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

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.

[1]  Marc Garbey,et al.  EVALUATION OF RICHARDSON EXTRAPOLATION IN COMPUTATIONAL FLUID DYNAMICS , 2002 .

[2]  Timothy G. Trucano,et al.  Verification and validation. , 2005 .

[3]  J. Z. Zhu,et al.  The finite element method , 1977 .

[4]  C. Bruneau,et al.  The 2D lid-driven cavity problem revisited , 2006 .

[5]  William L. Oberkampf,et al.  Methodology for characterizing modeling and discretization uncertainties in computational simulation , 2000 .

[6]  Jan Vierendeels,et al.  Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1, Reference solutions , 2005 .

[7]  S. Mergui,et al.  Wavy secondary instability of longitudinal rolls in Rayleigh–Bénard–Poiseuille flows , 2005, Journal of Fluid Mechanics.

[8]  M. Medale,et al.  Benchmark Solution for a Three-Dimensional Mixed-Convection Flow, Part 1: Reference Solutions , 2011 .

[9]  C. H. Marchi,et al.  The lid-driven square cavity flow: numerical solution with a 1024 x 1024 grid , 2009 .

[10]  Thomas J. R. Hughes,et al.  The Continuous Galerkin Method Is Locally Conservative , 2000 .

[11]  R. Sani,et al.  On pressure boundary conditions for the incompressible Navier‐Stokes equations , 1987 .

[12]  M. Garbey,et al.  A least square extrapolation method for improving solution accuracy of PDE computations , 2003 .

[13]  Christopher J. Roy,et al.  Review of code and solution verification procedures for computational simulation , 2005 .

[14]  S. Xin,et al.  Efficient Vectorized Finite-Difference Method to Solve the Incompressible Navier–Stokes Equations for 3-D Mixed-Convection Flows in High-Aspect-Ratio Channels , 2005 .

[15]  Timothy G. Trucano,et al.  Verification and Validation in Computational Fluid Dynamics , 2002 .

[16]  M. Medale,et al.  Benchmark solution for a three-dimensional mixed convection flow - Detailed technical report , 2011 .

[17]  P. Roache.  Perspective: A Method for Uniform Reporting of Grid Refinement Studies , 1994 .

[18]  Robert L. Lee,et al.  THE CONSISTENT GALERKIN FEM FOR COMPUTING DERIVED BOUNDARY QUANTITIES IN THERMAL AND/OR FLUIDS PROBLEMS , 1987 .

[19]  Samuel Paolucci,et al.  The thermoconvective instability of plane poiseuille flow heated from below: A proposed benchmark solution for open boundary flows , 1990 .