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

## TL;DR

This paper develops a numerical method for simulating two-phase liquid and gas flow in pipes, leveraging the independence of subsystems to simplify computations and improve efficiency.

## Contribution

It introduces a staggered variable approach for separate numerical treatment of gas and liquid flows in a coupled two-phase model.

## Key findings

- The staggered variable method simplifies the numerical solution process.
- Using Roe flux for both subsystems yields accurate results.
- Comparison with Nessyahu--Tadmor scheme demonstrates effectiveness.

## Abstract

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 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 parametrized 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 2x2 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 4x4 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.