Stability analysis of a growing horizontal thermal layer subject to sudden bottom heating

Abstract

The hydrodynamic stability considered here is formulated in terms of a two-point boundary-value, eigenvalue problem derived by assuming periodic small disturbances superimposed on a time-dependent base-state which is the quiescent growing conduction layer resulting from a step in surface heat flux. Since the local heat flux field is similar, a dimensionless similarity parameter arises. The assumed disturbance form is guided by previous stability investigations The stiff equations are integrated to obtain the neutral stability curve with an extreme at Gr = 202.19, α = 1.151, and C r = 0.0 Although the postulated disturbance form allowed for time-dependent periodic oscillation, the computed disturbance wave velocity is found to be identically zero for the entire curve. Thus, it is presumed that the assumed oscillatory component of the perturbation is not time dependent, during the instability onset, at least for the calculated range (202 ⩽ Gr ⩽ 1000). Thus, unlike analyses for steady base-states, these results prohibit ‘tracking’ disturbances along constant frequency paths. Nevertheless, experimental results suggest that the wave number should decrease with increasing Gr as the neutral curve is crossed by the perturbation.