Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation

Two strategies for the numerical resolution of a two-fluid model have been investigated. Both methods employ a Roe-type scheme. The first method (Roe4) solves the four-equation, one-pressure, isentropic two-fluid model directly. The second strategy (Roe5) is to add an evolution equation for the volume fraction. In the present case, that results in a five-equation two-pressure model, where it is necessary to employ pressure relaxation to calculate the typical two-phase problems that have been tested: The water faucet and two benchmark shock-tube problems known from the literature. The numerical calculations showed that the Roe4 and Roe5 schemes converge to the same solution when instantaneous pressure relaxation is employed in the Roe5 scheme. This is true both with and without the use of high-resolution flux-limiter functions. However, the Roe5 scheme was found to be significantly more diffusive than the Roe4 scheme. The diffusion is a strong function of the chosen time-step length, the grid size, whether a limiter function is employed or not, and also the liquid speed of sound. As the pressure-relaxation parameter in the Roe5 scheme was increased, the solution gradually approached that obtained using instantaneous pressure relaxation. Furthermore, the results indicate that the approach of two pressures and instantaneous pressure relaxation does not provide an easy way to overcome the problem of complex eigenvalues in the one-pressure two-fluid model.