Aerodynamic Methods for Advanced Sixth Generation Aircraft Design:

Forward and Inverse Transonic and Shear Flow Models for Close Ground Effect

Wilson C. Chin, Ph.D., M.I.T.

Table of Contents

Author Biography

Preface

Acknowledgements

Chapter 1 - Overview Ideas and Modeling Perspectives

Chapter Summary – Concepts in subsonic, supersonic and transonic flow with shocks are reviewed, as are limitations in potential flow modeling when background shear flows are present. The forward problem predicting pressures when shapes are given, and the inverse problem predicting shapes when pressures are given, are reviewed. Sixth Generation fighter jet objectives are discussed, e.g., blended wing-fuselage designs, flapless navigation using directed momentum jets, and the roles played by Kutta and trailing edge closure constraints, conservation laws and extended Cauchy-Riemann conditions, are summarized. A basic mathematics review is given, citing key approaches, ideas and limitations, and selected industry and educational software packages.

Keywords – Blended wing-fuselage, Euler solvers, finite differences, flapfree navigation, forward problem, inverse problem, Kutta condition, Navier-Stokes solvers, shockwaves, Sixth Generation aircraft, subsonic, supersonic, trailing edge closure, transonic flow.

1. Perspectives in Aerodynamic Flow Modeling

1.1 Conventional Potential Flow Analysis – A Foundational Building Block

1.2 Conventional versus Conservation Law Approaches

1.3 Author Background and Context

1.4 Avoiding Semantic Traps, Thinking Outside the Box

1.5 Mathematics and Numerical Methods Review

1.6 Delta Wings, Lambda Planforms, Vortex Roll-up, Euler and Navier-Stokes Solvers

1.7 References

Chapter 2 - New Analytical Methods for Thin Subsonic Airfoil Design

Chapter Summary – Constant density, irrotational flow formulations are derived for forward problems (solving for pressures when shapes are given) and inverse problems (solving for shapes when pressures are given). Laplace’s equations for potential and streamfunction variables are used. Closed form analytical solutions are derived from solutions to singular integral equations constructed from relevant distributions of source and vortex singularities. Dualities relating the forward problem for camber to the inverse problem for thickness, and the forward problem for thickness to the inverse problem for camber, are derived and discussed insofar as the analogies assist in effective airfoil design. Example forward and inverse solutions are derived to illustrate the methodology clearly.

Keywords – Cauchy Principal Value integrals, distributed sources, distributed vortexes, forward problem, inverse problem, Kutta condition, potential function, singular integral equations, streamfunction, trailing edge constraints.

2.1 Introduction and Objectives

2.2 Analytical Derivations

2.3 Discussion and Conclusions

2.4 References

Chapter 3 - Inverse Problems in Transonic Irrotational Supercritical Flow with Shockwaves

Chapter Summary – The Karman-Guderley nonlinear transonic small-disturbance potential flow forward equation is used as a starting point for this chapter on inverse methods for transonic supercritical flows with shocks. The mixed-type finite difference relaxation method of Murman and Cole is reviewed. The K-G equation is first recast in conservation form, from which extended Cauchy-Riemann conditions are identified to define a complementary streamfunction variable and its corresponding boundary value problem formulation, solvable using well-posed Neumann conditions related to pressure coefficient together trailing edge closure constraints. The method, illustrated with example calculations, is rapidly convergent and numerically stable. However, it does not apply to problems with background shear or to three-dimensional applications. These applications are developed in subsequent chapters.

Keywords – Cauchy-Riemann conditions, compressibility, conservation laws, forward problem, inverse problem, irrotational flow, Karman-Guderley equation, Murman and Cole type-differencing, shockwaves, supercritical flow, trailing edge closure constraints, transonic flow.

