On the selection of perturbations for thermal boundary layer control

Abstract

The convective instability of the natural convection boundary layers of air (Pr = 0.7) in the laminar-to-turbulent transition regime (Ra = 8.7 × 107-1.1 × 109) is investigated by stability analysis in the framework of direct numerical simulations. To understand the spatial and temporal evolution of the convective instability of the thermal boundary layers, small-amplitude random-mode numerical perturbations are first introduced into the boundary condition of the boundary layer flow. The prescribed full spectral perturbations (i.e., white noise) are mostly damped out immediately by a limited upstream boundary layer. A low-frequency band is initially distinct in the upstream near the leading edge but decays spatially as the instability propagates downstream. In contrast, a high-frequency band emerges to finally become the most dominant frequency band in the thermal boundary layer transition regime. To obtain further insights into the nature of the established high-frequency band, single-mode perturbations of various frequencies are then introduced into the boundary layer near the leading edge. It is found that a single-mode perturbation at the peak frequency within the high-frequency band excites the maximum response of the thermal boundary layer, suggesting that the peak frequency is in fact the characteristic frequency or resonance frequency of the thermal boundary layer. The dimensionless form of the dependence of the characteristic frequency on Ra is then found to be fc = 0.07Ra2/3. The single-mode perturbation numerical experiments also revealed the propagation speed of convective instability waves, which was significantly greater than the convection speed of the thermal boundary layer. The smaller the Ra, the larger the difference between the two propagation speeds. A semi-analytical scaling of the wave propagation speed in the form csc ∼Ra1/2y1/2Pr was derived (y denoting the streamwise location of the boundary layer), providing a predictive correlation that can be used for thermal boundary layer control.