A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

Abstract

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.

Approximate solutions of the Baer-Nunziato Model

“…Eventually we emphasize that more accurate schemes are necessary in order to achieve cheap enough and reliable flow simulations. Among recent proposals dedicated to the Baer-Nunziato model, we would like to point out the relaxation scheme introduced in [8,20]. In addition, for considering water-hammer flows in pipes, liquid-vapour phase transition has to be taken into account.…”

A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

“…The next section is devoted to the derivation of the relaxation model ( [6,16,[24][25][26][27][28][29][30][31][32][33]) in order to obtain the required approximate Riemann solver. The next section is devoted to the derivation of the relaxation model ( [6,16,[24][25][26][27][28][29][30][31][32][33]) in order to obtain the required approximate Riemann solver.…”

“…Then we obtain w eq at rest given by (33). Copyright (ii) Let w L and w R be two states in that satisfy…”

Validation of a two-fluid model on unsteady liquid–vapor water flows

“…A second algorithm was built using the VFRoe-ncv approximate Godunov solver detailed in [24], that relies on the non-conservative symmetrizing variable a v ; U k ; P k ; S k (see [16]). Eventually an efficient relaxation solver was proposed in [46,15], that enables to handle evanescent phases. Classical second-order extensions of the latter schemes are of course possible, as underlined for instance in [17].…”