Response of Propulsors to Turbulence 



The fluctuating aerodynamic forces acting on the various surface elements 

 are interdependent by virtue of induction effects as well as by virtue of spatial 

 and temporal correlation of the turbulent velocity fluctuations. Since the quanti- 

 ties involved in this problem are tensors, the index notation, including the sum- 

 mation convention in the case of repeated indices, constitutes the most conven- 

 ient choice. 



Directions are denoted by superscripts, whereas subscripts denote the 

 propulsor elements involved. For example u^(t') denotes the component of the 

 fluctuating velocity at time t ' in the direction /3 of the rotating reference frame 

 at the element k. Similarly, F°^(t,T') denotes the aerodynamic force acting on 

 the ith element in the direction a at the instant of time t caused by a velocity 

 fluctuation of unit magnitude in the direction /3 to which the kth element was sub- 

 jected at the instant of time t' . Finally, I °(t) indicates the aerodynamic force 

 acting on the ith element at time t in the direction a. In terms of these quan- 

 tities and neglecting higher order terms, the lift force is given by 



t 



ef(t) = r F«|^(t,r') u^^(t') dr' , _ ^. . (1) 



-°= 



where 



a, /5 =: 1, 2, 3 , 



■ ■'■ - ' i, k = 1, 2, . . . , m , ■ • "-'■ 



r' < t . 



In most cases the aerodynamic force tensor is time invariant and Eq. (1) 

 can be written as a convolution integral: 



^°(t) = r F^f(t) u^/3(t-r) dr 



(2) 



where 



t' > . 



The force acting on the entire propulsor at the instant of time t in direction a of 

 the roating reference frame is given by the sum of the the lift forces acting on 

 each individual element: 



L-(t) = 2 r«(t) . ■ , . (3) 



Since L^ct) is a random function, it is determined statistically by the complete 

 system of joint- probability distributions of the values of the function at any n 

 values of t, where n may take any integral value. Fortunately, from an 



293 



