that both r(r) and 6.(r) have been determined, either from design calcula- 



^ . ... 21,22 



tions or from a separate performance prediction. 



The induced velocity varies with angular position, <P , corresponding to 



a time dependence in a reference frame fixed to the hull, i.e., 



i_ = L^ 



8V Q. 9t 



By virtue of the propeller symmetry, the induced velocity may be resolved 

 into a time-averaged or steady velocity u(x,r) and harmonics in blade 

 passage frequency, Zfi. The steady component or zero harmonic which gives 

 rise to the thrust deduction, is given by 



2Tr 



(24) 



^ 1 f ^ 



u(x,r) =— - I u'(x,r,^) d«P 



2^ Jq 



The separate contributions from the bound and trailing vorticity can be 

 determined by introducing expressions of the type 



00 /•CO 



— = yi e cos n(«^ - V, ) / J (kr) J (kp)( 



1 ix.1 e = 1 n = 

 -k 1x1 n 



ID I „.0 •■ " - 2 n > 



P 



and performing the integration in equation (24) . The following results are 

 obtained: 



9 1 



Cummings, D.E., "Numerical Prediction of Propeller Characteristics," 



Journal of Ship Research, Vol. 17, No. 1 (Mar 1973). 



22 



Tsao, S.S., "Documentation of Programs for the Analysis of Performance 



and Spindle Torque of Controllable Pitch Propellers," Mass. Inst, of Tech- 

 nology, Dept of Ocean Engineering, Report No. 75-8 (May 1975). 



15 



