A generalized approach for sub- and super-critical flows using the Local Discontinuous Galerkin method

We present a generalized approach to solve sub- and super-critical flows using the Local Discontinuous Galerkin (LDG) method. The 5-equation model is used to capture multiple phases. This model converges to the Euler equation when only one phase is present. The formulation is independent of the equation of state (EOS) in each phase and thus allows for a more accurate representation of the different fluids by using their appropriate EOS. The multi-phase model includes the convection of the volume fraction, which is modeled by a Hamilton-Jacobi equation. This appears to be the first study that demonstrates the solution of the volume fraction in a two-phase model using the LDG method. A modified moment limiter is used to prevent oscillations close to discontinuities and shows that limiting using primitive instead of conservative variables is significantly bigger in two phase flows when compared to a single phase flow. An additional limiting technique to minimize the numerical diffusion is introduced, which is shown to be especially important at the interface. Canonical test cases are used to verify and validate the formulation in one, two, and three dimensions. Finally, the new two-phase LDG approach is applied to shock-bubble interaction studies and compared to available data.