Mass Conservation for Coupled Flow-Transport Problems

Abstract

In this chapter we examine mass-conservation properties of finite element discretizations of coupled flow-transport problems. The system under consid-eration is described by the unsteady incompressible Navier-Stokes equations and a time-dependent transport equation; see [GS00a, GS00b, Hir88, Hir90] for models where this combination arises. The incompressibility constraint implies that global mass is conserved in the weak solution of the transport equation. Since the discretized velocity only satisfies a discrete incompress-ibility constraint, global mass is in general conserved only approximately in the numerical scheme. We shall investigate conditions under which discrete global mass conservation can be guaranteed. 6.1 A Model Problem We consider the simplest case of a coupled flow-transport problem in a bounded domain Ω ⊂ R d , d = 2, 3. The system is described by the unsteady incompressible Navier-Stokes equations u t − νu + (u · ∇)u + ∇p = f in Ω × (0, T ], (6.1a) ∇ · u = 0 in Ω × (0, T ], (6.1b) u = u b on ∂Ω × (0, T ], (6.1c) u(0) = u 0 in Ω, (6.1d) and the time-dependent transport equation c t − εc + u · ∇c = g in Ω × (0, T ], (6.2a) (cu − ε∇c) · n = c I u · n on Γ − × (0, T ], (6.2b) ε∇c · n = 0 on Γ * × (0, T ], (6.2c) c(0) = c 0 in Ω. (6.2d) 530 6 Mass Conservation for Coupled Flow-Transport Problems Here u and p denote the velocity and the pressure of the fluid, ν and ε are small positive numbers, and T > 0 is the final time. The boundary ∂Ω is divided between the inflow boundary Γ − := {x ∈ ∂Ω : u · n < 0} and the remaining part of the boundary Γ * := ∂Ω \ Γ − , where n is the outward-pointing unit normal. Furthermore, c is the concentration of a species transported with the flow field and c I its concentration at the inflow boundary Γ − . We assume that the given velocity field u b on the boundary ∂Ω is the restriction of a divergence-free function that is also denoted by u b . The initial velocity u 0 satisfies the incompressibility constraint ∇ · u 0 = 0. Various discretization methods for the unsteady incompressible Navier-Stokes equations and the transport equation have been developed in previous chapters for the realistic and important cases where ν 1 and ε 1. Here we shall study the mass conservation of the discretized transport equation when using stabilized schemes. For simplicity of notation we confine our attention to the semi-discretization in space of the problems (6.1) and (6.2). The results can be extended to the fully discretized problems by using discontinuous Galerkin methods in time. 6.2 Continuous and Discrete Mass Conservation