A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Abstract

Abstract. The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict "perfectly" unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties ("perfect" representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.Mathematics Subject Classification. 65M99, 76M15, 76N15, 80A10.

A method to couple HEM and HRM two-phase flow models

“…We now want to highlight some properties of positivity of the regular solutions of ( (4)- (5), (10), (14)), or alternatively (15), (14). The demonstration of the following results can be found in Appendix 4 (section A).…”

“…We have introduced the source term Γ in (14). We can rewrite it in a form which will appear to be more convenient.…”

“…More precisely, the exact analytical form of the internal energy, for both the HEM and the HRM models, is such that the approximate Godunov scheme VFRoe-ncv (using variables U, P, ρ and U, P, ρ, C for HEM and HRM models respectively) perfectly preserves these pure contact waves on coarse and fine meshes (see 14 ). This is still true for the coupled HEM/HRM model, for similar reasons (see Appendix 3).…”

“…It refers to the work exposed in. 14 We formally set Γ = 0 in the system (4). A pure contact solution is an analytical solution of (4) (with Γ = 0) of the form:…”

Numerical simulations of gas-particle flows with combustion

“…Despite its simplicity, this property is not satisfied by all the classical finite volume schemes. The choice of the pressure law (6) is also important (see [13]). This property also appears to be important to compute more complicated configurations.…”