustrated which is also part of our knowledge base and can be implemented 1 DLR German Aerospace Research Establishment, Bunsenstr. 10, D-37073 Gottingen. Frontiers of Computational Fluid Dynamics 1998 Editors: David A. Caughey & Mohamed M. Hafez ©1998 World Scientific
SOBIECZKY
46
in modern and efficient optimization tools for airframe and turbomachinery design. Details of flow phenomena result in mathematical functions for their correct description. With the need to arrive at realistic three-dimensional configurations, the structure of desirable 3D flow elements should be aligned with the geometrical description of their boundary conditions. Geometry preprocessor tools may therefore transfer the fluid mechanic knowledge base to CAD systems supporting CFD analysis, experimental model production and component data definition for interdisciplinary work. A case study illustrates our present activity in this direction. It could serve for practical use in aerospace research, application and education.
3.2
T h e transonic knowledge base
In the past 50 years, until perhaps 30 years ago, the pioneers of transonics (Guderley, Oswatitsch, Frankl, Tricomi, Germain, ...) have developed models for the basic equations of flow motion so that is was possible to find either isolated or whole systems of particular solutions. These were amenable to explaining the hitherto strange and sometimes dangerous effects in the higher speed regime of approaching transonic flight Mach numbers. Thirty years ago, Potential theory was accepted as a good tool for modelling the basic aerodynamic component of airfoil flow. At this time the first (relatively) 'large scale' numerical simulations were made possible by the availability of digital computers. The problems of simulating transonic flow with mixed subsonic/supersonic domains were solved with the approach by Murman and Cole [5] to change the scheme of discretization from central to forward differences once supersonic flow appears imbedded in a subsonic flow model. From there on the analysis of transonic flow was a main goal in the development of CFD. The above-mentioned solutions to the basic differential equations also included 2D airfoil flows in high subsonic Mach numbers which did not have shocks and thus promised a higher lift/drag ratio. It took some time until refined technology, both in experiment and in analysis, could confirm this theoretical concept of shock-free flow, because the underlying mathematical solutions are mathematically isolated, required very accurate analysis and precise wind tunnel technology, for a while they were therefore not seen of practical value for the shaping of a real wing. But once proven useful for applied aerodynamics, systematic design methods were in demand to create transonic - 'supercritical' - airfoils and with their help, arrive at better wings with high aerodynamic efficiency in the transonic flight regime. The author, for three decades, has had the privilege to take part in the development of transonic flow design methodology, witnessing the advent of faster computers and refined software to analyze compressible flow.
GEOMETRY FOR FLUID DYNAMICS
47
Education with the above-mentioned pioneers' models led the author to the use of a simple analog electrostatic set-up to solve the potential boundary value problem in a mapped working plane (a hodograph plane). This approach, for instance, allowed for a relatively easy understanding of the mathematically elegant but difficult method by Garabedian [1] to design shock-free airfoils in a complex characteristics (4D) work space, and identify the 2D real part of the solution with the boundary condition in a 2D electrostatic network [6]. A replacement of the latter by a computational grid and solution by numerical relaxation was of course timely in the seventies, but the real profit of this early 'playing' with transonics was its educational value, giving us a 'feeling' for the occurring flow phenomena and a deeper understanding for the related work of others. Furthermore, the technique of solving mapped boundary value problems in the analog set-up was implemented in many of the numerical analysis methods based on potential theory operational by then and this way converted these to become design tools for shock-free 2D airfoils as well as 3D wings [7],[8]. This approach requires a temporary change of the ideal gas property once the static pressure within the flow field is falling below the critical value, a simple software modification. This semi-inverse technique therefore is termed 'Fictitious Gas' (FG) method and within the past years also has been implemented in modern Euler [4] and Navier-Stokes [9] analysis codes. With these tools available and many case studies in aircraft wing and turbomachinery blade redesign carried out, it became obvious that the reduction of shock losses which are observed for given initial configurations, results in systematic shape modifications which may allow a direct mathematical modelling applied to only local domains of the surface geometry. With geometry preprocessing being the starting point for any refined optimization in the various disciplines participating in an aerospace product design process, the knowledge about quality and quantity of surface changes for better aerodynamic efficiency may therefore greatly reduce the number of parameters to be varied in an optimization process.
3.3
G e o m e t r y generator
As an activity turning out to be of increasing importance for an economic production of CFD boundary conditions and grids, the development of a flexible geometry generator code with special input control for transonic design refinements was started 15 years ago and in the meantime has evolved to the preprocessor software as suggested above, complimenting and using the design knowledge base for not only transonic design and analysis but also supporting supersonic applications and shape definition for low speed aerodynamics and hydrodynamics. In a recent compilation the author tries to illustrate this link
48
SOBIECZKY
between gasdynamics and geometry in more detail [10], [11]. Recent developments in CFD show progress using both block-structured (hexahedral) and unstructured (tetrahedral) grids. CFD analysis of complex configurations like complete aircraft seems to be more successful with unstructured grids [2], discretizing most of the surrounding space and restricting the use of a structured grid to the viscous flow layer close to the surface. Such hybrid grids may currently already be generated by commercially available software [3] for which our geometry generator serves as a preprocessor: The analytical functions chosen from a catalog are able to model most of the complex details. Explicit and non-iterative evaluation for given surface metrics allows for very fast production of dense surface grids which are input for determining 3D boundaries for triangulation of space between all components and a far-field boundary surface. There are some important applications motivating the introduction of a time-like single 'superparameter', to represent a fourth dimension in the modelling of 3D configurations: Configurations may be considered changing their shape with time, like in a periodical motion occurring in nature with animal flight or in aeroelastics with fluid - structure interaction and in rotor aerodynamics. Or the shape varies in non-periodical mode in a sequence of controlled alterations defined by an evolutionary optimization process. This is paralleled by the mechanical adaptation process of experimental model or real configuration components. In our software, selected input parameters defining shape details are given within an interval, with the superparameter controlling their changes. A suitable data definition for such moving shapes is therefore very attractive for all unsteady processes, for optimization with a suitably controlled number of parameters and for the programming of microcomputers controlling actuators to adapt a wing geometry to varying operation conditions. Many innovative items in aircraft design call for tools like this geometry generator. In the following chapter the status of work on a newly defined test configuration is reported as an example for using geometrical modelling with a high degree of flexibility as needed to observe many details of the transonic knowledge base. It is proposed to serve as a testbed for various conceptual, computational and experimental studies aimed at refining the tools for innovative aircraft design.
3.4
Example: Generic transport aircraft configuration
Test cases in fluid dynamics and aerodynamics to verify theory and experiment always have been welcome and some of them have been used through many years. Their success in serving as a common base for various researchers and developers used to depend on a balance between
GEOMETRY FOR FLUID DYNAMICS
Figure 1 Generic high wing transport aircraft
mathematical simplicity needed to create the boundary conditions and geometrical complexity high enough to be attractive for the applications. After many years dealing with airfoils as test cases, we feel that the time has arrived to provide more sophisticated configurations, including whole aircraft with all components. Using the possibilities given by modern software and communication technology this goal seems reasonable. We envision tools available via the internet to create case studies and communicate data with interested partners. Suitable software tools based on the geometry generator are under development [13]and the following example is an exercise to refine them. Motivated by some of the current design activities and related collaboration between aircraft industry and our research group, the geometry generator was used for modelling a generic transport aircraft (Fig. 1). With these tools having been applied earlier to component junctures of mid-wing or low-wing mounted configurations, the task to model a high wing geometry with strong control of the shape details in the wing root area is an ambitious one, especially if the combined wing - body shape is to be optimized aerodynamically in transonic flow in the presence of the large wheel gear box. Our transonic knowledge base suggests application of the area rule for a smooth axial distribution of the cross section area. Fuselage cross sections are varying axially to contribute to the fillet shaping near the wing root leading edge and also provide the flat upper wing surface extending over the body roof (Fig. 2). Extending the basic wing-body geometry database with a high lift system requires providing the kinematic geometry of the motion of slats and flaps, the latter moving along a flat portion of the body to improve their aerodynamic efficiency. A tail and four propfan engines complete the aircraft shown here - almost, because the flap track fairings still need to be added. An experiment was planned for the relatively small Gottingen transonic wind tunnel (TWG), using only the data for fuselage and clean wing of this generic aircraft. Using the IGES format, the geometry preprocessor provided surface data for the CATIA CAD system, which allows for defining
SOBIECZKY
50
Figure 2 Details of the wing-body juncture
details of the construction and replaces the input data by surface patches for the numerically controlled milling of the model components. It became clear that only providing of a very dense supporting grid for the required accuracy of shape details was ensuring an acceptable surface quality. Dense data are generated, however, in a few seconds on a workstation because of the analytically explicit functions used to define the surfaces, without any iteration or interpolation, implemented in user-friendly interactive software. In the first experiment, a concept of using an incomplete model for only local flow quality analysis was brought to reality: Applying the classical aerodynamic knowledge about lifting wing theory, suitably shaped end plates on a large aspect ratio wing with its tips clipped should allow for maintaining the wing circulation on the remaining wing with its reduced span. This would ensure unchanged flow details in the plane of symmetry, namely in the area of the wing-body juncture. In the present example, the span was reduced to 60 percent of the original wing with aspect ratio 9 and "Circulation Control Splitter Blades" (CCSB) were mounted like winglets, Fig. 3. Their design using 2D swept wing theory and 3D CFD simulation is described in Ref. [14]. Reduced span of 80cm allows the model (DLR-F9) to fit in the l x l m transonic wind tunnel for the investigations of the flow details in the wing root area. This wind tunnel has adaptive upper and lower walls but closed and fixed sidewalls, the CCSB's should compensate not only for reduced lift-generating span but also from side wall interference. Fig. 4 shows the complete arrangement including model support: For the purpose of creating a future 'virtual wind tunnel database' with this test case, the additional components of sting and sword as given in the T W G were modeled also, their components consisting of shapes created either as 'wing' or 'body' type shapes, the only two types of surface topologies defined with the geometry generator so far. Wind tunnel side walls and a far field boundary above and below complete the flow space simulating the adaptive wind tunnel,
GEOMETRY FOR FLUID DYNAMICS
Figure 3
51
Configuration DLR-F9
later the data from wall adaptation will be modelled as flow boundary, too. Several runs with the unstructured grid code DLR-t [1] in Euler mode were carried out to estimate the needed adjustment range of the CCSB's. Fig. 5 shows a graphic postprocessing of this numerical simulation. Refinements of the CFD tools for application of a hybrid grid to simulate the flow with the N/S version of the code is the next step to go.. The experiments easily allow a variation of the angle of attack while the tunnel was running, but the adaptation of the CCSB's require a run interruption, so far. An on-line data processing technique was therefore developed and used [12] during the experiment, to select the flow conditions with an optimum angle of attack simulating a spanwise circulation distribution as occurring on the complete wing, from the root area close to the tip of the clipped wing with CCSB's. Four span stations with static pressure measurement were used to control spanwise load distribution (Fig. 6), the selected cases confirm a pressure distribution relatively undisturbed by the wing clipping up to at least 75% of the clipped wing span or up to nearly half of the full wing span. This experiment demonstrated that the model with clipped wings is well suited for detailed investigations in the wing-root area. We can start now
SOBIECZKY
52
Figure 4
DLR-F9 including model support in the transonic wind tunnel Gottingen
with parametric variations of the surface. Future tests will be devoted to obtaining data with an alternative juncture module. Later, the design of an adaptive module with an elastic or pneumatic surface element seems possible because of the large space available within the model fuselage for the necessary servo-equipment. Such purpose of the model might be useful to develop knowledge of a control system for surface adaptation to optimize flow quality with its pre-programming based on selected surface pressure measurements. Already the previously mentioned theoretical work to model shock-free flow by systematic surface redesign with the FG method suggested such an approach [8]. Numerical simulation will allow to study many more variations than experiments can afford, parameter alteration will provide the input conditions for optimization strategies and modelling wing deformations creates interface data for fluid - structure interactions. This way the 4D extension of this and
GEOMETRY FOR FLUID DYNAMICS
53
Figure 5 CFD simulation with DLR-r code: grid and isobar color isofringes in center plane
similar test configurations created by our geometry tools may become useful for development of more than just aerodynamic technologies.
