An Immersed Boundary Method for hyperbolic problems applied to the coupled Euler equations with moving boundaries

The Immersed Boundary Method (IBM) was initially developed for incompressible viscous flow, i.e. parabolic equations. Here, the boundary condition is the usual no-slip. This implies that the velocity is continuous across the boundary, and there is no ambiguity as to how to calculate the Dirac source term in the governing equations. However, the application of IBM to compressible high Reynolds number flows is not straightforward because the solution is primarily dominated by the hyperbolic subsystem of the governing equations. In the hyperbolic setting, one is not free to impose boundary conditions at either side of the boundary without consideration of the direction of propagation of information (characteristic directions). This complicates the construction of the IBM sources since continuity of the solution is not available. It was shown in Bermudez et. al (1994) that characteristic treatment of the hyperbolic equations with a source is possible when the source is a known function. The purpose of this research is to study a correct (accurate) and stable implementation of IBM when solving hyperbolic equations representing compressible flows with a Dirac-measure type of source. We propose a scheme that combines IBM and characteristic information to enforce boundary conditions. The method is tested against some exact solutions and other non-trivial moving boundary problems.