3.1 Fluid Dynamics Overview and Motivating Ideas

3.2 Streamfunction Formulation

3.3 Numerical Procedure

3.4 Calculated Results

3.5 Discussion and Closing Remarks

3.6 References

Chapter 4 - Class of Shockfree Airfoils Producing the Same Surface Pressure

Chapter Summary – The transonic inverse method in Chapter 3, which applies to planar, irrotational flows, solves a Neumann streamfunction problem subject to prescribed surface pressures together with trailing edge closure. In this chapter, a known shockfree surface pressure is used as input to the inverse formulation to successfully recover the original smooth shape at the assumed 0.72 Mach number. The inverse code is re-run for seven additional values up to 0.79 to produce eight airfoil sections. Interestingly, the smooth sections in our nonlinear problem become successively thinner as M_¥increases while shapes vary somewhat. This is consistent with Prandtl-Glauert similitude, which strictly applies to linear subsonic flows. This rule requires that, for a given airfoil shape, C_p’s increase in magnitude as M_¥ increases.

Keywords – Cathleen Morawetz, Karman-Guderley equation, Murman-Cole finite difference scheme, mixed subsonic and supersonic flow, Paul Garabedian, Prandtl-Glauert similitude rule, shockfree airfoils, supercritical flow, transonic inverse problem.

4.1 Motivating Ideas

4.2 Analysis Summary

4.3 Discussion and Conclusion

4.4 References

Chapter 5 - Inverse Formulations for Three-Dimensional Problems

Chapter Summary – In prior chapters, the inverse aerodynamic problem was solved using the streamfunction, e.g., ¶²y/¶x² + ¶²y/¶y² = 0, with Neumann conditions ¶y/¶y = - ½ C_p together with trailing edge closure. The method was extended to transonic flows with shocks. These problems were restricted to planar geometries where streamfunctions y(x,y) are defined. In 3D, scalar streamfunctions do not exist as they become vector potentials y(x,y,z). Here, restrictive vector calculus arguments are not invoked. A flexible formalism is developed for a scalar streamfunction-like y(x,y,z) satisfying ¶²y/¶x² + ¶²y/¶y² + ¶²y/¶z² = 0. This allows us to solve inverse problems in a direct manner, drawing on available algorithms and software for potential flow which bear strong math similarities. Effects of compressibility, applications to flows in fans and cascades, as well as to axisymmetric flows, are given.

Keywords – Cauchy-Riemann conditions, conservation laws, elliptic partial differential equations, inverse aerodynamic problems, mass conservation, potential, streamfunction, trailing edge constraints, vector identities, vector streamfunction.

5.1 Introduction

5.2 Constant density planar flows

5.3 Constant Density Flows Past Finite Wings

5.4 Compressible Flows Past Finite Wings

5.5 Flows in Fans and Cascades

5.6 Axisymmetric Compressible Flows

5.7 Sample Calculations

5.8 Closing Remarks

5.9 References

Chapter 6 - Methods for Planar, Inviscid, Incompressible Shear Flow

Chapter Summary – Classical aerodynamics assumes constant density, irrotational flow, leading to Laplace’s potential equation. Compressibility effects are modeled using the Prandtl-Glauert equation, while transonic effects are included by retaining an essential nonlinearity. All of these methods work with velocity potentials. When background shear flow effects are important, this usage breaks down and recourse to Euler’s or Navier-Stokes equations is needed. Here, we instead develop, using vorticity properties, "potential-like" and "streamfunction-like" formulations useful in rigorously solving forward and inverse problems. These general models are solved for practical airplane design problems and numerical solutions are explained. The methods build on solid mathematical foundations, however, they are restricted to constant density and two-dimensional flows – these restrictions are removed in Chapters 7, 8 and 9.

Keywords – Airfoil theory, mixed Dirichlet and Neumann boundary conditions, parallel shear flows, potential, ringwings, small disturbance theory, sources, streamfunction, vortexes.

