Stabilized finite element method based on local preconditioning for unsteady compressible flows in deformable domains with emphasis on the low Mach number limit application

Abstract

Flows with low Mach numbers represent a limit situation in the solution of compressible flows. The preconditioning of flow equations is one of the classical approaches proposed to capture the solution in the low Mach number limit. In this method, the time derivatives are premultiplied by a suitable preconditioning matrix in order to achieve a well-conditioned system by means of the scaling of the system eigenvalues. Hence, the modified equations have only steady-state solutions in common with the original system. For the application of these methods to unsteady problems, the dual time-stepping technique has emerged, where the physical time derivative terms are treated as source and/or reactive terms. The use of a preconditioning matrix defined to compute steady-state solutions may not be a good choice for unsteady problems, as showed by Vigneron et al. (European Conference on Computational Fluid Dynamics, 2006). However, such matrices can be adapted to perform the computation of transient flows by means of the appropriate redefinition of some coefficients. The application of a ?steady-state? preconditioning matrix to unsteady problems with an ALE (Arbitrary Lagrangian Eulerian) approach is presented. The equations are discretized in space using a stabilized Finite Element method and in time using finite differences. The preconditioning of the governing equations is not applied in the numerical scheme but is used, through the eigenvalues of the preconditioned system, to design appropriately the stabilization term. Several test cases are solved, including incompressible flows and the in-cylinder flow in a motored opposed-piston engine.