Absolute and convective instabilities of natural convection flow in boundary-layer regime

Abstract

The spatiotemporal instability of the buoyancy-driven flow adjacent to a vertically heated wall, which is immersed in thermally stratified medium, is studied theoretically and numerically. The temperature gradients ratio between the wall and the ambient fluid is shown to lead to rich scenario of absolute-convective instability transitions. The direct numerical simulations consistent with the theoretical prediction are presented. The supercritical steady state, found in previous simulations of the natural convection in vertically heated square cavity, is explained in terms of the convective instability, and the nonlinear effect on the convectively unstable waves is discussed as well.

Instability of the buoyancy layer on an evenly heated vertical wall

“…It follows (Shapiro & Fedorovich 2004b) that the wall heat flux is uniform, and that the solution also applies when the excess temperature boundary condition is replaced by one of a uniform normal temperature gradient (γ w , as indicated in figure 1). This flux condition is perhaps more natural, say, as a model of solar radiation (Tao, Le Quéré & Xin 2004a) or electrical resistance heating, or of radiative cooling to a clear night sky. It is thus the more natural condition (Manins & Sawford 1979;Skyllingstad 2003) for Prandtl's (1952, pp.…”

Nonlinear global instability in buoyancy-driven boundary-layer flows

“…It has been confirmed by direct numerical simulations (Tao, Le Quéré & Xin 2004b) that this similarity solution describes exactly the steady convection flows near the vertical walls dissipating uniform heat flux in a cavity. Since the velocity component includes the coordinate X, the boundary-layer flow is a slowly spatially developing flow except for a special case (a = 1).…”

“…The coupled disturbance equations (3.5) and (3.6) are discretized with a fourthorder finite difference scheme at uniformly distributed points in the η interval, and a fully spatio-temporal stability analysis is carried out in order to compare with direct numerical simulation results. For details of the solution methods refer to Tao & Zhuang (2000) and Tao et al (2004b).…”

Natural convection in air-filled differentially heated isoflux cavities: Scalings and transition to unsteadiness, a long story made short

“…The large difference between between the critical values resulting from linear stability theory of the boundary layer solution and the bifurcation to unsteadiness in the closed cavity, either isothermal or isoflux, has been logically ascribed to the difference between the concepts of convective and absolute instability [50], [59]. It has motivated several studies of wave properties [60] or convective-absolute stability analysis of various similarity solutions of boundary layers adjacent to vertical walls, [61], [62], [63], [64]. Although Tao [64] has been able to confirm quantitatively and relate the results from a convective-absolute linear stability theory to the existence of a global instability with sustained oscillations in the numerical simulation of a two dimensional flow developing along a vertical plate in a stratified environment, none of these studies have succeeded in thoroughly explaining the relationship between the instability studies of the buoyancy layer and the transition to unsteadiness in the isoflux cavity.…”