This article presents numerical results using a new finite-volume scheme on unstructured grids for the incompressible Navier-Stokes equations. The discrete unknowns are the components of the velocity, the pressure, and the temperature, colocated at the centers of the control volumes. The scheme is stabilized using an original method leading to local redistributions of the fluid mass, which simultaneously yields the control of the kinetic energy and the convergence of the scheme. Different comparisons with the literature (2-D and 3-D lid-driven cavity, backward-facing step, differentially heated cavity) allow us to assess the numerical properties of the scheme.

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International audienceComparison exercises have been carried out by different research teams to study the sensitivity of the natural convection occurring in a vertical asymmetrically heated channel to four sets of open boundary conditions. The dimensionless parameters have been chosen so that a return flow exists at the outlet. On the whole, results provided by the partners are in good agreement; benchmark solutions are then defined for each of the boundary conditions. Whilst the local and average Nusselt numbers based on the entrance temperature do not depend much on conditions applied in the aperture sections, the net fluid flow rates crossing the channel and the characteristics of the recirculation cells are highly influenced. But we proved that these modifications of flow patterns do not alter significantly the fluid flow rates leaving the channel through the exit section

A variational formulation of the standard MAC scheme for the approximation of the Navier-Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property allows us to give a mathematical proof of the convergence of the resulting approximate solutions, for the nonlinear Navier-Stokes equations in both steady-state and time-dependent regimes, without any small data condition. Numerical examples (analytical steady and time-dependent ones, inclined driven cavity) confirm the robustness and the accuracy of this method.