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We propose in this paper a discretization of the momentum convection operatorfor fluid flow simulations on quadrangular or hexahedral meshes. The spacediscretization is performed by the loworder nonconforming Rannacher-Turekfinite element: the scalar unknowns are associated to the cells of the mesh,while the velocities unknowns are associated to the edges or faces. Themomentum convection operator is of finite volume type, and its almost secondorder expression is derived by a MUSCL-like technique. The latter is ofalgebraic type, in the sense that the limitation procedure does not invoke anyslope reconstruction, and is independent from the geometry of the cells. Thederived discrete convection operator applies both to constant or variabledensity flows, and may thus be implemented in a scheme for incompressible orcompressible flows. To achieve this goal, we derive a discrete analogue of thecomputation ui ($\partial$t($\rho$ui)+div($\rho$uiu) = 1 2 $\partial$t($\rho$u2 i)+ 1 2 div($\rho$u 2 i u) (with u the velocity, ui one of its component,$\rho$ the density, and assuming that the mass balance holds) and discuss twoapplications of this result: firstly, we obtain stability results for asemi-implicit in time scheme for incompressible and barotropic compressibleflows; secondly, we build a consistent, semi-implicit in time scheme that isbased on the discretization of the internal energy balance rather than thetotal energy. The performance of the proposed discrete convection operator isassessed by numerical tests on the incompressible Navier-Stokes equations, thebarotropic and the full compressible Navier-Stokes and the compressible Eulerequations.

A MUSCL-like finite volumes approximation of the momentum convection operator for low-order nonconforming face-centred discretizations