The effect of wall heating on instability of channel flow

(Received ?? and in revised form ??) A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. ¿From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Rayleigh-Benard and Poiseuille modes for Grashof numbers upto ∼ 25000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary, destabilising significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows. One of the well-known methods for delaying a transition to turbulence, for example in boundary layers, has been to reduce the viscosity at the wall. Such a reduction could be brought about by heating or cooling the surface, for example. The objective of this paper is to study the effect of wall heating on the instability of a channel flow. It is shown that heat can have surprising effects on the different mechanisms of transition. We restrict ourselves here to routes based on the linear eigenmodes, a direct nonlinear interaction will be studied in future. The emphasis here is on delaying/advancing the onset of transition to turbulence, rather than drag reduction in full turbulence, as achieved by adding small quantities of polymer. The critical Reynolds number for linear instability in a plane Poiseuille flow is 5772.22 [Orszag (1971)]. However, experiments usually find fully developed turbulence at a much lower Reynolds number, around 1500 [see e. It is clear that routes to turbulence other than the traditional Tollmien-Schlichting (TS) mechanism are in operation. The background noise in the flow has a major influence in delaying/hastening transition to tur