An analytical and semi-empirical model for the viscous flow around a vortex cavity

Abstract

The minimum pressure in the core of a vortex, and therefore also the inception of cavitation, is considerably influenced by viscous effects. However, little information is available on the influence of viscous effects once the vortex cavity is fully developed. For that purpose an analytical solution for the azimuthal velocity distribution, including the boundary conditions, of a cavitating vortex in 2-D viscous flow is derived and investigated. The analytical solution, that can be designated as a cavitating Lamb–Oseen vortex, shows that with increasing cavity size the influence of viscosity decreases. For practical applications, the analytical model is extended to a second vortex model which allows for a region with vorticity roll-up. The resulting semi-empirical model for the azimuthal velocity distribution can be fitted well to experimental data for a wing-tip vortex for both non-cavitating and cavitating conditions. The vortex model has also been used to compare the relation between cavity size and cavitation number with experimental data. For the theoretical models, this relation can be presented such that it becomes independent of vortex strength and viscous core size.