3.5
Conclusion
The term "geometry" describes a discipline of science as well as the shape of a particular configuration, in the language of aerospace engineering. The title of this contribution is therefore ambiguous, but this is a welcome effect: Trying to place the discipline geometry into the position of mediating between important elements of the knowledge base of design aerodynamics and the practical applications, including knowledge transfer to the younger generation, is one message here. Trying also to be specific is suitably achieved with an example. A generic aircraft configuration is presented here as a test case for various techniques in numerical and experimental aerodynamics. The exciting options of worldwide cooperative work opened by the new methods of information and communication technology will enable us to use large data sets for ambitious goals in analysis and design in a truly multidisciplinary approach.
SOBIECZKY
54
Figure 6
Measured static pressure at four wing sections (Re = 2.3 Mill.)
REFERENCES . Bauer, F., Garabedian, P. R. k. Korn, D. G., Supercritical Wing Sections. Lecture Notes in Economics and Mathematical Systems, Vol. 66, Springer Verlag, 1972 . Friedrich, 0 . , Hempel, D., Meister, A. &c Sonar, Th., Adaptive Computation of Unsteady Flow Fields with the DLR-t-Code. 77th A G A R D F D P Meeting Conf. Proc. C P 578, 1995 . I C E M T M , C F D / C A E Manual set, Version 3.1.2.1, Ref. Manual by Control Data, 1993 . Li, P. & Sobieczky, H.,Computation of Fictitious Gas Flow with Euler Equations. Acta Mechanica (Suppl.) 4, 1994, pp. 251-257 . Murman, E. M. & Cole, J. D., Calculation of Plane Steady Transonic Flows, AIAA J. Vol. 9, 1971, pp 114-121 . Sobieczky, H., Related Analytical, Anolog and Numerical Methods in Transonic Airfoil Design. AIAA-79-1556, 1980 . Sobieczky, H., Fung, K-Y., Seebass A. R. & Yu, N. J., New Method for Designing Shock-free Transonic Configurations. AIAA J. Vol. 17, No. 7, 1979, pp. 722-729 . Sobieczky, H. & Seebass, A. R., Supercritical Airfoil and Wing Design. Ann. Rev. Fluid Mech. 16, 1984, pp. 337-63 . Sobieczky, H., Geissler, W. & Hannemann, M., Numerical Tools for Unsteady
G E O M E T R Y F O R F L U I D DYNAMICS
55
Viscous Flow Control. Proc. 15th Int. Conf. on Num. Meth. in Fluid Dynamics. Lecture Notes in Physics, ed. K. W. Morton, Springer Verlag, 1997 10. Sobieczky, H., Gasdynamic Knowledge Base for High Speed Flow Modelling, in: "New Design Concepts for High Speed Air Transport", CISM Lecture Series, Vol. 366, Springer Verlag, Vienna, 1997, pp 105-121 11. Sobieczky, H., Geometry Generator for Aerodynamic Design, in: "New Design Concepts for High Speed Air Transport", CISM Lecture Series, Vol. 366, Springer Verlag, Vienna, 1997, pp 137-158 12. Trapp, J., Pagendarm, H.-G. & Zores, R., Advanced D a t a Processing to Control Wind Tunnel Experiments on DLR-F9 Transport Aircraft Model. AIAA-97-2221, 1997 13. Trapp, J., Zores, R., Gerhold, T h . & Sobieczky, H., Geometrische Werkzeuge zur Auslegung Adaptiver Aerodynamischer Komponenten. Proc. DGLR J T 1996 14. Zores, R. & Sobieczky, H., Using Flow Control Devices in Small Wind Tunnels. AIAA-97-1920, 1997
4
Computation of an Axisymmetric Nozzle Flow L. Pamela Cook, 1 Elsa Newman, 2 Scott Rimbey 1 & Gilberto Schleiniger1
4.1
Introduction
In this paper we compute the axisymmetric, inviscid, compressible, steady flow from a nozzle. The Legendre, or contact, potential formulation is used in the hodograph plane. We have emphasized the axisymmetric case because the hodograph equation is then nonlinear, whereas it is linear for the more often studied two-dimensional flow. Results and formulation are compared and contrasted with those of earlier work by Alder [1]. The ultimate goals are to gain insight and provide a workable framework for dealing with axisymmetric flows that exhibit transonic behavior.
4.2
P r o b l e m Formulation
The equations governing inviscid, compressible, steady, isentropic, irrotational flows are irrotationality: u r - t > x = 0, 1 2
(4.1)
Department Mathematical Sciences, University of Delaware, Newark, DE 19716
Department of Mathematics, Marymount University, Arlington, VA 22207; This author's work was supported by the National Science Foundation under DMS grant #9796052 Frontiers of Computational Fluid Dynamics - 1998 Editors: David A. Caughey U Mohamed M. Hafez ©1998 World Scientific
COOK, NEWMAN, RIMBEY & SCHLEINIGER
58
Figure 1 Jet Flow
continuity: (a — u )ux — uv(ur + vx) + (a — v )vr + X—v r and the Bernoulli relation: u2 + v2
1 + 1 «-.
+7 - 1
2(7-l
=
0»
(4-2)
(4.3)
Here u and v are the a: and r components of velocity, respectively, a is the local speed of sound, a* is the speed of sound at sonic, and 7 is the ratio of specific heats (7 = 1.4 for air). For axisymmetric flow, x = 1 a n d ( z , r ) are cylindrical coordinates; for two-dimensional flow, x = 0 and (x,r) represent rectangular coordinates. The focus of this work is on jet flow as depicted in Figure 1. The gas occupies a region bounded by the nozzle surface OB, the free streamline BC, a uniform downstream state CD, the x axis DO and upstream infinity O. A typical streamline is shown. The nozzle wall OB makes an angle S with the horizontal and is prescribed by r = H — tan 0
(4.4) OB, as
x —> - 0 0 .
(4.5) (4.6)
AXISYMMETRIC NOZZLE FLOW
59
On the free streamline BC q = v u2 +v2=q0
and
— = —. ax u
The value of go is assumed to be at most a*. In this limiting case of the sonic jet Ovsiannikov [9] and separately Guderley [5] have shown for two-dimensional flow that a uniform sonic state is reached at a finite distance downstream of the nozzle exit. Cole [3] showed the same result for axisymmetric flow in the context of small disturbance theory (6 -¥ 0). Note that for subsonic jets the uniform state only occurs as x —¥ oo. If we denote by h the radius of the jet when this uniform state is achieved, then a contraction coefficient C c = h/H can be defined. This will prove useful for later comparisons. A principle drawback of the formulation above is the unknown location of the free streamline in the physical plane. This difficulty is overcome, however, in the hodograph, or velocity, plane. In this framework we have x(u,v) and r(u,v). A straightforward transformation of equations (4.1)-(4.3) to the hodograph plane gives xv-ru=
0,
(4.7) 2
(a 2 - u2)rv + uv{xv + ru) + (a 2 - v2)xu +
X
— J = 0,
(4.8)
where J = xurv — xvru is the jacobian. Note that for two-dimensional flow (x = 0), the system (4.7)-(4.8) is linear, but for axisymmetric flow (x = 1) it is nonlinear. The domain in the physical plane depicted in Figure 1 maps to the domain in the hodograph plane shown in Figure 2. All streamlines emanate from the origin (stagnation point) and flow into CD (uniform downstream state). The boundary condition on BC in the hodograph plane is determined as follows. Since u2 + v2 = q% =constant, udu = —vdv. Also, dr/dx = v/u requires v(xudu
+ xvdv) = u(rudu + rvdv)
Combining these yields v2xu - uv{ru +xv)
+ u 2rv = 0
on
q = qo-
(4.9)
The flow can be computed in several ways: (1) by solving the system (4.7)-(4.8) with the boundary conditions indicated above; (2) by using the stream function formulation; or (3) by using the Legendre, or contact,
60
COOK, NEWMAN, RIMBEY & SCHLEINIGER
Figure 2 Hodograph Plane potential formulation. The stream function setting was used by Rimbey [11] in the context of small disturbance theory and by Alder [1] for the more general situation when 8 is not assumed to be small. In the former case the equations are explicit and, after a variable transformation, quasilinear. The full stream function equation in two-dimensional flow is the well-known Chaplygin equation which is explicit and linear, but the full axisymmetric equation is nonlinear and, furthermore, has coefficients which can only be determined implicitly. The contact potential formulation is, however, explicit and is the method we have opted to pursue. We will now develop the appropriate equations. Equation (4.7) implies that there is a potential <j> such that
x = (/>„,
r = 4>v.
