Studies in the Motion and Decay of Vortices
Solutions of Navier-Stokes equations are constructed as an asymptotic expansion in terms of a small parameter related to the Reynolds number of the vortex. A general scheme is presented for the matching of the inner viscous core of the vortex to the outer inviscid solution. The singularities in the inviscid theory are removed and the condition of regularity in the flow field defines the velocity of the vortex line. This general scheme is applied to study vortices in two dimensional, axially symmetric and in three dimensional flow fields. Results for the outer solution which do or do not require the solution in the viscous core are obtained separately. Several examples are presented to show that the solution of this analysis can be identified with that of classical inviscid theory with the same initial vorticity distribution for the initial instant. They disagree afterwards because the inviscid theory ignores the diffusion of vorticity in the core. For a general three dimensional problem, the present scheme of analysis can be carried out provided the shape of the vortex line and the three dimensional flow field fulfill a constraint condition. This condition is automatically fulfilled for the two dimensional and axially symmetric problems. When the initial velocity of the vortex is prescribed to be different from that given by the analysis, the necessary modifications to the expansion scheme and the solution for the subsequent motion are presented.
KeywordsVortex Ring Similar Solution Vortex Line Three Dimensional Flow Vortex Point
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