Finite Element Solution of a Stream Function-Vorticity System and Its Application to the Navier Stokes Equations

Fattehallah Ghadi, Vitoriano Ruas, Mohamed Wakrim

Equipe Modélisation Mathématique et Simulation, Faculté des Sciences Université Ibn Zohr, Agadir, Maroc.
Institut Jean Rond d’Alembert/CNRS, UPMC Université, Paris, France.

DOI: 10.4236/am.2013.41A039                

Abstract

The finite element solution of a generalized Stokes system in terms of the flow variables stream function and vorticity is studied. This system results from a time discretization of the time-dependent Stokes system in stream function-vorticity formulation, or yet by the application of the characteristics method to solve the Navier-Stokes equations in the same representation. Numerical results presented for both cases illustrate the good behaviour of the adopted approach.

Keywords

Characteristics Method; Navier-Stokes; Stream Function; Vorticity

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	F. A. Ghadi, V. Ruas and M. Wakrim, “A Mixed Finite Element to Solve the Stokes Problem in the Stream Function and Vorticity Formulation,” Hiroshima Mathematical Journal, Vol. 28, No. 3, 1998, pp. 381-398.
[2]	R. Glowinski and O. Pironneau, “Numerical Methods for the First Bi-Harmonic Equation and for the Two-Dimensional Stokes Problem,” SIAM Review, Vol. 21, No. 2, 1979, pp. 167-212. doi:10.1137/1021028
[3]	F. A. Ghadi, V. Ruas and M. Wakrim, “A Mixed Method to Solve the Evolutionary Stokes Problem in the Stream Function and Vorticity Formulation,” Proceedings of LUXFEM, Luxemburg, 13-14 November 2003.
[4]	F. A. Ghadi, V. Ruas and M. Wakrim, “Numerical Solution to the Time—Dependent Incompressible Navier-Stokes Equations by Piecewise Linear Finite Elements,” Journal of Computational and Applied Mathematics, Vol. 215, No. 2, 2008, pp. 429-437. doi:10.1016/j.cam.2006.03.047