Vertical-Axis Propeller Efficiency 51 
= Re aX Bia A ste: oo ila Ae ae che tings 
(X,Y) = eg), fm n (sin Rp p+ 2p (3.3) 
m=0 0 
where 
Cm = (Sm + ing) = R (ua + sin 6 - art 1 cos ®). (3.4) 
Because the intensity (yw) of the bound vortex is a periodic function of w, the same holds 
for the free vortex density y(~), and besides this the mean of y(w) over a ail O0<W<2z, 
or over an interval of length 27pR in the x direction, is zero. 
Then it follows from (3.2) 
f(x+ 2mpR,y) - &x,y) = 0, ly] >R. (3.5) 
When, however, |y| < R we have for the mean vorticity between y = +R and y over the period 
27uR in the x direction 
+ M{ (2m Y) = r(yy} if 5 eee (3.6) 
Here by (3.2) we have 
H(x+ 2muR,y) - oxy) = W{T(2n-y) - Te}, lvl < R. @.7) 
Our aim is to determine the function I'(W) in such a way that the energy left behind in 
the wake becomes as small as possible. To this end we consider the kinetic energy E of 
the fluid in the strip 0 < x < 27pR, —0 < y <+00, which by Green’s theorem can be writtenas 
HAGE) (Bla be festa. oe 
The surface integral extends over a vertical strip of width 27yR, the line integral along 
the vertical boundaries of the strip and along both sides of the cycloids (Fig. 2, dashed 
lines). The normal derivative is directed into the enclosed regions. 
We consider first the integrals along both sides of the cycloids. These integrals for M - 
different cycloids have the same magnitude; hence we find by (2.14) for their combined con- 
tribution E, to the kinetic energy E, 
p, = - eft ij rey) 2S (yy J + u? + Qu cose ay (3.9) 
where the line of integration and the direction of the normal to this line are denoted in Fig. 
5. As in (3.9) we will not mention in this and the next section the order of accuracy of the 
formulas. This will be considered again in section 5. 
From (3.9) it follows that we have to determine the normal derivative 0¢6/dn. The unit 
normal of the cycloid is 
