Benchmark Solution for a Three-Dimensional Mixed-Convection Flow, Part 1: Reference Solutions

Abstract

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 = 5,000) and Prandtl number Pr = 0.7. The model is based on the Navier-Stokes equations with Boussinesq approximation. We propose reference solutions resulting from computations on large grids, Richardson extrapolation (RE), and cubic spline interpolations. The solutions obtained with one finite-difference, one finite-volume, and two finite-element codes are in good agreement, and reference values for the flow and thermal fields and for the heat and momentum fluxes are given with four to five significant digits.

Acknowledgments

Xavier Nicolas acknowledges Shihe Xin, from CETHIL, UMR 5008 CNRS/Insa-Lyon, France, for providing the finite-difference code, FD1, that was developed by him when he was at LIMSI, CNRS, UPR 3251, Orsay, France. The authors acknowledge Donna Calhoun for proofreading of the article and providing many useful suggestions. This work was supported by CNRS, which provided substantial computational resources on its NEC-SX5 vectorial supercomputer and on its IBM SP4 and SP6 parallel supercomputers at IDRIS, Orsay, France, under project numbers 06-1823 and 07-1823. Stéphane Glockner thanks the Aquitaine Regional Council for the financial support dedicated to a 256-processor cluster investment, located at I2M Institute.

Notes

¹the consistency order in space is the formal convergence order that is the leading order of the space discretization truncation error