Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows

Abstract

In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in Romenski et al. (J Sci Comput 42(1):68, 2009, Quart Appl Math 65(2):259–279, 2007). All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine–Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer–Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer–Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer–Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer–Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.

A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

“…Alternative relaxation methods leading to parabolic approximations of different variants of the NSK system were also considered by Rohde and collaborators in [51][52][53][54][55][56]. For alternative hyperbolic diffuse interface models for compressible multiphase flows, the reader may refer to this non-exhaustive selection of references [57][58][59][60][61][62][63][64][65][66][67][68][69][70].…”

High-order relaxation methods for nonequilibrium two-phase flow equations

“…In this study, we take into account the model developed in Zeidan (2016) and Zeidan and Slaouti (2009) for which the simplified two-fluid isentropic model is adopted as the basis for deriving the hyperbolic system of conservation laws in a mixture form, and we refer to Zeidan and Slaouti (2009) for more details on the derivation of this model. Note that the current model and those investigated in Romenski et al (2007Romenski et al ( , 2010 and Thein et al (2022) are hyperbolic and may be rewritten in a conservative form. The equations governing this model are reformulated in terms of state parameters.…”