  We propose in this paper a discretization of the momentum convection operator
for fluid flow simulations on quadrangular or hexahedral meshes. The space
discretization is performed by the loworder 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 almost second
order expression is derived by a MUSCL-like technique. The latter is of
algebraic type, in the sense that the limitation procedure 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 ui ($\partial$t($\rho$ui)+div($\rho$uiu) = 1 2 $\partial$t($\rho$u
2 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 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 and the compressible Euler
equations.
