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Submit a ProposalModern Aerodynamic Methods for Direct and Inverse Applications
 | By Wilson C. Chin Copyright: 2019 | Status: Published ISBN: 9781119580560 | Hardcover | 450 pages Price: $225 USD  |
One Line Description
A powerful new monograph from an aerodynamicist reviewing modern conventional aerodynamic approaches, this volume covers aspects of subsonic, transonic and supersonic flow, inverse problems, shear flow analysis, jet engine power addition, engine and airframe integration, and other areas, providing readers with the tools needed to evaluate their own ideas and to implement the newer methods suggested in this book.
Audience
Aerospace engineers and aerospace engineering students, mechanical engineers, industrial engineers
Aerospace engineers and aerospace engineering students, mechanical engineers, industrial engineers
Description
Just when classic subject areas seem understood, the author, a Caltech, M.I.T. and Boeing trained aerodynamicist, raises profound questions over traditional formulations. Can shear flows be rigorously modeled using simpler “potential-like” methods versus Euler equation approaches? Why not solve aerodynamic inverse problems using rapid, direct or forward methods similar to those used to calculate pressures over specified airfoils? Can transonic supercritical flows be solved rigorously without type-differencing methods? How do oscillations affect transonic mean flows, which in turn influence oscillatory effects? Or how do hydrodynamic disturbances stabilize or destabilize mean shear flows? Is there an exact approach to calculating wave drag for modern supersonic aircraft?
This new book, by a prolific fluid-dynamicist and mathematician who has published more than twenty research monographs, represents not just another contribution to aerodynamics, but a book that raises serious questions about traditionally accepted approaches and formulations – and provides new methods that solve longstanding problems of importance to the industry. While both conventional and newer ideas are discussed, the presentations are readable and geared to advanced undergraduates with exposure to elementary differential equations and introductory aerodynamics principles. Readers are introduced to fundamental algorithms (with Fortran source code) for basic applications, such as subsonic lifting airfoils, transonic supercritical flows utilizing mixed differencing, models for inviscid shear flow aerodynamics, and so on – models they can extend to include newer effects developed in the second half of the book. Many of the newer methods have appeared over the years in various journals and are now presented with deeper perspective and integration.
This book helps readers approach the literature more critically. Rather than simply understanding an approach, for instance, the powerful “type differencing” behind transonic analysis, or the rationale behind “conservative” formulations, or the use of Euler equation methods for shear flow analysis when they are unnecessary, the author guides and motivates the user to ask why and why not and what if. And often, more powerful methods can be developed using no more than simple mathematical manipulations. For example, Cauchy-Riemann conditions, which are powerful tools in subsonic airfoil theory, can be readily extended to handle compressible flows with shocks, rotational flows, and even three-dimensional wing flowfields, in a variety of applications, to produce powerful formulations that address very difficult problems. This breakthrough volume is certainly a “must have” on every engineer’s bookshelf.
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Just when classic subject areas seem understood, the author, a Caltech, M.I.T. and Boeing trained aerodynamicist, raises profound questions over traditional formulations. Can shear flows be rigorously modeled using simpler “potential-like” methods versus Euler equation approaches? Why not solve aerodynamic inverse problems using rapid, direct or forward methods similar to those used to calculate pressures over specified airfoils? Can transonic supercritical flows be solved rigorously without type-differencing methods? How do oscillations affect transonic mean flows, which in turn influence oscillatory effects? Or how do hydrodynamic disturbances stabilize or destabilize mean shear flows? Is there an exact approach to calculating wave drag for modern supersonic aircraft?
This new book, by a prolific fluid-dynamicist and mathematician who has published more than twenty research monographs, represents not just another contribution to aerodynamics, but a book that raises serious questions about traditionally accepted approaches and formulations – and provides new methods that solve longstanding problems of importance to the industry. While both conventional and newer ideas are discussed, the presentations are readable and geared to advanced undergraduates with exposure to elementary differential equations and introductory aerodynamics principles. Readers are introduced to fundamental algorithms (with Fortran source code) for basic applications, such as subsonic lifting airfoils, transonic supercritical flows utilizing mixed differencing, models for inviscid shear flow aerodynamics, and so on – models they can extend to include newer effects developed in the second half of the book. Many of the newer methods have appeared over the years in various journals and are now presented with deeper perspective and integration.
This book helps readers approach the literature more critically. Rather than simply understanding an approach, for instance, the powerful “type differencing” behind transonic analysis, or the rationale behind “conservative” formulations, or the use of Euler equation methods for shear flow analysis when they are unnecessary, the author guides and motivates the user to ask why and why not and what if. And often, more powerful methods can be developed using no more than simple mathematical manipulations. For example, Cauchy-Riemann conditions, which are powerful tools in subsonic airfoil theory, can be readily extended to handle compressible flows with shocks, rotational flows, and even three-dimensional wing flowfields, in a variety of applications, to produce powerful formulations that address very difficult problems. This breakthrough volume is certainly a “must have” on every engineer’s bookshelf.
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Supplementary Data
• Is an informative critical monograph on modern aerodynamic issues drawing on rigorous mathematics and analysis
• Explains strengths and weaknesses of conventional methods and provides credible alternative methods
• Develops new approaches to aerodynamic inverse and shear flow analysis problems, written by the inventor of Boeing’s supersonic drag analysis models and United Technology’s jet mixer, actuator disc, and engine and airframe integration methods
• Addresses important questions, such as: How do transonic oscillations affect the mean flow? How are strong shear flows modeled using simpler superpotentials without Euler’s equations? Can we solve aerodynamic inverse problems in a rapid direct manner? What is the effect of engine power addition on wing aerodynamics and how are actuator disc models developed to simultaneously handle power and rotational effects?
