Modern Aerodynamic Methods for Direct and Inverse Applications
Description
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, providing 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. These are 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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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.
Content
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
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'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
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
Chapter 1
Basic Concepts, Challenges and Methods
The fluid dynamics world is inundated with thousands of books on the subject, volumes on theory, numerical and engineering niches to no end. Within the specialty of computational fluids, hundreds of thousands of papers have appeared within the past two decades. And in the subset dubbed "aerodynamics," tens of thousands may be found authored by specialists from dozens of countries. This being the case, we will not offer still another "first principles" derivation of governing equations. We will cite relevant subjects and refer readers to readily available literature where excellent presentations are already available. But it will be the author's responsibility to develop and critique significant areas of fluids research that deserve further investigation. And, just as important, introduce ambitious students to key ideas quickly and rigorously, in the least amount of time, with minimal formal course work but with objectivity and honest speculation - to prepare him to understand, contribute and write software to evaluate new ideas. To this end, we have developed a fast-paced presentation style combining "simple numerics" with modern ideas in aerodynamics. With these disclaimers said and done, we now begin discussions on many exciting subjects.
1.1 Governing Equations - An Unconventional Synopsis
The equations governing fluid motions are numerous, for example, as developed in excellent books by Batchelor (1967), Schlichting (2017), Yih (1969) and others. They cover constant density and compressible fluids; liquids and gases; inviscid and viscous motions; one, two and three dimensions; steady and unsteady flows; irrotational and rotational limits; and rectangular, polar, spherical and curvilinear coordinates. For the most part, we will deal with a special subset of these properties to develop the great majority of our ideas. In two dimensions, assuming Cartesian or rectangular coordinates, the momentum and mass conservation equations governing constant density, constant viscosity flows can be written concisely in the form
These represent a highly simplified version of the Navier-Stokes equations. Generalizations of the above have appeared for special applications. For example, in high-speed aerodynamics, the density ? is variable, and equations of state and energy conservation laws apply (we will describe some transonic applications in Chapters 2, 3 and 4). The viscosity µ shown above is constant, but in gas dynamics, it may well be a function of temperature; in meteorology and oceanography, additional dependencies of pressure on properties like humidity and salinity will appear, implying more complicated mathematical descriptions and solutions. Sometimes the stress terms on the right are replaced by an anisotropic tensor; this author has developed models of fluid flow in petroleum reservoirs in a number of books (refer to "About the Author" for further publication information). For our purposes, it suffices to note how Equations 1.1.1a,b,c and similar high-order models (with high-order derivatives) require "Navier-Stokes solvers," which are a challenge to develop, and computationally expensive and resource-intensive to run.
A simpler limit is found by eliminating µ at the very outset, leading to what we call "Euler's equations," a low-order system, namely
The above applies at constant density only and the great majority of applications appears in flows, for instance, with oncoming velocity shear. The reader may recall words of caution. For irrotational flows, Bernoulli's equation "p/? + 1/2 (u2 + v2) = constant" applies, where the constant, fixed for the entire flowfield, is known from upstream conditions. But for rotational flows, this "constant" is only so along a streamline; in fact, it varies from streamline to streamline. What happens when the flow about a jet engine is to be modeled? The external flow, uniform upstream, is irrotational and satisfies a simple Laplace and Bernoulli equation model; however, the flow behind the actuator disk, which imparts radial position-dependent work, is sheared and requires "Euler solvers" with complicated streamline tracking. Algorithm development combining potential with Euler solvers is no small task.
Investigators have developed sophisticated Euler equation solvers requiring equally sophisticated users. And all because "potential flow solvers" for ?xx + ?yy = 0 (or "?xx + ?rr + 1/r ?r = 0," axisymmetrically) will not apply. Every student of fluid mechanics understands how potentials only apply to flows without shear. But what if potentials did apply? What if it were possible to solve ?xx + ?rr + (1/r - 2Um/Um) ?r = 0 valid for mean background flows with strong Um(r) velocity profiles? Simple potential flow codes would, through minor modification, address new classes of important flow problems. In fact, the mathematical basis behind "superpotentials" is developed in Chapter 4 with examples.
Now let's digress and turn to "analysis problems" described by classic potential formulations, that is, solving ?xx + ?yy = 0 subject to tangency conditions for the normal derivative ?y along y = 0, plus a requirement on a "potential jump" [?] related to Kutta's condition at the trailing edge. This formulation, which determines the surface pressure due to a prescribed geometry, as old as aerodynamics itself, has been solved straightforwardly in numerous ways: Glauert's series, panel methods, finite differences, finite elements and so on. But the complementary "inverse problem," searching for the geometry that induces a prescribed pressure, is more subtle and also known as the "indirect" problem. And for good reason. Often, the above analysis solver is run over and over, varying all sorts of empirically defined parameters in endless ways, until some type of convergence is achieved. Is there a simple but "direct approach to indirect problems?"
The answer is, "Yes." Enter the streamfunction, the "black sheep" of modern computational fluid-dynamics. We will show that the airfoil shape is described by the ordinate y(x) = - ?(x,0) where ?xx + ?yy = 0 is solved, subject to normal specifications for ?y(x,0) = - 1/2 U8Cp(x) along y = 0, plus a requirement on a jump [?] related to the degree of trailing edge closure. In other words, given the surface pressure coefficient Cp(x), the shape can be directly (meaning non-iteratively) solved using any potential flow algorithm for analysis problems already available!
Then again, pessimists might argue that the method is limited because it could not be extended to, say transonic supercritical problems. In developing our model, we drew upon Cauchy-Riemann conditions (from complex variables) which strictly apply to complementary equation pairs like ?xx + ?yy = 0 and ?xx + ?yy = 0. And so, "no constant density assumption, no streamfunction inversion." Correct? Incorrect. To solve the problem, we developed a completely rigorous "engineer's
Cauchy-Riemann transform" that allowed us to create a compressible, mixed subsonic and supersonic extension of ?xx + ?yy = 0 to solve inverse formulations in a single pass. Not quite a pure partial differential equation, but one with an integral coefficient that could be just as easily solved. So that's another success story, where we've solved indirect problems directly, initially with a lots of speculation and then some luck.
Next consider compressible flow extensions of Equations 1.1.2a,b,c, which are interesting in a very different way. At steady cruise conditions, under the assumption of irrotationality, these equations (as we will show) lead to a potential flow model not unlike () ?xx + ?yy = 0 where () is somewhat tricky. At Mach zero, or flight speeds say 300 mph or less, this reduces to a purely subsonic (scaled) equation not unlike Laplace's "?xx + ?yy = 0." Dozens of classical texts, conformal maps and singular integral equation methods are and have been available for decades. Near 550 mph or so, fluid particles accelerate so rapidly around leading edges that flows become locally supersonic. Most of the time, they terminate abruptly at shockwaves - where sudden discontinuous increases in pressure lead to losses and unstable wing oscillations. Such are typical of problems suggestive of a "sonic barrier" just several decades ago. But computational methods were non-existent until the 1970s, when Murman and Cole (1971) published a pioneering "type-dependent" numerical algorithm for mixed elliptic and hyperbolic equations. Their idea was simple: use "upwind differencing" for supersonic points and central for subsonic to proper account for...
