A stabilized finite element formulation to solve high and low speed flows. (English) Zbl 1096.76024

Costa, Gustavo K.; Lyra, Paulo R. M.

Commun. Numer. Methods Eng. 22, No. 5, 411-419 (2006).

Summary: It is well known that numerical methods designed to solve the compressible Euler equations, when written in terms of conservation variables, behave poorly in the incompressible limit, that is, when density variations are negligible. However, a change to pressure-based variables seem to, partly, eliminate the problem by making Jacobian matrices fully invertible whatever the flow regime may be. Despite this apparent benefit, the stabilization matrix plus discontinuity capturing operator (for the compressible regime) still need attention, since they tend to be ill behaved for either conservation or pressure variables. In this paper, we introduce a simple way of balancing two stabilizing matrices, one of them suitable for low Mach number flows and the other one for supersonic flows, so that a wide range of flow regimes is covered with only one formulation. Comparison between conservation and pressure variables is made, and numerical examples are shown to validate the method.

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
76N15	Gas dynamics (general theory)

Keywords:

Euler equations; SUPG; stabilization matrix

References:

[1]	Nigro, Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 154 pp 203– (1998) · Zbl 0957.76032
[2]	Turkel, Frontiers of Computational Fluid Dynamics pp 215– (1994)
[3]	Choi, The application of preconditioning in viscous flows, Journal of Computational Physics 105 pp 207– (1993) · Zbl 0768.76032
[4]	Moussaoui, A unified approach for inviscid compressible and nearly incompressible flow by least-squares finite element method, Applied Numerical Mathematics 44 pp 183– (2003) · Zbl 1019.76024
[5]	Hauke, A unified approach to compressible and incompressible flows, Computer Methods in Applied Mechanics and Engineering 113 pp 389– (1994) · Zbl 0845.76040
[6]	Wong, The solution of the compressible Euler equations at low Mach numbers using a stabilized finite element algorithm, Computer Methods in Applied Mechanics and Engineering 190 pp 5719– (2001) · Zbl 1044.76036
[7]	Mittal, A unified finite element formulation for compressible and incompressible flows using augmented conservation variables, Computer Methods in Applied Mechanics and Engineering 161 pp 229– (1998) · Zbl 0943.76050
[8]	Zienkiewicz, Compressible and incompressible flow; an algorithm for all seasons, Computer Methods in Applied Mechanics and Engineering 78 pp 105– (1990) · Zbl 0708.76099
[9]	Shakib, A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 89 pp 141– (1991)
[10]	Aliabadi, Parallel fluid dynamics computations in aerospace applications, Computer Methods in Applied Mechanics and Engineering 21 pp 783– (1995) · Zbl 0862.76033
[11]	Lyra, A review and comparative study of upwind biased schemes for compressible flow computation. Part III: multidimensional extension on unstructured grids, Archives of Computational Methods in Engineering 9-3 pp 5719– (2002)