A generalized model for transitional blade tip vortices

A mathematical model is proposed to represent the induced velocity of a rotor blade tip vortex at any vortex Reynolds number. Rotating-wings operate at conditions where the tip vortex Reynolds number is in the intermediate (transitional) regime when the vortex is neither fully laminar nor turbulent. An analytical model for a transitional vortex has been developed using an eddy viscosity intermittency function in such a way that this function smoothly and continuously models the eddy viscosity variation across the vortex from its inner rotational region into the outer potential flow region. This intermittency function is developed based on Richardson number concept, which brings in the effects of swirling flow (rotation) on turbulence present inside the vortex boundaries. This model is then incorporated into the Navier-Stokes equations governing the development of an axisymmetric vortex flow. A unique aspect of the proposed model is the Reynolds number dependency of the final solutions. Furthermore, because the solutions satisfy the Navier-Stokes equations, vortex velocity profiles can be solved for any given vortex Reynolds number. The model is shown to correctly reduce to the solution for a laminar (Lamb-Oseen) model for very low Reynolds numbers. The proposed model is validated using tip vortex flow measurements for a hovering rotor.