Motion of a vortex pair at high and low Reynolds numbers

Abstract eng:
We establish a high- and a low-Reynolds-number asymptotics for the speed of a counter-rotating vortex pair, traveling in a viscous incompressible fluid. At a high Reynolds number, the solution of the Navier-Stokes equation is constructed by use of the matched asymptotic expansions in a small parameter, a measure of the core radius to the half distance between the vortices. The correction to the traveling speed, originating from finite-thickness effect, arises at fifth order. We drastically simplify it in a form including solely strength of the quadrupole field of second order. We derive a two-dimensional analogue, applicable in the entire range of the Reynolds number, of the Helmholtz-Lamb formula for an axisymmetric vortex ring. At a low Reynolds number, the vorticity field obeying the Stokes equation is substituted into this formula. Thereby we describe the whole life of the motion a vortex pair.

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