# Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions

Patrick Le Quéré; Catherine Weisman; Henri Paillère^{[1]}; Jan Vierendeels^{[2]}; Erik Dick; Roland Becker^{[3]}; Malte Braack^{[3]}; James Locke^{[4]}

- [1] CEA Saclay, DEN/DM2S/SFME,91191 Gif-sur-Yvette Cedex, France.
- [2] Ghent University, 9000 Gent, Belgium. ; ; Ghent University, B-9000 Gent, Belgium.
- [3] Heidelberg University, Germany
- [4] U. Warwick and British Energy Generation Ltd.

- Volume: 39, Issue: 3, page 609-616
- ISSN: 0764-583X

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topQuéré, Patrick Le, et al. "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 39.3 (2005): 609-616. <http://eudml.org/doc/245595>.

@article{Quéré2005,

abstract = {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heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of 0.6, $\rm Ra=10^6$ (constant property and variable property cases) and $\rm Ra=10^7$ (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.},

affiliation = {CEA Saclay, DEN/DM2S/SFME,91191 Gif-sur-Yvette Cedex, France.; Ghent University, 9000 Gent, Belgium. ; ; Ghent University, B-9000 Gent, Belgium.; Heidelberg University, Germany; Heidelberg University, Germany; U. Warwick and British Energy Generation Ltd.},

author = {Quéré, Patrick Le, Weisman, Catherine, Paillère, Henri, Vierendeels, Jan, Dick, Erik, Becker, Roland, Braack, Malte, Locke, James},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {natural convection; non-Boussinesq; low Mach number},

language = {eng},

number = {3},

pages = {609-616},

publisher = {EDP-Sciences},

title = {Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions},

url = {http://eudml.org/doc/245595},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Quéré, Patrick Le

AU - Weisman, Catherine

AU - Paillère, Henri

AU - Vierendeels, Jan

AU - Dick, Erik

AU - Becker, Roland

AU - Braack, Malte

AU - Locke, James

TI - Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 3

SP - 609

EP - 616

AB - 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, $\rm Ra=10^6$ (constant property and variable property cases) and $\rm Ra=10^7$ (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.

LA - eng

KW - natural convection; non-Boussinesq; low Mach number

UR - http://eudml.org/doc/245595

ER -

## References

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- [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. Zbl0623.76097
- [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. Zbl0538.76075
- [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. Zbl0538.76076
- [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. Zbl0328.76066
- [8] P. Le Quéré, Accurate solutions to the square differentially heated cavity at high Rayleigh number. Comput. Fluids 20 (1991) 19–41. Zbl0731.76054
- [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. Zbl0763.76061
- [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. Zbl0433.76071
- [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. Zbl0233.76163
- [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. Zbl0955.76067

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