Bénard-Marangoni Convection
We consider Benard-Marangoni convection in confined containers (here circular) where a thin (d) fluid layer is heated from below (q). This problem has previously been studied extensively, both experimentally as well as computationally. An intriguing feature of this problem is the formation of hexagonal convection cells from random initial conditions. It has been found that the onset of convection can originate from two different effects; it can be caused by buoyancy effects due to the fact that the density is a function of the temperature, or it can be due to the variation in the surface tension, i.e., thermocapillary forces. Both effects can also be present at the same time. In fact, the concurrent presence of the two effects was a source of confusion for years. Benard himself had an incorrect interpretation of which effect was the dominant one in his original experiments, which lead to Rayleigh's subsequent stability analysis also being done under the assumption of buoyancy-driven convection. It took several decades before this misunderstanding was cleared up.
Through numerical simulations 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 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 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.
This work has been funded by the Research Council of Norway.