Oscillatory two- and three-dimensional thermocapillary convection

Abstract

The character and stability of two- and three-dimensional thermocapillary driven convection are investigated by numerical simulations. In two dimensions, Hopf bifurcation neutral curves are delineated for fluids with Prandtl numbers ( Pr ) 10.0, 6.78, 4.4 and 1.0 in the Reynolds number ( Re )–cavity aspect ratio ( A x ) plane corresponding to Re [les ]1.3×10 4 and A x [les ]7.0. It is found that time-dependent motion is only possible if A x exceeds a critical value, A xcr , which increases with decreasing Pr . There are two coexisting neutral curves for Pr [ges ]4.4. Streamline and isotherm patterns are presented at different Re and A x corresponding to stationary and oscillatory states. Energy analyses of oscillatory flows are performed in the neighbourhood of critical points to determine the mechanisms leading to instability. Results are provided for flows near both critical points of the first unstable region with A x =3.0 and Pr =10. In three dimensions, attention is focused on the influence of sidewalls, located at y =0 and y = A y , and spanwise motion on the transition. In general, sidewalls have a damping effect on oscillations and hence increase the magnitude of the first critical Re . However, the existence of spanwise waves can reduce this critical Re . At large aspect ratios A x = A y =15, our results with Pr =13.9 at the lower critical Reynolds number of the first unstable region are in good agreement with those from infinite layer linear stability analysis.