This Legendre potential 4> is linked to the physical potential $ , where u = $x and v = $ r , by the relationship = xu + rv — $ . In terms of <j> the continuity equation (4.8) becomes a generalized MongeAmpere equation: (a 2 - u2)<j)vv + 2uv4>uv + (a 2 - v2)4>uu + X-r-{uuw - 4>lv) = 0
(4.10)
AXISYMMETRIC NOZZLE FLOW
61
This equation is fully nonlinear so one may question what advantage has been gained by using the contact potential formulation. However, it enjoys several special properties that are pointed out in Courant and Hilbert [4]. In particular, it behaves essentially as a quasilinear equation with type determined by the linear terms: L[4>] = (a2 - u2)vv + 2uv4>uv
+ ( a 2 - v2)4>uu.
(4.11)
Thus the equation is respectively elliptic or hyperbolic as the Mach number M = q/a < 1 or M > 1. Moreover, uniqueness is assured for the Cauchy problem in a hyperbolic region and for the Dirichlet problem in an elliptic region [8] [10]. Despite these advantages, the equation appears to have received scant attention [6] [2], at least from a computational point of view. For the problem at hand, jet flow, it is convenient to replace the (u,v) coordinates with (q, 9) where tan 9 = v/u and q = \/u2 + v2. This yields L[4] + XN[] = 0
(4.12)
where
L[4>] = q2<j>qq + f{q){q4>q n l
+ 4>ee),
q2 _ (1 + l)(g*2-q2) a2 (7 + l ) a * 2 - ( 7 - l ) 9 2
and N
\]
q sin 6 I . / 1 =-jT-Thnr < 4> \ 4>M, < M- -\4>[qe4>- v«- ] = s i n 0 ^ , H
2
z+') f = °>
0.
Note that x = 4>u — cos0cf>q
smt e,
r = 4>„ = smdcl>q + ^-». q
(4-13) (4.14)
From equations (4.4)-(4.6) and (4.9) it follows that <j>„ = 0 <j>e = q0h
on
9 = 0,-6
when
9 = 0,q = q0
9e + qoq = 0 o n q = qo cf> = 0 on q = 0
(4.15) (4-16) (4.17) (4.18)
These boundary conditions are indicated in Figure 3. The homogeneity of the condition on 9 = -8 is achieved by allowing the value of x0 to vary with
COOK, NEWMAN, RIMBEY & SCHLEINIGER
62
the wall angle 6. The Dirichlet condition at q = 0 results from the source like nature of the flow at that point. As x —► —oo (q -* 0), the flow is axisymmetric incompressible source flow: §(x,r) ~ {x2 + r2)'1!2. For the contact potential this requires ~ y/q as g -> 0. The unusual second-order boundary condition on the right warrants special mention. Although there is a second order derivative, it is a tangential derivative, thus actually giving less (not more) information than if (j> were to replace it (a Robin condition).
4.3
Numerical M e t h o d
Finite differences are used to solve the boundary value problem in Figure 3. A uniform grid in (q, 6) coordinates is established and a system of difference equations for the values cj>ij of the potential at the mesh nodes results. Note that only the nonhomogeneous condition in the upper right-hand corner prevents this system from having a trivial solution. This condition involves the unknown quantity h, which is assigned a value of one. This choice sets the size H of the jet opening. For the two-dimensional problem, we solve L[4>ij] = 0. The boundary condition on the right-hand side causes the coefficient matrix to be nonpositive definite, so a direct, rather than iterative, method is implemented. We achieve good results using a fairly crude 24 x 24 grid. For the nonlinear axisymmetric problem, we use the results from the linear calculation, > L) where the cumulative nonlinear effect can cause acoustic signature distortion, while the atmospheric stratification may result in wave attenuation and refraction. However, the notion of a farfield, which was used by some earlier workers to denote an asymptotic/limit state of the wavefield at large distance, such as occurring in an isothermal atmosphere, needs not be considered in the context of the present investigation. Central to the sonic boom theory and predication method is the analysis of the midfield structure. The analysis simplified for the midfield breakdowns in approaching the aircraft, where a proper matching with the nearfield structure
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
Figure 1
69
Coordinates for superboom problem
is necessary for determining the initial data or F-function for the midfield (wave propagation) calculation. An issue arising from the nearfield matching is whether a nonlinear (Euler) calculation of the nearfield may actually raise the accuracy level of the midfield analysis [4]. This will not however be the main concern of this paper. A second and more important issue is the midfield breakdown with the aforementioned super/transition boom. The analysis and the numerical simulation of this problem are the main focuses of this paper. The wave propagation problem will be analyzed in a Galilean coordinate system, fixed to the vehicle (the moving acoustic source), unlike the rayacoustics treatise underlying most existing sonic boom programs [2, 14, 22, 28, 29, 32]. The formulation employs a time-dependent characteristic surface (the Mach conoid), as a coordinate reference, and will reveal wavefield properties not apparent from the ray-acoustic approach. It also facilitates considerably the matching analyses at both ends of the midfield. The Cartesian system (x, y, z, t) and cylindrical polar coordinate system (x, r,w, t) will be alternatively used, in which the origin is fixed to the apex (nose) of the aircraft and the positive z-axis is chosen to point towards the ground (see Figs, la, b for orientation and origin for z and w). 5.2.2 5.2.2.1
The Second-Order Small-Disturbance Equation Vorticity
Correction
Although not apparent from the ray-acoustic method, a consistent description of the midfield structure must account for the baroclinically generated vorticity in the presence of an atmospheric entropy nonuniformity. With the first-order vorticity determined from the Kelvin-Helmholtz equation, the second-order small-disturbance equation governing the midfield can be corrected to read
D D a'VV-m DtDt
• +H0
7-1 D_ |W|2 + 2a 2 Dt
D
(5.1)
CHENG & HAFEZ
70
where <j> and D/Dt are a potential associated with the perturbation velocity u', corrected for the vorticity, and the convective derivative
*=v*-w^
^ = l +(c7+tr)
(5.2)
respectively. In above the undisturbed velocity vector is U = Ui + Vj + Wk, Hp and Ha signify scale heights based on with the basic atmospheric density and entropy gradients H;1
^fjnp, -
n : ^ ^
=
^lnp-^lnp
respectively. Note that the vorticity component of u in Eq. (5.2), with p' = —pD/dt, readily satisfies the Kelvin-Helmholtz vorticity equation to the leading order, and that the last term on the left of Eq.(5.1) results from combining the contributions from pressure gradient and entropy gradient. For the midfield analysis, Hp,Ha and Hp = (dlnp/dz)*1 will be taken to be comparable to a reference scale height A. The subscripts "A" on p and other thermodynamic variables indicate values pertaining to the quiescent stratified atmosphere at the location z. 5.2.2.2
Time-Dependent,
Horizontal, Rectilinear Flight (V = W = 0)
In the following, we analyze the time-dependent sonic boom problem in the Galilean system, not attempted in our previous studies. The analysis is restricted to cases involving only a horizontal, rectilinear vehicle motion (V = W = 0), but suffices for demonstrating the critical nature of acceleration/deceleration influence on the super/transition boom wavefield. The second-order P D E in this case becomes o?V2S-U2
2U~4>xt — 4>u — ( -77"bx +
— dt Mach Conoid of the Linear
U— dx
HD
(5.3)
Theory
To capture the slowly evolving nonlinear midfield, the analysis will employ the Mach conoid (the characteristic surface) in the linear theory x—xlc(y, z,t) = 0 as a reference surface which corresponds to a Mach cone distorted by the time and altitude dependence in U(t) and a(z), respectively. The function
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM xlc
{y,z,t) as
71
is governed by a nonlinear first-order P D E which can be written
1 + |V x z lc | 2 - W l - ^T) =0
(5.4)
where | V ± x l c | 2 = {x\cf + ( x ^ ) 2 , and M = U(t)/a(z). The equation is solvable by integrating the ODE set along its own bi-characteristics (following Courant and Hilbert's terminology), with p 2 = x\c>Pz = x\c, and p 4 = x]c: dy
dz
P2
P3
dt {U-p^a-2
dp2
dp3
0
-{l/2)a-i{daldz){Uiz)(U-Pi)-2p 4 ) 2
dpi 2 {dU/dt){U Pi)a- ' For convenience, the bi-characteristics will be referred to as "bi-char." along the bi-char x*c and [xj c - U(t)] are invariant and will retain respective initial values prescribed in the limit r = y/y2 + z2 -} 0 at t Namely, x\yc = p., x]c - U(t) = (x]c)t - U(U) s - U „ Whereas,
c — ^ic x\ =
(5.5) Thus their = £;, (5.6)
p-j can be evaluated from Eq. (5.4) as x?l r
c
= ±VR = ±V(U,/a( 2 )) 2 -l-p 2
(5.7)
Henceforth, all initial values will be denoted with the subscript asterisk. Obviously, the vanishing of R identifies the transition boundary, beyond which the hyperbolic wavefield of Eq. (5.3) can not exist. The function xlc(y,z,t) lc is thus obtained from Eq. (5.5) in the form of x ((z,p«,t + |v±^|27+ 2a 2
^c\2
(5.9)
' n'v
The time scale characterizing the nonlinear evolution of the outgoing wave in the midfield is expected to be comparable to A/a*, and the length scales for £, y' and z' can be taken to be L, A and A, respectively. We stipulate that the actual s-length of the waveform V can not be far greater than that of the aircraft, L; this is verified a posterior. In above, V i and Vj_ are the transverse field gradients in (y,z) and (y',z'), respectively. It follows that terms on the second line of Eq. (5.9), as well as those operated by d/dt' on the first line, may all be deleted as being relatively small. The resulting midfield equation is best studied in the variables set [v-idiii 2,i; xlc,y,z,i] after normalizing [,(,y',z',t'; xlc,y,z,t] by the scale set [T'U0L,L, A, A, A / a „ ; A, A, A/a*], respectively. Equation (5.3) for the time-dependent midfield then assumes the normalized form T(z,U)u-=u+xlc
dy
+jjLQ(z,t*,p*) ■u
d . U.a* d „ 1 =lc -KZU U+ X + zU + f OZ dt 2 d2 dy- 9
=
XZ LHP '
+
\dy*
+
2 Ql
A d '
dz*
+.
