Generalized Viscous Vortex Model for Application to Free-Vortex Wake and Aeroacoustic Calculations

The development of a generalized viscous vortex model is described for application in rotor aeroacoustic analyses. A brief summary is given of previous models for the viscous core growth and induced velocity field of a trailing tip vortex. The assumptions made for the vortex core growth are shown to influence the predictions of the rotor wake geometry, as well as the induced velocities, airloads, and performance, thereby emphasizing the need for a more universal and physically representative vortex model. In the present approach, the viscous core growth was modeled using an extension of the classic Lamb-Oseen model for the diffusion of laminar vortices. Turbulence in the tip vortex affects the diffusion of vorticity, and these effects were incorporated using an empirically validated correction for the average apparent or “eddy” viscosity. The vortex induced velocity profiles measured in experiments were found to exhibit strong self-similarity when using the vortex core radius as a length-scale, suggesting that a generalized model is possible. A family of algebraic models for the swirl, axial, and radial components of the velocity induced by a viscous trailing tip vortex is proposed. These velocity components were found from a solution to a simplified form of the incompressible Navier-Stokes equations. The velocity profiles are written in terms of a single integer exponent with the viscous core growth being given by a semi-empirical relation for the turbulent viscosity that scales as a function of the vortex Reynolds number. The model is compared with velocity field measurements for both fixed-wing and rotating-wing tip vortices, with good agreement.