A stabilized finite element formulation to solve high and low speed flows

Abstract

SUMMARYIt 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 the Jacobian matrices fully invertible whatever the ow regime may be. Despite this apparent beneÿt, 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 ows and the other one for supersonic ows, so that a wide range of ow regimes are covered with only one formulation. Comparison between conservation and pressure variables is made and numerical examples are shown to validate the method.

Simulation of Low-Speed Buoyant Flows with a Stabilized Compressible/Incompressible Formulation: The Full Navier–Stokes Approach versus the Boussinesq Model

“…For definitions of the τ matrix for low-speed flows based on primitive variables, see [20,29], and for compressible flows, [30][31][32][33]. Other proposals for τ for a low-speed limit can be found in [34]. As shown in Codina et al [14], the buoyancy terms in C need not intervene into τ; therefore, the above definitions for incompressible flows can be used directly for the Boussinesq model.…”

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

“…to stabilize the numerical scheme considering only the advective part of the preconditioned system. The numerical diffusivity introduced by the SUPG method in the inviscid case is [22] K num v =Ã vsvÃv (28) wheres v is the matrix of intrinsic time scale in the viscous variables basis. In the conservative variables basis this matrix is expressed as…”

“…Therefore, the resulting variational formulation that we solve is the standard variational formulation for compressible viscous flows. The difference is in the definition of the stabilization matrix s. This kind of finite elements scheme was proposed and applied in several works [27,28], in which a proper definition of the s matrix was given. Then, the procedure we followed in this study could be a way to find a suitable definition of the stabilization parameter.…”