Benchmark solution for a three-dimensional mixed convection flow - Detailed technical report
Résumé
A solution to a benchmark problem for a three-dimensional mixed convection flow in a horizontal rectangular channel heated from below and cooled from above (Poiseuille-Rayleigh-Bénard flow) is proposed. This flow is a steady thermoconvective longitudinal roll flow in a large aspect ratio channel at moderate Reynolds and Rayleigh numbers (Re=50, Ra=5000) and Prandtl number Pr=0.7. The model is based on the Navier-Stokes equations with Boussinesq approximation. We propose a reference solution resulting from computations on large grids, Richardson extrapolation (RE) and cubic spline interpolations. The solutions obtained with finite difference, finite volume and finite element codes are in good agreement and reference values for the flow fields and the heat and momentum fluxes are given up to 4 to 5 significant digits. Some difficulties in the use of RE are highlighted due to the use of mixed Dirichlet and Neumann thermal boundary conditions on the same wall. The observed convergence orders of the numerical methods with RE are then discussed from the viewpoint of this singularity. A correction to the Taylor expansion involved in the RE formalism is proposed to take into account the singularity and to explain the majority of the RE behaviors observed. The results of the present study are published in two papers in Numerical Heat Transfer, Part B [1, 2].
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