Free boundary method for calculating compressible viscous flows on unfitted meshes

Abstract

We further develop the Free Boundary Method (FBM) for calculating compressible flow equations on Cartesian unfitted grids which has been recently proposed for the Euler model of inviscid equations. The method belongs to the class of immersed boundary methods. The effect of the solid surface is taken into account by means of the compensating flux applied on the element of solid surface in the cut cell. This flux is introduced to compensate losses in mass, momentum, and energy which occur when we virtually remove the solid element from the cut cell. The method is extended to the Navier-Stokes equations. We show how the viscous compensating flux should be introduced to assure the adequate solution in the flow domain, discuss its discrete approximation and propose the numerical method for solving the system of discrete equations. Calculation of the compensating flux requires a few data about the solid geometry in the cut cell, namely fluid volume fraction, area of the solid element and its unit normal, and baricentric coordinates of the fluid sub-element. All the data can be obtained from the level set method for setting the solid geometry. Numerical results concern testing the FBM on the solution of several benchmark viscous problems of aerodynamics.