Linear and nonlinear dynamics of a differentially heated slot under gravity modulation

Abstract

In this paper we consider the effect of sinusoidal gravity modulation of size ε on a differentially heated infinite slot in which a vertical temperature stratification is imposed on the walls. The slot problem is characterized by a Rayleigh number, Prandtl number, and the imposed uniform stratification on the walls. When ε is small, we show by regular perturbation expansion in ε that the modulation interacts with the natural mode of the system to produce resonances, confirming the results of Farooq & Homsy (1994). For ε ∼ O(1) we show that the modulation can potentially destabilize the longwave eigenmodes of the slot problem. This is achieved by projecting the governing equations onto the least-damped eigenmode, and investigating the resulting Mathieu equation via Floquet theory. No instability was found at large values of the Prandtl number and also low stratification, when there are no travelling modes present.To understand the nonlinear saturation mechanisms of this growth, we consider a two-mode model of the slot problem with the primary mode being the least-damped travelling parallel-flow mode as before and a secondary mode of finite wavenumber. By projecting the governing equations onto these two modes we obtained the equations for temporal evolution of the two modes. For modulation amplitudes above critical, the growth of the primary mode is saturated resulting in a stable weak nonlinear synchronous oscillation of the primary mode. An unexpected and intriguing feature of the coupling is that the secondary mode exhibits very high-frequency bursts which appear once every cycle of the forcing frequency.

Nonlinear dynamics of two-dimensional convection in a vertically stratified slot with and without gravity modulation

“…(Figure 8 contains results not only for γ = 10, Ra = 10 5 for comparison with earlier results, but also γ = 2, Ra = 5 × 10 5 , which is relevant to studies in a later section.) The flow is found to be less stable for the larger γ, in general agreement with the results of Farooq & Homsy (1996).…”

“…Thus our main conclusions from this section are: (i) the features observed by Farooq & Homsy (1996) based on a Galerkin expansion are consistent with the solutions of the partial differential equations, but are quantitatively different;…”

Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

“…the blow-up in finite time is firmly established by our computations. This nonlinear-like response of the system under consideration is an innate feature of the problem and has been first discovered by Farooq and Homsy [9,11] using analysis in normal modes and comparing the most dominant mode with the forcing function.…”

“…The problem is well described in the literature (refer [8,9] and Figure 1 for a definition sketch), and the notation we use is standard:…”

g -Jitter induced free convection near a stagnation point

“…There have also been a number of studies to investigate the effects of g-jitter on¯uid motion of a viscous¯uid in cavities and channels, e.g. Biringen and Peltier [2], Biringen and Danabasoglu [3], Alexander et al [4], Farooq and Homsy [5,6], Li [7], Pan and Li [8], and Suresh et al [9]. These studies have shed some light on the basic nature of g-jitter effects and have provided a thrust to devise useful mechanisms by which the g-jitter induced convective¯ows may be suppressed.…”