Hervé Pabiou

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

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).

Linear Stability Analysis for the Wall Temperature Feedback Control of Planar Poiseuille Flows

Active control of a planar Poiseuille flow can be performed by increasing or decreasing the wall temperature in proportion to the observed wall shear stress perturbation. In continuation with the work of H. H. Hu and H. H. Bau (1994, Feedback Control to Delay or Advance Linear Loss of Stability in Planar Poiseuille Flow, Proc. R. Soc. London A, 447, pp. 299-312), a linear stability analysis of such a feedback control is developed in this paper The Poiseuille flow control problem is reduced to a modified Orr-Sommerfeld equation coupled with a heat equation. By solving numerically the coupled equations with a finite element method, many numerical results about the stability of the flow control are obtained. We focus our attention on the interpretation of the numerical results. In particular, the role of two essential parameters-the Prandtl number Pr and the control gain K-is investigated in detail. When Pr>1.31, stabilizing K is negative; while, when Pr <1.31, stabilizing K is positive. And when Pr =1.31, the flow cannot be stabilized by a real K. A comparison between symmetric two-wall control and non-symmetric one-wall control is also made.