A Generalized Model for Transitional Blade Tip Vortices

A new mathematical model is proposed to represent the induced velocity of a rotor blade tip vortex at any vortex Reynolds number. A complete description of the tip vortex requires a solution to the Navier-Stokes (N-S) equations, but solutions can only be obtained by reducing these equations under certain assumptions and approximations. Such reductions give a completely laminar vortex model when the Reynolds number is low, or a completely turbulent model at very high vortex Reynolds numbers. However, many rotating-wings operate at conditions where the vortex Reynolds number is in the intermediate (transitional) regime when the vortex is neither fully laminar nor fully turbulent. This is particularly true at model scale, where most experimental measurements exist. A new analytical model for a transitional vortex has been developed using an intermittency function for the eddy viscosity 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 flow rotation on the development of turbulence present inside the vortex boundaries, and is incorporated into the N-S equations governing the development of an axisym-metric vortex flow. A unique aspect of the proposed model is the Reynolds number dependency of the final solutions. 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 measurements made on hovering rotors.