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 × 10 7 -1.1 × 10 9 ) 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 = 0.07Ra 2/3 . 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 ∼ Ra 1/2 y 1/2 Pr was derived (y denoting the streamwise location of the boundary layer), providing a predictive correlation that can be used for thermal boundary layer control.

Experimental investigation of a turbulent natural convection flow in a cubic cavity with an inner obstacle partially heated

“…More recently, developing perturbations along isothermal vertical plates were studied by Zhao et al [26,27,28,29]. They have also observed a growing perturbation of high frequency (that does not depend on the vertical position along the plate) which is acknowledged as the characteristic frequency.…”

Scaling laws for natural convection boundary layer of a Pr > 1 fluid on a vertical solid surface subject to a sinusoidal heating flux in a linearly stratified ambient

“…In practical applications, particularly in solar thermal technologies, the heating is time‐varying as the solar radiation on surfaces is nearly sinusoidal 1 . Nevertheless, presently there have been very limited investigations on the unsteady NCBL flow on a vertical solid surface heated by a heat flux or temperature which changes with time 11,16–18 …”

“…1 Nevertheless, presently there have been very limited investigations on the unsteady NCBL flow on a vertical solid surface heated by a heat flux or temperature which changes with time. 11,[16][17][18] We obtained, through a thorough scaling analysis, the scalings for the key variables which characterize the behavior of unsteady NCBL flow of a homogenous Pr > 1 fluid on a vertical solid surface subject to sinusoidal heat flux. 11 These variables include the wall temperature, the temperature boundary-layer (TBL) thickness, the velocity boundary-layer (VBL) thickness, the maximum vertical velocity within the VBL, the Nusselt number, and the corresponding time scales typifying the various flow development stages.…”

“…[28][29][30] There are also some other recent studies on the unsteady NCBL flow behavior subject to different timedependent sinusoidal temperature or heat flux for Newtonian and non-Newtonian fluids, such as References [31][32][33][34][35]. Zhao et al 17 studied the convective instability of NCBL flow during its transition to the turbulent regime by introducing time-dependent sinusoidal temperature perturbations.…”