Transition to oscillatory flow in a differentially heated cavity with a conducting partition

Abstract

AbstractNumerical evidence is presented for previously unreported flow behaviour in a two-dimensional rectangular side-heated cavity partitioned in the centre by vertical wall with an infinite conductivity. In this flow heat is transferred between both sides of the cavity through the conducting wall with natural convection boundary layers forming on all vertical surfaces. Simulations have been conducted over the range of Rayleigh numbers $\mathit{Ra}= 0. 6\text{{\ndash}} 1. 6\ensuremath{\times} 1{0}^{10} $ at Prandtl number $\mathit{Pr}= 7. 5$ and at aspect ratios of $H/ W= 1\text{{\ndash}} 2$ where $H$ and $W$ are the height and width of the cavity. It was found that the thermal coupling of the boundary layers on either side of the conducting partition causes the cavity flow to become absolutely unstable for a Rayleigh number at which otherwise similar non-partitioned cavity flow is steady but convectively unstable. Additionally, unlike the non-partitioned cavity, which eventually bifurcates to a multi-modal oscillatory regime, this bifurcation is manifested as a single mode oscillation with ${f}^{+ } = f{\nu }^{1/ 3} / \mathop{ (g\ensuremath{\beta} \mrm{\Delta} \theta )}\nolimits ^{2/ 3} \approx 0. 0145$, where $ \mrm{\Delta} \theta $ is the temperature difference between the hot and cold walls, $g$ is the gravitational acceleration, $f$ is the oscillation frequency and $\nu $ and $\ensuremath{\beta} $ are the fluid viscosity and coefficient of thermal expansion respectively. The critical Rayleigh number for this transition occurs between $\mathit{Ra}= 1. 0\text{{\ndash}} 1. 2\ensuremath{\times} 1{0}^{10} $ for $H/ W= 2$ and $\mathit{Ra}= 1. 2\text{{\ndash}} 1. 4\ensuremath{\times} 1{0}^{10} $ for $H/ W= 1$, indicating that the instability has an aspect ratio dependence.