Commun. Comput. Phys., 11 (2012), pp. 893-924. |
On Pattern Selection in Three-Dimensional Benard-Marangoni Flows Arne Morten Kvarving ^{1}, Tormod Bjontegaard ^{1}, Einar M. Ronquist ^{1*} 1 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway.Received 28 June 2010; Accepted (in revised version) 6 April 2011 Available online 28 October 2011 doi:10.4208/cicp.280610.060411a Abstract In this paper we study Benard-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. AMS subject classifications: 65C20, 76D05, 76E06, 65N35Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Benard-Marangoni, spectral elements, pattern formation. *Corresponding author. Email: arnemort@math.ntnu.no (A. M. Kvarving), bjontega@math.ntnu.no (T. Bjontegaard), ronquist@math.ntnu.no (E. M. Ronquist) |