A centered limited finite volume approximation of the momentum convection operator for low-order nonconforming face-centered discretizations

Abstract

We propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low-order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two-step technique: $\left(i\right)$ computation of a tentative flux, here, with a centered approximation of the velocity, and $\left(ii\right)$ limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation $u_i\left({\partial }_t\right(\rho u_i)+\text{div}(\rho u_{i}u)=\frac{1}{2}{\partial }_t(\rho u_i^2)+\frac{1}{2}\text{div}(\rho u_i^{2}u)$ (with $u$ the velocity, $u_i$ one of its component, $\rho$ the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi-implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi-implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.