A constant shear stress core flow model of the bidirectional vortex

Abstract

In this paper, we discuss the merits of two models for the swirl velocity in the core of a confined bidirectional vortex. The first is piecewise, Rankine-like, based on a combined-vortex representation. It stems from the notion that a uniform shear stress distribution may be assumed in the inner vortex region of a cyclone, especially at high Reynolds numbers. Thereafter, direct integration of the shear stress enables us to retrieve an expression for the swirl velocity that overcomes the inviscid singularity at the centreline. The second model consists of a modified asymptotic solution to the problem obtained directly from the Navier–Stokes equations. Both solutions we present transition smoothly to the outer, free-vortex approximation at some intermediate position in the chamber. This position is deduced from available experimental data to the extent of providing an accurate swirl velocity distribution throughout the chamber. By scaling the constant shear radius to the core layer thickness, the constant of proportionality is readily calculated using the method of least squares. Interestingly, the constant of proportionality is found to be invariant at several vortex Reynolds numbers, thus helping to achieve closure. The combined-vortex representation is validated against a large body of experimental measurements and through comparisons to a laminar core model that is enhanced through the use of an eddy viscosity. Other heuristic schemes are discussed and the two most suitable models to capture realistic flow behaviour at high vortex Reynolds numbers are identified. Our two models are first derived analytically and then anchored on the available experimental measurements.