Modelling compressible dense and dilute two-phase flows
Résumé
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. Emails: Richard.Saurel@univ-amu.fr; Ashwin.Chinnayya@ensma.fr; Quentin.Carmouze@rs2n.eu 2 I. Introduction It 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 two-phase 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 differential equations it means that the Riemann problem must have a solution, and the Riemann problem is correctly posed only if the equations are hyperbolic. However, only a few two-phase flow models are hyperbolic in the whole range of parameters. The Baer and Nunziato (1986) model seems to be the only formulation able to deal with such requirement. Its symmetric extension (Saurel et al., 2014) facilitates the Riemann problem resolution as the corresponding model involves 7 wave's speeds (instead of 6 in the original version). See also Ambroso et al. (2012) for similar conclusions. However, in the dilute limit at least, the acoustic properties of this model seem inconsistent (Lhuillier et al., 2013). Indeed with this model, the dispersed phase sound speed corresponds to the one of the pure phase, while this phase is not continuous and unable to propagate sound in reality, at least at a scale larger than particle's one. When the phase is not continuous (dispersed drops in a gas, dispersed bubbles in a liquid), the associated sound speed should vanish, such effect being absent in the formulation. In the low particles concentration limit, the Marble (1963) model is preferred. This model corresponds to the Euler equations with source terms for the gas phase and pressureless gas dynamic equations for the particle phase (see also Zeldovich, 1970). This model is thermodynamically consistent and hyperbolic as well, except that the particle phase equations are hyperbolic degenerate. In this model, contrarily to the BN model, sound doesn't propagate in the particles phase, this behaviour being more physical in this limit. 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. There are thus fundamental differences between these two models:-The volume occupied by the condensed phase is considered in BN while it is neglected in the dilute model, restricting its validity to low dispersed phase volume fractions.-Condensed phase compressibility is considered in BN while incompressible particles are assumed in the dilute formulation.-Acoustic properties of the BN model are well accepted in the dense domain but seem inappropriate in the dilute limit. Even if these two models can be used in the entire space of two-phase flow variables without yielding computational failure (this is characteristic of thermodynamically consistent hyperbolic models) validity of their results is questionable when they are used out of their range of physical validity. This issue has been clearly understood in Lhuillier et al. (2013), McGrath et al. (2016) and Houim and Oran (2016) where various attempts to build new formulations are reported. In Lhuillier et al. (2013) discussion on the volume fraction equation is done, but no explicit flow model is given. In McGrath et al. (2016) a model is given with conditional hyperbolicity. Same issue is present with different cause in Houim and Oran (2016). The aim of the present paper is to build an alternative to the BN model with improved acoustic properties, while remaining unconditionally hyperbolic and thermodynamically consistent. The new model is derived from number density and particle radius (or bubble radius) evolution equations resulting in a volume fraction evolution equation expressed in conservation form with
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