Non-parallel linear stability analysis of the vertical boundary layer in a differentially heated cavity

Abstract

A non-parallel linear stability analysis which utilizes the assumptions made in the parabolized stability equations is applied to the buoyancy-driven flow in a differentially heated cavity. Numerical integration of the complete Navier–Stokes and energy equations is used to validate the non-parallel theory by introducing an oscillatory heat input at the upstream end of the boundary layer. In this way the stability properties are obtained by analysing the evolution of the resulting disturbances. The solutions show that the spatial growth rate and wavenumber are highly dependent on the transverse location and the disturbance flow quantity under consideration. The local solution to the parabolized stability equations accurately predicts the wave properties observed in the direct simulation whereas conventional parallel stability analysis overpredicts the spatial amplification and the wavenumber.

Thermal receptivity of free convective flow from a heated vertical surface: Linear waves

“…Rather, each streamwise location in the boundary layer has associated with it its own most dangerous frequency, at least in terms of the heat transfer response to otherwise identical disturbances. We have also seen that this frequency increases with increasing distance, which is the opposite to the conclusion of Brooker et al [15] who state that the frequency decreases. However, there are significant differences between the two respective studies: the present paper deals with the external flow of air in a uniformly cold environment, while Brooker et al consider water in a differentially heated square cavity where the core temperature field is stratified.…”

“…suction/blowing slots and temperature perturbations generate disturbance waves inside boundary layer. Some relevant works to this category are Fasel and Konzelmann [14], Brooker et al [15], Herwig and You [16].…”

Long-term behavior of cooling fluid in a vertical cylinder

“…Xin and Le Quéré [21] investiged numerically chaotic natural convection in a differentially heated air-filled cavity with adiabatic horizontal walls. Brooker et al [22] conducted a non-parallel linear stability analysis of the vertical boundary layer in a differentially heated cavity. Kwak et al [23] conducted a numerical study on the transient natural convective cool-down process of a fluid in a cylindrical container, with emphasis on the flow patterns when the maximum density temperature is experienced.…”

Boundary layer instability of the natural convection flow on a uniformly heated vertical plate

“…Direct numerical simulations of the natural convection cavity flow have been reported in [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and of the unbounded plate with stratified ambient in [24][25][26]. The cavity studies were mainly concerned with the occurrence of global …”

“…Thus while the linear stability analysis of the boundary layer can provide some insights into the cavity flow, further analysis of the cavity flow will not provide the detailed information on the semi-infinite plate boundary layer that is sought here. Brooker et al [21,22] investigated the stability of the natural convection boundary layer using direct numerical simulation and the non-parallel linear parabolized stability theory, as well as the standard parallel method, to in particular examine the importance of the non-parallel effects. They found excellent agreement for the critical Rayleigh number predictions between the parallel and non-parallel approaches for the higher unstable frequencies, but not for the lower unstable frequencies.…”