# The Returning Wake: Loewy’s Problem

Theodorsen’s theory has represented an isolated 2-D thin-airfoil with the wake convected downstream to infinity. For rotorcraft work, this is perhaps a questionable as­sumption because the rotor blade sections may encounter the wake vorticity from previous blades as well as the returning wake from the blade in question. This fact was acknowledged by Loewy (1957) and by Jones (1958) who set up a model of a 2-D blade section with a returning shed wake, as shown in Fig. 8.14.

This returning wake can be modeled with planar 2-D vortex sheets, just as in Theodorsen’s method, but now with a series of sheets below the airfoil with vertical separation, h, that depend on the mean induced velocity through the rotor disk and the number of rotor blades. Loewy (1957) has shown that in this case the lift on the blade section can be expressed by replacing Theodorsen’s function by

c(k. h) = -______ + _________ , (8.33)

to J Hf(k) + iH^(k) + 2(Ji(k) + iJ0(k))W

where C'(k) is known as the Loewy function, with argument of reduced frequency k. For a single blade, the complex valued W function is given by w(^—, — ^ = (eM/bei2n(.a>/a) _ jj-i If a)/ £2 = an integer, then all the shed wake effects are in phase. Notice from Eqs. 8.33 and 8.34 that as h —oo then W■ —► 0 and C'(k) -> C(k), and Loewy’s function approaches

Theodorsen’s result, as it should. For a rotor with Nb blades, the W function is modified to read

W(t’ I’ AVf’ = – l)~’ , (8.35)

where the parameter aj/NbSl now controls the wake phasing.

The wake spacing ratio h/b can be determined from the spacing of the helical vortex sheets that are laid down below the rotor. If an average induced velocity u, = XQR is assumed, then during a single rotor revolution the shed wake generated by a single blade will be at a distance h = (2я/ Q)v, below the rotor. For multiple blades, the spacing will be (27t)Vi/(QNb), that is, h X£IR2ti AX

b QNbb о with a as the rotor solidity. For = 0, which means that the only phase shift in the wake vorticity results from the spacing between the blades, then W(~t’ ЇІ’ °’ = (ekh/bei27t(a>/NbQ) – 1) 1 .