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

In a previous paper (Int. J. Thermal. Sci., vol. 47, pp. 1382-1392, 2008), the authors performed a detailed numerical investigation of the linear instability of the thermal boundary layer flow over a vertical surface by introducing unsteady thermal disturbances near the leading edge and by solving numerically the fully elliptic linearized stability equations. The main aim of the present paper is to extend those results into the nonlinear regime by seeding the boundary layer with similar disturbances of finite amplitude. The ensuing nonlinear waves are found to exhibit a variety of behaviours, depending on the precise amplitude and period of the forcing. When the amplitude is sufficiently small, the linearized theory of the previous work is reproduced, but for larger amplitudes, cell splitting or cell merging may occur as waves travel downstream. Cell splitting takes place when disturbance frequencies are somewhat smaller than the most strongly amplified nondimensional disturbance frequency of 0.4 for which the boundary layer response, is at its greatest in terms of the surface rate of heat transfer (see Fig. 8 in previous paper). Cell merging takes place at frequencies what are approximately double that of the most strongly amplified disturbance frequency. Attention is focussed on fluids with a unit Prandtl number.