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 V 1/2 and V, respectively. The converse may be said of the peak velocity which increases with V 1/2. 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.

Sidewall Boundary Layers of the Bidirectional Vortex

“…These inviscid solutions resembled Bloor and Ingham's in exhibiting a singularity at the centerline. Shortly thereafter, viscous corrections were derived to overcome the swirl velocity's singularity at the origin and the no slip at the sidewall in the tangential direction [28]. Thus, with the exception of Bloor and Ingham [20], no other bidirectional vortex model has been advanced despite its relevance to both the propulsion and particle separation industries.…”

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

“…For a complete physical description of V, the reader may consult with Majdalani & Chiaverini (2009). The injection mass flow rate appears asṁ i while the density and viscosity are given by ρ and μ.…”

“…For more detail on the pressure relations, the reader is referred to Majdalani & Chiaverini (2009) and Maicke & Majdalani (2009). Figure 11 depicts a normalized form of ∆p in the vortex chamber.…”

On the compressible bidirectional vortex in a cyclonically driven Trkalian flow field

“…The sidewall boundary layers were also examined by Majdalani & Chiaverini (2009), while a compressible analogue with a viscous core was conceived by Maicke & Majdalani (2008). At the heart of these approaches stood the vorticity-streamfunction approach, an extended form of the classical technique that related the vorticity to the streamfunction by way of the vorticity transport equation.…”

“…Over time, their model was refined to include increasingly more realistic effects. For example, a treatment of the viscous core was first added by Majdalani & Chiaverini (2009)…”