An Energy Dissipative Spatial Discretization for the Regularized Compressible Navier-Stokes-Cahn-Hilliard System of Equations

Abstract

We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the potential body force. The sought total density, mixture velocity and concentration of one of the components are defined at nodes of one and the same grid. The results of computer simulation of several 2D test problems are presented. They demonstrate advantages of the constructed discretization including the absence of the so-called parasitic currents.

An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

“…Moreover, for the main sought functions, we apply staggered meshes (C-mesh according to the Arakawa classification [3]) which is implemented firstly for the regularizations of considered type. For the same equations, alternative discretizations on non-staggered meshes for all the main sought functions having the total energy dissipativity property have quite recently been constructed in the simplified 1D [5] and 3D periodic [7] statements. It turned out that the discretizations of the Navier-Stokes viscosity tensor and regularizing terms are simpler in the case of the staggered main meshes.…”

“…It includes the regularizing momentum m (rather than a regularizing velocity following [7,35] but in contrast to the standard approach) with the regularization parameter τ = τ (ρ, C) > 0. The tensor Π = Π N S − Q + Π τ consists of the Navier-Stokes viscous stress tensor Π N S , the capillary stress tensor Q and the regularizing tensor Π τ :…”

“…(The theoretical constructions are valid for general Ψ sep (C) but numerical simulations are accomplished namely for its form (7).) The parameters ω 1 > 0 and ω 2 > 0 depend on the mixture temperature and the component properties; hereafter we assume that they are constant.…”

“…In [5,7], to construct an energy dissipative spatial discretizations, the following non-divergent form of terms ∇p + div Q in (2) and (4) has recently been applied…”

“…Finally, recall the equilibrium solutions ρ = ρ S (x) > 0, u = 0 and 0 < C = C S (x) < 1 of the considered system. Similarly to [7] it can be checked that under the boundary condition ∂ n µ| ∂Ω = 0 they satisfy the following system of equations…”