c\-r>
(5.10) where {/* = dU(tt)/dt*,
r=
A\
and the coefficients T and Q are 7 + 1
fUtU0
Q(z,U,P*)
=
U»a» dtr
dV
and U0 is a constant reference velocity. The RHS of (5.10) shows all terms of an order L / A higher than those on the LHS and will be omitted in the subsequent analysis. The "..." represents terms not written out and includes terms comparable to ( L / A ) 2 under the assumption of a nonvanishing T ' L / A . In above, the variables (y,z,t) and (y,z,i) can be interchanged. Equation (5.10) with the RHS terms omitted, generalizes the midfield equation of Cheng et al for the steady case [4, 6-9].
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM ODE solution along
73
bi-characteristics
The first-order quasilinear P D E for u can again be solved by an ODE system along its bi-char £ - £c(y,z,i) = £„, corresponding x - xc(y,z,t) = x„, d£ _ dy _ dz
di
_
du
N ~ if ~ i f u.a.a-» - _i j^pn + v2 ±f ic + u, Q] a
(5 n)
-
Its solution may be expressed formally as line integrals along the bi-char which will become crucial in the subsequent analysis of the behavior near the transition boundary i-L=
[fuR-'^dz, Jo
y= [p.R-'^dz, Jo
u = U(£„U,p.)J^exp
t-U = \Jtat
\ ~ \ JQ~Kd~z ~\jj
I Jo
J Xdz\
a-2R-l'2dz
(5.12)
where 2
,1. ,i,
K.Vi^/^,
- _ I U*a*\ fdtt
*
^[^)|f)/J
,,.
and the bi-char properties x | c = pt, and x\c = VR have been used. In terms of the variable z, parameters £»,£* and pt, which are in fact the characteristic variables, the solution u is identical to that of the linear theory, valid also in the time-dependent case. In re-expressing u in x,y,z and t through the characteristic variables, the only change from the linear theory is to be found in a forward displacement of the bi-characteristics through
{* = £ - I" TuR-l,2dz. Jo Shock jumps The weak solution to the midfield P D E admits shock discontinuity surfaces £ — £D{y, z,i) = 0 satisfying the jump relations
dyD p-x/2 rfylc diD Utat 1/2_dP° dz d« d£ a2 dz where the superscript " C " , not to be confused with the superscript " l c " refers to the bi-char of the nonlinear system and the symbol < > signifies the arithmetical mean across the jump.
CHENG & HAFEZ
74
Solution examples from computer program based on the above ODE system by C.J. Lee compare well with results obtained by an accurate method of characteristics [13] and from a variant of the PC Boom program based on the ray-acoustics approach [28, 32], as reported by Cheng et al [4, 5, 8].
5.3
The Superboom Problem
A thorough investigation of the phenomenon in question must address analytically the midfield solution behavior at the breakdown boundary, and derive the rescaled governing equations. In the following, we shall examine the midfield solutions for the sources of the difficulties and the nature of their breakdown (Section 3.1), with which the governing equation and boundary condition consistent with the asymptotic matching principle can be derived (Section 3.2). A major part of this section (Section 3.3) will be devoted to our computational study of the reduced equation system towards an improved numerical simulation for the problem, which Earll and colleague had undertaken twenty-seven years earlier [31]. Onyeonwu [27] has given a semi-analytic treatment of the wave focusing problem produced by aircraft maneuver, based on the ray-acoustic approach. The various explicit midfield behavior brought out in Sections 3.1.1 and 3.1.2 below are not apparent however from Ref. [27]. 5.3.1
Midfield Breakdown: A n Analytical S t u d y
Whereas the phenomenon in question is undoubtedly with wave coalescence or ray focusing, it is instructive and useful to identify the source of the difficulty and the types of breakdown behavior from the midfield solution of Section 2.2. 5.3.1.1
Midfield Singularity at Transition Boundary and Elsewhere
Two sources of midfield breakdown For vehicles in steady, horizontal, rectilinear motion, the midfield solution u is expected to breakdown only at the hyperbolic-elliptic transition boundary R = ( U * / a ) 2 - 1 - pi = 0 . With a nonuniform vehicle motion (U ^ 0), however, another type of singularity in u can occur elsewhere in the midfield which is to be identified with the occurrence of a "limit line" [11,33]. These sources of nonuniformity are traced to the integrands of two (line) integrals within the exponential factor of the solution form Eq. (5.12):
B = VI*-/4 C and x S ( ^ )
(§)
/*J«
(5.13)
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
75
One must note that neither V\xlc nor dU/di can be readily determined from the O D E solution of Eq. (5.5) together with x\c,x\c and x\c along a single bi-char trajectory, along which the parameter pair (t*,p») is fixed. [In our computer solution to this ODE system for the steady case [6], C.J. Lee succeeded in obtaining accurate numerical values of Vj_x l c , hence K, by employing and differencing x\c and x\c values at neighboring bi-chars.] Evaluation
of V\xlc
and
dit/di
To achieve the objective, one must express V^_5 lc and dit/&t in terms of known value along the bi-char, from which the breakdown behavior may become apparent. The transverse Laplacian of well as dtt/dt, requires data along neighboring bi-chars and can be evaluated from combining the yand 2-partial derivatives on both sides of Eq. (5.4), i.e., of
(4C)2 + (^c)2 - ( § " l ) = °'
(5 14)
-
and taking into account the dependence of U 2 on + < - ^ £ (* IV-tt+ ¥ J - L
(5.16)
K
St*
U , I + p*K
dt
a*
(5.17)
A
where A is the Jacobian determinant of the transformation (not to be confused with the scale height A) A = — (I + p , K ) - ^ [ ( N - J ) ( I + p , K ) - p 2 L 2 ]
(5.18)
and corresponds to the occurrence of a "limit line" in the classical theory of a compressible fluid flow [11,33]. The I, J , etc. above are integrals of known functions of z, it and p* 1 =
f'Z dzt
2
Jo v ^ 7 '
f
*Jo
f* dzj
dh al^R,'
K
- i o
liJil3/2'
Their behavior near the transition boundary R = 0 where z = zt* is crucial to the u behavior of interest. Let m = —(dR/dz)tt,z" = ztt — z, we have, for a sufficiently small z" R ~ (dR/dz)„{z
- z„) =
mz".
In the limit z" —>■ 0, I and J remain finite, but K, L and N become infinite: K - m - 3 / V T 1 / 2 + CK,
L ~ 2 U 2 a - 2 m - 3 / V | " 1 / 2 + CL,
N ~ 2 U t a - 4 m - 3 / 2 | z " | - 1 / 2 + CN -1 2
(5.20)
therefore are unbounded like | z " | / . It follows from Eqs. (5.13) and the dependence of V\xlc and dit/di on R and A that the midfield solution u can be unbounded only at locations where either R or A vanishes. Three classes of u breakdown may be anticipated: (a) A = 0, and R is far from zero, (b) R = 0, and A is far from zero, (c) A = 0, and R is small. It follows from Eqs. (5.18) and (5.19) that A can vanish with an acceleration {U* > 0), and the significant role of U „ / U * in controlling these three types of occurrence is obvious. For type (a) to occur, a large acceleration U » / U , = 0(1) would be called for. In the following analysis, we will be more concerned, however, with the problem of critical influence on the super/transition boom by the small | U , / U „ | , which will be seen to belong to types (b) and (c) above. In passing, we may remark that, for a unitorder U » / U * acceleration leading to type (a), an algebraic type breakdown (in variable z"' = zttt — z, where z = ztt* at A = 0) is anticipated.
