A simple method for compressible multiphase mixtures and interfaces

Abstract

SUMMARYWe develop a Godunov-type scheme for a non-conservative, unconditional hyperbolic multiphase model. It involves a set of seven partial di erential equations and has the ability to solve interface problems between pure materials as well as compressible multiphase mixtures with two velocities and nonequilibrium thermodynamics (two pressures, two temperatures, two densities, etc.). Its numerical resolution poses several di culties. The model possesses a large number of acoustic and convective waves (seven waves) and it is not easy to upwind all these waves accurately and simply. Also, the system is non-conservative, and the numerical approximations of the corresponding terms need to be provided. In this paper, we focus on a method, based on a characteristic decomposition which solves these problems in a simple way and with good accuracy. The robustness, accuracy and versatility of the method is clearly demonstrated on several test problems with exact solutions.

A Godunov-type method for the seven-equation model of compressible two-phase flow

“…7 with the ones proposed by Saurel and Abgrall [37] (referred to as G HLL ), by Andrianov et al [4] (referred to as G ASW ) and by Schwendeman et al [40] (referred to as G 1 ) on Fig. 8.…”

“…We begin by considering Riemann problems, for which exact solutions are available, and compare the numerical results with the ones provided by alternate methods available in the literature. More precisely, we will consider the HLL-type method proposed by Saurel & Abgrall [37], the strategy presented by Andrianov, Saurel & Warnacke [4], the VFRoe scheme derived by Gallouët, Hérard & Seguin [21], and at last the Godunov scheme of Schwendeman, Wahle & Kapila [40]. Note from now on that the comparison will be based on simulations and pictures proposed in these papers only.…”

“…As far as the other methods are concerned, we first observe that the G HLL method proposed by Saurel and Abgrall [37] is the most diffusive on this test case. The G ASW method of Andrianov et al [4] is less diffusive but does not provide us with a very good approximation of the constant states of the u I coupling wave. On the contrary, we can observe that our approach behaves very well in the vicinity of this coupling wave (while still presenting overshoots and undershoots next to the extreme waves of the solution).…”

“…The Riemann initial data is the same as for Test 2. So far, we dealt with χ = 1 for the sake of comparison with the results presented in [37], [4], [40]. We now also consider the natural values χ = 0 and χ = 0.5.…”

“…In particular, the following is certainly not exhaustive. Let us first mention that Saurel & Abgrall [37] and Andrianov, Saurel & Warnacke [4] for instance (see also Saurel & Lemetayer [39] for a multidimensional framework) take into account the nonconservative terms by means of a free streaming physical condition associated with uniform velocity and pressure profiles. The discretization technique of [37] is improved by the same authors in [38].…”