OVERVIEW

The vorticity transport equation for barotropic fluids or for fluids whose density is uniform throughout the domain of flow [Eq. (3.8.9)] shows that the rate at which vorticity is produced vanishes at every point within an irrotational flow, and this suggests that vorticity may enter the flow only by means of diffusion across the boundaries. Once vorticity has entered the flow, it is convected by the velocity field while diffusing with a diffusivity that is equal to the kinematic viscosity of the fluid, and intensifies or attenuates due to vortex stretching. An unbounded flow does not have any vorticity entrance ports, and therefore, if it is irrotational at the initial instant, it will remain irrotational at all subsequent times.

Under certain circumstances, the distance across which the vorticity penetrates the fluid from the boundaries is small compared to the overall size of the boundaries, and the bulk of the flow remains nearly irrotational. This occurs, in particular, when the rate of the diffusion of vorticity into the flow is comparable to the rate of convection of vorticity by tangential and normal motions. Balancing the orders of magnitude of the rate of convection and diffusion of vorticity yields the formal requirement that the Reynolds number, defined with respect to the typical size of the boundaries, be sufficiently large. When the necessary conditions are met, the vorticity will be convected along thin layers that wrap around the boundaries, and will then either be channeled into slender wakes or deposited into regions...