A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows

Abstract

We consider the fluid subsystem of a variationally derived hyperbolic thermodynamically compatible (HTC) model of compressible multiphase flows. Its total energy is in general not convex in the natural state variables, so a scalar numerical viscosity acting on them need not produce non-negative entropy. Reformulating the equations in terms of the volume fractions separates changes in the amount of each phase from changes in its intrinsic state. This guides the construction of a configurational numerical viscosity in phase variables, in which the energy is convex, while the scheme retains the structure of the HTC model. At the semi-discrete level, the new schemes satisfy: I) local conservation of the phase masses, the mixture momentum and the total energy; II) a cell entropy inequality whose production is non-negative face by face and phase by phase, with no convexity assumption in the evolved variables and no clipping; III) positivity of the partial densities and volume fractions under explicit time-step conditions; IV) exact preservation of contact equilibria; V) equivalence with a scheme for the density-weighted volume fractions. The key ingredients are: i) an exact decomposition of the jump of each phase across a face into a configurational part, dissipated with the material speed, and an intrinsic part, dissipated with the acoustic speed; ii) an exchange form of this dissipation between neighbouring cells, with non-negative weights; iii) a side-wise transfer of the dissipation to the evolved variables through the chain rule of each cell, which turns the energy defects into Bregman divergences of the single-fluid energies; iv) a thermodynamic correction of the reversible part built in the phase variables. Properties I)-V) are proved and verified numerically, including a shock tube in which the Rusanov viscosity produces negative entropy and a shock-water column interaction.