The Stability of Water Flow Over Heated and Cooled Flat Plates

Abstract

The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.

Boundary-Layer Stability and Transition

“…It is curious that for boundary layers in water, the effect of cooling is destabilizing while the effect of heating is stabilizing (Wazzan, Okamura and Smith 1967). Because the viscosity of water decreases sharply with increase in temperature, heating yields a fuller velocity profile while cooling tends to give an inflected velocity profile.…”

Linear stability analysis of miscible two-fluid flow in a channel with velocity slip at the walls

“…[13][14][15] Wall and Wilson 15 studied the influence of continuous viscosity variation due to a temperature gradient in a rigid channel and showed that the Péclet number has little influence on the stability and that the base state viscosity influences the Tollmien-Schlichting (TS) mode. The experiments [16][17][18][19][20] on miscible two-fluid flow in different geometries revealed interesting instabilities.…”

On the Breakdown of the Wave Packet Trailing a Turbulent Spot in a Laminar Layer.

“…Farther downstream, the boundary layer became unstable, causing amplification and eventual breakdown of the waves. Since the stability of the boundary layer to small disturbances is greatly affected by the pressure gradient (Wazzan et al 1968), the same may hold for the amplification of the wave packet accompanying the spot. Some aspects of the experiment were therefore repeated in the presence of a slightly favorable pressure gradient for the purpose of delaying the breakdown process.…”