Theory of the Ducted Propeller — A Review 



In (110), emphasis is placed on the 

 net characteristics such as the lift and 

 the moment of the body and of the ring 

 airfoil. But essentially the theory can 

 also be used to calculate the entire flow 

 field. Numerical results are presented 

 for four families of bodies: (1) (infinite) 

 cylinder, (2) ellipsoid, (3) a body with a 

 blunt nose and a sharp tail, and (4) the 

 same body reversed (sharp nose, blunt 

 tail). In what follows we shall mention 

 only a few of the theoretical and numeri- 

 cal results. 



Fig. 5 - (See (106)) Ring 

 airfoil with central body 



The lift and the moment of the wing 

 are determined by the first-order Fou- 

 rier coefficient gi(<f ) of the wing vortex distribution (2.15) (if = o corre- 

 sponds to the "lowest" point on the circumference), otherwise there may be a 

 change of sign and/or gj must be replaced by hj. gi(<f) is determined by the 

 integral equation 



(Tx + T^^)g, 



- 2FB(^)a„ 



1 < <f < 1 



where T^ is the basic first-order operator and T^g is defined by 



1 1 



TlBgl = -^FbC^')/ gi(^')df'+^/ B(^,^-) gj(^')d^' 



(2.25) 



(2.26) 



The function 



Fr(^) 



'B ^B'" 



(2.27) 



and the kernel B(^,f ') are continuous functions that depend only on the geom- 

 etry of the configuration. 



The lift coefficient of the body (referred to the same area 277Ec as the lift 

 coefficient c^^ of the wing) can be expressed in the form 



1 



(2.28) 



The moment coefficient can be written in a similar manner. 



The function F3(t) is plotted in Fig. 6 for three ellipsoidal bodies with dif- 

 ferent values of X.3 = 1b/(2R). For a cylindrical body the kernel B(^,^') can be 

 written in the form B(>.^- \^') = B(7]) = B(-^); this is plotted in Fig. 7. 



The lift ratio L^^,+b/L;^ of the wing-body combination and the (cylindrical) 

 wing alone is shown in Fig. 8 for an ellipsoidal body in several axial positions 

 and for the cylindrical body and several values of k in Fig. 9. The lift ratio has 

 been plotted over the whole range ^ p^ < l oi p^, although the theory assumes 



1223 



