AN APPROXIMATE RIEMANN SOLVER FOR A 2-PHASE FLOW MODEL WITH NUMERICALLY GIVEN SLIP RELATION

In this paper an approximate Riemann solver is constructed for solving one class of two-phase models. These models describe the gas-liquid h ow in a long tube where the flow behaviour perpendicular to the tube a xis is averaged, so that the model is essentially one-dimensional in t he direction of the axis. The model consists of equations for the cons ervation of mass for each of the phases and the conservation of moment um of the mixture. In addition, an equation is supplied which relates the velocities of the two phases at any point, the slip relation, whic h may have a different shape for each of the flow regimes of interest and change in time. Generally, a slip relation will not be known entir ely in algebraic form, but given partly in numerical form. Under certa in restrictions the resulting system of conservation laws is hyperboli c and allows discontinuous solutions. The general idea is that the num erical algorithm for solving this model must be able to handle any val id slip relation. As the slip relation affects the Jacobian of the flu x function to a large extent, this means that flux vector or flux diff erence splittings cannot be based on algebraic manipulation of the Jac obian. We propose to use a first-order upwind scheme of Roe-type, wher e the construction of the approximate Riemann solver is fully numerica l. This basic scheme, however, has the same limitation as the original Roe method, namely, that for systems the positivity of the solution i s not guaranteed. The modification of the basic scheme to ensure posit ivity of the solution, is based on the HLL Riemann solver. (C) 1998 Sh ell International Oil Products B.V. Published by Elsevier Science Ltd. All rights reserved.