Rotational inviscid flow with circulation zones in a channel of variable cross section

Abstract

The development of the theory of rotational motion of inviscid fluids for the purposes of describing channel flow encounters certain difficulties in connection with the appearance of viscosity effects near the walls. In the potential-rotational model [1], in which the vorticity is nonzero only in a closed circulation zone surrounded by potential flow, it is assumed that the separation and attachment points are known in advance. For example, for flow around a cavity these points coincide with the extreme corner points of the contour. The problem of determining the vorticity in a closed zone for the potential-rotational model has been investigated in a number of studies [2, 3], etc. In the case of an incompressible fluid the vorticity in the circulation zone is constant for two-dimensional flow and proportional to the distance from the axis for axisymmetric flow. The value of the constant is found from the steady-state condition for the adjoining viscous layers. If the channel walls have a smooth profile without corner points, then for determining the boundaries of the circulation zones additional conditions must be used. This study employs another scheme, in which the vorticity is formed outside the region of flow and in a particular problem is specified in the form of a boundary condition. An analytic solution describing the rotational flow of an inviscid fluid in a channel with a slightly varying cross section is obtained. Three types of entrance flow nonuniformity are considered: 1) uniform shear flow, 2) wake-type flow, and 3) potential flow with a narrow wall boundary layer. Streamline patterns with circulation zones are constructed for flows in diffuser channels with the above-mentioned types of entrance nonuniformity. A model of flow separation in a channel with a turbulent boundary layer on the walls is discussed.