Local Preconditioning Techniques Coupled with a Variational Multiscale Method to Solve Compressible Steady Flows at Low Mach Numbers

In this work we evaluate a combination of the Weiss-Smith and Choi-Merkle local preconditioners coupled with the density-based Nonlinear Multiscale Viscosity (NMV) finite element method for solving steady compressible flows at low Mach numbers. The multiscale formulation is based on the strategy of separating scales, in which the subgrid scale space is spanned by bubble functions, allowing to use a static condensation procedure in the local matrix system to define the resolved scale problem. Also, a residual-based nonlinear viscosity operator is added to the Galerkin formulation in order to obtain a stabilized formulation. As density-based methods do not work well in problems with Mach numbers tending to zero, resulting in a degradation of the solution accuracy, the resulting numerical method gathering those two approaches allows to solve compressible flows in the incompressible limit. We evaluate this methodology simulating a steady flow over the NACA 0012 airfoil under some regimes of inflow Mach numbers. The numerical result exhibits promising solutions to compressible flow problems in the incompressible limit.