Simulation numerique par nne methode d' elements finis optimale des equations de N avier-Stokes en formulation
Keywords:
Driven cavity, finite element, Navier-Stokes equations, stream -function, vorticity, mixed approach, backstep problemAbstract
This paper present numerical results for the Navier-Stokes problem in terms of the stream-function and the vorticicty, obtained by our mixte approach [GHA 95] based on the decomposition of the vorticity function, in the case k = l,for the problem of the driven cavity and the backstep problem. In order to prove the performance of this method a comparision with the existing results is done.
Downloads
References
[ACH 92] ACHDOU Y., GLOWINSKI R., PIRONNEAU 0., « Tuning the mesh of
mixed method for the Stream function Vorticity formulation of the NavierStokes
equations», Numer. Math, 63, 145-163, 1992.
[BER 92] BERNARDI C., GIRAULT V., MADAY Y., « Mixed Spectral element
approximation of the Navier-Stokes equations in the Stream-function and
Vorticity formulation», IMA. J Numer. Ana. 12, 565-608, 1992.
[BRE 80] BREZZI F., RAPPAZ J., RAVIART P. A.,« finite Dimensional Approximation
of Nonlinear problems», Numer. Math. 36, 1-25, 1980.
[BRE 91] BREZZI F., FORTIN M., Mixed and hybrid finite element methods,
Springer-Verlag. 1991.
[BRE 76] BREZZI F., RAVAIRT P. A.,« Mixed finite Element Methods for 4th Order
Elliptic Equations», Topics in Numerical Analysis Ill (J.J.H. Miller ed.) p. 33-56,
London, Academic Press, 1976.
[BUF 91] BUFF AT M., Etude de Ia simulation numerique par une methode
d'elements finis des ecoulements internes subsonique instationnaires bi et tridimensionnels,
These de Doctorat es Sciences, Lyon, France, 1991.
[BUR 66] BURGGRAF 0. R., «Analytical and Numerical studies of the structure of
steady separated flows», J Fluid Mech Vol. 24, p. 113-152, 1966.
[CIA 78] CIARLET P. G., The finite element method for Elliptic problems
North-Holland publishing company. Amsterdam, New-York, Oxford, 1978/79.
[DHA 81] DHATT G., TOUZOT G., Une presentation de Ia methode des elements
finis, Les presses de I'Universite Laval Quebec Maloine S. A. editeur Paris, 1981.
[GHA 94] GHADI F., Resolution par Ia methode des elements finis des equations
de Navier-Stokes en formulation {7/J -w), These de I'Universite de Saint-Etienne,
France, 1994.
[GHA 95] GHADI F., RUAS V., WAKRIM M., «A mixed finite method to solve the
Stokes problem in {7/J - w) formulation », Hiroshima Mathematical Journal
Vol. 28, N°3, November, 1998.
[GLO 73] GLOWINSKI R., « Approximations extemes, par elements finis de
lagrange d'ordre un et deux du probleme de Dirichlet pour l'operateur biharmonique.
Methode iterative de resolution des problemes approches», topics in
Numerical Analysis (J .J .H. Miller, editor), 123-171, Academic Press, London,
[GLO 79) GLOWINSKI R., PIRONNEAU 0., « Numerical methods for the first
bihannonic equation and for the two dimensional Stokes problem », SIAM
Review 121, 167-212, 1979.
[GHI 89] GHIA K. N., OSSWALD G. A., GHIA U., « Analysis of incompressible
massively separeted viscous flows using unsteady Navier-Stokes equations »,
Int. J for Numer. Methods in Fluids, Vol. 9, I 025-1050, 1989.
[GIR 86] GIRAULT V., RAVIART P. A.,finite Element Method for Navier-Stokes
equations, Springer, Berlin Heidelberg New-York, 1986.
[HUG 93) HUGHES T. J. R., HAUKE G., JANSEN K., JOHAN Z., « Current
reflections on stabilized finite element methods for computational fluid
mechanics », Proceedings of the VIII international confrerence on
finite elements in fluids, 20-23, September 1993, p. 44-63, Barcelona,
CIMNE (Pineridge Press).
[KIM 85] KIM J., MOIN P., «Application of fractional-step method to incompressible
Navier-Stokes Equations», J Comput. Phys, 59, 308-323, 1985.
[MER 74] MERCIER B.,« Numerical solution of the bihannonic problem by mixed
finite elements of class C0», Boll. UMI. 10, (4), 133-149, 1974.
[MIY 73] Ml YOSHI T., « A finite element method for the solutions of fourth order
partial differential equations», Kunamoto. J Sci.(Math), 9, 87-116, 1973.
[ODE 93) 0DEN J. T., Wu W., LEGAT V., « An hp adaptive strategy for finite
element approximation of the Navier-Stokes equations », Proceedings of the
VIII international conference on finite elements in fluid, 20-23 September 1993,
p. 32-43, Barcelona, CIMNE (Pineridge Press).
[QUA 93] QUARTAPELLE L., Numerical solution of the incompressible NavierStokes
equations, Birkhiiuser, Basel, 1993.
[ROA 72] ROACHE P. J., Computational Fluid Dynamic, Hennosa, Albuquerque,
N. M., 87108, 1972.
[RUA 93] RUAS V., GHADI F.,« Numerical solution ofthe incompressible NavierStokes
equations by optimally approximation the Stream function and the vorticity
», Proceedings of the VIII international conference on finite elements in
fluids, 20-23 September 1993, p. 156-165, Barcelona, CIMNE (Pineridge Press).
[RUA 91] RUAS V., «Variational approaches to the two-dimensional Stokes system
in tennes of the vorticity », Mech. Research Communications, Vol. 18(6), 359-
, 1991.
[RUA 94] RUAS V., «On Fonnulation of Vorticity systems for a viscous incompressible
flow with numerical applications», ZAMM Z. Angew, Math. Mech. 74, 1,
-55, 1994.
[RUA 951] RUAS V ., « Iterative Solution of the Stationary Incompressible NavierStokes
Equations in Stream Function-Vorticity Fonnulation », CRAS de Paris,
t. 321,381-386, 1995.
[RUA 952] RUAS V., «Approximating Harmonic Bases in a Decoupled Solution of
Viscous Flow Equations in 'ljJ-w Formulation», ZAMM Z Angew, Math. Mech.
, 5, 407-408, 1995.