6.1 Introduction to Shear Flow Aerodynamics

6.2 Planar Flows with Constant Vorticity: Inverse Problems

6.3 Planar Flows: Direct Formulations

6.4 Some Planar Analytical Solutions

6.5 Analogy To Ringwing Potential Flows

6.6 Source and Vortex Interactlons for Ringwings

6.7 Airfoils in General Parallel Shear Flow

6.8 Numerical Results

6.9 Closing Remarks

6.10 References

Chapter 7 - Unified Formulations for Inviscid Transonic, Supercritical and Rotational Flows

Chapter Summary – Conventional aerodynamics offers "exact" potential flow solutions without geometric limitations, e.g., panel methods, finite element and volume, but these are actually limited. Why? All solutions are grid dependent and require calibration. Potential flows do not address rotation, important in wind shears, ground effect boundary layers, flight behind large aircraft or wind wakes behind aircraft carriers. Euler and Navier-Stokes methods "solve" such problems, but are research heavy, seldom used, with solutions slow and far from accurate. Here, new rigorous "potential-like" and "streamfunction-like" methods handle background shear, transonic nonlinearities, 3D and ground effects, providing rapid, stable and accurate solutions. Forward and inverse formulations motivated by AI/ML "LLMs" like Deepseek-R1 are recast as partial differential equations and solution strategies are outlined for Sixth Generation aircraft.

Keywords – AI/ML, Deepseek-R1, Euler methods, finite differences, aerodynamic forward and inverse models, Large Language Models, Navier-Stokes solvers, potential, small disturbance model, streamfunction, supercritical transonic flow.

7.1 Compressible Irrotational Flow and Deepseek-R1 Evaluation

7.2 Deepseek-R1 Insights on Rotational Streamfunction Shear Flow Models

7.3 Simple Integrated Formulations for Transonic Shear Flow (Non-Euler Equation Models)

7.4 Transonic Forward and Inverse Formulations in Parallel Shear Flow

7.5 Closing Remarks and Research Summary

7.6 References

Chapter 8 - Aerodynamic Modeling Concepts and Algorithm Development

Chapter Summary – All encompassing "potential-like" and "streamfunction-like" formulations are developed for aerodynamic "forward" and "inverse" problems, solving for pressures when shapes are given and vice-versa. The methods model general background parallel shear flows, transonic flow with shockwaves, full three-dimensionality, operating rapidly, accurately, and stably. Boundary value problems are clearly stated and detailed finite difference relaxation algorithms with trailing edge constraints are given for 2D/3D applications for Sixth Generation fighter planforms. The methods apply to blended wing-fuselages, flapless control (with opened trailing edge navigation by using fluid momentum jets), flight in wind shears, close ground proximity, following aircraft and wind wakes of aircraft carriers. Gridding strategies are outlined for modern fighter aircraft, e.g., F-35, F-47, J-20, J-36, Rafale, MIG, GCAP and others.

Keywords – Aircraft carrier take-off and landing, aircraft carrier wakes, blended wing-fuselage, flapless wings, forward problem, ground effect, inverse problem, potential, Sixth Generation aircraft, streamfunction, wind shear, Wing in Ground WIG applications.