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM 5.3.1.2
Midfield behavior near transition boundary: critical dependency vehicle acceleration
77 on
The analysis will examine the several cases of vehicle acceleration effects on the wavefield in the vicinity of (or not far from) the transition boundary, where
0 < z" = «„ -z while, for A = 0 occurring at z = z** +* * 0*> Here, A will not vanish, but in view of the K behavior near the boundary, Eq. (5.20), it may become unbounded and in fact help reducing the singularity strength of V2j_xlc. The result on the function k of Eq. (5.12), without restricting the relative order of magnitude of p2, and \/R,
I
ndz ~ -Zn|z"| — In
z" + 2 p 2 / m 3 / 2 I „
+ 0(1)
which reduces u near the transition boundary in the steady case to -
frit
« \ [P* V P \y/z" +
Co(z„,pt) 2pi/mi/zItt\1/'
where Co is the exponential of the finite part of the integrals in Eq. (5.12). The result recovers that deduced earlier in Hayes' [20, 21] superboom study, for which p\ = [ ( U , / a * ) 2 — 1]COSUJ* is necessarily small compared to \fzT\ at least in the symmetry plane. The result has extended its applicability domain to one not restricted to a small p», hence not restricted to a small [ ( U t / a , ) 2 — 1]. Interestingly, for a nonvanishing pi, u is seen to be bounded at the transition boundary, but the derivative of u and the midfield equation must still undergo a modification in view of the singularity in x\c in this limit.
CHENG & HAFEZ
78
(ii) Effect of vehicle deceleration: U , / U * < 0 For vehicle decelerating at an earlier time, i.e., U , / U * < 0, at t = i», A can not vanish identically in the vicinity of the transition boundary. Taking into account of the behavior of K, L and N , the line integral of R and \ can be evaluated for small R as
I
adz
^2
I « + p . -.u,
'u.
m3/2A
U, U
ln\z"\
a.
P2t. I —Inz m \ a* / A** \ V Jo \a** This leads to an exponential decay towards R — 0 caused by an earlier deceleration at t = i , Xdz
u~
_x/2v / p 7
P* C3(Zttt,tt,pt)
U{£t,tt,pt]
9...
(5.21)
b"|-+l/4
where C3 is the finite parts of the integrals in the exponential factor of Eq. (5.12), and A=
1 m3/2A
U, U,
. , - - (
l
u, /u.
—
U. l a ,
1+? ait
This result for deceleration is not restricted to a small \U*/U*\, as long as U. U.
4
1+
U2
and a A which does not vanish elsewhere (due to acceleration at an earlier time). Whereas, the exponential decay in this case indicates that the midfield description may not breakdown near the transition boundary, there may still exist a neighhood of R = 0 where the governing equation for an exponentially small u must be modified from that of Eq. (5.10). The sensitivity of the u behavior to a weak deceleration for a small p2, is more clearly exhibited from an alternative expression of the factors depending on z" in Eq. (5.21), namely,
z"\-1/i™p(-filn\Z"\-\\\z»\-^
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
79
which reduces to the Hayes [20] result for negligibly small U / U and p2,, but becomes exponentially small in the range of z close enough to ztt where
> > Vz
(Hi) Effect of an earlier vehicle
acceleration
With U « / U * > 0, A can vanish before reaching the transition boundary (R = 0). Taking into consideration of the singular behavior of K , L and N noted in Eqs. (5.20), the determinant A becomes in the vicinity of z = z„* where R = 0 I — )1 a*
U, [\jt
m3/2Vz"
where S*» = J»* — CN — CKU\/«*»> II _
II
_
4~* // a* "*
f-2
\
I
I , , - ^ - ^ . .
a*
and can vanish at
u_
u~rt
t 2 —p* a.
(5.23) The expression Eq. (5.22) for A can be written alternately in terms of the variable z'" = ztt* — z which vanishes with A . Namely, 1 U
A
* T
1
Z a*
V^ZT
The functions « and x assume the forms, for small A and small R = mz" =
pi a, A \ V S + \R\3'2
m
U. 1 /
2R U« 1 J / U
I,
3/2
u, A [ V<w l-R| 2
1
2 /U, m a
v *»
v^
77
- v
u. i?
2p*
1
«M / a. V 2
1+
7
^
+ ™3/2
^ ( v / i 7 7 - xAST)
Carrying out the integration in a manner similarly as before, after making use of the definition of z" , leads to
I
a
Rdz ~
4
VmI«»
-\ln\z''\+-^—(Vz^+Jz^ln\Vz^-./zJZ\)-ln
/ v>i
CHENG & HAFEZ
80
T
A**
¥T
m U* a*
These yield a breakdown behavior of ti under the acceleration influence
/p7
C4(z*»,£*,p*)
(5.24)
where, to be sure z" = z"## + 2'" > z'"tt, and e» = y/rn
ul(qj{a*J
\/*t" + m*/>)
for a sufficiently small acceleration at t = i, with «"„ < < 1 [cf. Eq, (5.23)]. Accordingly, et is small like the square of ([/*/£/*). In the limit U —> 0, Eq. (5.24) recovers that in the steady case. In the range of z1' comparable to z"rt however, the acceleration effect takes over the u behavior, and gives u an entirely distinct algebraic behavior characterized by
i4U(i/4)+e-iv^-v/^r1+e*>>iNote that y/z~" - Jz1^ 5.3.2
~ {l/2)JzJ^-
z'" for \z'"\ «
0, and is therefore restricted. Its extension to allow for the elliptic domain can be verified by a derivation directly from the original second-order P D E using the variables x,y,z and $ under the same set of assumptions. The 2-D version of Eq. (5.29) is the nonlinear Tricomi equation which has been used as a model of the superboom problem and studied analytically by Hayes [20, 21] and Seebass [30], and also computationally by Seebass, Murman, Krupp [31]. Equation (5.29) above represents a version which accounts for 3-D aspects of the wave refraction. We may infer from Eq. (5.25) that the transverse length scale of the domain is small compared to the midfield scale A but is much larger however than the nearfield scale L. The weak solution of Eq. (5.27) admits shock jumps as in the transonic small-disturbance theory. In the hyperbolic region, the first-order P D E of the midfield for $ j reproduces itself from Eq. (5.27):
!■
(+2(ii'|+1"a)*«-^i"-*«=°
provided the transformation Eq. (5.28) is used with the same xlc (in reverse). The match with the midfield is thereby assured, but a more appropriate matching in the accepted sense of an asymptotic theory [11, 34] must be or large z, as shown below.
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM The far boundary
83
condition
With the behavior of xlc near z" = 0, and K or K being seen to behave as —p 1/2; the exponential function in u then becomes being proportional to |.z"| - 1 ' 4 - Hence the midfield approaches the transition boundary as z" —» 0 as
« ~ ( - ) \rlP/
U((,pt)Co(pt,z„)\z"\-^\ **
where U can be related directly to the F-function [4-6]. The magnitude of u thus approaches the level ( A / L ) 1 / 6 , befitting the descriptive term "superboom". This behavior has been numerically substantiated [5, 6]. In terms of the proper super/transition boom variables, this expression becomes u = ^
~ ( | )
(r'jj
(5)"1/4
( 7 + l ) t f ( f . , p » ) C 7 . ( p „ * o ) (fy
(5.30) and furnishes the required far field behavior for the P D E (5.29), assuring a good match with the midfield. The common domain of the two solutions to be matched must be 1 «
*«
{L/A)-2'3,
or
(L/A)2/3 «
\z"\ «
1.
