Numerical modelling of the Riemann problem for a mathematical two-phase flow model

We report preliminary results obtained using Godunov methods of a centred type for hyperbolic systems of conservation laws for which the analytical solution of the Riemann problem is too difficult to develop. For this purpose, we consider a mathematical model used for modelling an unsteady compressible non-equilibrium mixture two-phase flow which results in a well-posed initial value problem in a conservative form. The mathematical two-phase flow model consists of six first-order partial differential equations that represent one-dimensional mass, momentum and energy balances for a mixture of gas-liquid and gas volume concentration, gas mass concentration and the relative velocity balances. This system of six partial differential equations has the mathematical property that its six characteristic roots are all real with a complete set of independent eigenvectors which form the basic structure of the solution of the Riemann problem. The construction of the solution of the Riemann problem for the model poses several difficulties. Since the model possesses a large number of non-linear waves, it is not easy to consider each wave separately to derive a single non-linear algebraic equation for the unknown region, the star region, between the left and right waves. We propose numerical techniques of a centred type specifically developed for high speed single-phase gas flows. A main feature of centred methods is that they do not explicitly require the solution of the Riemann problem. This is a desirable property which guarantees that the methods can handle the solution of the Riemann problem numerically and resolve both rarefaction and shock waves for the model in a simple way with good accuracy. Finally, we present some numerical simulations for the gas-liquid two-phase flow Riemann problem to illustrate the efficiency of the proposed schemes.