Yamazaki 



Here, in each component of the induced velocity, the first term of the right- 

 hand side represents the velocity component induced by the trailing vortex 

 along the helix, and the second term represents the velocity component induced 

 by the bound and shed vortices arranged in the radial direction. The function 

 ^g*('^'.Vi,-Sk/)/3v (-i<vi<i) is sometimes called "the bound vortex." More- 

 over, for V 1 > 1, we can get 



g*i(^''^i'-Sk') = gik".v', - [e(^')(Vj-v') + S^,)]} dv' 



1 



gl{^',v', - [9(^')(v^-l) + S^, + 0(<f')(l-v')]}dv' 



. ■'' = g* {^', 1,- [0(^')(v^- 1) + S^,]} .' , ■ '^ 



= g*{^', 1,5m(^') + ^(^') -f%(^') + ^(^') Vi + S^,]}, (106) 



and this function is rewritten as g*('S',S'), where 0' = 6u(^') +d(<^')vi + s^,. 

 That is, the function g*(^,£^) represents the strength of the helical trailing vortex 

 at the point 



Next we consider the velocity potential and the induced velocity in the domain of 

 a propeller wake, i.e., f g < ^ < i and Vj > i . In this domain the velocity po- 

 tentials and velocity components except ^p^ , wp^ ^, Wp^ ^, and Wp^^ are continu- 

 ous. The quantities 0pj , wp^^, Wp^^., and wp^^ are discontinuous at any point on 

 the surfaces of trailing vortices which satisfies Eqs. (107) and are continuous at 

 all remaining points in the domain of propeller wake. In general, the closed 

 vortex can be replaced hydrodynamically by doublet distributions. Accordingly 

 the helical surfaces of trailing vortices are equivalent to the surfaces of doublet 

 distributions. We denote the increments of values of 4-*^, wp^^, Wp^^., and Wp^^ 

 at the discontinuous surfaces when a point passes through the point (^,<^,^) of 

 Eqs. (107) so as to increase 6 keeping i constant or decrease C maintaining d 

 constant by A0pj , Aw|j^, Awpj^., and Awpj,^ respectively. Then using the classical 

 potential theory we get 



A0pe = g (^,9) , Awpj^ = — , Aw*j^ ^- - ^ 



(108) 



^ 3g*(^,^) 



^Pte 



^2 + y2 ^0 



We must take Eqs. (108) into account in the actual calculations of wp^^, wp^^., 

 and wpjg in the domain of the propeller wake. 



54 



