On the Boundary Layers of the Bidirectional Vortex

Abstract

To complement former studies of the bidirectional vortex, our principal focus here is to resolve the viscous boundary layers forming in both the axial and radial directions at the sidewall of a vortex chamber. The analysis is initiated by the formulation of the laminar boundary layer equations via an order of magnitude reduction of the incompressible NavierStokes equations at the wall. Next, asymptotic concepts are applied to linearize and rigorously truncate the governing equations, thus converting them, when possible, from PDEs to manageable ODEs. Scaling transformations are then applied to resolve the rapid changes near the sidewall. Due to the nature of the outer solutions, additional transformations of the dependent variables are undertaken to permit securing the problem’s multiple boundary conditions. After some algebra and matched-asymptotic expansions, we recover nearly identical boundary layer structures in all three orthogonal directions: the axial and radial components presented here, and the wall-tangential boundary layer formulated previously. This behavior is not surprising given that the resultant velocity is dominated by its tangential component and that the tangential boundary layer is axially invariant. This forces the axial layer to remain uniform in the streamwise direction. Corroborating assumptions include an axially independent pressure distribution and consistency in the asymptotic assumptions made, the linearization techniques, and the governing equations that apply to all three cases. We remark that although curvature terms are retained initially, they are found to be so small that the problem is reducible to the case of two-dimensional layer analysis. It can be seen that all viscous corrections at the wall are strongly dependent on the vortex Reynolds number, V. With the newly obtained solutions, essential flow characteristics, such as pressure, vorticity, swirling intensity, and wall shear stresses, are evaluated and discussed. We find the axial and tangential boundary layers to be of the same size, approximately twice the thickness of the radial layer.