Convective instabilities in a rotating vertical Hele-Shaw cell

Abstract

Convective instabilities driven by vertical buoyancy in a Boussinesq fluid in a rotating vertical Hele-Shaw cell, a long channel with rectangular cross section of finite height h and small width ⌫h with ⌫ 1, are investigated both analytically and numerically. The problem is characterized by the Taylor number T, the Rayleigh number R, and the aspect ratio ⌫. Explicit asymptotic solutions describing convective instabilities are derived for ⌫T 1/6 1, where T is assumed to be large compared to unity. Comparison between the asymptotic and fully numerical solutions shows a satisfactory quantitative agreement. It is found that an overall condition for convective instabilities becomes optimal when ⌫T 1/6 = O͑1͒. Direct three-dimensional simulations for strongly nonlinear convection are also carried out in the regime 0 Ͻ ͑R − R c ͒ / R c Յ O͑10͒, where R c denotes the critical Rayleigh number. As a consequence of both the geometric and dynamic constraints imposed by the narrow channel ͑geometric͒ and rapid rotation ͑dynamic͒, the nonlinear flow remains temporally stationary and spatially simple and is comprised primarily of vertically long thin convection cells that transport heat all the way from the bottom to the top of the channel.

Geometry effects on Rayleigh-Bénard convection in rotating annular layers

“…Instead they combine to form a nearly steady pattern of convection in the form of rolls oriented nearly perpendicular to the sidewalls as shown in Fig. 7 In the limit of an infinite radius corresponding to a straight channel both sidewalls are equivalent and a steady convection pattern must be expected [9,10,17]. In the present configuration the dominance of the mode at the outer wall gives rise to a pattern drifting steadily in the retrograde direction with a frequency somewhat larger than the sum of the two sidewall mode frequencies.…”