On the ground plane, an impermeability condition is to be imposed. In compu tation, one is justified also to match it directly with ( L / A ) 1 ' 3 ( r ' A / L ) ( 7 + l)u provided it is done in the above z-range. Relation to Hayes' and Guiraud's
similitudes
Equation (5.29) with the far field condition (5.30), the condition on the ground plane, and the requirement of vanishing disturbances in the up- and downstream, constitute a system which contains two similarity parameters, namely,
where zgr identifies the distance of the ground plane from where B = 0. This system is equivalent to Hayes' [20] in which a similar parameter with a factor {L/R)1!6 appears in the reduced P D E instead in the far boundary condition, where R is a radius of curvature in the caustics geometry, undetermined a priori. The works following Hayes by Gill and Seebass [15], and Plotkin [28, 29] did not consider (or correctly account for) the ground influence which should have been an important aspect of the sonic-boom impact analysis. Whether there is a need for a re-orientation of the ground plane in the coordinate system
CHENG & HAFEZ
84
of Eq. (5.29) for the more general case involving vertical motion is also open to question. In Guiraud's [16] study, the factor (L/A)1'6 also appears in u, which cannot be strictly correct for vehicles undergoing acceleration (see below). The condition in Eq. (5.30) indicates that for a bounded (T'A/L) assumed here, $£ may be considered asymptotically small like (L/A)1'6, under which the leading approximation to Eq. (5.27) would become a 3-D version of the genuinely linear Tricomi equation [33]. The limit of a vanishing (L/A)1/6 was considered by Guiraud [16] in which the formalism of the resulting linear system allows the (L/A)1'6 to be completely scaled out. This formalism ignores, however, the role of the nonlinearity in a several distinct regions along the transition boundary z = 0, which in turn produces a reflected wave structure not contained in the simplified similitude, to be elucidated further later. However, such a treatment based on an asymptotically small ( L / A ) 1 / 6 may not be strictly necessary for the purpose of computation, since the order (L/A)1'6 terms may still be included along with the unit-order terms in the analysis, inasmuch as the remainder of the equation belongs to the higher order ( L / A ) 2 / 3 . Further discussion of the wavefield structure will be made in a broader context later at the end of Sec. 3.2.2. 5.3.2.2
Accelerated Superboom: Horizontal, Rectilinear
Motion
The more complicated algebraic behavior shown earlier in Section 3.1 for the time-dependent case calls for a departure from the standard procedure of matched asymptotic expansion. The following analysis applies to the problem involving only horizontal, rectilinear vehicle motions, for which the midfield breakdown behavior has been rather thoroughly analyzed in Section 3.1. Introducing the gauge
functions
As seen from the preceding subsection for the steady case, crucial is the identification of a transverse spatial scale which will elevate the Laplacian V j_y> from the remainder group of the midfield P D E to the rank of firstorder importance. The analysis will be concerned with small distances from the singular boundary at z = zttt which is identified with R = 0 or A = 0 as explained in Section 3.1 earlier. We shall employ the variables introduced earlier in Section 3.1
x"' = x - x „ » ,
z'" = zttt
- z,
y'" =
y-yt„
Note that, if the breakdown occurs at z = 2»*» < zttf, we have z" = | „ — z = z'" + (z*t - z»*,). As in Section 3.1, we permit A to vanish before R near the hyperbolic-elliptic transition boundary where R and z" are small, without restricting the relative magnitudes of z",z'" and z",„ = ztt — zt„. This will allow an unambiguous delineation of the transition from a steady-state
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
85
superboom to one dominated by acceleration. To accomplish the task similar to that in Section 3.2.1, a particular scale is needed which, however, are not apparent owing to the lack of simple algebraic expressions for the midfield behavior. To overcome this difficulty, we introduce two gauge functions: one is "J" which gauges the transverse dimensions of this zone and is required to vanish asymptotically with ( L / A ) , and the other is "g" which gauges dxlc /dz or ft of the time-dependent reference Mach conoid (and may vanish, bounded, or unbounded in the limit (L/A) -4 0). The behavior of dxlc/dz or ft in the vicinity of the focus (\z'"\ = \z — z***| < < 1) is (assumed being) known/established by analytical or accurate numerical means, as shown earlier in Section 3.1. This implies that q as a function of 5 will assume the same form as dxlc/dz as a function of fi{\z'"\). We may therefore take / i as q, writing
ft = ^ ! ~ < K | z " ' | ) ,
and q{\z'"\) = 0{q(S)}
(5.31)
More important is the assumption that = 0[Sq(S)]
(5.32)
This rule is obviously satisfied by the xlc behavior in most cases, which is algebraic in \z'"\- We note that q(z'") is not required to vanish identically at z"' = 0. According to Section 3.1, the breakdown behavior of it can be characterized by an algebraic function u ~ g*„v(z ) where gttt is independent of z'", therefore in the super/transition domain, the u must be rescaled as u = 0\g.„u(S)], (5.33) allowing the numerical factor g*„ to be much different from 1. The behavior of v(5) for the accelerated transition, for example, has been established as an algebraic function of z" and z"tlf J-T^c,
I/
= |z"|"
1/4
(5.34)
and can be converted to one in z'" = z" - (z»* - zt„) = z" - z"*«- Thus far, the magnitude of xlc has not been assured. Interestingly, if one were to conjecture that the magnitude of xlc is to be of the physically acceptable order, namely L/A, Eq. (5.33) may then establish the magnitude for the scale for z'" ,5, from 5q(S) = |
(5.35)
86
CHENG & HAFEZ
Since q ~ y/xm + z", the rule Eq. (5.31) applied to Eq. (5.35) in this case yields a cubic algebraic equation for 8, which can be conveniently expressed as
(£),+(£)'--«••'-©■ It gives k = z'l^/S
as an algebraic function of y, = (z" t „)(L/A) k-3+k-2
where £ and i are assumed to be unaffected by the rescaling, and G „ » = f0g***d£. The time-dependent midfield P D E Eq. (5.10) for the horizontal rectilinear motion may now be reduced to
A ^ J 5 l ( ^ + ^J^°fe'rA^'J where
1
^B) ^-
(5 38)
'
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
87
and the triple asterisks signify the condition at the transition location. It is now clear that, in the range of \z'"\ = 0(8) with S determined from Eq. (3.3)
S-q(5) = ~ the Laplacian on the RHS of Eq. (5.38) will emerge from the midfield remainder group to rank equally with terms on the left, while the stratification and unsteady corrections in the (time-dependent) midfield equation drop to the level of the remainders on the RHS of Eq. (5.38). The parametric form of the remainder in Eq. (5.38) indicated that the validity of Eq. (5.38) will require r L , 5 5 = —- < < 1, and - < < 1, qA q The second requirement in above results from the neglect of the timedependent term from the original midfield equation. More remarks on the scale choice The reasons for the assumptions made on the rescalings of y'" and t in the derivation of Eq. (5.38) require more scrutiny. Examination of ODE system Eq. (5.11) as well as the P D E Eq. (5.14) indicates that the y should share the same scale factor as for z, i.e., 5; indeed, the disturbances can not be prevented from spreading transversely once they are in the transonic wavefield, unless special type of boundary conditions were imposed to prevent such an occurrence. Without allowing same scale for y and z, the lateral extent of the superboom domain can never be correctly recovered. Next, a causal inspection of the ODE system could suggest that dt would have to be scaled and reduced by a factor S/q. But this suggestion is false because the dt along the bi-car refers merely to the signal traveling time (which will remain, in fact, even in the steady limit) and is therefore irrelevant. The relevant time scale is determined by the acceleration ratio Ut/Ut, which appears in the change rate of the acoustic source strength and is much longer than that raised by the factor 5/q. Therefore, the time scale is not change from that of the midfield. Reduction to Tricomi-type
boundary-value
problem
The P p E (5.38) in variables £, y and z, with the remainder group on RHS omitted, is but an extended version of the transonic smalldisturbance equation in disguise. Recast into Cartesian form via the inverse transformation x =( + xlc(y,z)
88
CHENG & HAFEZ
with y and z unchanged, as in Eq. (5.28) in the steady case, the second-order PDE for the superboom domain becomes d_ 1 dx
,
M - l „
(|r + |r)
^
with a{z)
q
Whereas parameters A and q are constant fixed by Eqs. (5.31,5.32), (5.33) and (5.35), once U and U are known at an earlier time it, the M in this PDE, unlike that in the transonic small-disturbance equation [11], must vary with z in a manner to be consistent with the scales and Mach conoid properties within the unit-order range of z. Here, the quotient (M2 — l)/q2 is recognizable as a variant of the transonic similarity parameter K, namely M2 - 1 ^ mz" _ z + k =
q2
~ !f~ ~ T+k
where k = z"„/8 has been shown to be an algebraic function of fi = £»*»(■£/A) -3 / 3 , according to Eq. (5.37). In the limit k -> 0, —Jf becomes 5 and the nonlinear, 3-D version of the Tricomi equation [20,21,30] is recovered. In the opposite (acceleration dominated) limit, —K approaches unity and Eq. (5.39) is found to be the standard transonic small-disturbance equations with a particular value of K = — 1 (in the supersonic range). The hyperbolic-elliptic transition boundary of the Tricomi equation (normally at z = 0) is seen to be displaced by the acceleration and nonlinearity to z = —k — A(^i(l + k) At z less than this value, Eq. (5.39) is to be analytically continued to the elliptic domain; its validity is verified by an analysis of the fuller smalldisturbance equation, employing the variables x,y,z and i. Shock wave properties deduced from the weak solution of Eq. (5.39) follow those in the transonic small-disturbance theory [11]. Boundary
conditions
The mixed, elliptic-hyperbolic problem will be solved as a quasi-steady for each t, with the far-boundary condition on the hyperbolic side y ~ u(5z)/u(8)
one
(5.40)
to be satisfied over Kj+\ - 2j + 4>j-i . 4>k+i - 2k + ^fc_i _ + Ay2 Az2 fi+1/2 - /i —1/2 Ax
€
'
fi-1/2 - fi-3/2 Ax
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM -min. (£;,£;_!, £i_ 2 )
fi-l/2
-min. (e,-,e;_i,ej_2,ej_ 3 )
— fi-3/2 Ax
fi-3/2 ~ fi-S/2 Ax
95
_ / ; —3/2 ~ /i—5/2 Ax fi-5/2 ~ fi-7/2 Ax
The left hand side is a central difference approximation; the first term on the right hand side leads to a first-order upwind scheme and the second- and third-terms are higher order corrections. Note that the second- and third-order corrections are not numerically conservative. (iv) Modifications of Murman's scheme Murman's scheme admits an expansion shock. A simple modification of the sonic point operator is useful in this regard. We propose the following /rt n —1
Wxx
, n t_n • — Kx and n is the level of iteration. A central difference approximation is used for the y-derivatives, while the ^-derivatives are approximated by upwind differences. Across the sonic line, 10, provided a downwind shift in the waveform is also made to account for the local sweep of the impact zone which cannot be rationally determined by existing 2-D models. Beyond y = 10, the nonlinear and 3-D effects are decisive in establishing the over pressure behavior in the "lateral cut-off zone." Importance of grid refinement; distinct refraction/interaction
features
While the comparisons made in Fig. 4a and in Fig. 6 for a fixed grid and a fixed computational (x, z) domain may suffice to indicate the adequacy of the modified upper boundary condition (which utilizes a shorter span), and of the 2-D hypothesis in the symmetry plane, they fall short in indicating correctly the limit P D E solution, as the mesh size (grid-point spacing) vanishes. A series of 2-D calculations can be used to indicate how the waveform near the
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
Figure 6
99
Comparison of 2-D and 3-D pressure distributions
(linearized) transition boundary at z = 0 and on the ground (z = min z) may change, as the grid is made finer and finer; this process indeed brings out features of relevance to the superboom phenomena. Here in Figs. 7a,b, we repeat the 2-D results of Fig. 6 for three sets of grid with mesh size Ax being reduced successively by decreasing the z-range, without changing the numbers of grid points in x- and z-ranges. The latter is fixed at (800 x 201). Figure 7a compares the coarse-grid results (with —0.1 < x < 25) to those of fine grid (with —0.1 < x < 16), and Fig. 7b compares the fine-grid results to the those of a finer grid (with —0.1 < x < 9). Several features which have been the familiar characteristics of the superboom waveform, particularly the overpressure peak at the front and the overshoot behind the real shock,
CHENG & HAFEZ
Comparison of Cp Distribuflons
I
0.0
5.0
IOD
x
Figure 7 Effect of grid resolution on 2-D pressure distributions
sharpen progressively as the mesh size reduces. The overpressure peak is now seen to be more than four times than that at the upper boundary. In terms of physical variables, the boom intensity in this example reaches 4.4-5.0 pound per square (psf) foot at its maxima. Solutions to the subcritical cases corresponding to a flight Mach number below the threshold value have also be obtained and reveal a more pronounced tendency towards the U-shape over-pressure profile (the inverted U-shape for the waveform in #,), with a lesser intensity and smoother distribution, however, on the ground. The pressure overshoots, resembling rabbit's ears raise from both ends of the reflected waveform are believed a feature distinctly associated with the presence of the hyperbolic-elliptic transition boundary, and the interaction of the (nearly) linear hyperbolic wavefield with the nonlinear, mixed layerlregion straddling the transition boundary (of the linear theory).
