Modelling compressible dense and dilute two-phase flows

Abstract

Many two-phase flow situations, from engineering science to astrophysics, deal with transition from dense (high concentration of the condensed phase) to dilute concentration (low concentration of the same phase), covering the entire range of volume fractions. Some models are now well accepted at the two limits, but none is able to cover accurately the entire range, in particular regarding waves propagation. In the present work an alternative to the Baer and Nunziato (1986) (BN for short) model, initially designed for dense flows, is built. The corresponding model is hyperbolic and thermodynamically consistent. Contrarily to the BN model that involves 6 wave speeds, the new formulation involves 4 waves only, in agreement with the Marble (1963) model based on pressureless Euler equations for the dispersed phase, a well-accepted model for low particle volume concentrations. In the new model, the presence of pressure in the momentum equation of the particles and consideration of volume fractions in the two phases render the model valid for large particle concentrations. A symmetric version of the new model is derived as well for liquids containing gas bubbles. This model version involves 4 wave speeds as well, but with different wave's speeds. Last, the two sub-models with 4 waves are combined in a unique formulation, valid for the full range of volume fractions. It involves the same 6 wave's speeds as the BN model, but at a given point of space 4 waves only emerge, depending on the local volume fractions. The non-linear pressure waves propagate only in the phase with dominant volume fraction. The new model is tested numerically on various test problems ranging from separated phases in a shock tube to shock -particle cloud interaction. Its predictions are compared to BN and Marble models as well as against experimental data.Key words: hyperbolic, two-phase, compressible, shocks Emails: Richard.Saurel@univ-amu.fr; Ashwin.Chinnayya@ensma.fr; Quentin.Carmouze@rs2n.eu 2 I. IntroductionIt is well accepted that hyperbolic models are mandatory to deal with phenomena involving wave propagation. This is the case for multiphase flow mixtures in many situations such as in particular shocks and detonations propagation in granular explosives and in fuel suspensions, as well as liquid-gas mixtures with bubbles, cavitation and flashing, as soon as motion is intense and governed by pressure gradients. This is thus the case of most unsteady two-phase flows situations. Wave propagation is important as it carries pressure, density and velocity disturbances. Sound propagation is also very important as it determines critical (choked) flow conditions and associated mass flow rates. It has also fundamental importance on sonic conditions of detonation waves when the twophase mixture is exothermically reacting (Petitpas et al., 2009). Hyperbolicity is also related to the causality principle, meaning that initial and boundary conditions are responsible of time evolution of the solution. When dealing with first-order partial different...

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

“…However, the Marble model has a limited range of validity as the volume of the dispersed phase is neglected, this assumption having sense only for low (less than per cent) condensed phase volume fraction. Recently, the gap between these two models has been filled (Saurel et al, 2017a). Modifying the volume fraction equation in the BN model resulted in a flow model where sound propagates only in the carrier phase.…”

“…The two-phase flow model of Saurel et al (2017a), considering a dispersed phase 1 in a carrier fluid 2 is recalled hereafter. Pressure and velocity relaxation terms only are considered as interaction effects:…”

“…These relations will be used in the RSIR derivation. Numerical resolution of System (III.1) has been done in Saurel et al (2017a) with the help of Rusanov (1961) solver. This solver being quite diffusive, the aim is now to build an improved solver.…”

Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

“…Note that Equations (18)- (21) reduce to the NASG expressions if p ∞,1 = 0 and b 1 = 0. In addition, inserting Equation (18) into (21), the internal energy expresses…”

“…Material interface problems [15], chemical reactions [16], phase change [17], surface tension [18], solid-fluid [19], plastic transformation [20], dense and dilute flows [21], and shallow water flows [22] can be cited for instance. In these flow models, compressibility of each phase is responsible for the hyperbolic character of the equations and an appropriate and convex EOS is required for each fluid.…”