APPENDIX B 

 SINK DISK REPRESENTATION OF A MODERATELY LOADED PROPELLER 



The circumferential mean axial and radial velocities induced by a 

 moderately loaded propeller (outside the slipstream) can be related to the 

 flow generated by a sink disk. Consider the velocity potential associated 

 with a distribution of sources, a(r,<p), located at the propeller plane 

 X = 0, £ r £ R, 



, /"Zir /"R , ^ .. , , , . . 



*(x,r,^) = - ; r f ^i^-^^)-- ^ ^^. .^ 



-'O *'0 V^ +r +r' -2rr' cQs(iP-<P') 



If the source strengths are independent of angular position, (p , the integral 

 may be rewritten as 



,,)=4f\(o /; 



•R . _k I I 



<f,(x,r) = --T j <^(r') J dk JQ(kr) J^Ckr^e '""'dk dr' 



Integrating by parts yields 



*(->=Yr--ir-'-r- 



J„(kr) J (kr-) .^ ,1 



-- — , e ' ' dk dr 



^ 



where it is required that a(R) = 0. The axial and radial velocities are 

 then given by 



|(x,r) - ±\l "-'"JP--/ Jo*"'' •'l^'"'' ^'^ '"' '"' '*"^' " 5 " 



|iu.„=i/\-M5.a r.^ao.^ck.-) 



-k X 

 ^ ' dk dr' all x 



75 



