A positivity-preserving high-order weighted compact nonlinear scheme for compressible gas-liquid flows

TL;DR

This paper introduces a high-order finite difference scheme for simulating compressible gas-liquid flows that preserves positivity and boundedness, ensuring physical realism and robustness under extreme conditions.

Contribution

It develops a positivity-preserving, high-order weighted compact nonlinear scheme for the five-equation model of two-phase flows, with novel limiting procedures ensuring physical bounds.

Findings

The scheme maintains positive densities and sound speeds in simulations.

It demonstrates high accuracy and robustness in tests with water and air.

The method is extendable to other conservative schemes.

Abstract

We present a robust, highly accurate, and efficient positivity- and boundedness-preserving diffuse interface method for the simulations of compressible gas-liquid two-phase flows with the five-equation model by Allaire et al. using high-order finite difference weighted compact nonlinear scheme (WCNS) in the explicit form. The equation of states of gas and liquid are given by the ideal gas and stiffened gas laws respectively. Under a mild assumption on the relative magnitude between the ratios of specific heats of the gas and liquid, we can construct limiting procedures for the fifth order incremental-stencil WCNS (WCNS-IS) with the first order Harten-Lax-van Leer contact (HLLC) flux such that positive partial densities and squared speed of sound can be ensured in the solutions, together with bounded volume fractions and mass fractions. The limiting procedures are discretely conservative for all conservative equations in the five-equation model and can also be easily extended for any other conservative finite difference or finite volume scheme. Numerical tests with liquid water and air are reported to demonstrate the robustness and high accuracy of the WCNS-IS with the positivity- and boundedness-preserving limiters even under extreme conditions.