Vertical-Axis Propeller Efficiency 57 
where ¥/(z) is a bounded function for z + €(9,), which is of Q(T). Making use of (3.10) we 
find for the normal component v, ,(9,s) of this velocity, 
1 FC9,) In (s- a) 
vv (9,8) = — - —_ 
Lin 21 
RV1 + uw? + 2 cos on 
‘S- a2 
+ O(r) “S! wg) In (“Z*) + Ory . (5.4) 
It will be assumed that the profile is infinitely thin and cambered. On the profile we 
assume a vortex layer with density I, ,(9,s) which represents the sum of bound and free 
vorticity. Hence we have the following equation: 
att Feo (0:2) do 
1 a 2 os 
°° ri J Tata aay TEN Gag YOO) law) + POO ls ce "(oye 
- wo) In (757) + 6) (5.5) 
where Q@(Q) is the angle of incidence of the profile, f(s) is the camber, and 6(9) represents 
the angle of the profile with respect to a line of fixed direction, for instance the x axis (Fig. 
6). The angle of incidence (9) can be defined as the angle between the tangent to the pro- 
file at the trailing edge and the velocity V(~). The angles a, 0, and 0; to be defined in 
(5.6) are positive in the counterclockwise direction. The angle @(¢) can be split into two 
parts: 
e. =p. “4 sin Q 
ADS Ore Os CN EA Sera gt (5.6) 
For small values of ¢ Eq. (5.5) can be written in the form 
1 att Poel O97) do , , , 
on | ERS SoS V( 9) [a( @) + f (s)] + [wO (9) + wa (9)] Ss 
- wo) In (“F#)+ OF). 65.7 
The inversion of this integral equation is well known [4, section 88]: 
a+ 
oe A(9) 2 { : : 
Pe(@s) = —— S++ v f 
oo 0S) Stee Wan = SW eee ere ae | (9) [a(9) + #()] 
= 949) + wa"(o)}o + wo) In (“Fe ) 
+ orp Yea eee vatt-o de | (5.8) 
(so) 
