A Two-Dimensional Relaxation Scheme for the Hybrid Modelling of Two-Phase Flows

Abstract

Recently, a new relaxation scheme for hybrid modelling of two-phase flows has been proposed. This one allows to obtain stable unsteady approximations for a system of partial differential equations containing non-smooth data. This paper is concerned with a two-dimensional extension of the present method, in which two alternative relaxation schemes are compared. A partial analysis of continuous stability is given.

A relaxation scheme for hybrid modelling of gas-particle flows

“…and we can see that imposing ϑ 1 , ϑ 2 > 0 in (18) leads to the same constraints. The solution of (27) provides us with "natural" conditions to choose a physically relevant lower bound for a 0 .…”

“…Some energy estimates for relaxation systems corresponding to (A1) and (A2) are given in Appendix B, in a one-dimensional framework (and in [18,16] in a 2D framework). These should not be confused with entropy estimates for barotropic Euler equations for instance, such as those provided in [4], [35] and [9] among others.…”

“…Assume that initial conditions are physically relevant (ρ l,r > 0 and R l,r > 0). Assume moreover that a 0 > a min 0 = max(r 1 , r 2 , 0) ≥ 0 with constant a 0 and r 1 , r 2 given by (16), (17). Then the Riemann problem associated with the homogeneous part of (7) that is (9), initial conditions (14) and Ψ = a 2 0 ϑ, admits a unique solution (t, x) → Z exact (x/t; Z l , Z r ), composed of four constant states Z l , Z 1 , Z 2 , Z r separated by three LD waves.…”

“…We check at once that formulae (18), (19), (20) are the unique solutions of (22), (23), (24). Eventually, the constructed analytical solution to the Riemann problem associated with (9) and initial conditions (14) can be sketched as:…”

“…Assume that |λ max |∆t/h < 1/2 and that the parameter a 0 guarantees the positivity of the density in the intermediate states defined by (18): ϑ 1,i+1/2 , ϑ 2,i+1/2 > 0, i ∈ Z. Then the evolution step (39) preserves the positivity of the cell values of the density and the whole algorithm computes ρ n+1 i ≥ 0 (since the relaxation step (33) preserves cell densities).…”