Sidewall Boundary Layers of the Bidirectional Vortex

Abstract

This study seeks to resolve the sidewall boundary layers forming in the axial and radial directions of a bidirectional vortex chamber. Our analysis is initiated by the formulation of the laminar boundary-layer equations via an order of magnitude reduction of the incompressible Navier-Stokes equations at the wall. Asymptotic theory is then applied to linearize and systematically truncate the governing equations, thus converting them from partial differential equations to more manageable ordinary differential equations. Scaling transformations are additionally applied to resolve the rapid changes arising near the sidewall. Because of the spatial character of the outer solutions, further transformations of the dependent variables are undertaken to secure the axially changing outer conditions. Through the use of matched-asymptotic expansions, we recover similar boundary-layer structures in all three orthogonal directions: the axial and radial components presented here, and the wall-tangential boundary layer obtained previously. This behavior is consistent with the resultant velocity being dominated by its tangential component and with the tangential boundary layer being axially invariant. These factors cause the axial layer to remain uniform in the streamwise direction. Based on the ensuing asymptotic results, viscous corrections at the wall are seen to be mainly dependent on the vortex Reynolds number, V. The latter combines the swirl number, the Reynolds number, and the chamber aspect ratio. Having obtained the three components of the velocity, essential flow characteristics, such as pressure, vorticity, swirling intensity, and wall shear stresses, are evaluated and discussed.