Benchmark Solution for a Three-Dimensional Mixed-Convection Flow, Part 2: Analysis of Richardson Extrapolation in the Presence of a Singularity

Abstract

A reference solution to a benchmark problem for a three-dimensional mixed-convection flow in a horizontal rectangular channel differentially heated (Poiseuille-Rayleigh-Bénard flow) has been proposed in Part 1 of the present article (Numer. Heat Transfer B, vol. 60, pp. 325–345, 2011). Since mixed Dirichlet and Neumann thermal boundary conditions are used on the horizontal walls of the channel, a temperature gradient discontinuity is generated. The aim of this article is to analyze the consequences of this singularity on Richardson extrapolation (RE) of the numerical solutions. The convergence orders of the numerical methods used (finite difference, finite volume, finite element), observed from RE of local and integral quantities are discussed with an emphasis on singularity influence. With the grids used, it is shown that RE can increase the accuracy of the discrete solutions preferentially with the discretization methods of low space accuracy order, but only in some part of the channel and for a restricted range of the extrapolation coefficient. 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.

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 and Sergey Kudriakov 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

*the consistency order is the formal convergence order, that is, the leading order of the space discretization truncation error.