(A) hyperbolic solution method for two-phase flow using surface tension effect

Abstract

The difficulty of mathematically ill-posed initial value problem that the system of equations for two-phase flow has is now removed by introducing the concept of surface tension thickness at the interface. The pressure discontinuity at the interface is split in this process into the two phasic components, that are the jumps from the phasic pressure to the representative interfacial pressure. The two bulk moduli, $L^l$ and $L^g$, obtained from the surface tension concept, have played a crucial role in making the equation system mathematically a complete hyperbolic type. The new hyperbolic equation system produces five different sets of eigenvalues by different combinations of the bulk moduli, among which three sets turn out to coincide with the existing two-phase flow regimes. The present theory gives expression for the surface tension thickness which has been only qualitatively predicted in the physical chemistry. Being hyperbolic type, the present governing equation system can now be numerically solved by such a high-order upwind scheme as the flux vector splitting method well treated in the gas dynamics. As a results, the numerical diffusion or artificial dissipation related with poor accuracy and stability property of the conventional upwind scheme is significantly removed. The precise agreement of the present result and experiment suggests that the theoretical and numerical formulation developed in this study has presented a totally new working methodology for the two fluids, two-phase flow.