An acoustic-convective splitting-based approach for the Kapila two-phase flow model

TL;DR

The paper introduces an acoustic-convective splitting-based scheme for the Kapila five-equation two-phase flow model, decoupling pressure/ acoustic waves from convective transport. The acoustic part is solved in Lagrangian coordinates with an HLLC-type Riemann solver, while the convective part uses a standard upwind scheme, enabling a simple method that accommodates a general equation of state. The method preserves conservation of mass, momentum, energy and partial mass and maintains positivity of volume and mass fractions, achieving accurate shock-capturing and good agreement with reference results across multiple shock-tube tests. Compared to the unsplit HLLC approach, the splitting method often allows larger time steps and can reduce numerical diffusion, especially in subsonic or mixed-flow regimes, with robust performance for multiphase interfaces and cavitation scenarios.

Abstract

In this paper we propose a new acoustic-convective splitting-based numerical scheme for the Kapila five-equation two-phase flow model. The splitting operator decouples the acoustic waves and convective waves. The resulting two submodels are alternately numerically solved to approximate the solution of the entire model. The Lagrangian form of the acoustic submodel is numerically solved using an HLLC-type Riemann solver whereas the convective part is approximated with an upwind scheme. The result is a simple method which allows for a general equation of state. Numerical computations are performed for standard two-phase shock tube problems. A comparison is made with a non-splitting approach. The results are in good agreement with reference results and exact solutions.

Figures (26)

• Figure 1: The different states $\mathbf{Q}^{\mathcal{L}ag}_L, \mathbf{Q}^{\mathcal{L}ag,*}_L, \mathbf{Q}^{\mathcal{L}ag,*}_R, \mathbf{Q}^{\mathcal{L}ag}_R$ and wave speeds $-a_{j+1/2}, 0, a_{j+1/2}$ in the Riemann problem.
• Figure 2: Translating interface problem - density profile - Exact solution "-", splitting approach "$\boldsymbol{\circ}$" and direct approach "$\bold{+}$"at $t=0.1$.
• Figure 3: Translating interface problem - velocity profile - Exact solution "-", splitting approach "$\boldsymbol{\circ}$" and direct approach "$\bold{+}$"at $t=0.1$.
• Figure 4: Translating interface problem - pressure profile - Exact solution "-", splitting approach "$\boldsymbol{\circ}$" and direct approach "$\bold{+}$"at $t=0.1$.
• Figure 5: Translating interface problem - volume and mass fraction profiles - Exact solution "-", splitting approach "$\boldsymbol{\circ}$" and direct approach "$\bold{+}$"at $t=0.1$.