Pseudo-Unsteady Methods for Inviscid or Viscous Flow Computation

  • R. Peyret
  • H. Viviand


This paper presents a review of various pseudo-unsteady techniques based upon different principles used for the numerical calculation of inviscid or viscous flows.

Two types of approaches to devise pseudo-unsteady methods are considered. The first, which relates to the mathematical formulation of the time-dependent problem, consists essentially in using artificial time-derivative terms associated with the steady equations, in order to obtain an unsteady (nonphysical) system with better properties (from the computational point of view) than the exact unsteady system. The second type, which relates to the numerical algorithm, consists in using numerical schemes that are not consistent with the unsteady equations but become consistent with the steady equations at convergence. Various criteria exist for such a choice and they are based on numerical considerations, such as stability, damping, and simplicity of implementation.

In the first part of the paper, the general concepts for constructing pseudo-unsteady systems are discussed and some examples presented for viscous flows. Then, some usual nonconsistent schemes are reviewed and their properties analyzed for model equations.

In the second part, a general family of pseudo-unsteady systems for the computation of inviscid compressible isoenergetic flows or incompressible flows is presented. These systems are required to be hyperbolic with respect to time. After a general discussion of this condition, examples are presented and it is shown, in particular, that there exist systems that are optimum for the stability condition of CFL type encountered with explicit numerical schemes.


