Linear stability of natural convection in a differentially heated shallow cavity submitted to high-frequency horizontal vibrations

Abstract

This study concerns the linear stability of buoyant convection induced by lateral heating inside a shallow cavity. It highlights the effects caused by submitting the flow to horizontal high-frequency vibrations. The steady-state profiles are first derived using a parallel flow approximation and studied for two types of boundaries, either thermally insulating or thermally conducting. The basic flow is found to be attenuated when subjected to horizontal high-frequency vibrations, with a faster decay in the case of thermally insulating walls than in the case of thermally conducting walls. The effects of vibrations and thermal boundary conditions are then investigated for various types of instability that may arise in such a situation, depending on the Prandtl number, such as shear, oscillatory, and thermal instabilities. It is observed that horizontal high-frequency vibrations have a stabilizing effect on all instabilities developing in such a situation and that this stabilization is generally more efficient in the case of insulating walls, for which the basic flow is attenuated more rapidly. We finally analyze the physical mechanisms that trigger these instabilities through fluctuating energy budgets at the critical thresholds.