Compressible solver for two-phase flows with sharp interface and capillary effects preserving accuracy in the low Mach regime

Abstract

We present a sharp interface approach for compressible two-phase flows that preserves its accuracy in the low Mach regime. The interface between both fluids is captured by a Level Set function via the Ghost Fluid method. In the low Mach regime, classic Finite Volume compressible solvers lose accuracy on quadrangle or hexahedral meshes and a low Mach correction is then necessary to reduce the numerical dissipation. The sharp interface that separates both fluids, may induce an important jump in the medium properties that induces additional challenges for the design of a numerical scheme. Indeed, the well-known low Mach fixes in the literature could lead to significant truncation errors when the flow involves large density and sound speed ratios between both phases. To preserve the accuracy of the numerical method, we propose a specific low Mach fix for this interface problem that yields a uniform truncation error with respect to the Mach number. Numerical results show significant evidences of the efficiency and accuracy of the proposed low Mach correction.

An acoustic-transport splitting method for the barotropic Baer-Nunziato two-phase flow model

“…Overall Discretization After injecting (30) in ( 29) one obtains the complete update procedure from t n to t n+1 for the conservative variables:…”

“…(4) Thanks to (30), the CFL condition (32) ensures that (ρ k ) ♯ j > 0 and 0 < (α k ) ♯ j < 1 for j ∈ Z. The CFL condition (32) yields that (ρ k ) n+1 j and (α k ) n+1 j are convex combinations of (ρ k ) ♯ l and (α k ) ♯ l respectively for l = j ± 1, j and therefore (ρ k ) n+1 j > 0 and 0 < (α k ) n+1 j < 1.…”

“…The idea is then to propose a time-implicit treatment of the acoustic waves, in order to get rid of a too restrictive CFL condition, together with an explicit treatment of the contact waves in order to preserve accuracy. This strategy has already been exploited earlier within the framework of Euler equations and Shallow Water equations (see for instance [8,10,17]) using a Lagrange-Projection approach, but also for two-phase flows models (see [6,9,16,24,28,30]).…”