The Euler equations for multiphase compressible flow in conservation form simulation of shock–bubble interactions

The Euler equations, together with an equation of state, govern the motion of an inviscid compressible fluid. Here, a new equation of state for volumes containing both gas and liquid is derived; this allows the Euler equations for two substances, here air and water, to be expressed in pure conservation form. This in turn allows simulation of shocks in water interacting with small bubbles of air as the meniscus no longer needs to be tracked explicitly. Extension to three space dimensions is shown to be straightforward. A test case showing how a shock wave in water interacts with a small (two-dimensional) air bubble is presented. Simulations of a shock wave interacting with two air bubbles, and a small multiphase region (comprising 50% water and 50% air by volume) are then given.