Vortex regions in a potential stream with a jump of Bernoulli's constant at the boundary: PMM vol. 35, no. 5, 1971, pp. 773–779

Abstract

The problem of “splicing” of a vortex flow in a certain finite region of an incompressible fluid with the surrounding potential stream along a fluid streamline is considered in the case in which the Bernoulli constant is subject to discontinuity of a given magnitude along the streamline separating these two flows. A solution is found in the form of integrals containing two unknown functions for the definition of the contour and the vortex sheet intensity. A system of two nonlinear integral equations is derived for the determination of these parameters and the results of certain computer calculations are presented. Some of the recent models of incompressible fluid flow with zones of separation at high Reynolds numbers [1, 2] show that the limit solution of the Navier-Stokes equations defines a flow with a constant vortex in the separation zone (in the case of plane flow) bordering on the external potential stream. This has prompted a number of investigations of vortex and potential flows in contact along a fluid streamline. The problem of such flow in a given finite region is considered in [3]. A similar problem of flow in an unbounded region is considered in [4 – 6], and an application of this solution to the investigation of flows past bodies with stationary separation zones at high Reynolds numbers is presented in [7]. The problem of “splicing” of vortex and potential flows in the presence of a body when the Bernoulli constant becomes discontinuous at the vortex zone boundary is examined in [8] in an approximate manner. Below we present a solution of the exactly formulated problem of “splicing” in the presence of a jump of Bernoulli's constant in a flow without rigid boundaries, which according to [7] corresponds to infinitely great Reynolds numbers and special boundary conditions in the separation zone.