Preprint 2003-072

Weakly Implicit Numerical Schemes for the Two-Fluid Model

Steinar Evje and Tore Flåtten

Abstract: The aim of this paper is to construct semi-implicit numerical schemes for a two-phase (two-fluid) flow model, allowing for violation of the CFL-criterion for sonic waves while maintaining a high level of accuracy and stability on volume fraction waves. Based on the results of a previous work \cite{evj02c}, we here present a general framework for constructing such {\it weakly implicit} schemes without making use of any Riemann solver nor referring to any calculation of flux jacobians.

One important step of the proposed methods is the introduction of a pressure evolution equation. This equation, which is discretized at cell-interfaces, naturally defines a consistent numerical flux for the discretization of the pressure term of the two momentum equations. This step is crucial for the stability of the solutions when the CFL-criterion for sonic waves is violated. Another major step is the decomposition of the numerical mass fluxes $F_k$, assumed to be consistent with the physical mass flux $f_k=\rho_k\alpha_k v_k$, into two components $F_k^D$ and $F_k^A$ respectively. The purpose of the $F_k^D$-component is to ensure stability (non-oscillatory behavior) of solutions when the time step is dictated by the fluid velocity and not the sonic velocity, whereas the $F_k^A$-component is designed such that accurate resolution of volume fraction waves is ensured. Our techniques, which we refer to as "Mixture Flux" (MF) methods, are based on the above two steps, but give room for different choices in the discretization of the pressure evolution equation as well as the construction of the $F_k^D$ and $F_k^A$ flux components. Particularly, by using an AUSMD type of discretization for the $F_k^A$-component (originally proposed for the Euler equtions in \cite{wad97}) we obtain a Weakly Implicit Mixture Flux AUSMD scheme.

We present several numerical simulations, all of them indicating that the CFL-stability of the resulting WIMF-AUSMD scheme is largely governed by the velocity of the volume fraction waves and not the rapid sonic waves. Comparisons with an explicit Roe scheme indicate that the scheme presented in this paper is highly efficient, robust and accurate on slow transients. In fact, by exploiting the possibility to take much larger time steps it outperforms the Roe scheme in the resolution of the volume fraction wave for the classical water faucet problem. On the other hand it is more diffusive on pressure waves. Although conservation of positivity for the masses is not proven, we demonstrate that a transition fix may be applied making the scheme able to handle transition to one-phase flow while maintaining a high level of accuracy on volume fraction fronts.

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Steinar Evje, <>
Tore Flåtten, <>
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<> December 9 2003.

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