Closed-form solution for the edge vortex of a revolving plate

Abstract

Flapping and revolving wings can produce attached leading edge vortices (LEVs) when the angle of attack is large. In this work, a low order model is proposed for the edge vortices that develop on a revolving plate at 90 degrees angle of attack, which is the simplest limiting case, yet showing remarkable similarity with the generally known LEVs. The problem is solved analytically, providing short closed-form expressions for the circulation and the position of the vortex. A good agreement with the numerical solution of the Navier-Stokes equations suggests that, for the conditions examined, the vorticity production at the sharp edge and its subsequent three-dimensional transport are the main effects that shape the edge vortex.

Versatile reduced-order model of leading-edge vortices on rotary wings

“…We follow the path of a selected Lagrangian element of the line vortex as its distance from the axis of rotation r (τ ) increases in time τ due to the spanwise advection with velocity V r = dr/dτ . We define the spanwise vorticity transport coefficient K sp = V r /( r ) that can be assumed independent of r [12,15]. The solution of dr/dτ = K sp r, up to a constant of integration r 0 , is r = r 0 exp K sp τ .…”

“…The solution closely follows the procedure described in Ref. [15] for the special case α = 90 • , with the difference that it now accounts for a nonzero velocity component parallel to the wing. The inflow complex velocity is defined as…”

“…The vorticity transport coefficient K sp was introduced in Ref. [15] as a parameter that measures the efficiency of spanwise vorticity drainage, and it was related to the rate at which the spanwise flow of the fluid trapped in the recirculation bubble evacuated the LEV vorticity. Since the width of the bubble scales with the effective angle of attack β as c sin β, it is plausible to assume that…”

“…The incompressible Navier-Stokes equations are solved using a commercial finite-volume-based code ANSYS CFX (version 14.5). The CFD model for simulation is similar to our previous work [15]; however, in the present study we use a shorter rectangular-plate wing that has length 4c that is inclined to the vertical axis as shown in Fig. 1(a).…”

“…A special case of this problem at α = 90 • was scrutinized previously [15]. Vorticity production due to flow separation and its three-dimensional transport including spanwise outboard-directed drainage were found sufficient to describe the LEV equilibrium state in that case.…”