Non-uniqueness Problems in Transonic Flows

  • M. Hafez
Part of the Fluid Mechanics and its Applications book series (FMIA, volume 73)


In this review paper, multiple numerical solutions of potential, Euler and Navier-Stokes equations for steady transonic flows over symmetric airfoils are presented. Nonuniqueness problems of three dimensional and unsteady flows are also discussed. No experimental data is available for validation of the calculations.


Euler Equation Transonic Flow Inviscid Flow Wavy Surface Asymmetric Solution 
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  1. 1.
    C. J. Lee and H.K. Cheng: An airfoil theory of bifurcating laminar separation from thin obstacles, J. of Fluid Mech., 216, 255–284, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    H. Bailey and R. Beam: Newton’s method applied to Finite Difference approximations to the steady-state compressible Navier-Stokes equations, J. of Comp. Physics 93, 108–127, 1991ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    J. Steinhoff and A. Jameson: Multiple solutions for the transonic potential flow past an airfoil, AIAA J. 20, 1521, 1982ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    M. Bristeau, O. Pironneau, R. Glowinski, J. Periaux, P. Perrier and G. Poirier: On the numerical solution of nonlinear problems in fluid dynamics by least squares, Computer Methods in Applied Mechanics and Engineering, 51, 363–394, 1985MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    M. Salas, A. Jameson and R. Melnik: A comparative study of the nonuniqueness problem of the potential equation, AIAA Paper 83–1988Google Scholar
  6. 6.
    A. Jameson: Nonunique solutions to the Euler equations, AIAA Paper 91-1625Google Scholar
  7. 7.
    K. McGraten: Comparison of transonic flow models, AIAA J., 30, 2340–2343, 1992ADSCrossRefGoogle Scholar
  8. 8.
    C. Chan and P. Garabedian: Complex analysis of Transonic Flow, Frontiers of Computational Fluid Dynamics, World Scientific, 1998Google Scholar
  9. 9.
    M. Hafez and A. Dimanling: Simulations of compressible inviscid flows over stationary and rotating cylinders, Acta Mech [Suppl] 4, 241–249, 1994Google Scholar
  10. 10.
    M. Hafez and A. Dimanling: Some anomalies of the numerical solutions of the Euler equations, Acta Mech 119, 131–140, 1996MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    M. Hafez and W. Guo: Nonuniqueness of transonic flows, Acta Mech, 138, 177–184, 1999zbMATHCrossRefGoogle Scholar
  12. 12.
    M. Hafez and W. Guo: Some anomalies of the numerical solutions of shock waves, part I: Inviscid flows, Computers and Fluids 28, 701–720, 1999MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    M. Hafez and W. Guo: Some anomalies of the numerical solutions of shock waves, part II: Effect of artificial and real viscosity, Computers and Fluids 28, 721–739, 1999MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. Fornberg: Steady viscous flow past a circular cylinder up to Reynolds number 600, J. of Comp. Physics 61, 297–320, 1985MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    M. Pandolfi and F. Larocca: Transonic flow about a circular cylinder, Computers and Fluids 17, 205–220, 1989CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • M. Hafez
    • 1
  1. 1.Department of Mechanical and Aeronautical EngineeringUniversity of CaliforniaDavisUSA

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