Nonequilibrium wake flows

Abstract

In order to study the memory of the larger eddies in turbulent shear flow, experiments have been conducted on plane turbulent wakes subjected to external pressure gradients. In some of these experiments the pressure gradient was effectively an impulse function, obtained by changing the free stream velocity over a relatively short distance, and allowed observation of the wake during transition from an initial (carefully prepared) equilibrium state to a different final one. It is shown that under the conditions obtaining in the experiment the equations of motion possess self?preserving solutions in the sense of Townsend (1956), but the observed behaviour of the wake is appreciably different, as the flow goes through a slow relaxation process before achieving final equilibrium. Measurements of the Reynolds stresses in the nonequilibrium region show that to a good approximation the final approach to equilibrium is exponential, with a relaxation length of the order of 10² momentum thicknesses. It is shown that consideration of a model equation for the stress explicitly incorporating a characteristic relaxation time (Narasimha 1969) limits the self?preserving solutions severely — indeed it appears very likely that only when the pressure gradient is very small or zero can the wake be in equilibrium. Calculations made using a simple version of this model equation show that the model provides an excellent description of not only the relaxing flows mentioned above, but also of wakes subjected to more general pressure gradients or to constant pressure distortions (Keffer 1965). As used in these calculations, the model involves only two empirical constants, one of which is related to the relaxation length and the other to equilibrium (constant pressure) wake growth; both can in principle be determined from a single relaxation experiment. Finally a simple integral method for calculating wake development using the model equation is proposed, and its accuracy is demonstrated by an example.