On generalized-vortex boundary layers

Abstract

We define a generalized vortex to have azimuthal velocity proportional to a power of radiusr⁻ⁿ. The properties of the steady laminar boundary layer generated by such a vortex over a fixed coaxial disk of radiusaare examined. Though the boundary-layer thickness is zero a t the edge of the disk, reversals of the radial component of velocity u must occur, so that an extra boundary condition is needed at any interior boundary radiusr_Eto make the structure unique. Numerical integrations of the unsteady governing equations were carried out forn= − 1, 0, ½ and 1. Whenn= 0 and − 1 solutions of the self-similar equations are known for an infinite disk. Assuming terminal similarity to fix the boundary conditions atr=r_Ewhenu_r> 0, a consistent solution was found which agrees with those of the self-similar equations whenr_Eis small. However, ifn= ½ and 1, no similarity solutions are known, although the terminal structure forn= 1 was deduced earlier by the present authors. From the numerical integration forn= ½, we are able to deduce the limit structure forr→ 0 by using a combination of analytic and numerical techniques with the proviso of a consistent self-similar form asr_E→ 0. The structure is then analogous to a ladder consisting of an infinite number of regions where viscosity may be neglected, each separated by much thinner viscous transitional regions playing the role of the rungs. This structure appears to be characteristic of all generalized vortices for which 0.1217 <n< 1.