Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-16T08:47:28.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling of Natural Convection Flowswith Large Temperature Differences:A Benchmark Problem for Low Mach Number Solvers. Part 1. Reference Solutions

Published online by Cambridge University Press:  15 June 2005

Patrick Le Quéré ,
Catherine Weisman ,
Henri Paillère ,
Jan Vierendeels ,
Erik Dick ,
Roland Becker ,
Malte Braack  and
James Locke
Patrick Le Quéré
Affiliation:
LIMSI, BP 133, 91403 Orsay Cedex, France. plq@limsi.fr; weisman@limsi.fr
Catherine Weisman
Affiliation:
LIMSI, BP 133, 91403 Orsay Cedex, France. plq@limsi.fr; weisman@limsi.fr
Henri Paillère
Affiliation:
CEA Saclay, DEN/DM2S/SFME,91191 Gif-sur-Yvette Cedex, France. henri.paillere@cea.fr
Jan Vierendeels
Affiliation:
Ghent University, B-9000 Gent, Belgium. Jan.Vierendeels@UGent.be; erik.dick@UGent.be
Erik Dick
Affiliation:
Ghent University, B-9000 Gent, Belgium. Jan.Vierendeels@UGent.be; erik.dick@UGent.be
Roland Becker
Affiliation:
Heidelberg University, Germany. malte.braack@iwr.uni-heidelberg.de
Malte Braack
Affiliation:
Heidelberg University, Germany. malte.braack@iwr.uni-heidelberg.de
James Locke
Affiliation:
U. Warwick and British Energy Generation Ltd.
Get access

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heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of 0.6, Ra = 106 (constant property and variable property cases) and Ra = 107 (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.

Keywords

Natural convectionnon-Boussinesqlow Mach number.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 3: Special issue on Low Mach Number Flows Conference , May 2005 , pp. 609 - 616
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Becker, R., Braack, M., Solution of a stationary benchmark problem for natural convection with high temperature difference. Int. J. Thermal Sci. 41 (2002) 428–439. CrossRef
Chenoweth, D.R. and Paolucci, S., Natural Convection in an enclosed vertical air layer with large horizontal temperature differences. J. Fluid Mech. 169 (1986) 173–210. CrossRef
de Vahl Da, G.vis, Natural convection of air in a square cavity: a benchmark solution. Int. J. Numer. Methods Fluids 3 (1983) 249–264. CrossRef
de Vahl Da, G.vis and I.P. Jones, Natural convection of air in a square cavity: a comparison exercice. Int. J. Numer. Methods Fluids 3 (1983) 227–248. CrossRef
FEAT User Guide, Finite Element Analysis Toolbox, British Energy, Gloucester, UK (1997).
Gray, D.D. and Giorgini, A., The Validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transfer 15 (1976) 545–551. CrossRef
Le Quéré, P., Accurate solutions to the square differentially heated cavity at high Rayleigh number. Comput. Fluids 20 (1991) 19–41. CrossRef
Le Quéré, P., Masson, R. and Perrot, P., Chebyshev, A collocation algorithm for 2D Non-Boussinesq convection. J. Comput. Phys. 103 (1992) 320–335. CrossRef
W.L. Oberkampf and T. Trucano, Verification and validation in Computational Fluid Dynamics. Sandia National Laboratories report SAND2002-0529 (2002).
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.
S. Paolucci, On the filtering of sound from the Navier-Stokes equations. Sandia National Laboratories report SAND82-8257 (1982).
Patterson, J.C. and Imberger, J., Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100 (1980) 65–86. CrossRef
Polezhaev, V.L., 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. CrossRef
Vierendeels, J., Riemslagh, K. and Dick, E., Multigrid, A semi-implicit line-method for viscous incompressible and low-Mach number flows on high aspect ratio grids. J. Comput. Phys. 154 (1999) 310–341. CrossRef