Flow Control pp 79-109 | Cite as

Quasi-Analytical Shape Modification for Neighboring Steady-State Euler Solutions

  • J. S. Brock
  • W. F. Ng
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 68)


Aerodynamic inverse design methods which are governing equation consistent are generally limited to the Full Potential equations. Consistent design methods use identical governing equations for all fluid dynamic segments of the algorithm, including shape modification. This ensures that all relevant physical information is included within each design estimate, and therefore, a minimum number of analysis/design iterations are required. This report presents a new, and consistent, shape modification method for future use within a direct-iterative inverse design algorithm. The method is simple, being developed from a truncated quasi-analytical Taylor’s series expansion of the global governing equations. The method is general, since it may use either the Euler or Navier-Stokes equations, any combination of numerical techniques, and any number of spatial dimensions. The proposed method also includes a unique iterative algorithm, and new geometry/grid constraints, to solve the over-determined design problem. An upwind, cell-centered, finite-volume formulation of the two-dimensional Euler equations is used within the present effort. The method is evaluated within a symmetric channel where the design variable is a mid-channel ramp angle which is nominally θ = 5°. Tests were conducted for three target ramp angle perturbations, Δθ = 2%, 10%, and 40%, and three inlet Mach numbers, M = 0.30, 0.85, and 2.00. For a single design estimate, using design-like test conditions, the new method is demonstrated to accurately predict geometry shape changes. This includes the transonic test case with an extreme 40% design variable perturbation where the target geometry was predicted with 95% accuracy.


