Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model

Abstract

For single phase fluid models, like the Euler equations of compressible gas dynamics, upwind finite volume schemes suffer from a loss of accuracy when computing flows in the near incompressible regime. Preconditioning of the numerical dissipation is necessary to recover results consistent with the asymptotic behaviour of the continuous model. In this paper, we examine this situation for a two phase flow model. We show that as in the single phase case, the numerical approximation has to be done carefully in the near incompressible regime. We propose to adapt the preconditioning strategy used for single phase problems and present numerical results that show the efficiency of this approach

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

“…The Roe-Turkel scheme uses then the expression of A ± Q in (4) with V Q i+1/2 replaced by the preconditioned dissipation term in (6). In practice, in the implementation of the algorithm we still use the standard expression of A ± Q in Sec.…”

“…The behavior of solutions of the 5-equation model as the Mach number tends to zero was first studied by Murrone-Guillard in [6], by using asymptotic expansions in powers of the Mach number M W * , where here M W * = | u * | c W * is a reference Mach number based on the Wood's sound speed. In particular, the authors in [6] show that, analogously to the results found for the Euler equations, at the continuous level in the low Mach number limit pressure fluctuations scale with the square of the Mach number, p( …”

“…We perform a two-phase liquid-gas flow numerical experiment analogous to channel flow tests considered by several authors in the literature, e.g. [2,6]. We simulate a flow of liquid water initially containing a uniformly distributed small amount of gas, α g0 = 10 −3 , in a channel of length = 4 m and height = 1 m with a bump defined by y = (1 − cos((x − 1)π ))/10, if x ∈ [1,3], and y = 0 otherwise.…”

Numerical simulations of low Mach compressible two-phase flows: Preliminary assessment of some basic solution techniques

“…The successful extension of the approaches among them to multiphase flow problems include the preconditioned method of Murrone and Guillard [28] and the constrained interpolation profile method of Yabe et al [52]. Our goal, in this work, is to explore further some of these low speed approaches in the hope to come up with a robust and accurate numerical method for general weakly compressible multiphase flow with real materials.…”

“…This method yield a class of finite volume compressible flow solvers constituting a viable approach when the Mach number goes to zero. Lastly, in the work of Guillard and coworkers [11,28] a preconditioning technique has been used to construct the flux functions that can also lead to the correct asymptotic behavior of the solution in the zero Mach number limit; this is an extension of the approach given previously by Turkel for steady flows [46,47].…”