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM Distinct wavefield structure dominated by a linear Tricomi
101
domain
A unique concept on the superboom waveform structure brought out by this study is the existence of two distinctly different, neighboring wavefields: a large spatial region governed by the linear Tricomi equation and a less extensive region where the nonlinearity becomes crucial to the transition from the hyperbolic to elliptic behavior. This concept can be inferred from the discussions at the end of Section 3.2.1 and the remark following Eq. (5.38) for a relative small Su/q. This is substantiated by the computational study in details. For this purpose, we examine first the relative importance of the nonlinear term 4>x(f>Xx as compared to the Tricomi term K'(0)z ■ cf>XI in the governing equation; their ratio is x/K'(Q)z and is shown in Fig. 8a as a function of z for three x's. The unimportance of nonlinearity over an extensive z-range of the Tricomi domain, 1 < z < 5.925, is evident from the exceedingly small value of the ratio shown for the 3-D calculation at the span station y = 4.5. Similar behavior is found at other span stations and for the corresponding 2-D solution. It is essential to note that the upper boundary location at i w 5.925 has been chosen by an estimate of the location where the midfield analysis breaks down in accordance with the theory, which is independent of the nonlinearity. The significance of this concept is brought out more directly in Fig. 8b for the 2-D case, where the noticeable difference between the linear and nonlinear solutions can be found only in the less extensive range, z < 1. The result indicates the feasibility for developing a more effective computational procedures as well as field-structure study based on asymptotic methods. The very fine calculations shown in Figs. 6a,b and Figs. 7a,b have indicated, on the other hand, the distinct difference between the linear and nonlinear results in regions near the reflected/refracted oblique shocks which are unaccountable by a purely linear version of the Tricomi equation. Shock-fitting
problem
Even though grid refinement has proved helpful in bringing out those distinct features in the superboom domain, it does not work well on the more oblique part of the shocks removed from the transition boundary. This is due to the inherent inadequacy of the shock-point operator used. The basic problem has been overcome by adopting the shock-fitting algorithm of Hafez and Cheng [18], which proves to be difficult to implement, however, wherever multiple shocks occur and become too close together, as in the case of shock reflecting from a wall. It also becomes ineffective, if the shock is too weak to be distinguished from a compression wave. A procedure is being developed, in which the program will switch from a shock-fitting to a shock-point operator scheme when the shock discontinuity becomes too weak, or when the wall
Plot of mllo.vs. Z at y = 4.5
Comparfson of Cp DlsbibuUots
lo00 X 81 X 1 0 1 . rcmdadn.ltI--OZaI)
(LWX2DlZDl.raordadn.r(r)--02~)
0.0
a
-
-1-8.812284 8
4.0
-
2.0
-
6.WW
-.-I 7.819774 r
I
- 4 0.0
--.
-5-c-30
-4-1.0
1.0
5.0
8.0
-0
ww-O.z~W
5.0
10.0
X
Figure 8 Importance of nonlinearity
vicinity is reached.
5.4
Closing Remarks
We have discussed the principal features in the recent work on sonic boom propagation which underlies our study on the super/transition boom phenomena. The analysis employs a Galilean frame moving with the aircraft and uses the Mach conoid of the linear theory as a coordinate reference (allowing time dependency). The theory delineates explicitly the singular midfield behavior near the transition boundary and the crucial influence of the horizontal vehicle acceleration and deceleration on the wavefield behavior, not specifically contemplated heretofore. A matching strategy has been devised to allow proper scaling and reduction/recovery of a modified Tricomi-type equation in the embedded nonlinear 3-D domain. The reduced mixed, elliptic-hyperbolic problem lends itself to numerical solutions by relaxation methods well developed for the transonic smalldisturbance equation. Apart from the 3-D wavefield description, with which the ground impact area may now be identified, the solution allows for
ANALYSIS AND SIMULATION OF THE SUPERBOOM PROBLEM
103
the important ground and sonic-boundary interaction near/at the threshold condition, and satisfies shock jumps correctly (via shock capturing/fitting algorithm), not considered in the existing (focal-zone) model. The details of a computational analysis for a flight Mach number near the threshold value have revealed several anticipated as well as unsuspected features, including the substantiation for the adequacy of the 2-D hypothesis, an assessment of its accuracy in the symmetry plane, and the existence of a distinctly linear, hyperbolic domain, together with a few isolated, relatively small, nonlinear mixed-type zones. The principal conclusion of practical consequence from the study is that the influence of vehicle acceleration on the superboom structure and similitude is significant. Nevertheless, provided the scalings and the far-boundary condition are chosen in accordance with the theory, the same numerical algorithm and procedure developed for the steady case are expected to be equally applicable. The need of analyses for problems involving climb and vertical acceleration is obvious for space launch applications, which may alter the governing equation and boundary conditions for the modified Tricomi equation. Much of the results presented have been the fruit of collaboration with C.J. Lee, C.Y. Tang and W.H. Guo, to whom we wish to express our most grateful appreciation. We would like to thank M. Downing, J. Edwards, G. Haglund, H. Lam, K. Plotkin and R. Seebass for many stimulating discussions and helpful advice. The support from DoD SBIR program monitored by AF Armstrong Laboratory OEBN Noise Effect Branch during 03/19/96-09/18/96 is acknowledged.
REFERENCES 1. Ashley, H. and Landahl, M., (1965) Aerodynamics of Wings and Bodies, AddisonWesley. 2. Carlson, H.W. and Maglieri, D.J., (1972) "Review of Sonic-Boom Generation Theory and Prediction Methods," J. Acoustical Soc. America, vol. 51, No. 2 (pt. 3), pp. 675-685. 3. Cheng, H.K. and Hafez, M.M., (1975), "Transonic Equivalence Rule: A Nonlinear Problem Involving Lift," J. Fluid Mech., vol. 72, pt. 1, pp. 161-187. 4. Cheng, H.K., (1996) "Supersonic Wave-Field Analyses in Support of HSCT SonicBoom Impact Studies," Proc. 1995 NASA High-Speed Res. Program: Sonic Boom Workshop, at NASA Langley Res. Center, Sept. 12-13, 1995, vol. 1, pp. 136-150. 5. Cheng, H.K. Hafez, M.M. Lee, C.J. and Tang, C.Y., (1996) "Assessment of an Improved Theory for Further Program Development of Sonic-Boom Prediction and Submarine Impact Study," Final Report Submitted to AF Armstrong Laboratories, OEBN Noise Effects Branch, HKC Research Nov. 28 1996. 6. Cheng, H.K. and Lee, C.J., (1996) "Problems in Sonic Boom Theory Studied as Issues of ,Singular Perturbation," AIAA paper 96-2165.
104
CHENG & HAFEZ
7. Cheng, H.K., Lee, C.J., Hafez, M.M. and Guo, W.H., (1995) "The C F D Problem of Sonic-Boom Propagation in a Stratified Atmosphere," First Asian CFD Conference, Hong Kong Univ. of Science and Technology, Hong Kong, J a n . 16-19. 8. Cheng, H.K., Lee, C.J., Hafez, M.M., and Guo, H., (1996) "Sonic Boom Propagation and Its Submarine Impact: A study of Theoretical and Computational Issues," AIAA paper 96-0755 (1996). 9. Cheng, H.K., Lee, C.J., Hafez, M.M. and Tang, C.Y., (1996) "Focused Sonic Boom in a Nonisothermal Atmosphere as a Mixed Hyperbolic-Elliptic, Nonlinear, 3-D Problem:," Proc. Second Asian CFD Conf., Tokyo, Dec. 1996, vol. 2, p p . 337-342. 10. Cheng, H.K. and Meng, S.Y., (1980) "The Oblique Wing as a Lifting-Line Problem in Transonic Flow," J. Fluid Mech., vol. 97, p t . 3, p p . 531-556; also see Cheng, H.K., (1982), "Computational Study of Asymptotic Flow Structure of A High Aspect Ratio Wing in Transonic Flow," Transonic Shock and Multidimensional Flows, (ed. R.E. Meyer), Acad. Press, p p . 107-146. 11. Cole, J . D . , (1968) Perturbation Methods in Applied Mathematics, Blaisdell P u b . Company. 12. Courant, R. and Hilbert, D., (1965) "Method of Mathematical Physics," vol. II, Partial Differential Equations (by R. Courant), Interscience P u b . p p . 75-88, 173-174. 13. Darden, C M . , (1984) "An Analysis of shock Coalescence Including ThreeDimensional Effects with Application to Sonic Boom Extrapolation," NASA TP2214. 14. Darden, C M . , (1988) (ed.) "Status of Sonic Boom Methodology and Understanding," NASA Conf. Pub. 3027. 15. Gill, P.M. and Seebass, A.R., (1975) "Nonlinear Acoustic Behavior At a Caustic: An Approximation Solution," AIAA Progress in Astr. and Aero., ed. H.T. Nagamatsu, M I T Press. 16. Guiraud, J.P., (1965) "Acoustic geometrique, bruit ballistique des avions supersoniques et focalisation," J. de Mecanique, vol. 4, p p . 215-267, 17. Haering, E.A., Ehernbergen, L.J. and Whitmore, S.A., (1995) "Preliminary Airborne Measurement for the S R - 7 1 Sonic Boom Propagation Experiment," NASA Tech. Memo. 104307. 18. Hafez, M.M. and Cheng, H.K., (1977) "Shock-Fitting Applied to Relaxation Solutions of Transonic Small Disturbance Equations," AIAA J., vol. 15, no. 6, pp. 786-793. 19. Hayes, W . D . , (1968) "Geometric Acoustics and Wave Theory," Proc. 2nd Conf. on Sonic Boom Research (ed. I. R. Schwartz) NASA SP-180, p p . 158-. 20. Hayes, W.D., (1969) "Similarity Rules for Nonlinear Acoustic Propagation Through a Caustic," 2nd Conf. Sonic Boom Research, NASA SP-180, p p . 165— 171. 21. Hayes, W.D., (1971) "Sonic Boom," Annual Rev. Fluid Mech., vol. 3, p p . 269-290. 22. Hayes, W . D., Haefeli, R. C , and Kulsrud, H. E., (1969) "Sonic Boom Propagation in a Stratified Atmosphere with Computer Program," NASA CR1299. 23. Jameson, A., (1975) "Transonic Potential Flow Calculation, Using Conservation Form," Proc. 2nd AIAA CFD Conf, Hartford, J u n e 1975, p p . 148-161.
