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On Pattern Selection in Three-Dimensional Bénard-Marangoni Flows

Published online by Cambridge University Press:  20 August 2015

Arne Morten Kvarving ,
Tormod Bjøntegaard  and
Einar M. Rønquist
Arne Morten Kvarving*
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway
Tormod Bjøntegaard*
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway
Einar M. Rønquist*
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway
*
Email:arnemort@math.ntnu.no
Email:bjontega@math.ntnu.no
Corresponding author.Email:ronquist@math.ntnu.no
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Abstract

In this paper we study Bénard-Marangoni convection in confined containers where a thin fluid layer is heated from below. We consider containers with circular, square and hexagonal cross-sections. For Marangoni numbers close to the critical Marangoni number, the flow patterns are dominated by the appearance of the well-known hexagonal convection cells. The main purpose of this computational study is to explore the possible patterns the system may end up in for a given set of parameters. In a series of numerical experiments, the coupled fluid-thermal system is started with a zero initial condition for the velocity and a random initial condition for the temperature. For a given set of parameters we demonstrate that the system can end up in more than one state. For example, the final state of the system may be dominated by a steady convection pattern with a fixed number of cells, however, the same system may occasionally end up in a steady pattern involving a slightly different number of cells, or it may end up in a state where most of the cells are stationary, while one or more cells end up in an oscillatory state. For larger aspect ratio containers, we are also able to reproduce dislocations in the convection pattern, which have also been observed experimentally. It has been conjectured that such imperfections (e.g., a localized star-like pattern) are due to small irregularities in the experimental setup (e.g., the geometry of the container). However, we show, through controlled numerical experiments, that such phenomena may appear under otherwise ideal conditions. By repeating the numerical experiments for the same non-dimensional numbers, using a different random initial condition for the temperature in each case, we are able to get an indication of how rare such events are. Next, we study the effect of symmetrizing the initial conditions. Finally, we study the effect of selected geometry deformations on the resulting convection patterns.

Keywords

65C2076D0576E0665N35Bénard-Marangonispectral elementspattern formation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 893 - 924
Copyright
Copyright © Global Science Press Limited 2012

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