One and one half wing bone-end horizontal  axis wind turbine
 

This design applies the curveset:

X = If [ f <= 1 , Cos [ pi + f pi ] , Cos [ 2 pi + ( f - 1 ) pi ] ] ;

Y = If [ f <= 1 , -Sin [ pi + f pi ] times Sin [ pi / 3 + f pi / 3 ]^n times Sin [ pi / 6 + f pi / 6 ]^n , (* plot underside for f values 0 to 1 *)

-Sin [ 2 pi + ( f - 1 ) pi ] times Sin [ 2 pi / 3 + ( f - 1 ) pi / 3 ]^n times Sin [ 2 pi / 6 + ( f - 1 ) pi / 6 ]^n (* plot upperside for f values 1 to 2 *) ] .

Here, n is varied across the radius : from 1/2, (at 1/2), to 6, (at 6, where the longer wing closes ; the shorter closes at 3) .

This helps to balance lift forces along the wing, (having the effect of shallowing the profile as it moves out from center), together with the narrowing of the wing across its length, (width is halved as radius doubles).

The pitch of the wing is shallowed from 45° at radius 1, where the wing is hoped to rotate at the same speed as the incoming air, to 7.5° at 6 ; where the wing is hoped to be rotating at 6 times the air speed.

Closing the short wing at 3 is done to reduce the chance that the upwind part of the long wing's wake will return to hit the short wing, (which in a steady wind i'd expect at around pi length).

Closing the long wind at 6 is, similarly, done to reduce the chance it will hit its own wake, (expected at around the length 2 pi).

The short wing will need to have added mass to counterbalance the long.

As shown above, the rotation is counter-clockwise.

 
29 April 2010
www.tendingtodivinity-theshapeofmathematics.com