ANALYSIS A N D SIMULATION O F T H E S U P E R B O O M P R O B L E M
105
24. M u r m a n , E.M., and Cole, J.D., (1971) "Calculation of Plane Transonic Flow," AIAA J., vol. 9, p p . 114-121. 25. M u r m a n , E.M., (1974) "Analysis of Embedded Shock Waves Calculated by Relaxation Method," AIAA J., vol. 12, p p . 626-632. 26. Hafez, M.M. and Cheng, H.K., (1977) "Convergence Acceleration of Relaxation Solution for Transonic Computations," AIAA J., vol. 15, no. 3, p p . 329-336. 27. Onyeonwu, R.C., (1973) "A Numerical Study of the Effects of Aircraft Maneuvers of t h e Focussing of Sonic Boom," Institute for Aerospace Studies, University of Toronto, U T I A S Report No. 192. 28. Plotkin, K.J., (1985) "Evaluation of a Sonic Boom Focal Zone Prediction Model," Wyle Research Lab. Report W R 84-43, Feb. 1985. 29. Plotkin, K . J . , Downing, M. and Page, J.A., (1994) "USAF Single-Event Sonic Boom Prediction Model: P C Boom 3," High-Speed Research NASA Sonic Boom Workshop, NASA Conf. Pub. 3279, p p . 171-184. 30. Seebass, R., (1970) "Nonlinear Acoustic Behavior at a Caustic," Proc. 3rd. Conf. Sonic Boom Research, NASA SP-255, p p . 87-120. 31. Seebass, R., M u r m a n , E.M. and Krupp, J.A., (1970) "Finite Difference Calculation of the Behavior of a Discontinuous Signal Near a Caustics," Proc. 3rd Conf. Sonic Boom Research, Oct. p p . 29-30, 1970 (ed. L B . Schwartz), NASA SP-255, p p . 361-371. 32. Thomas, C.L., (1972) "Extrapolation of Sonic Boom Pressure Signatures by t h e Waveform Parameter Method," NASA TND-6832. 33. Tricomi, F-G., (1957) "Equazioni a Derivate Parziali," Cremonese, Rome; see also Ferrari, C. and Tricomi, F-G., Transonic Aerodynamics, (Transl. R.H. Cramer) Academic Press, p p . 105-117 (1968). 34. Van Dyke, M.D., (1975) Perturbation Methods in Fluid Mechanics, Acad. Press (1964); Parabolic Press, annotated edition, pp. 106-120. 35. Haglund, George T., private communication (1996). 36. Downing, M. and Zamot, (1995) "USAF Flight Test Investigation of Focused Sonic Booms," J. Acoustics Soc. Am., vol. 97, no. 5, pt. 2, Proc. 129th Meetings Acoustics Soc. Am., May 1995.
6
Complex Analysis of Transonic Flow Connie K. Chen 1 k Paul R. Garabedian 1
6.1
Introduction
It is almost thirty years since the method of complex characteristics led to the design of shockless airfoils that looked like practical wing sections [1,10]. At about the same time Murman and Cole invented their simple, efficient upwind difference scheme to calculate transonic flows [9]. Soon computational fluid dynamics became the favorite tool of supercritical wing technology [2]. However, it was Murman's analysis concept that really worked, while only a few specialists were able to extract from the design method examples as successful as the Korn airfoil. The goal of this paper will be to describe essential features of the new FLOW code, which provides a more robust implementation of the method of complex characteristics. First we shall unravel some of the mysteries of the domain of two complex variables and show how the numerical calculations are organized. Then examples will be given to illustrate the kind of progress that has been made recently. Perhaps the most important contribution to be discussed is a filter that exploits analyticity of solutions of the flow equations to define more accurate profile coordinates.
1
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 Frontiers of Computational Fluid Dynamics 1998 Editors: David A. Caughey & Mohamed M. Hafez ©1998 World Scientific
CHEN & GARABEDIAN
108 6.2
The hodograph transformation
Since we are concerned with shockless flows, and since the change in entropy when the fluid passes through a shock front is of the third order in the shock strength, we assume that the entropy is conserved, which means that the flow is isentropic. Hence a velocity potential 1. Our goal will be to obtain shockless airfoils on which the speed distribution is partially prescribed. To achieve this we need to solve for the transonic flow field of the airfoil t o be designed, so we use t h e hodograph method to transform (6.1) into a linear system. To facilitate the computation, we introduce a conformal mapping of the flow region onto an ellipse in the complex characteristic plane. Then we calculate a complete set of special solutions on a designated region to represent
(£, f?i) and 4>((i, rf) are given along two complex Mach lines rj = r]i and £ = £i- An efficient finite difference scheme can be applied to construct a complete system of solutions in the Cartesian product of an ellipse of the £-plane and its reflected image with £ = fj in the r\plane when the initial data are defined as functions of ( and rj by Chebyshev polynomials [4]. With an appropriate constant of integration (6.5) implies that M2 = 1 when a + (5 = 0. This defines a two-dimensional sonic locus in the fourdimensional complex domain C 2 . Its intersection with the real physical plane corresponds to the sonic line. The method of complex characteristics breaks down on the sonic locus because of a singularity in the transformation (6.5) to characteristic coordinates. The difference scheme we use to obtain numerical solutions becomes singular at points where M2 = 1 and is ill conditioned at nearby points. Therefore all computational grids must be constructed so as to avoid the sonic locus.
6.3
Singularities
We introduce the fundamental solution of second order partial differential equations in general to analyze the specific behavior of the singularities of transonic flow representing sources, sinks and dipoles in the hodograph plane.
CHEN & GARABEDIAN
110
Let us consider linear elliptic differential operators of the form ![*] = j A * - A*,
(6.6)
where A is a real analytic function of x and y. A fundamental solution of L[F] — 0 is defined to be a solution of the form F = F(x,y;x,y)
= R{x,y;x,y)\og
- + Q(x,y;x,y), (6.7) r where r = yj(x — x)2 + (y — y)2, where x, y are parameters, and where R and Q are regular in some neighborhood of the parameter point (x,y). The coefficient R will turn out to be the Riemann function with R(x, y; x, y) = 1. Let us perform an analytic continuation of F to complex values of the variables x, y, x and y and let us introduce the complex substitution £ = x + iy, T] = x -iy,
| = x + iy, fj = x -iy.
(6.8)
The complex variables £ and 77 are actually characteristic coordinates for (6.6). One easily verifies that ! [ * ] = 9(n - A * = 0
(6.9)
in terms of these characteristic coordinates. We now use a multiplicative substitution \1/ = cr{q)ip to simplify the flow equation for the stream function tf> in the hodograph plane so that it acquires the canonical form (6.9). If we insert the fundamental solution and note that r 2 = (£ — £)(?7 —fj), we obtain the identity
w^-^-w^+w-0-
(6 10)
-
The first three terms in (6.10) are singular. They do not cancel each other easily because the logarithmic function is multiple-valued, but the poles are not. In order to satisfy (6.10) we must have both L[R] = 0 and
Riv) = a,
Rviln,Lv)
= 0-
(6.11)
But these are precisely the properties of the Riemann function. We can solve for R by finite differences or successive approximations. When we do so, the sum of the three singularities in (6.10) becomes regular and the remaining term Q can be calculated, too. In the previous section, we introduced the map function / between the hodograph plane and an ellipse in the £-plane. We need to choose points inside the ellipse to represent the velocity at infinity for airfoils or to represent inlet and outlet flow for a cascade of blades. These points are singularities specifying a dipole for an airfoil or a source and a sink for a cascade of blades. It is clear that such flows can be calculated analytically in terms of the fundamental solution.
COMPLEX ANALYSIS OF TRANSONIC FLOW
6.4
111
Nonlinear b o u n d a r y value problem
We use an inverse method to design shockless airfoils. This involves a free boundary problem in which the profile of the airfoil will be determined as the streamline $ = 0, but the speed distribution q = q(s) is to be prescribed along the boundary, where s is the arc length. When the velocity is high enough, the flow will become transonic and the partial differential equations will be of mixed type. We determine the velocity potential tp along the profile, and the circulation T, by the rules