An exact inverse method for subsonic flows
Author:
Prabir Daripa
Journal:
Quart. Appl. Math. 46 (1988), 505-526
MSC:
Primary 76G99; Secondary 35R30
DOI:
https://doi.org/10.1090/qam/963587
MathSciNet review:
963587
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A new inverse method for aerodynamic design of airfoils is presented for subcritical flows. The pressure distribution in this method can be prescribed in a natural way, i.e., as a function of arclength of the as yet unknown body. This inverse problem is shown to be mathematically equivalent to solving only one nonlinear boundary value problem subject to known Dirichlet data on the boundary. The solution to this nonlinear problem determines the airfoil, free stream Mach number ${M_\infty }$ and the upstream flow direction ${\theta _\infty }$. The existence of a solution to a given pressure distribution is discussed. The method is easy to implement and extremely efficient. We present a series of results for which comparisons are made with the known airfoils. This method will be extended to design supercritical airfoils in the future.
A. B. Arlinger, An exact method of two-dimensional airfoil design, SAAB Tech. Notes TN 67 (1970)
- Lipman Bers, Mathematical aspects of subsonic and transonic gas dynamics, Surveys in Applied Mathematics, Vol. 3, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096477
F. Bauer, P. R. Garabedian, and D. Korn, Supercritical wing sections III, Lecture Notes in Economics and Mathematical Systems 150, Springer-Verlag, New York, 1977
L. A. Carlson, The inverse transonic flow calculations using experimental pressure distributions, AIAA J. 12 (4), 571–572 (1976)
P. Daripa, The Direct and Inverse Problems in Gas Dynamics, Ph.D. Thesis, Brown University, Providence, R.I., 1985
- Prabir K. Daripa and Lawrence Sirovich, Exact and approximate gas dynamics using the tangent gas, J. Comput. Phys. 62 (1986), no. 2, 400–413. MR 828141, DOI https://doi.org/10.1016/0021-9991%2886%2990136-1
- Prabir K. Daripa and Lawrence Sirovich, An inverse method for subcritical flows, J. Comput. Phys. 63 (1986), no. 2, 311–328. MR 835821, DOI https://doi.org/10.1016/0021-9991%2886%2990196-8
W. H. Davis, Technique for developing design tools from the analysis methods of computational aerodynamics, AIAA Pap. 79–1529, 1979
- K. Y. Fung, H. Sobieczky, and R. Seebass, Shock-free wing design, AIAA J. 18 (1980), no. 10, 1153–1158. MR 586600, DOI https://doi.org/10.2514/3.50865
P. R. Garabedian, Design methods for supercritical airfoils, Proceedings on ICIDES, University of Texas, Austin, 133–143 (1984)
P. R. Garabedian and D. G. Korn, Numerical design of transonic airfoils, In Numerical Solution of Partial Differential Equations. II, ed. Bert Hubbard, Academic Press, New York, 1971, p. 253
P. R. Garabedian and G. McFadden, Computational fluid dynamics of airfoils and wings, Transonic, shock and multidimensional flows: advances in scientific computing, ed. by R. E. Meyer, Academic Press, New York, 1982, pp. 1–16
A. H. Glauert, Aerofoil and Airscrew Theory, Cambridge University Press, 1926
S. Goldstein, A Theory of Aerofoils of Small Thickness, Rep. ARC. 6225, London, 1942
N. D. Halsey, Methods for the design and analysis of jet-flapped airfoils, AIAA Pap. 74–188, 1974
A. A. Hassan, A method for the design of shock-free slender bodies of revolution, AIAA J. 24 (5) (1986)
A. A. Hassan, H. Sobieczky, and A. R. Seebass, Subcritical and supercritical airfoils for a given pressure distribution, AIAA Pap. 81-1233, 1981
A. A. Hassan, H. Sobieczky, and A. R. Seebass, Subsonic airfoils with a given pressure distribution, AIAA J. 22 (9) (1984)
R. Hicks and G. Vanderplaats, Application of numerical optimization to the design of supercritical airfoils without drag creep, Society of Automotive Engineers Pap. 770440, 1977