Incompressible Flow AIAA Paper Compressible Flow Flow Computation Compressible Fluid 
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.
    L. Crocco, A suggestion for the numerical solution of the steady Navier-Stokes equations, AIAA J. 3, 1824–1832 (1965).MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. C. Du Fort and S. P. Frankel, Stability conditions in the numerical treatment of parabolic differential equations, Math. Tables Aids Comp. 7, 135–152 (1953).CrossRefGoogle Scholar
  3. 3.
    G. W. Evans, R. Brousseau and R. Kierstead, Stability considerations for various difference equations derived from the linear heat conduction equation, J. Math. Phys. 34, 267–285 (1956).MATHGoogle Scholar
  4. 4.
    R. Peyret and H. Viviand, Calcul de l’écoulement d’un fluide visqueux compressible autour d’un obstacle de forme parabolique, in: Lecture Notes in Physics, Vol. 19, pp. 222–228, Springer-Verlag, Berlin-Heidelberg-New York (1973).Google Scholar
  5. 5.
    G. D. Raithby and K. E. Torrance, Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow, Comput. Fluids 2, 191–206 (1974).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    F. H. Harlow and J. E. Welsh, Numerical calculation of time-dependent viscous incompressible flow, Phys. Fluids 8, 2182–2189 (1965).MATHCrossRefGoogle Scholar
  7. 7.
    A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22, 745–762 (1968).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    R. Temam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II), Arch. Ration. Mech. Anal. 33, 377–385 (1969).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow, Springer-Verlag, Berlin-Heidelberg-New York (1983).MATHGoogle Scholar
  10. 10.
    A. J. Chorin, Numerical Study of Thermal Convection in a Fluid Layer Heated from Below, New York University, A.E.C. Research and Development Report. No. NYO-1480-61 (August, 1966).Google Scholar
  11. 11.
    A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys. 2, 12–26 (1967).MATHCrossRefGoogle Scholar
  12. 12.
    R. Temam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I), Arch. Ration. Mech. Anal. 32, 135–153 (1969).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    R. Temam, Navier-Stokes Equations, North-Holland Publishing Company, Amsterdam (1977).MATHGoogle Scholar
  14. 14.
    M. Fortin, R. Peyret, and R. Temam, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible, J. Méc. 10, 357–390 (1971).MathSciNetMATHGoogle Scholar
  15. 15.
    E. Elsaesser and R. Peyret, Méthodes hermitiennes pour la résolution des équations de Navier-Stokes, in: Méthodes Numériques dans les Sciences de l’Ingénieur (E. Absi and R. Glowinski, eds.), pp. 249–258, Dunod, Paris (1979).Google Scholar
  16. 16.
    E. Elsaesser and R. Peyret, Hermitian methods for the solution of the Navier-Stokes equations in primitive variables, in: Recent Advances in Numerical Methods in Fluids, Vol. 3: Viscous Flows Computational Methods (W. G. Habashi, ed.), Pineridge Press, Swansea (to appear).Google Scholar
  17. 17.
    X. Aubert and M. Deville, Steady viscous flows by compact differences in boundary-fitted coordinates, J. Comput. Phys. 49, 490–522 (1983).MATHCrossRefGoogle Scholar
  18. 18.
    N. N. Yanenko, The Method of Fractional Steps, Springer-Verlag, Berlin-Heidelberg-New York (1971).MATHCrossRefGoogle Scholar
  19. 19.
    T. D. Taylor and E. Ndefo, Computation of viscous flow in a channel by the method of splitting, in: Lecture Notes in Physics, Vol. 8, pp. 356–364, Springer-Verlag, Berlin-Heidelberg-New York (1971).Google Scholar
  20. 20.
    H. J. Wirz, Relaxation methods for time-dependent conservation equations in fluid mechanics, in: AGARD-VKI Lecture Series, No. 86, Von Karman Institute for Fluid Dynamics, Rhode-St-Genèse (1977).Google Scholar
  21. 21.
    S. M. Richardson and A. R. H. Cornish, Solution of three-dimensional flow problems, J. Fluid Mech. 82, 309–319 (1977).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    G. J. Hirasaki and J. D. Heliums, Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics, Q. Appl. Math. 28, 293–296 (1970).MATHGoogle Scholar
  23. 23.
    G. J. Hirasaki and J. D. Heliums, A general formulation of the boundary conditions on the vector potential in three-dimensional hydrodynamics, Q. Appl. Math. 26, 331–342 (1968).MATHGoogle Scholar
  24. 24.
    G. D. Mallinson and G. I. De Vahl Davis, The method of false transient for the solution of coupled elliptic equations, J. Comput. Phys. 12, 435–461 (1973).MATHCrossRefGoogle Scholar
  25. 25.
    P. Bontoux, Contribution à l’étude des écoulements visqueux en milieu confiné, Thèse Doctorat d’Etat, IMFM, Université d’Aix-Marseille (1978).Google Scholar
  26. 26.