• Provides commented Fortran source code for key problems, such as lifting airfoil, mixed-type transonic supercritical flow, inverse aerodynamic analysis problems, and others, with detailed explanations
• Presents numerous model extensions which the reader can further develop using computational methods introduced here
• Is concise, written by a well-respected M.I.T. trained scientist that “gets to the point†rapidly, teaches with a minimal of jargon, and addresses subtle and profound issues in aerodynamic analysis in a manner not offered by conventional textbooks
• Is an informative critical monograph on modern aerodynamic issues drawing on rigorous mathematics and analysis
• Explains strengths and weaknesses of conventional methods and provides credible alternative methods
• Develops new approaches to aerodynamic inverse and shear flow analysis problems, written by the inventor of Boeing’s supersonic drag analysis models and United Technology’s jet mixer, actuator disc, and engine and airframe integration methods
• Addresses important questions, such as: How do transonic oscillations affect the mean flow? How are strong shear flows modeled using simpler superpotentials without Euler’s equations? Can we solve aerodynamic inverse problems in a rapid direct manner? What is the effect of engine power addition on wing aerodynamics and how are actuator disc models developed to simultaneously handle power and rotational effects?
• Provides commented Fortran source code for key problems, such as lifting airfoil, mixed-type transonic supercritical flow, inverse aerodynamic analysis problems, and others, with detailed explanations
• Presents numerous model extensions which the reader can further develop using computational methods introduced here
• Is concise, written by a well-respected M.I.T. trained scientist that “gets to the point†rapidly, teaches with a minimal of jargon, and addresses subtle and profound issues in aerodynamic analysis in a manner not offered by conventional textbooks
Author / Editor Details
Wilson C. Chin, PhD, earned his M.Sc. at Caltech and Ph.D. from M.I.T. both in aerospace engineering. Early on, he had served as Senior Research Aerodynamicist at Boeing and Turbomachinery Manager at Pratt & Whitney Aircraft, authoring two dozen papers on supersonic and transonic flow, panel methods, hydrodynamic stability, computational fluid-dynamics and inverse formulations. Mr. Chin would later turn his efforts to the petroleum geosciences, writing more than twenty monographs with Wiley-Scrivener and other publishers on several areas in oil and gas exploration. He has also more than a hundred papers and four dozen patents to his credit. He is the recipient of five prestigious awards from the United States Department of Energy and is a well-regarded software developer to domestic and international petroleum companies. In the present book, Mr. Chin focuses on key aerodynamic issues he had addressed, which have grown in importance, and contributes numerous insights to modern analysis methods now key to the resurgence of new types of aircraft on the drawing boards.
Wilson C. Chin, PhD, earned his M.Sc. at Caltech and Ph.D. from M.I.T. both in aerospace engineering. Early on, he had served as Senior Research Aerodynamicist at Boeing and Turbomachinery Manager at Pratt & Whitney Aircraft, authoring two dozen papers on supersonic and transonic flow, panel methods, hydrodynamic stability, computational fluid-dynamics and inverse formulations. Mr. Chin would later turn his efforts to the petroleum geosciences, writing more than twenty monographs with Wiley-Scrivener and other publishers on several areas in oil and gas exploration. He has also more than a hundred papers and four dozen patents to his credit. He is the recipient of five prestigious awards from the United States Department of Energy and is a well-regarded software developer to domestic and international petroleum companies. In the present book, Mr. Chin focuses on key aerodynamic issues he had addressed, which have grown in importance, and contributes numerous insights to modern analysis methods now key to the resurgence of new types of aircraft on the drawing boards.
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
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
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
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
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-- 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
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
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
4.8 Inviscid Steady Flow Past Turbofan Mixer Nozzles . . . 361
4.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 361
4.8.2 Analytical Formulation . . . . . . . . . . . . . . . . . . 361
4.8.3 Calculated Results and Closing Remarks . . . . . . . 363
4.8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 365
5. Engine and Airframe Integration Methods . . . . . . . . . 366
5.1 Big Picture Revisited . . . . . . . . . . . . . . . . . . . . . . . 367
5.2 Engine Component Analysis . . . . . . . . . . . . . . . . . . . 371
5.3 Engine Power Simulation Using Actuator Disks . . . . . . . 374
5.4 Mixers and Supersonic Nozzles . . . . . . . . . . . . . . . . . 375
5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Cumulative References . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
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Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
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
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
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
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-- 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
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
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
4.8 Inviscid Steady Flow Past Turbofan Mixer Nozzles . . . 361
4.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 361
4.8.2 Analytical Formulation . . . . . . . . . . . . . . . . . . 361
4.8.3 Calculated Results and Closing Remarks . . . . . . . 363
4.8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . 365
5. Engine and Airframe Integration Methods . . . . . . . . . 366
5.1 Big Picture Revisited . . . . . . . . . . . . . . . . . . . . . . . 367
5.2 Engine Component Analysis . . . . . . . . . . . . . . . . . . . 371
5.3 Engine Power Simulation Using Actuator Disks . . . . . . . 374
5.4 Mixers and Supersonic Nozzles . . . . . . . . . . . . . . . . . 375
5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Cumulative References . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
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BISAC SUBJECT HEADINGS
TEC002000 : TECHNOLOGY & ENGINEERING / Aeronautics & Astronautics
SCI084000 : SCIENCE / Mechanics / Aerodynamics
MAT003000 : MATHEMATICS / Applied
BIC CODES
TGMF1: Aerodynamics
TRP: Aerospace & aviation technology
PDE: Maths for engineers
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