R. Hicks, G. M. Vanderplaats, E. M. Murman and R. R. King, Airfoil section drag reduction at transonic speeds by numerical optimization, NASA TMX. 73097, 1976
F. T. Johnson, A general panel method for the analysis and design of arbitrary configurations in incompressible flows, NASA CR. 3079, 1980
F. T. Johnson and P. E. Rubbert, Advanced panel-type influence coefficient methods applied to subsonic flows, AIAA Pap. 75–50, 1975
M. J. Langley, Numerical methods for two-dimensional and axisymmetric flows, ARA memo. 143, 1973
- H. W. Liepmann and A. Roshko, Elements of gasdynamics, Galcit aeronautical series, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. MR 0092515
M. J. Lighthill, A new method of two-dimensional aerodynamic design, ARC, R&M 2112, 1945
- Geoffrey B. McFadden, AN ARTIFICIAL VISCOSITY METHOD FOR THE DESIGN OF SUPERCRITICAL AIRFOILS, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–New York University. MR 2628595
- José M. Sanz, A well posed boundary value problem in transonic gas dynamics, Comm. Pure Appl. Math. 31 (1978), no. 6, 671–679. MR 502825, DOI https://doi.org/10.1002/cpa.3160310604
J. W. Sloof, A survey of computational methods for subsonic and transonic aerodynamics design, Proceedings on ICIDES, University of Texas, Austin, 1–68 (1984)
H. Sobieczky, Related analytical, analog and numerical methods in transonic airfoil design, AIAA Pap. 79–1556, 1979
H. Sobieczky, Analytical tools for systematic transonic design, Proceedings of ICIDES, University of Texas, Austin, 85–112 (1984)
H. Sobieczky and A. R. Seebass, Supercritical airfoil and wing design, Ann. Rev. Fluid Mech. 16, 337 (1984)
T. Strand, Exact method of designing airfoils with given velocity distribution in incompressible flow, J. Aircraft 10, 651–659 (1973)
B. S. Stratford, Rep. & Mem. Aero. Res. Coun. 3002, London, 1954
- B. S. Stratford, The prediction of separation of the turbulent boundary layer, J. Fluid Mech. 5 (1959), 1–16. MR 104431, DOI https://doi.org/10.1017/S0022112059000015
B. S. Stratford, An experimental flow with zero skin friction throughout its region of pressure rise, J. Fluid Mech. 5, 17–35 (1959)
P. N. Swarztrauber and R. A. Sweet, Efficient Fortran subprograms for the solution of elliptic equations, NCAR TN/IA 109, 138 (1975)
T. L. Tranen, A rapid computer aided transonic airfoil design method, AIAA Pap. 74–501, 1974
G. Volpe, The inverse design of closed airfoils in transonic flow, AIAA Pap. 83–0504, 1983
G. Volpe and R. E. Melnik, The role of constraints in the inverse design problem for transonic airfoils, AIAA Pap. 81–1233, 1981
N. J. Yu, Efficient transonic shock-free wing redesign procedure using a fictitious gas method, AIAA J. 18, 143 (1980)
A. B. Arlinger, An exact method of two-dimensional airfoil design, SAAB Tech. Notes TN 67 (1970)
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958
F. Bauer, P. R. Garabedian, and D. Korn, Supercritical wing sections III, Lecture Notes in Economics and Mathematical Systems 150, Springer-Verlag, New York, 1977
L. A. Carlson, The inverse transonic flow calculations using experimental pressure distributions, AIAA J. 12 (4), 571–572 (1976)
P. Daripa, The Direct and Inverse Problems in Gas Dynamics, Ph.D. Thesis, Brown University, Providence, R.I., 1985
P. Daripa and L. Sirovich, Exact and approximate gas dynamics using the tangent gas, J. Comput. Phys. 62, 400–413 (1986)
P. Daripa and L. Sirovich, An inverse method for subcritical flows, J. Comput. Phys. 63, 311–328 (1986)
W. H. Davis, Technique for developing design tools from the analysis methods of computational aerodynamics, AIAA Pap. 79–1529, 1979
K.-Y. Fung, H. Sobieczky, and A. R. Seebass, Shock-free wing design, AIAA J. 18, 1153–1158 (1980)
P. R. Garabedian, Design methods for supercritical airfoils, Proceedings on ICIDES, University of Texas, Austin, 133–143 (1984)
