Turbulent Boundary-layer Flow Beneath A Vortex. Part 2. Power-law Swirl

The problem formulated in Part 1 (Loper, J. Fluid Mech., vol. 892, 2020, A16) for flow in the turbulent boundary layer beneath a vortex is solved for a power-law swirl: , where is cylindrical radius and is a constant parameter, with turbulent diffusivity parameterized as and the diffusivity function either independent of axial distance from a stationary plane (model A) or constant within a rough layer of thickness adjoining the plane and linear in outside (model B). Model A is not a useful model of vortical flow, whereas model B produces realistic results. As found in Part 1 for , radial flow consists of a sequence jets having thicknesses that vary nearly linearly with . A novel structural feature is the turning point , where the primary jet has a minimum height. The radius is a proxy for the eye radius of a vortex and is a proxy for the size of the corner region. As decreases from , the primary jet thickens, axial outflow from the layer increases and axial oscillations become larger, presaging a breakdown of the boundary layer. For small , and . The lack of existence of the turning point for and the acceleration of the turning point away from the origin of the meridional plane as provide partial explanations why weakly swirling flows do not have eyes, why strongly swirling flows have eyes and why a boundary layer cannot exist beneath a potential vortex.