Modern Aerodynamic Methods for Direct and Inverse Applications

439 Pages·2019·6.98 MB·English

by  Wilson C Chin

Modern Aerodynamic Methods
for Direct and Inverse
Applications
Scrivener Publishing
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Beverly, MA 01915
Publishers at Scrivener
Martin Scrivener ([email protected])
Phillip Carmical ([email protected])
Modern Aerodynamic Methods
for Direct and Inverse
Applications
Wilson C. Chin
This edition first published 2019 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA
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Library of Congress Cataloging-in-Publication Data
ISBN 978-1-119-58056-0
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Cover design by Kris Hackerott
Set in size of 11pt and Times New Roman PSMT
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1. Basic Concepts, Challenges and Methods . . . . . . . . . 1
1.1 Governing Equations - An Unconventional Synopsis . . . . 1
1.2 Fundamental “Analysis” or “Forward Modeling”
Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Basic “Inverse” or “Indirect Modeling” Ideas . . . . . . . . 15
1.4 Literature Overview and Modeling Issues . . . . . . . . . . . 20
1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2. Computational Methods: Subtleties, Approaches and
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Coding Suggestions and Baseline Solutions . . . . . . . . 33
2.1.1 Presentation Approach . . . . . . . . . . . . . . . . . . 33
2.1.2 Programming Exercises . . . . . . . . . . . . . . . . . 35
2.1.3 Model Extensions and Challenges . . . . . . . . . . . 36
2.2 Finite Difference Methods for Simple Planar Flows . . . 39
2.2.1 Finite Differences - Basic Concepts . . . . . . . . . . 39
2.2.2 Formulating Steady Flow Problems. . . . . . . . . . . 45
2.2.3 Steady Flow Problems . . . . . . . . . . . . . . . . . . 46
2.2.4 Wells and Internal Boundaries. . . . . . . . . . . . . . 55
2.2.5 Point Relaxation Methods . . . . . . . . . . . . . . . . 62
2.2.6 Observations on Relaxation Methods. . . . . . . . . . 64
v
2.3 Examples - Analysis, Direct or Forward Applications . . 75
2.3.1 Example 1 - Thickness Solution, Centered Slit
in Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.3.2 Example 2 - Half-Space Thickness Solution . . . . . 91
2.3.3 Example 3 - Centered Symmetric Wedge Flow. . . . 98
2.3.4 Example 4 - General Solution with Lift,
Centered Slit . . . . . . . . . . . . . . . . . . . . . . . . 101
2.3.5 Example 5 - Transonic Supercritical Airfoil with
Type-Dependent Differencing Solution, Subsonic,
Mixed Flow and Supersonic Calculations . . . . . . . 119
2.3.6 Example 6 - Three-Dimensional, Thickness Only,
Finite, Half-Space Solution. . . . . . . . . . . . . . . . 129
2.4 Examples - Inverse or Indirect Applications . . . . . . . . 138
2.4.1 Example 1 - Constant Pressure Specification and
Symmetric Thin Ellipse . . . . . . . . . . . . . . . . . . 138
2.4.2 Example 2 - Inverse Problem, Pressure Specification,
Centered Sit, Trailing Edge Closed vs Opened . . . . 145
2.4.3 Example 3 - Inverse Problem, Pressure Specification,
Three-Dimensional Half-Space, Closed Trailing Edge,
Nonlifting Symmetric Section . . . . . . . . . . . . . . 158
3. Advanced Physical Models and Mathematical
Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.1 Nonlinear Formulation for Low-Frequency Transonic
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 170
3.1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.1.3 Discussion and Summary . . . . . . . . . . . . . . . . . 174
3.1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.2 Effect of Frequency in Unsteady Transonic Flow . . . . . 176
3.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 176
3.2.2 Numerical Procedure . . . . . . . . . . . . . . . . . . . 177
3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 180
3.2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 181
vi
3.3 Harmonic Analysis of Unsteady Transonic Flow . . . . . 182
3.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 182
3.3.2 Analytical and Numerical Approach . . . . . . . . . . 183
3.3.3 Calculated Results . . . . . . . . . . . . . . . . . . . . . 184
3.3.4 Discussion and Closing Remarks . . . . . . . . . . . . 185
3.3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.4 Supersonic Wave Drag for Nonplanar Singularity
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 189
3.4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.4.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3.4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.5 Supersonic Wave Drag for Planar Singularity
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 195
3.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.5.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 206
3.5.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.6 Pseudo-Transonic Equation with a Diffusion Term . . . 208
3.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 209
3.6.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 212
3.6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 212
3.7 Numerical Solution for Viscous Transonic Flow . . . . . 213
3.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 213
3.7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
3.7.3 Numerical Approach . . . . . . . . . . . . . . . . . . . 216
3.7.4 Sample Calculation . . . . . . . . . . . . . . . . . . . . 217
3.7.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . 218
3.7.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.8 Type-Independent Solutions for Mixed Subsonic and
Supersonic Compressible Flow . . . . . . . . . . . . . . . . 221
3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 221
3.8.2 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . 221
3.8.3 Numerical Approaches . . . . . . . . . . . . . . . . . . 223
vii
3.8.3.1 Horizontal Line Relaxation. . . . . . . . . . . 223
3.8.3.2 Vertical Column Relaxation . . . . . . . . . . 224
3.8.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.8.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.9 Algorithm for Inviscid Compressible Flow Using the Viscous