P. R. Garabedian and D. G. Korn, Numerical design of transonic airfoils, In Numerical Solution of Partial Differential Equations. II, ed. Bert Hubbard, Academic Press, New York, 1971, p. 253
P. R. Garabedian and G. McFadden, Computational fluid dynamics of airfoils and wings, Transonic, shock and multidimensional flows: advances in scientific computing, ed. by R. E. Meyer, Academic Press, New York, 1982, pp. 1–16
A. H. Glauert, Aerofoil and Airscrew Theory, Cambridge University Press, 1926
S. Goldstein, A Theory of Aerofoils of Small Thickness, Rep. ARC. 6225, London, 1942
N. D. Halsey, Methods for the design and analysis of jet-flapped airfoils, AIAA Pap. 74–188, 1974
A. A. Hassan, A method for the design of shock-free slender bodies of revolution, AIAA J. 24 (5) (1986)
A. A. Hassan, H. Sobieczky, and A. R. Seebass, Subcritical and supercritical airfoils for a given pressure distribution, AIAA Pap. 81-1233, 1981
A. A. Hassan, H. Sobieczky, and A. R. Seebass, Subsonic airfoils with a given pressure distribution, AIAA J. 22 (9) (1984)
R. Hicks and G. Vanderplaats, Application of numerical optimization to the design of supercritical airfoils without drag creep, Society of Automotive Engineers Pap. 770440, 1977
R. Hicks, G. M. Vanderplaats, E. M. Murman and R. R. King, Airfoil section drag reduction at transonic speeds by numerical optimization, NASA TMX. 73097, 1976
F. T. Johnson, A general panel method for the analysis and design of arbitrary configurations in incompressible flows, NASA CR. 3079, 1980
F. T. Johnson and P. E. Rubbert, Advanced panel-type influence coefficient methods applied to subsonic flows, AIAA Pap. 75–50, 1975
M. J. Langley, Numerical methods for two-dimensional and axisymmetric flows, ARA memo. 143, 1973
H. W. Liepmann and A. Roshko, Elements of Gas Dynamics, John Wiley and Sons, Inc., 1957
M. J. Lighthill, A new method of two-dimensional aerodynamic design, ARC, R&M 2112, 1945
G. B. McFadden, An artificial viscosity method for the design of supercritical airfoils, Ph.D. Thesis, N.Y. University, New York, N.Y., 1979
J. Sanz, A well posed boundary value problem in transonic gas dynamics, Comm. Pure Appl. Math. 31, 671–679 (1978)
J. W. Sloof, A survey of computational methods for subsonic and transonic aerodynamics design, Proceedings on ICIDES, University of Texas, Austin, 1–68 (1984)
H. Sobieczky, Related analytical, analog and numerical methods in transonic airfoil design, AIAA Pap. 79–1556, 1979
H. Sobieczky, Analytical tools for systematic transonic design, Proceedings of ICIDES, University of Texas, Austin, 85–112 (1984)
H. Sobieczky and A. R. Seebass, Supercritical airfoil and wing design, Ann. Rev. Fluid Mech. 16, 337 (1984)
T. Strand, Exact method of designing airfoils with given velocity distribution in incompressible flow, J. Aircraft 10, 651–659 (1973)
B. S. Stratford, Rep. & Mem. Aero. Res. Coun. 3002, London, 1954
B. S. Stratford, The prediction of separation of the turbulent boundary layers, J. Fluid Mech. 5, 1–16 (1959)
B. S. Stratford, An experimental flow with zero skin friction throughout its region of pressure rise, J. Fluid Mech. 5, 17–35 (1959)
P. N. Swarztrauber and R. A. Sweet, Efficient Fortran subprograms for the solution of elliptic equations, NCAR TN/IA 109, 138 (1975)
T. L. Tranen, A rapid computer aided transonic airfoil design method, AIAA Pap. 74–501, 1974
G. Volpe, The inverse design of closed airfoils in transonic flow, AIAA Pap. 83–0504, 1983
G. Volpe and R. E. Melnik, The role of constraints in the inverse design problem for transonic airfoils, AIAA Pap. 81–1233, 1981
N. J. Yu, Efficient transonic shock-free wing redesign procedure using a fictitious gas method, AIAA J. 18, 143 (1980)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76G99,
35R30
Retrieve articles in all journals
with MSC:
76G99,
35R30
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society
