A class of unsteady self-similar flows without an ?essential? singularity and a mechanism of whirlwind formation

Abstract

This article is a continuation of reference [1], in which the author represented the known self-similar incompressible flows by means of a single table. The table incorporated more than 50 problems of practical interest. Although such a representation can never be complete, the method of representation itself also proved useful, since it drew attention to the existence of interesting uninvestigated self-similar flows corresponding to the “empty” boxes of the table. For example, it turned out that the problem of unconfined flow resulting from the instantaneous generation of finite momentum at a point had been considered in the case of turbulent viscosity [2], but the analogous formulation for an ideal fluid was not known, although on the basis of dimensional considerations such a self-similar flow can exist and, in fact, was subsequently found [3]. A class of self-similar flows comprising more than 20 problems which can be combined in five different formulations is considered. These flows can have an “essential” singularity only at the initial instant of time (for example, a powerful explosion) and subsequently do not contain singularities of the energy or mass source type. Flows caused by the motion of bodies are excluded from consideration, i.e., cases in which the flow is bounded by fixed or free boundaries with zero pressure on the latter are considered. In the case of an ideal fluid vortex singularities at which no energy is released are admitted.