8.1 Basic Challenges in Laplace Equation Solutions

8.2 Motivation – Solution Algorithm for Constant Density and More Complicated Flows

8.3 Shear Flow Modeling Ideas, Models and Formulas

8.3.1 Classical transonic irrotational model without shear flow

8.3.2 On potentials and streamfunctions at nonzero Mach numbers

8.3.3 Constant density flows with strong parallel background shear

8.3.4 Compressible flows with strong parallel background shear

8.3.5 Dimensionless numbers and boundary value problems

8.4 Closing Comments Prior to Example Calculations

8.4.1 Note 1 – Perspectives on fluid dynamics education

8.4.2 Note 2 – Potential flow "forward" analysis and inverse extensions

8.4.3 Note 3 – Inverse formulations

8.4.4 Note 4 – Computational details

8.5 References

Chapter 9, Validations and Applications in Multiple Physical Limits^*

Chapter Summary – Unified "potential-like" and "streamfunction-like" formulations for 2D/3D transonic flow, background shear flow, close "Wing in Ground" (WIG) application, wing planform analysis for 6^th Generation fighter aircraft are summarized, outlining theory, capabilities, CFD details, specific applications to J-36, F-35, MIG, Rafale, Bohai Sea Monster and more. Applications chapter includes more than thirty detailed "mini-thesis" projects with detailed C_p curves (for forward problems) and surface shapes (for inverse problems), treating different combinations of M_¥, ground proximity, transonic nonlinearity, positive and negative background shear, demonstrating the versatility and power of the forward and inverse models. Detailed visuals explain all algorithm details, formulations and jet fighter applications. Three applications classes (2D forward, 2D inverse, and 2D/3D forward and inverse) are presented for rectangular, delta and lambda wings. All calculations are listed with code names for readers interested in separately available software.

Keywords – Aircraft carrier take-off and landing, aircraft carrier wakes, blended wing-fuselage, delta wings, flapless wings, forward problem, ground effect, inverse problem, lambda wings, potential low, Sixth Generation aircraft, streamfunction, wind shear, "Wing in Ground" WIG applications.

^{*Chapter 9 applications examples are used with Stratamagnetic Software, LLC permission, obtained from marketing materials associated with software support, licensing and distribution.

9-1. Forward 2D Analysis Problems, Validations and Calculated Solutions
Forward, Analysis or Direct 2D Problems

Example 9-1. Flat plate camber line in large and smaller boxes, first irrotational (MOD-16), then with background parallel shear flow, all runs Mach 0.0 (MOD-16-SHR).
Example 9-2. Thickness distribution in large and smaller boxes, no parallel shear, all runs Mach 0.0 (MOD-15).
Example 9-3. Airfoil C_L versus height in ground effect, without and with parallel shear flow with applications to "Wing in Ground" (WIG) aircraft design, all runs Mach 0.0

(MOD-16, MOD-16-SHR).
Example 9-4. Centered unpitched biconvex and flat plate airfoils – pressure distributions, shockwave formation and movement, numerical stability for Mach numbers 0-0.95
(MOD-18).
Example 9-5. Centered biconvex airfoil, transonic flow with parallel shear U(y)
(MOD-18-SHR).
Example 9-6. Runs 1-4 . Unpitched biconvex, pitched flat plate, Mach 0 and 0.85, with and without shear in infinite media, with and without close ground effect (MOD-19, which combines prior codes).
Example 9-7. Discriminant plot, subsonic and supersonic zone definition (MOD-19).
Example 9-8. Trailing edge requirements, Kutta conditions and trailing edge constraints (MOD-19).
Example 9-9. Integrated forward, potential-like numerical algorithm in two-dimensional, nonlinear transonic flow in the presence of parallel shear velocity U(y) function (MOD-22).
Example 9-10. Prandtl-Glauert pressure similitude (MOD-22).

9-2. Inverse or Indirect 2D Problems, Validations and Calculated Solutions

Inverse, Design or Indirect 2D Problems
Example 9-11. Exact solution validations - Inverse symmetric and antisymmetric C_p inputs at M_¥ = 0 without shear (MOD18-SF-16B).

Example 9-12. Inverse shape prediction, C_p = - 0.2 constant upper and lower surfaces, constant density flow and strong background parallel shear flow at Mach 0
(MOD18-SF-16B).
Example 9-13. Inverse calculations for symmetric C_p surface distributions with different degrees of trailing edge closure at Mach 0 without shear (MOD18-SF-16B).
Example 9-14. Inverse irrotational calculations for symmetric C_p = -0.2 surface distributions with trailing edge closure at different Mach numbers (MOD18-SF-16B).
Example 9-15. Inverse irrotational calculations for symmetric C_p = -0.2 at Mach 0, with proximity to ground plane varied (for Wing in Ground, "WIG" applications) (MOD18-SF-16B).
Example 9-16. Inverse airfoil design, irrotational, subsonic to high transonic with "shape shock," symmetric C_p = - 0.2 fixed, chord centered vertically in a large box (MOD18-SF-16B).

