Conservative Immersed Boundary Methods on Cartesian grids for inviscid compressible flows simulation

Abstract

This work introduces three conservative methods based on the Immersed Boundary Method. These methods make use of cut-cells to ensure the conservation properties in the numerical solution. However, since some cut-cells can be very small, they can significantly restrict the time step of an explicit time integration scheme. To circumvent this limitation, a semi-implicit treatment of the small cells is employed. The first method relies on a straightforward flux redistribution procedure that globally restores conservation on the cut-cells grid. The other two methods employ the local conservative discretization form of the finite volume method, along with optimization procedures, to ensure local conservation of the numerical solution within each cell. These methods have been tested on two-dimensional inviscid compressible flow problems, demonstrating results comparable to those obtained with the standard Cut-Cells method in terms of accuracy and conservation. Furthermore, the methods are stable and can be effectively used with an explicit time integration scheme without encountering any stability issues related to the small cut-cells.