Wavy secondary instability of longitudinal rolls in Rayleigh–Bénard–Poiseuille flows

Abstract

An experimental investigation of the stability of longitudinal rolls in a horizontal layer heated from below in the presence of a Poiseuille flow is carried out. This study follows on from the theoretical work of Clever & Busse (J. Fluid Mech., vol. 229, 1991, p. 517) who detected a wavy instability for a range of relatively low Rayleigh and Reynolds numbers depending on the Prandtl number. In the present study, an air flow is circulating in a rectangular channel of transverse aspect ratio 10 for Rayleigh numbers of 6300 and 9000 and Reynolds numbers from 100 to 174. The system exhibits a wavy pattern only if the flow is continuously excited. The amplitude of the waves grows as they propagate downstream and the frequency of the oscillations is equal to the frequency of the imposed disturbance. The bifurcation from steady longitudinal rolls to unsteady wavy rolls is thus a convective instability. A mode by mode study is performed by measuring the wave velocity and the spatial growth of the instability along the channel for a large range of the imposed frequency. The phase velocity is found to depend only on the Reynolds number, and is nearly equal to the bulk velocity of the flow for all the modes in the range of parameters under study. The maximum spatial growth rate corresponding to the most unstable mode as well as the corresponding frequency decrease with decreasing Reynolds number or Rayleigh number, providing a decrease in the wavelength. This feature is in agreement with the theoretical results of Clever & Busse (1991).