Instabilities in the Rayleigh-Bénard-Eckart problem

Abstract

This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width H(b) for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, A(c), has a minimum for a beam width H(b) ≈ 0.32. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [A(x) = (length/height) = 10]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter A on the stability threshold is investigated for various values of the beam width H(b) and different Prandtl numbers Pr. It is shown that, for not too small values of the Prandtl number (Pr > Pr(l)), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter A. The critical curves display two behaviors. For small or moderate values of A, the critical Rayleigh number Ra(c) increases with A up to a maximum. Then, when A is further increased, Ra(c) undergoes a decrease and eventually goes to 0 at A = A(c), i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value Pr(l) (which depends on H(b)), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.