Response of Propulsors to Turbulence 



1 



H*(a)) H.(a^) % H*(aj) H^(a;) = \Hj(co)\^ % (iTTpV^h^)- 



1 + 277 



cohn 



^R:SR 



(13) 



At this stage the major steps of the theory have been stated. The remainder 

 consists of formal mathematical operations. We substitute Eqs. (11), (12) and 

 (13) in (10) and obtain an expression for the real part of the thrust force spec- 

 trum. After some algebraic manipulation this reduces to 



where 



Re G^\r) = a 



1 + eF U2 + r2 



r2 



\2 + r2 



g(M/RA) 



r - i^ 



(14) 



a = 77R^u^UM [2hjp ^ 



1 + 



TT R_ 



L J = 77U/nR = advance ratio , ■.-.••; .- . " .. - '■' 



R = tip radius of the propeller . 



The function g(M/R,X) relates two characteristic length scales of our prob- 

 lem and is given by 



g(M/RA) = JJe"'^"''''^ RiRj dR.dRj , 



(15) 



where the integration is applied over all the elements of the propeller. In the 

 case of a two-bladed propeller this function can be readily integrated; we obtain 



g,(M/RA) 



4 / M 



3 \\R 



1 - 3 e^^/"^ — 1 + — cosh sinh- — 



\R/\ >vR/\ M \R M 



For propellers containing a larger number of blades Eq. (15) has been evaluated 

 numerically and the results are plotted in Fig. 4. 



299 



