On steady rotational cyclonic flows: The viscous bidirectional vortex

Abstract

This study is focused on the tangential boundary layers of a bidirectional vortex, specifically those forming at the core and the sidewall of a swirl-driven cyclonic chamber. Our analysis is based on the regularized, tangential momentum equation which is rescaled in a manner to capture the forced vortex near the chamber axis and the no slip requirement at the sidewall. After identifying the coordinate transformations needed to resolve the rapid changes in the regions of nonuniformity, two inner expansions are constructed. These expansions are then matched with the outer, free vortex solution that is sandwiched between the core and the hard wall. By combining inner and outer expansions, uniformly valid approximations are subsequently obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by a dimensionless grouping that we refer to as the vortex Reynolds number, V. This keystone parameter appears as a ratio of the mean flow Reynolds number and the product of the swirl number and the chamber aspect ratio. Based on V, several fundamental features of the bidirectional vortex are quantified. Among them are the thicknesses of the viscous core and sidewall boundary layers; these decrease with V1/2 and V, respectively. The converse may be said of the peak velocity which increases with V1/2. In the same vein, the angular speed of the rigid-body rotation of the forced vortex is found to be linearly proportional to V. Our laminar swirl velocity is reminiscent of Sullivan’s two-cell vortex except for its additional dependence on the aspect ratio of the chamber. For the purpose of verification, theoretical predictions are compared to particle image velocimetry measurements and Navier–Stokes simulations at high vortex Reynolds numbers. By properly accounting for the turbulent eddy viscosity in the analytical model, local agreement is obtained with both laboratory measurements and computer simulations.