Double-diffusive Marangoni convection in a rectangular cavity: Onset of convection

Abstract

Double-diffusive Marangoni convection in a rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal Marangoni effects are of equal magnitude ͑solutal to thermal Marangoni number ratio R =−1͒. In this case a no-flow equilibrium solution exists and can remain stable up to a critical thermal Marangoni number. Linear stability analysis and direct numerical simulation show that this critical value corresponds to a supercritical Hopf bifurcation point, which leads the quiescent fluid directly into the oscillatory flow regime. Influences of the Lewis number Le, Prandtl number Pr, and the cavity aspect ratio A ͑height/length͒ on the onset of instability are systematically investigated and different modes of oscillation are obtained. The first mode is first destabilized and then stabilized. Sometimes it never gets onset. A physical illustration is provided to demonstrate the instability mechanism and to explain why the oscillatory flow after the onset of instability corresponds to countersense rotating vortices traveling from right to left in the present configuration, as obtained by direct numerical simulation. Finally the simultaneous existence of both steady and oscillatory flow regimes is shown. While the oscillatory flow arises from small disturbances, the steady flow, which has been described in the literature, is induced by finite amplitude disturbances.

Transition to chaos in thermocapillary convection

“…Thermocapillary flow for silicone oil 1.5 cSt with the depth of 2 mm follows the Feigenbaum route, which undergoes 2 period-doubling bifurcations. The proportional relationship between oscillatory frequency and Ma obtained in our experimental study is in agreement with numerical results by Yok-Sheung Li et al [21,22].…”

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

“…Validations of both the linear stability analysis and direct numerical simulation procedures were detailed in Chen et al, 17 where these two methods were successfully used to investigate the onset of double-diffusive Marangoni convection in a similar configuration.…”

“…However, the nature of the primary bifurcation point had long remained unclear. Very recently, this problem was studied systematically by Chen et al, 17 where it was shown by both linear stability analysis and direct simulation that the first primary instability of the equilibrium corresponds to a supercritical Hopf bifurcation. The flow arising from small disturbance is oscillatory rather than steady.…”

Energy Budget of Internal Solitary Waves in South China Sea from Continuously Stratified Nonhydrostatic Model

“…Details of the numerical implementations of the above procedures can be found in Ferziger and Peric (2002). This numerical code has been successfully used to study a series of double-diffusive convection problems (Chen et al, 2010a(Chen et al, , 2010b(Chen et al, , 2012 …”

“…In order to study such a wide range of geophysical flows, considerable effort has been spent on upgrading the existing hydrostatic model to a nonhydrostatic one, e.g., the hydrostatic ROMS (Regional Oceanic Modeling System) now has a nonhydrostatic version (Kanarska et al, 2007). In the present work, an in-house fully nonlinear, nonhydrostatic numerical code that is originally developed for the study of double-diffusive convection (Chen et al, 2010a(Chen et al, , 2010b(Chen et al, , 2012(Chen et al, and 2013Li et al, 2010;Zhan et al, 2010) is now employed for simulation of ISWs generated by tidal flows over a Gaussian sill. The energy budget of these ISWs is then analyzed.…”