How to preserve the mass fractions positivity when computing compressible multi-component flows

Abstract

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A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

“…This equation differs from the mass balance for chemical species in compressible multicomponent flows studied in [18] by the addition of a non-linear term of the form ∇ · ρϕ(y) u r , where y is the unknown, ϕ(·) is a regular function such that ϕ(0) = ϕ(1) = 0 and u r is a general (in particular non-necessarily divergence free) velocity field. We prove the existence and uniqueness of the finite volume approximation together with the fact that it remains within physical bounds, i.e.…”

“…within the interval [0, 1]. As in [18], the necessary condition for this L ∞ stability is that the discretization of the convection operator must be such that it vanishes for constant y, which amounts to demanding that a particular discrete mass balance equation be satisfied. The second ingredient of this scheme is a discretization of the non-linear term based on the notion of monotone flux functions [13].…”

“…This suggests to build a numerical scheme which reproduces these features at the discrete level. This is performed by discretizing the mass balance equation (1.1a) with an upwind finite volume scheme and combining a monotone flux approach [13, section 21] for the term ∇ · (ρ φ (y) u r ) in the advection diffusion equation (1.1c) with the argument introduced by Larrouturou [18]: the discrete counterpart of the advection operator ∂ ρ y/∂t + ∇ · (ρ y u) satisfies a maximum principle provided that this operator applied to a constant value of y vanishes, i.e. that a discrete version of the mass balance is satisfied.…”

Nonlinear projection methods for multi-entropies Navier-Stokes systems

“…A discrete entropy inequality for {ρs 1 } n+1,= i , distinct from (3.7), enters the following statement and will be motivated just hereafter: Proposition 3.1. Under the CFL condition (3.5), the Lax entropy pair (ρs 1 , ρs 1 u) obeys the following discrete entropy inequality: [14] in (3.9) by the following formula with j = 1, 2:…”

“…where after Larrouturou, the right-hand side is nothing but a convex decomposition of ((s j ) n i−1 , (s j ) n i , (s j ) n i+1 ) under the CFL restriction (3.5) (see again [14] for the details). We immediately infer from this convex decomposition the next inequality:…”