A centered limited finite volumes approximation of the momentum convection operator for low-order nonconforming face-centred discretizations
Résumé
We propose in this paper 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 to the cells of the mesh while the velocities unknowns are associated to 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: (i) computation of a tentative flux, here, with a centered approximation of the velocity, and (ii) 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 (∂t(ρ u_i)+div(ρ u_i u) = 1/2 ∂t(ρ u_i/2)+ 1/2 div(ρ u_i/2 u) (with u the velocity, u_i one of its component, ρ the density, and assuming that the mass balance holds) and discuss two applications of this result: firstly, we obtain stability results for a semi-implicit in time scheme for incompressible and barotropic compressible flows; secondly, 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.
Mots clés
Compressible flows
Consistency
Convection operator
Fluid flows
Incompressible flows
Kinetic energy balance
Monotonicity
Stability
Staggered meshes
2020 AMS Subject Classification. 65M08 and 65M12 and 76M12 Fluid flows incompressible flows compressible flows staggered meshes convection operator kinetic energy balance monotonicity stability consistency
2020 AMS Subject Classification. 65M08 and 65M12 and 76M12 Fluid flows
incompressible flows
compressible flows
staggered meshes
convection operator
kinetic energy balance
monotonicity
stability
consistency
Domaines
Analyse numérique [math.NA]
Fichier principal
Numerical Methods in Fluids - 2024 - Brunel - A centered limited finite volume approximation of the momentum convection.pdf (3.72 Mo)
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