Approximate solutions of the Baer-Nunziato Model

Abstract

Abstract. We examine in this paper the accuracy of some approximations of the BaerNunziato two-phase flow model. The governing equations and their main properties are recalled, and two distinct numerical schemes are investigated, including a classical secondorder extension relying on symmetrizing variables. Shock tube cases are considered, and two simple Riemann problems based on well-balanced initial data are detailed. These enable to recover the expected convergence rates. However, it is shown that these simple cases are indeed very difficult and that the accuracy of basic schemes is rather poor.Résumé. Approximation des solutions du modèle de Baer-Nunziato On examine ici la précision des approximations obtenues pour le modèle diphasique de BaerNunziato. Leséquations du modèle et ses principales propriétés sont rappellées. Deux schémas distincts sont proposés, et des extensions classiques au second-ordre sont considérées, utilisant les variables de symétrisation. Des cas tests de tubeà choc sont analysés, notamment deux cas utilisant des conditions initiales enéquilibre. Les taux de convergence attendus sont retrouvés, mais on montre que la précision des approximations de certains problèmes de Riemann est assez médiocre.

A relaxation scheme for a hyperbolic multiphase flow model

“…Another natural generalization is the extension of the scheme to higher order. A formally order two scheme can be obtained by considering a classical minmod reconstruction on the symmetrizing variable and a second order Runge-Kutta time scheme (see [13] for the two phase model). Such a procedure however does not ensure the preservation of the discrete energy inequality.…”

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

“…For all numerical simulations, an explicit CFL condition enforces the time step in order to stabilize numerical approximations. As emphasized in [32,17], a h 1=2 (respectively h 2=3 ) rate of convergence is retrieved when the mesh size h tends to 0, when focusing on one-dimensional Riemann problems involving shocks, contact and rarefaction waves, which is precisely what is expected for first-order schemes (respectively second-order schemes) when a contact discontinuity occurs in the exact solution. One should keep in mind the fact that the two-fluid model described in Section 2.1 involves two (or even three when nðWÞ ¼ m l =ðm l þ m v Þ) contact discontinuities.…”

“…We recall that the present fractional step method involves several steps that comply with the whole entropy inequality (see Section 2.2). For more details on steps that are not described herein, we thus refer to [17,24,[30][31][32]34]. We nonetheless recall below very briefly all steps of the fractional step method, which contains an evolution step, which deals with convective contributions, and a relaxation step, which handles relaxation source terms (mass transfer, drag effects, energy transfer and pressure relaxation terms).…”

“…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]. Obviously, many other schemes may be used, such as the relaxation solvers introduced in [3] or [4], or other approximate Riemann solvers (see [47,51] among others).…”

“…are of particular interest (see [17]). Eventually, we wish to emphasize once more that all these verifications are made possible due to the fact that exact unsteady solutions including complex waves with discontinuities can be exhibited, which is of course not true for some other two-fluid models, as recently pointed out in [7] for instance.…”