Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 3, p. 609-616

There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of 0.6, $\mathrm{Ra}={10}^{6}$ (constant property and variable property cases) and $\mathrm{Ra}={10}^{7}$ (variable property case). These reference solutions were produced after a first international workshop organized by CEA and LIMSI in January 2000, in which the above authors volunteered to produce accurate numerical solutions from which the present reference solutions could be established.

DOI : https://doi.org/10.1051/m2an:2005027
Classification:  65M50,  76M10,  76M12,  76M20,  76M22,  76R10
Keywords: natural convection, non-Boussinesq, low Mach number
@article{M2AN_2005__39_3_609_0,
author = {Qu\'er\'e, Patrick Le and Weisman, Catherine and Paill\ere, Henri and Vierendeels, Jan and Dick, Erik and Becker, Roland and Braack, Malte and Locke, James},
title = {Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {3},
year = {2005},
pages = {609-616},
doi = {10.1051/m2an:2005027},
zbl = {1130.76047},
mrnumber = {2157153},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_3_609_0}
}

Quéré, Patrick Le; Weisman, Catherine; Paillère, Henri; Vierendeels, Jan; Dick, Erik; Becker, Roland; Braack, Malte; Locke, James. Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 3, pp. 609-616. doi : 10.1051/m2an:2005027. http://www.numdam.org/item/M2AN_2005__39_3_609_0/`

[1] R. Becker, M. Braack and R. Rannacher, Numerical simulation of laminar flames at low Mach number with adaptive finite elements. Combustion Theory and Modelling, Bristol 3 (1999) 503-534. | Zbl 0951.76035

[2] R. Becker, M. Braack, Solution of a stationary benchmark problem for natural convection with high temperature difference. Int. J. Thermal Sci. 41 (2002) 428-439.

[3] D.R. Chenoweth and S. Paolucci, Natural Convection in an enclosed vertical air layer with large horizontal temperature differences. J. Fluid Mech. 169 (1986) 173-210. | Zbl 0623.76097

[4] G. De Vahl Davis, Natural convection of air in a square cavity: a benchmark solution. Int. J. Numer. Methods Fluids 3 (1983) 249-264. | Zbl 0538.76075

[5] G. De Vahl Davis and I.P. Jones, Natural convection of air in a square cavity: a comparison exercice. Int. J. Numer. Methods Fluids 3 (1983) 227-248. | Zbl 0538.76076

[6] FEAT User Guide, Finite Element Analysis Toolbox, British Energy, Gloucester, UK (1997).

[7] D.D. Gray and A. Giorgini, The Validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transfer 15 (1976) 545-551. | Zbl 0328.76066

[8] P. Le Quéré, Accurate solutions to the square differentially heated cavity at high Rayleigh number. Comput. Fluids 20 (1991) 19-41. | Zbl 0731.76054

[9] P. Le Quéré, R. Masson and P. Perrot, A Chebyshev collocation algorithm for 2D Non-Boussinesq convection. J. Comput. Phys. 103 (1992) 320-335. | Zbl 0763.76061

[10] W.L. Oberkampf and T. Trucano, Verification and validation in Computational Fluid Dynamics. Sandia National Laboratories report SAND2002-0529 (2002).

[11] H. Paillère and P. Le Quéré, Modelling and simulation of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers, 12th Séminaire de Mécanique des Fluides Numérique, CEA Saclay, France, 25-26 Jan., 2000.

[12] S. Paolucci, On the filtering of sound from the Navier-Stokes equations. Sandia National Laboratories report SAND82-8257 (1982).

[13] J.C. Patterson and J. Imberger, Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100 (1980) 65-86. | Zbl 0433.76071

[14] V.L. Polezhaev, Numerical solution of the system of two-dimensional unsteady Navier-Stokes equations for a compressible gas in a closed region. Fluid Dyn. 2 (1967) 70-74. | Zbl 0233.76163

[15] J. Vierendeels, K. Riemslagh and E. Dick, A Multigrid semi-implicit line-method for viscous incompressible and low-Mach number flows on high aspect ratio grids. J. Comput. Phys. 154 (1999) 310-341. | Zbl 0955.76067