Effects of thermal boundary conditions and cavity tilt on hydrothermal waves: suppression of oscillations.

Hydrothermal waves are longitudinal modes responsible for the onset of oscillations of low-Prandtl number flows inside end-heated cavities. We consider the flow induced by the hydrothermal wave in a rectangular enclosure whose differentially-heated side is tilted alpha degrees from the vertical position. An analytical approximation to the neutral curve and dispersion relation obtained by the Galerkin procedure is shown to quantitatively agree with the exact numerical solution of the stability problem. The analytical expressions are then used to dissect the effect of the Prandtl and Biot numbers and the inclination on the wave stability. In conducting walls the critical Rayleigh R(cr) and wave number m(cr) tend to a constant value at low Pr, while the critical frequency f(cr) approximately Pr(-1/12). In adiabatic walls all these critical parameters increase like Pr(1/2). The boundaries can be considered to be poorly insulated if Bi>Pr, and in this case the critical parameters increase like Bi(1/2). On the other hand, R(cr) and m(cr) reach a minimum value at intermediate inclinations, while the critical frequency steadily increases with alpha. A closed equation for the frequency is also derived. This equation correctly forecasts the critical frequency in the unbounded domain and also the fundamental frequency measured in confined flows, as revealed by comparison with previous experiments and hereby presented numerical calculations for varying alpha. An important conclusion of the study is that for any arbitrarily small value of Pr the hydrothermal wave can be suppressed by heating the cavity above a theoretically predicted (Pr-dependent) angle. This prediction is of great relevance in the application domain (viz. the crystal growth from melts by the Bridgman technique).