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TEXAS A&M UNIVERSITY

DETERMINATION OF AERODYNAMIC SF_NSITIVITY COEFFICIENTS

TexasNo. 5802-92-02

............... sp6-nsored by NASA Grant No. NAG-1793 ...........................- _ , --_7 - _ T:--_: ,_-:

TEXAS ENGINEERING EXPERIMENT STATION


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DETERMINATION OF AERODYNAMIC SENSITIVITY COEFFICIENTS

FOR WINGS IN TRANSONIC FLOW

by

Leland A. Carlson and Hesham M. El-BannaAerospace Engineering Department

Texas A&M UniversityCollege Station, Texas 77843-3141

May 1992

Texas A&M Research Foundation ReportNo. 5802-92-02

Sponsored by NASA Grant No. NAG- 1793


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DETERMINATION OF AERODYNAMIC SENSITIVITY COEFFICIENTSFOR WINGS IN TRANSONIC FLOW

Leland A. CarlsonProfessor, Aerospace Engineering

and

Hesham M. ElbannaGraduate Research AssistantTexas A _ M University

College Station, Texas 77843

ABSTRACT

The quasianalytical approach is applied to the three-dimensional full potential equationto compute wing aerodynamic sensitivity coefficients in the transonic regime. Symbolic ma-nipulation is used to reduce the effort associated with obtaining the sensitivity equations,and the large sensitivity system is solved using "state of the art" routines. The quasian-alytical approach is believed to be reasonably accurate and computationaUy efficient forthree-dimensional problems.

INTRODUCTION

To design transonic vehicles using codes which utilize optimization techniques requiresaerodynamic sensitivity coefficients, which are defined as the derivatives of the aerodynamicfunctions with respect to the design variables. In most cases, the main contributor to theoptimization effort is the calculation of these derivatives; and, thus, it is desirable to havenumerical methods which easily, efficiently, and accurately determine these coefficients forlarge complex problems. The primary purpose of the present study is to investigate theapplication of the quasianalytical method [1,2] to three-dimensional transonic flows using asthe fundamental flow solver the three-dimensional transonic full potential fully conservativecode, ZEBRA [3].

PROBLEM STATEMENT

Application of the quasianalytical method to the full potential equation yields the sen-sitivity equation

( a_,,ji,kk _ ( aR_.j.kOXD ] =-\ O--'X-D) (1)


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where XD is the vector of design variables and the residual expression, R,.i_, of the fullpotential equation in the computational plane, X, Y, z, in terms of backward _fferences is

(2)

Here, the retarded density _ and the contravariant velocity components U,V, and w, arelengthy functions of the reduced potential function, . The boundary conditions for Eq.(2)w = u_--_ +re the surface condition, a, v_-_y, the Kutta condition along the wing semispan,r = A4, xTs < x _


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approach, which calculates sensitivities by perturbing a design variable from its previousvalue, obtaining a new solution, using the differences between the new and old solutionsto obtain the sensitivity coefficients. While this direct technique is computer intensive andinefficient, it should serve as a check on the present method.

Based upon the present results, it is concluded that the quasianalytical method is aviable and efficient concept for the determination of three-dimensional transonic aerodynamicsensitivity coefficients. In addition, use of symbolic manipulation to evaluate the elementsof the sensitivity equation is believed to be an efficient approach to the development ofsuch methods. Finally, further studies are needed to determine the accuracy and range ofapplicability of the quasianalytical approach.

ACKNOWLEDGMENT

This project was supported by the National Aeronautics and Space Administration undergrant No. NAG 1-793, Dr. E. Carson Yates, Jr., and Dr. Woodrow Whitlow, Jr., NASALangley Research Center, as technical monitors.

REFERENCES

1. Sobieski, J.S. "The Case for Aerodynamic Sensitivity Analysis", Paper presented toNASA/VPI&SU Symposium on Sensitivity Analysis in Engineering, September 25-26,1986.

2. Elbanna, H.M. and Carlson, L.A., "Determination of Aerodynamic Sensitivity Coef-ficients Based on the Transonic Small Perturbation Formulation", Journal of Aircraft,Vol 27, No.6, June 1990, pp 507-515.

3. Carlson, L.A. and %Veed, R.A., "Direct-Inverse Transonic Wing Analysis-Design Methodwith Viscous Interaction", Journal of Aircraft, VoI 23, No.9, Sept.1986, pp 711-718.

4. Bau, H.H., "Symbolic Computation - An Introduction for the Uninitiated", SymbolicComputation in Fluid Mechanics and Heat Transfer, The American Society of MechanicalEngineers, Vol 97, 1988, pp 1-10.

5. Roach P. and Steinberg S., "Symbolic Manipulation and Computational Fluid Dynam-ics", AIAA Journal, Vol 22, No.10, I984, pp 1390-1394.

6. Steinberg S. and Roach P., "Automatic Generation of Finite Difference Code", SymbolicComputation in Fluid Mechanics and Heat Transfer, The American Society of MechanicalEngineers, Vol 97, 1988, pp 81-86.

7. Saad, Y. and Schultz, M.H., "GMRES: A Generalized Minimum Residual Algorithmfor Solving Nonsymmetric Linear Systems", SIAM Journal of Scientific and StatisticalComputing, Vol 7, No.3, 1986, pp 856-869.

8. Sonneweld, Wesseling, and De Zeeuv, Multigrid and Conjugate Gradient Methods asConvergence Acceleration Techniques in Multigrid Methods for Integral and DifferentialEquations, pp 117-167, Edited by Paddon, D.J. and Holstein, M., Oxford UniversityPress (Claredon), Oxford.

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