Riemann solver with internal reconstruction (RSIR) for compressible single-phase and non-equilibrium two-phase flows

Abstract

A new Riemann solver is built to address numerical resolution of complex flow models. The research direction is closely linked to a variant of the Baer and Nunziato (1986) model developed in Saurel et al. (2017a). This recent model provides a link between the Marble (1963) model for two-phase dilute suspensions and dense mixtures. As in the Marble model, Saurel et al. system is weakly hyperbolic with the same 4 characteristic waves, while the system involves 7 partial differential equations. It poses serious theoretical and practical issues to built simple and accurate flow solver. To overcome related difficulties the Riemann solver of Linde (2002) is revisited. The method is first examined in the simplified context of compressible Euler equations. Physical considerations are introduced in the solver improving robustness and accuracy of the Linde method. With these modifications the flow solver appears as accurate as the HLLC solver of Toro et al. (1994). Second the two-phase flow model is considered. A locally conservative formulation is built and validated removing issues related to nonconservative terms. However, two extra major issues appear from numerical experiments: The solution appears not self-similar and multiple contact waves appear in the dispersed phase. Building HLLC-type or any other solver appears consequently challenging. The modified Linde (2002) method is thus examined for the considered flow model. Some basic properties of the equations are used, such as shock relations of the dispersed phase and jump conditions across the contact wave. Thanks to these ingredients the new Riemann solver with internal reconstruction (RSIR), modification of the Linde method, handles stationary volume fraction discontinuities, presents low dissipation for transport waves and handles shocks and expansion waves accurately. It is validated on various test problems showing method's accuracy and versatility for complex flow models. Its capabilities are illustrated on a difficult two-phase flow instability problem, unresolved before.

Numerical investigations of two-phase finger-like instabilities

“…In the present paper, the origin of such finger-like instabilities is investigated with the help of numerical experiments. To the authors' knowledge, such instabilities have been successfully computed for the first time in Carmouze et al (2019) [13] with the help of the dense-dilute model of [14].…”

“…The use of triangular cells enables clear definition of the particle ring. No mixed cell is present initially, meaning that no initial wavelength or perturbation is introduced (see Carmouze et al (2019)[13]). Initial data are reported in…”

“…Stiff pressure relaxation is considered. Results are obtained with the RSIR solver of Carmouze et al (2019)[13] embedded in the MUSCL-type scheme[24] with Sweby's limiter and Φ = 1.5 (Eq. 3.17 of[31]).…”

Shock-induced interfacial instabilities of granular media

“…For the gas phase, the volume-averaged governing equations, (2.1)–(2.3), constructed in the Eulerian frame are based on a five-equation transport model, i.e. a simplified form of the Baer–Nunziato model (Baer & Nunziato 1986), which has been modified to account for compressible multiphase flows ranging from dilute to dense gas– particle flows (Carmouze et al 2020; Chiapolino & Saurel 2020): …”

Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations

“…In Carmouze et al (2019) [8] an alternative is given and a new Riemann solver with internal reconstruction (RSIR) is designed. It relies on the following observation.…”