Loewy’s Problem: Returning Wake Problem

The theory of Theodorsen is developed for an airfoil whose wake extends to undisturbed farfield. On the other hand, more complex motions of an airfoil can be studied by the aid of the Theodorsen function. A representative example for that is the study of a helicopter blade or a blade of a propeller. Loewy and Jones sepa­rately studied this problem with the parameters N being the number of blades and h being the distance between the blade and the returning wake. Now, let us give the related formulas for the modified version of the Theodorsen function for a single blade and the multi-blade rotors.

(i) Single blade: The modified Theodorsen function is given in terms of X being the rotational speed of the blade in radians per second and h:

C'(kAh) «T(k>+J.(k>W and

V X ‘ Hf>(k> + /H02>(k>+2[/1(k> + i70(k>]W

where W, – = (ekh/be‘2p-/X) _ 1>-

b X

Here, in Eq. 3.34a, b if we let h go to infinity we recover the Theodorsen function as expected. In addition, if the ratio given by XX is an integer, which means the oscillation frequency of the profile is multiples of rotational speed of the blade then the vortices shed are in phase according to 3.34a, b.

(ii) N-blades: For this case W as function is altered with number of blades N and AW as follows

—, – AW, N = ekh/bei2nx/NX)e(AWx/n) – 1 bX

If we take DW = 0 and study the phase difference only for the distance between the blades the form of W becomes

w(kh > X’ DW; N) = (ekh/bei2rao/wX) – l) -.

Loewy’s approach applied to a single blade rotor causes the unsteady lift to increase or decrease depending on the reduced frequency. In Fig. 3.10 given is the change in the amplitude of the Loewy function with h/b and k.

So far we have examined the response of a simple harmonically oscillating airfoil in a free stream or in a returning wake. Now, we can study the unsteady aerodynamic response of an airfoil to its arbitrary motion or to an arbitrary external excitation.