Computational Fluid Dynamics AIAA Paper Normal Matrix Jacobian Matrice Shape Modification 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Slooff, J.W., Computational Methods for Subsonic and Transonic Aerodynamic Design, AGARD-R-712, May 1983, pp. 3.1–3.40.Google Scholar
  2. [2]
    Dulikravich, G.S., Aerodynamic Shape Design, AGARD-R-780, May 1990, pp. 1.1–110.Google Scholar
  3. [3]
    Volpe, G., Transonic Shock Free Wing Design, AGARD-R-780, May 1990, pp. 5.1–516.Google Scholar
  4. [4]
    Tranen, T.L., A Rapid Computer Aided Transonic Airfoil Design Method, AIAA Paper 74-0501, 1974.Google Scholar
  5. [5]
    Shankar, V., A Full Potential Inverse Procedure for Wing Design Based on a Density Linearization Scheme, NASA CR-165991, October 1982.Google Scholar
  6. [6]
    Volpe, G. and Melnik, R.E., Method for Designing Closed Airfoils for Arbitrary Supercritical Speed Distributions, Journal of Aircraft, Vol. 23, No. 10, 1986, pp. 775–782.CrossRefGoogle Scholar
  7. [7]
    Gaily, T.A. and Carlson, L.A., Inviscid Transonic Wing Design Using Inverse Methods in Curvilinear Coordinates, AIAA Paper 87-2551, 1987.Google Scholar
  8. [8]
    Bauer, F., Garabedian, P., and Korn, D., Supercritical Wing Sections III, Springer-Verlag, New York, 1977.MATHGoogle Scholar
  9. [9]
    Sobieczky, H., Yu, N.J., Fung, K.Y., and Seebass, A.R., New Method for Designing Shock-Free Transonic Configurations, AIAA Journal, Vol. 17, No. 7, July 1979, pp. 722–729.MATHCrossRefGoogle Scholar
  10. [10]
    Davis, W.H., Technique for Developing Design Tools from the Analysis Methods of Computational Aerodynamics, AIAA Journal, Vol. 18, No. 9, 1980, pp. 1080–1087.MATHCrossRefGoogle Scholar
  11. [11]
    Campbell, R.L. and Smith, L.A., A Hybrid Algorithm for Transonic Airfoil and Wing Design, AIAA Paper 87-2552, 1987.Google Scholar
  12. [12]
    Lee, J. and Mason, W.H., Development of an Efficient Inverse Method for Supersonic and Hypersonic Body Design, AIAA Paper 91-0395, 1991.Google Scholar
  13. [13]
    Meauze, G., An Inverse Time Marching Method for the Definition of Cascade Geometry, ASME Journal of Engineering for Power, Vol. 104, July 1982, pp. 650–656.Google Scholar
  14. [14]
    Schmidt, E. and Berger, P., Inverse Design of Supercritical Nozzles and Cascades, International Journal For Numerical Methods in Engineering, Vol. 22, 1986, pp. 417–432.MATHCrossRefGoogle Scholar
  15. [15]
    Giles, M.B. and Drela, M., Two-Dimensional Transonic Aerodynamic Design Method, AIAA Journal, Vol. 25, No. 9, 1987, pp. 1199–1206.CrossRefGoogle Scholar
  16. [16]
    Dedoussis, V., Chaviaropoulos, P., and Papailiou, K.D., Rotational Compressible Inverse Design Method for Two-Dimensional, Internal Flow Configurations, AIAA Journal, Vol. 31, No. 3, 1993, pp. 551–558.MATHCrossRefGoogle Scholar
  17. [17]
    Taylor, A.C., Korivi, V.M. and Hou, G.W., Sensitivity Analysis Applied to the Euler Equations: A Feasibility Study with Emphasis on Variation of Geometric Shape, AIAA Paper 91-0173, 1991.Google Scholar
  18. [18]
    Taylor, A.C., Hou, G.W., and Korivi, V.M., An Efficient Method for Estimating Neighboring Steady-State Numerical Solutions to the Euler Equations, AIAA Paper 91-1680, 1991.Google Scholar
  19. [19]
    Taylor, A.C., Hou, G.W., and Korivi, V.M., A Methodology for Determining Aerodynamic Sensitivity Derivatives with Respect to Variation of Geometric Shape, AIAA Paper 91-1101, 1991.Google Scholar
  20. [20]
    Baysal, O. and Eleshaky, M.E., Aerodynamic Design Optimization Using Sensitivity Analysis and Computational Fluid Dynamics, AIAA Paper 91-0471, 1991.Google Scholar
  21. [21]
    Taylor, A.C. Korivi, V.M. and Hou, G.W., Approximate Analysis and Sensitivity Analysis Methods for Viscous Flow Involving Variation of Geometric Shape, AIAA Paper 91-1569, 1991.Google Scholar
  22. [22]
    Korivi, V.M., Taylor, A.C., Hou, G.W., Newman, P.A., and Jones, H.E., Sensitivity Derivatives for a 3D Supersonic Euler Code Using the Incremental Iterative Strategy, AIAA 11th Computational Fluid Dynamics Conference, Orlando, Florida, July 1993.Google Scholar
  23. [23]
    Thomas, J.L. and Walters, R.W., Upwind Relaxation Algorithms for the Navier-Stokes Equations, AIAA Journal, Vol. 25, No. 4, 1987, pp. 527–534.MATHCrossRefGoogle Scholar
  24. [24]
    Van Leer, B., Flux-Vector Splitting for the Euler Equations, ICASE Report 82-30, September 1982.Google Scholar
  25. [25]
    Korivi, V.M., Taylor, A.C., Newman, P.A., Hou, G.W., and Jones, H.E., An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives, NASA TM-104207, February 1992.Google Scholar
  26. [26]
    Golub, G.H. and Van Loan, C.F., Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, 1989, pp. 193–259.MATHGoogle Scholar
  27. [27]
    Volpe, G. and Melnik, R.E., The Role of Constraints in the Inverse Design Problem for Transonic Airfoils, AIAA Paper 81-1233, 1981.Google Scholar
  28. [28]
    Volpe, G., Geometric and Surface Pressure Restrictions in Airfoil Design, AGARD-R-780, May 1990, pp. 4.1–414.Google Scholar
  29. [29]
    Reddy, J.N., An Introduction to the Finite Element Method, 1st ed., McGraw-Hill, New York, 1984, pp. 173–174.Google Scholar
  30. [30]
    Riggins, D.W. and Walters, R.W., The Use of Direct Solvers for Compressible Flow Computations, AIAA Paper 88-0229, 1988.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • J. S. Brock
    • 1
  • W. F. Ng
    • 1
  1. 1.Mechanical Engineering DepartmentVirginia Polytechnic Institute and State UniversityBlacksburgUSA

Personalised recommendations