Example 9-17. Inverse airfoil design, M = 0.3 and positive shear level both fixed, symmetric C_p = - 0.2 fixed, distance to ground decreases from chord centered vertically in large box (top) to near ground proximity (bottom) (MOD18-SF-16B).

Example 9-18. Inverse airfoil design, M = 0.4 and vertically centered all runs, symmetric C_p = - 0.2 fixed, positive shear level increases downward (MOD18-SF-16B).

9-3. Forward and Inverse 3D Problems, Special Projects and Calculated Solutions

Forward and Inverse – Transonic, Three-Dimensional, Shear and Ground Effect
Example 9-19. Three-dimensionality, nonlinearities and ground effect models
(MOD22-3D-9).
Example 9-20. High aspect ratio wing at Mach 0 (MOD22-3D-9).
Example 9-21. High aspect ratio wing at Mach 0.9 showing shock capture (MOD22-3D-9).
Example 9-22. High aspect ratio wing calculations, at high Mach number with shock, positive wind shear, and strong ground effect (MOD22-3D-9).
Example 9-23. Simplified grids for swept delta and NASA oblique wings (MOD22-3DSWP-1).
Example 9-24. Inverse shape prediction – 3D modeling concepts, rectangular planform baseline calculation, closed trailing edges (MOD18-SF3D-5).
Example 9-25. Inverse shape prediction – Delta wings, background shear flow, close ground effect, opened trailing edges with mass outflow (MOD18-SFSWP-1).
Example 9-26. Inverse shape prediction – Delta wing, irrotational, close ground effect, very high Mach number, opened trailing edge with mass outflow (MOD18-SFSWP-1).
Example 9-27. Inverse problem – High sweep delta wing, irrotational, centered, high Mach number, strong supersonic zone, "shape shock" and closed trailing edge
(MOD18-SFSWP-1).

Example 9-28. Model problem – "Wing in Ground" (WIG), proximity, shear and nonlinear effects (MOD-22).
Example 9-29. Meshing strategy – H-20 and similar bomber planforms (MOD18-SFSWP-1).

Example 9-30. Cylindrical flow strategies – Ringwings, inlet/nacelles and mixers (calculated results from modified rectangular wing algorithms).

Example 9-31. Forward 3D problems for simple delta and lambda planforms at Mach 0.7, Lockheed Vectis, China H-20 Stealth Bomber and CH-7 Stealth Drone (MOD22-3DSWP-2).
Example 9-32. Inverse 3D problems for simple delta and lambda planforms at Mach 0.7, Lockheed Vectis, China H-20 Stealth Bomber and CH-7 Stealth Drone (MOD18-SFSWP-2).

9-4. Development Philosophy and Closing Remarks
9-5. References
9-6. Appendixes
Appendixes 4 and 5 derive the linear disturbance streamfunction equation with transonic compressibility and shear. Appendixes 5, 6 and 7 derive the transonic small disturbance equation with the correct nonlinear terms for irrotational flows. Calculated examples solving a unified model are provided in Chapter 9. All results are new to the aerodynamics literature.
Appendix 1, Three-Dimensional Constant Density Flows
Appendix 2, Planar Compressible Shear Flow of a Gas
Appendix 3, Deepseek-R1, Conversational Query-4 and Modeling Results
Appendix 4, Deepseek-R1, Conversational Query-5
Appendix 5, Deepseek-R1, Transonic Small Disturbance Streamfunction for Irrotational Flow
Appendix 6, Deepseek-R1, Revisited, Llama-4-Scout-Nitro and GPT-4.1 Transonic Small

Disturbance Streamfunction for Irrotational Flow

Appendix 7, ChatGPT - Transonic Small-Disturbance Streamfunction Irrotational Equation
10. Glossary and Focused Tutorials
11. Cumulative References}