Sevik 



engineering standpoint the correlation tensor (L°(t) L^(t + r)) and its Fourier 

 transform are the most significant quantities; consequently only a knowledge of 

 the second-order two- point- product mean values of the turbulence is required. 



In Eq. (2) u represents the velocity of the fluid relative to a rotating ele- 

 ment of the propulsor. Referring to Fig. 1, let y^ denote the location of the ith 

 element relative to the unprimed reference frame. As was stated this unprimed 

 reference frame is fixed to the propeller and rotates with it. 



Suppose that at time t the fixed and rotating frames coincide momentarily; 

 then the position vector of a fluid particle will be the same in both frames. If n 

 denotes the angular velocity vector of the rotating frame, the velocity compo- 

 nents of the fluid relative to the propulsor element considered are given by 



-.. . - u.^ = a^° uj" - e^°^ fi° y/ , (4) 



where e is the permutation symbol. 



Note that all terms in Eq. (4) are time dependent: could, for instance, 

 represent fluctuations in angular velocity resulting from a torsional vibration of 

 the propeller shaft. If the propulsor rotates at a steady speed, we obtain 



In forming average values of the forces and fluid velocities we assume that 

 the random processes are stationary and ergodic. The mathematical expectation 



1 



E[L°(t) L^(t + T)] = lim :J: f L^Ct) L/5(t + T) dt = <D°/3(t) 



(5) 



is equal to the sum of the auto- and crosscorrelation functions of the forces 

 acting on the constituent segments of the propulsor. This can be shown by sub- 

 stituting Eq. (3) in Eq. (5): 



T 



^~*°° i j i j ' 



The crosscorrelation tensor <J'6°fc -(t) can be expressed in terms of the 

 aerodynamic force functions and the velocity correlation tensor: 



T 



^6°^ (-^) = lim Y r e:°(t) e./3(t + r) dt 



CO 00 T 



= f FiY(^l) drj [ F^'cr^) dr, lim | J u^^ ( t - r^ ) u/ ( t + r - r ,) dt 



(Cont) 

 294 



