Numerical investigations into a model of partially incompressible two-phase flow in pipes

We consider a model for flow of liquid and gas in a pipe. We assume that the gas is ideal and that the liquid is incompressible. Under this assumption the resulting system of equations, expressing conservation of mass and momentum, splits into two subsystems such that the gas flow is independent of the liquid flow, and the liquid flow is described by a conservation law parameterized by the mass fraction of gas. When solving these equations numerically, we propose to stagger the gas and liquid variables with respect to each other. The advantage of this is that in finite volume methods one can use numerical flux functions designed for 2×2 systems of hyperbolic conservation laws to solve both the gas flow and the liquid flow, rather than a much more complicated numerical flux for the whole 4×4 system. We test this using the Roe numerical flux for both subsystems, and compare the results with results produced by using the second-order Nessyahu–Tadmor scheme for the second subsystem