The Decay of a Turbulent Trailing Vortex

Abstract

The decay of a trailing vortex containing a turbulent core is examined according to the argument that the turbulence is partly sustained by, and interacts with, the irrotational flow through viscous diffusion of eddies across the core boundary. The region separating the core from the outer irrotational flow, in which the interaction between the turbulence and the mean flow occurs, expands with time like (time)1/2, characteristic of viscous diffusion. On the basis of a simple representation of the radial distribution of mean circumferential velocity, the decay of the vortex is calculated and is found to behave in a way similar to what Squire predicted, on the assumption of a constant eddy viscosity, in 1954. The essential difference from Squire’s solution found here is that the apparent viscosity is not simply proportional to the initial circulation about the vortex as he proposed, but depends on the Reynolds number defined with respect to the circulation. Wind tunnel and flight observations of vortex decay can be more closely reconciled than Squire’s theory allowed; those differences that exist are attributed to the vastly different ages of the vortices in the laboratory and flight experiments. The principal conclusion of practical interest is that the maximum circumferential velocity in a trailing vortex generated by an aeroplane in flight is where Γ is the circulation about large circuits enclosing the vortex, v the kinematic viscosity of the air and t the (sufficiently great) age of the vortex.