Weissinger and Maass 



Vo=T-^fl/f^Q,.(^)- ■ (^•^) 



with the standard operator T^ . 



Equations (5.6) and (5.9) can be solved by the methods of Sec. 2, after nu- 

 merical evaluation of the right-hand side integrals for a given function r(r). If 

 r(Rp) ^ 0, the right-hand side includes an explicit term similar to that of (4.16). 



By both the THERM work sheets (151) and the NSRDC computer program 

 (134) only the steady part of the distributions can be calculated. In (151), y^ is 

 determined from (5.9) with r(r) specified as the optimum propeller circulation 

 of Betz, including a tip correction for wall effects derived by Goodman (22). In 

 (134), Eq. (5.6) is solved with an arbitrary distribution r(r). The tangent of the 

 propeller hydrodynamic pitch angle determined by Lerbs' theory of moderately 

 loaded propellers (48) is used for j(r), i.e., the propeller-induced velocity is 

 taken into account in determining j (r) (to some extent), while the velocity in- 

 duced by duct and hub is neglected. These are neglected in (151), too. 



The program (134) includes also a design program based on an iterative 

 procedure with three steps in each cycle. First, r(r) is determined for the un- 

 ducted propeller from a slight modification of Lerbs' theory. Because of the 

 modification, small additional axial and azimuthal velocities such as those in- 

 duced by a duct can be taken into account. Second, the steady radial (Eq. (5.7)) 

 and axial velocity components induced on the duct by the propeller are computed. 

 Then, (5.6) and the corresponding equations of ring-airfoil theory, as far as 

 camber and thickness are concerned, are solved and the velocity induced at the 

 propeller by these duct distributions computed. Then the next cycle can be 

 started. The computation starts with the determination of r ( r ) for the unducted 

 propeller. The iteration is repeated until the inflow velocity at the propeller 

 converges to four significant figures. Usually, this accuracy can be obtained in 

 less than six cycles. 



The propeller can be designed on the basis of thrust or shaft horsepower 

 and for a prescribed blade circulation or pitch distribution. The viscous effects 

 of the propeller are taken into account by giving as input the blade -section drag 

 coefficient and the propeller blade outline. The viscous drag on the duct, which 

 includes both the skin-friction and pressure drag, can also be calculated on the 

 computer. The frictional drag on the duct is computed by giving as input the 

 frictional drag as presented by Gertler (20), where the Reynolds number is 

 based on the duct length. The computer program calculates the pressure drag 

 on the duct according to the method developed by Granville (23). 



At THERM (139, 142) and NSRDC (128, 129), similar theories have been de- 

 veloped for determining the unsteady part of y, too, i.e., the harmonics of non- 

 zero order in the expansion 



00 ■' ' 



7(x,0) = V {gp + ^ [gjx) cosmN0 + hjx) sinmN0]} . (5-10) 



1240 



