The decay of a viscous vortex pair

Publisher Website
Google Scholar
Abstract
The evolution of a viscous vortex pair is investigated through the use of a heuristic model. The model is based on the linear superposition of two Oseen vortices of opposite circulation spaced a distance 2b apart. The vortices are allowed to evolve through viscous diffusion and their mutual induction. The motion is unforced and as a consequence the total hydrodynamic impulse is exactly conserved for all time. In the model the total circulation in the upper half plane is assumed to remain initially constant. This constraint is applied up to a finite time when the model solution reaches its asymptotic form corresponding to a drifting Stokes dipole dominated by interdiffusion of vorticity across the plane of symmetry. The drift velocity of the vortex pair is determined by the condition that the integrated pressure force vanishes on the line of symmetry at all times. At large time this leads to an asymptotic value of the drift velocity which scales with the similarity properties of the Stokes solution. To provide a more rigorous foundation for the drift, the asymptotic behavior of the flow for large time is investigated through an expansion of the solution in inverse powers of the time. First the second‐order pressure is determined as a solution of a Poisson equation with the source term generated by the first‐order flow field. Surprisingly, the solution turns out to be independent of the drift. Nevertheless, an exact condition for the drift is found by considering the limiting form of the second‐order pressure at infinity where the flow is irrotational and the pressure can be computed directly from the first‐order velocity field using Bernoulli’s equation. In this latter approach the far field pressure is determined up to an unknown function of time which upon comparison with the Poisson solution is identified as the drift. The exact drift obtained in this fashion differs by only 10% from the value obtained using the pressure field of the heuristic model. Finally, it is shown that the existence of the complete second‐order asymptotic solution of the Navier–Stokes equations requires the inclusion of the same drift in the first‐order solution that was found from the examination of the pressure. The second‐order vorticity and streamfunction are determined; the latter contains a free constant to accommodate conditions at earlier times. Prospects for the existence of higher‐order asymptotic terms are discussed.

This publication has 6 references indexed in Scilit: