Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model

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We describe two-phase flows by a six-equation single-velocity two-phase compressible flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid–gas mixtures such as cavitating flows. For the numerical approximation of the homogeneous hyperbolic portion of the model equations we have previously developed two-dimensional wave propagation finite volume schemes that use Roe-type and HLLC-type Riemann solvers. These schemes are very suited to simulate the dynamics of transonic and supersonic flows. However, these methods suffer from the well known difficulties of loss of accuracy and efficiency encountered by classical upwind finite volume discretizations at low Mach number regimes. This issue is particularly critical for liquid–gas flows, where the Mach number may range from very low to very high values, due to the large and rapid variation of the acoustic impedance. In this work we focus on the problem of loss of accuracy of standard schemes related to the spatial discretization of the convective terms of the model equations. To address this difficulty, we consider the class of preconditioning strategies that correct at low Mach number the numerical dissipation tensor. First we extend the approach of the preconditioned Roe–Turkel scheme of Guillard–Viozat for the Euler equations [Computers & Fluids, 28, 1999] to our Roe-type method for the two-phase flow model, by defining a suitable Turkel-type preconditioning matrix. A similar low Mach number correction is then devised for the HLLC-type method, thanks to a novel reformulation of the HLLC solver. We present numerical results for two-dimensional liquid–gas channel flow tests that show the effectiveness of the proposed preconditioning techniques. In particular, we observe that the order of pressure fluctuations generated at low Mach number regimes by the preconditioned methods agrees with the theoretical results inferred for the continuous relaxed two-phase flow model by an asymptotic analysis.