Transonic Equation . . . . . . . . . . . . . . . . . . . . . . . 228
3.9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 228
3.9.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
3.9.3 Sample Calculations. . . . . . . . . . . . . . . . . . . . 231
3.9.4 Summary and Conclusions . . . . . . . . . . . . . . . . 232
3.9.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 233
3.10 Inviscid Parallel Flow Stability with Nonlinear Mean
Profile Distortion . . . . . . . . . . . . . . . . . . . . . . . . . 234
3.10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 235
3.10.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.10.3 Discussion and Conclusion. . . . . . . . . . . . . . . 239
3.10.4 References . . . . . . . . . . . . . . . . . . . . . . . . 240
3.11 Aerodynamic Stability of Inviscid Shear Flow Over
Flexible Membranes . . . . . . . . . . . . . . . . . . . . . . . 242
3.11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 242
3.11.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.11.3 Specific Examples . . . . . . . . . . . . . . . . . . . . 245
3.11.4 Discussion and Concluding Remarks. . . . . . . . . 247
3.11.5 References . . . . . . . . . . . . . . . . . . . . . . . . 248
3.12 Goethert’s Rule with an Improved Boundary
Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
3.12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 249
3.12.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 250
3.12.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.12.4 References . . . . . . . . . . . . . . . . . . . . . . . . 253
3.13 Some Singular Aspects of Three-Dimensional Transonic
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
3.13.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 254
3.13.2 Discussion and Summary. . . . . . . . . . . . . . . . 257
3.13.3 References . . . . . . . . . . . . . . . . . . . . . . . . 259
viii
4. General Analysis and Inverse Methods for Aerodynamic
Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.1 On the Design of Thin Subsonic Airfoils . . . . . . . . . . 264
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 264
4.1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.1.3 First-Order Problem . . . . . . . . . . . . . . . . . . . . 266
4.1.4 Second-Order Problem . . . . . . . . . . . . . . . . . . 269
4.1.5 Discussion and Conclusion. . . . . . . . . . . . . . . . 271
4.1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 273
4.2 Airfoil Design in Subcritical and Supercritical
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
4.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 274
4.2.2 Streamfunction Formulation . . . . . . . . . . . . . . . 278
4.2.3 Numerical Procedure . . . . . . . . . . . . . . . . . . . 281
4.2.4 Calculated Results .. . . . . . . . . . . . . . . . . . . . 284
4.2.5 Discussion and Closing Remarks . . . . . . . . . . . . 285
4.2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 290
4.3 Direct Approach to Aerodynamic Inverse
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
4.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 292
4.3.2 Theory and Examples . . . . . . . . . . . . . . . . . . . 295
4.3.2.1 Constant Density Planar Flows . . . . . . . . 295
4.3.2.2 Constant Density Flows Past Three-Dimensional
Finite Wings . . . . . . . . . . . . . . . . . . . 299
4.3.2.3 Compressible Flows Past Finite Wings . . . 301
4.3.2.4 Flows in Fans and Cascades . . . . . . . . . . 302
4.3.2.5 Axisymmetric Compressible Flows . . . . . 303
4.3.3 Sample Calculations . . . . . . . . . . . . . . . . . . . . 304
4.3.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . 307
4.3.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . 310
4.4 Superpotential Solution for Jet Engine External Potential
and Internal Rotational Flow Interaction . . . . . . . . . 312
4.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 313
4.4.2 Rotational Flow Equations . . . . . . . . . . . . . . . . 314
4.4.3 The Linearized Problem . . . . . . . . . . . . . . . . . 316
ix
4.4.4 Application to Jet-Engine External Potential and Internal
Rotational Flow Interaction. . . . . . . . . . . . . . . . 318
4.4.5 Calculated Results and Closing Discussion . . . . . . 321
4.4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 325
4.5 Thin Airfoil Theory for Planar Inviscid Shear
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
4.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 327
4.5.2 Planar Flows With Constant Vorticity . . . . . . . . . 330
4.5.2.1 Planar Flows: Inverse Problems . . . . . . . . 330
4.5.2.2 Planar Flows: Direct Formulations . . . . . . 331
4.5.2.3 Some Planar Analytical Solutions. . . . . . . 332
4.5.2.4 Analogy To Ringwing Potential Flows. . . . 333
4.5.2.5 Source and Vortex Interactions for
Ringwings. . . . . . . . . . . . . . . . . . . . . 334
4.5.3 Airfoils in General Parallel Shear Flow . . . . . . . . 335
4.5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . 339
4.5.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . 341
4.5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . 343
4.5.7 Appendix I, Three-Dimensional Constant Density
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
4.5.8 Appendix II, Planar Compressible Shear Flow
of a Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
4.6 Class of Shock-free Airfoils Producing the Same Surface
Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
4.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 348
4.6.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
4.6.3 Discussion and Conclusion. . . . . . . . . . . . . . . . 351
4.6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 353
4.7 Engine Power Simulation for Transonic Flow-Through
Nacelles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
4.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 355
4.7.2 Analytical and Numerical Approach . . . . . . . . . . 356
4.7.3 Numerical Results and Closing Remarks . . . . . . . 357
4.7.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 360
x