The generation and diffusion of vorticity in three-dimensional flows: Lyman's flux

We examine Lyman's (App]. Mech. Rev., vol. 43, issue 8, 1990, pp. 157-158) proposed definition of the boundary vorticity flux, as an alternative to the traditional definition provided by Lighthill (Introduction: boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), chap. 2, 1963, pp. 46-109. Oxford University Press). While either definition may be used to describe the generation and diffusion of vorticity, Lyman's definition offers several conceptual benefits. First, Lyman's definition can be interpreted as the transfer of circulation across a boundary, due to the acceleration of that boundary, and is therefore closely tied to the dynamics of linear momentum. Second, Lyman's definition allows the vorticity creation process on a solid boundary to be considered essentially inviscid, effectively extending Morton's (Geophys. Astrophys. Fluid Dyn., vol. 28, 1984, pp. 277-308) two-dimensional description to three-dimensional flows. Third, Lyman's definition describes the fluxes of circulation acting in any two-dimensional reference surface, enabling a control-surface analysis of three-dimensional vortical flows. Finally, Lyman's definition more clearly illustrates how the kinematic condition that vortex lines do not end in the fluid is maintained, providing an elegant description of viscous processes such as vortex reconnection. The flow over a sphere, in either translational or rotational motion, is examined using Lyman's definition of the vorticity flux, demonstrating the benefits of the proposed framework in understanding the dynamics of vortical flows.