• ~ • Cox 



Finally, if the propeller is assumed to operate in homogeneous inflow, 

 there appears to be little advantage in not assuming a constant induced advance 

 ratio k^. This will permit some simplifications of the lifting- surface equations. 



3. PROPOSED LIFTING LINE - INDUCED VELOCITIES 



Prior to the use of lifting- surface theory, it is necessary to determine the 

 induced advance ratio ki{r) and bound vorticity y{T,e). The determination of 

 y{T,0) is dependent on the total radial circulation distribution r(r), where 



r(r) = J y(T,0)dd , 



since the chordwise spreading of 7{r,e) is a matter of choice. For a finite- 

 bladed subcavitating propeller this information, including the lift distribution 

 CL(r)[c(r)/D], is Obtained using the lifting-line concept for a propeller blade. 

 This is a simplification which ignores chordwise effects. Its major purpose is 

 to adjust radial circulation for the effect of a finite number of blades. Like- 

 wise, the use of this concept is required for super cavitating propeller design, 

 but recognizing the effect of the blade cavities. 



In order to assist the design problem, and in analogy with the lifting- line 

 concept for subcavitating propellers [14], the blades will be represented by 

 lifting lines to account for loading. In other words, as a tentative first step, 

 an initial procedure will be formulated where the mathematical model for each 

 blade is considered to be a lifting line with associated free trailing vortices, 

 together with a pressure source distribution on the trailing vortex sheet to 

 represent the cavity. 



By the use of the Dirac delta function, 7{r,e ){r^ + k-^^) ^"^ is replaced by 

 r(r) S(^) in Eqs. (2), (3), (4), and (7). In addition, d* is equated to zero in all 

 but Eq. (7), since the induced velocity components are only required at the lift- 

 ing line. For an unskewed blade, it should also be noted that 0L(r) = for the 

 pressure source integrals. Hence, 



u^(e,r*,0) = ^ 2] j J J S(r,d)k.(r' + k.^)''' 



q=i 



r(r) 



f 



(e - k.^r) dr 



d^di 



r r(r-r* 

 J K 



(9) 



cos i//) dr 



/ '-^ f 



r ( r - r* cos \p) dr 



dr 



936 



