Family of Similarity Flows with Vortex Sheets

Abstract

Prandtl's two-dimensional, time-dependent similarity solutions are corrected and extended to cases of N-branched vortex sheets possessing a certain central symmetry. For both N = 1 (Prandtl's case) and N = 2 it is found that the similarity parameter m is restricted to the range m ≤ 0.5. When N = 1, multiple solutions are found for a given value of m provided |m| is sufficiently small compared with one. A property of these multiple solutions is that the tangential velocity to the concave side of a vortex sheet is 0[exp( −4π) smaller than the tangential velocity to the convex side. All values of m are found to be admissible for the case N = 3. In all cases it is found that certain features of the flows resemble a classical (but time-dependent) circular vortex. For example, it is found that the circulation in a disk of radius r is proportional to r2. There are also important differences. The flow fields for N = 1, 2, and 3 are described in some detail.