Motion and Decay of a Vortex in a Nonuniform Stream

Abstract

The motion of a vortex in a two-dimensional incompressible nonuniform stream is studied by including the viscous effects in the inner core of the vortex. A systematic procedure is presented by the introduction of two different sets of length and time scales with the larger scales identified with the typical scales of the outer nonuniform flow. The ratios of the scales are powers of the small parameter ε which is inversely proportional to the square root of the Reynolds number of the vortex. It is shown that the leading term of the solution to the Navier-Stokes equations are composed of the classical inviscid solution matched with the solution of a decaying axially symmetric vortex. Thus the singularity in the center of the vortex associated with the classical inviscid theory is removed. The next-order solution shows that the time average of the velocity of the center of the vortex over a period of the order of the larger time scale should agree at least to the order of ε2 with the local space mean of the velocity of the outer inviscid flow which has been assumed to be the velocity of the center of the vortex in the classical inviscid theory of vortex motion. The nonuniformity of the outer flow will induce a nonaxially symmetric solution for the inner viscous region of the order ε2 or higher, which in turn will introduce an additional term to the outer flow of the order of ε4 or higher.