Numerical simulation of two-fluid two-phase flows by HLL scheme using an approximate Jacobian matrix

A compressible two-fluid two-phase flow model based on two sets of governing equations is formulated. We solve, using the second-order Harten, Lax, and van Leer (HLL) scheme, five disparate two-phase benchmark problems such as shock propagation in the two-phase medium, the cavitating flow, and the gravity-induced phase separation. In the conventional HLL scheme, the simple sonic speeds evaluated from the two single phases are used in lieu of the fastest speeds in the two phases, since their accurate estimation is difficult. To improve the method, we here propose utilization of the analytic eigenvalues from the temporary 6 x 6 Jacobian matrix, reduced by dropping the interfacial transfer terms. The total sound speed of the two-phase flow evaluated by these eigenvalues agrees very well with the existing experimental data. The second-order HLL scheme using these analytic eigenvalues is proved efficient, robust, and accurate in comparison with other available methods.