    P. Bontoux, B. Gilly, and B. Roux, Analysis of the effect of boundary conditions on numerical stability of solutions of Navier-Stokes equations, J. Comput. Phys. 15, 417–427 (1980).MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Douglas and J. E. Gunn, A general formulation of alternating direction methods, Numer. Math. 6, 428–453 (1964).MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    O. D. Burggraf, Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech. 24, 113–151 (1966).CrossRefGoogle Scholar
  29. 29.
    D. Gottlieb and B. Gustafsson, Generalized Du Fort-Frankel methods for parabolic initial-boundary-value problems, SIAM. J. Numer. Anal. 13, 129–144 (1976).MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    D. Gottlieb and L. Lustman, The Du Fort-Frankel-Chebyshev Method for Parabolic Initial-Boundary-Value Problems, ICASE, Report No. 81-42 (December, 1981).Google Scholar
  31. 31.
    J. S. Allen and S. I. Cheng, Numerical solution of the compressible Navier-Stokes equations for the laminar near wake, Phys. Fluids 19, 37–52 (1970).CrossRefGoogle Scholar
  32. 32.
    I. Yu. Brailovskaya, A difference scheme for numerical solution of the two-dimensional nonstationary Navier-Stokes equations for a compressible gas, Sov. Phys. Dokl. 10, 107–110 (1965).Google Scholar
  33. 33.
    R. Peyret and H. Viviand, Calcul numérique de l’écoulement supersonique d’un fluide visqueux sur un obstacle parabolique, La Rech. Aérosp., No. 1972-3, 123–131 (1972).Google Scholar
  34. 34.
    P. Garabedian, Estimation of the relaxation factor for small mesh size, Math. Tables Aids Comp. 10, 183–185 (1956).MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    H. Lomax and J. L. Steger, Relaxation methods in fluid mechanics, in: Annual Review in Fluid Mechanics, Vol. 7, pp. 63–88, Annual Review Inc., Palo Alto (1975).Google Scholar
  36. 36.
    R. Peyret and H. Viviand, Computation of viscous compressible flows based on the Navier-Stokes equations, AGARDograph No. 212 (1975).Google Scholar
  37. 37.
    C. P. Li, Numerical solution of viscous reacting blunt body flows of a multicomponent mixture, AIAA paper No. 73-202 (1973).Google Scholar
  38. 38.
    C. P. Li, Hypersonic nonequilibrium flow past a sphere at low Reynolds numbers, AIAA paper No. 74-173 (1974).Google Scholar
  39. 39.
    R. W. MacCormack, The effect of viscosity in hypervelocity impact cratering, AIAA paper No. 69-354 (1969).Google Scholar
  40. 40.
    H. McDonald and W. R. Briley, Computational fluid dynamics aspects of internal flows, AIAA paper No. 79-1445 (1979).Google Scholar
  41. 41.
    E. L. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, New Jersey (1966).MATHGoogle Scholar
  42. 42.
    S. J. Shamroth and H. J. Gibeling, The prediction of a turbulent flow field about an isolated airfoil, AIAA paper No. 79-1543 (1979).Google Scholar
  43. 43.
    S. J. Shamroth and H. J. Gibeling, A compressible solution of the Navier-Stokes equations for turbulent flow about an airfoil, NASA CR-3183 (October, 1979).Google Scholar
  44. 44.
    W. R. Briley and H. McDonald, Solution of the three-dimensional compressible Navier-Stokes equations by an implicit method, in: Lecture Notes in Physics, Vol. 35, pp. 105–110, Springer-Verlag, Berlin-Heidelberg-New York (1975).Google Scholar
  45. 45.
    A. Lerat, J. Sides, and V. Daru, An implicit finite-volume method for solving the Euler equations, Lecture Notes in Physics, Vol. 170, pp. 343–349, Springer-Verlag, Berlin-Heidelberg-New York (1982).Google Scholar
  46. 46.
    H. Viviand, Pseudo-unsteady methods for transonic flow computations, in: Lecture Notes in Physics, Vol. 141, pp. 44–54, Springer-Verlag, Berlin-Heidelberg-New York (1981).Google Scholar
  47. 47.
    A. Jameson, W. Schmidt, and E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, AIAA Paper No. 81-1259 (1981).Google Scholar
  48. 48.
    H. Viviand, Systèmes pseudo-instationnaires pour les écoulements stationnaires de fluide parfait, Publication ONERA No. 1983-4 (1983).Google Scholar
  49. 49.
    J. Brochet, Calcul numérique d’écoulements internes tridimensionnels transsoniques, La Rech. Aérosp., No. 1980-5, 301-315 (1980).Google Scholar
  50. 50.
    H. Viviand and J. P. Veuillot, Méthodes pseudo-instationnaires pour le calcul d’écoulements transsoniques, Publication ONERA No. 1978-4 (1978).Google Scholar
  51. 51.
    A. Rizzi and L. E. Eriksson, Transfinite mesh generation and damped Euler equation algorithm for transonic flow around wing-body configurations, AIAA Paper, No. 81-0999 (1981).Google Scholar

Copyright information

© Plenum Press, New York 1985

Authors and Affiliations

  • R. Peyret
    • 1
    • 2
  • H. Viviand
    • 1
  1. 1.Département de MathématiquesUniversité de NiceNice CedexFrance
  2. 2.ONERAChatillonFrance

Personalised recommendations