Time-dependent thermocapillary convection in a rectangular cavity: numerical results for a moderate Prandtl number fluid

Abstract

The present numerical simulation explores a thermal–convective mechanism for oscillatory thermocapillary convection in a shallow rectangular cavity for a Prandtl number 6.78 fluid. The computer program developed for this simulation integrates the two-dimensional, time-dependent Navier–Stokes equations and the energy equation by a time-accurate method on a stretched, staggered mesh. Flat free surfaces are assumed. The instability is shown to depend upon temporal coupling between large-scale thermal structures within the flow field and the temperature sensitive free surface. A primary result of this study is the development of a stability diagram presenting the critical Marangoni number separating the steady from the time-dependent flow states as a function of aspect ratio for the range of values between 2.3 and 3.8. Within this range, a minimum critical aspect ratio near 2.3 and a minimum critical Marangoni number near 20 000 are predicted, below which steady convection is found.

Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: Past history and current status

“…For instance, in the case A=1, microgravity conditions and cylindrical shape, the critical Marangoni number for Silicon melt is Ma c 1 =12. 43. If the gravity-deformation of the shape is considered the critical Marangoni number (Ma c 1 =8.40) is significantly decreased with respect to the cases where the cylindrical shape is assumed (∆Ma c 1 =O(40%)).…”

“…For instance, the nearly two-dimensional nature of the hydrothermal wave for high Prandtl numbers is probably the reason why two-dimensional numerical computations (see, e.g., Peltier and Biringen [43]) easily capture oscillatory flow in open containers and/or finite extended layers. Xu and Zebib [44] found two-dimensional supercritical oscillatory bifurcations over a wide range of Prandtl numbers (1<Pr<13.9) and aspect ratios A (ratio of the length to the height of the finite layer).…”

Objective function choice for control of a thermocapillary flow using an adjoint-based control strategy

“…For Pr ¼ 14:8 (Shevtsova and Legros, 2003) the strength of the co-rotating vortices increases towards the hot wall (for low Prandtl numbers, see, Ben Hadid and Laure et al, 1990). Upon increasing the thermocapillary driving force beyond a critical value the flow becomes time-periodic (Peltier and Biringen, 1993;Xu and Zebib, 1998) with the cells and the associated temperature variations moving back and forth parallel to the free surface. Movie 1 shows an example for the uncontrolled oscillatory flow.…”

Numerical Study of Effect of Thermocapillary Convection on Melting Process of Phase Change Material subjected to Local Heating

“…This model is based on the fact that the mushy zone is porous medium filled with liquid by an up-scaling procedure. Here, A mush is a constant (1.0×10 6 (kg/(m 3 s))) accounting for the mushy region morphology. ε is a represented small value (0.001) introduced to prevent the numerical singularity at β = 0, where β is the fraction of liquid in the cell (0: liquid, 1: solid; details will be explained in later), which behaves effectively as a porous media (28) .…”