Sevik 



APPLICATION TO A PROPELLER 



As an illustration consider a propeller of low solidity with blades of high 

 aspect ratio operating in a turbulent flow. The turbulence considered is homo- 

 geneous and isotropic. The unsteady aerodynamic forces correspond to two- 

 dimensional theory applied to stripwise elements of the blades. 



When the turbulence is homogeneous and isotropic, the velocity correlation 

 tensor adopts a simple form. In terms of the distance r between two points and 

 the mean square value of the velocity fluctuations u ^ it is given by 



C(r) = u^ 



1 ",/3liWf , I ^] s°/5 



2r Br \ 2 Br 



(9) 



The function f (r) has been measured by Stewart and Townsend (6) and is 

 shown in Fig. 2 together with its approximate representation used in this paper, 

 namely, 



f(r) = 



,-X/M 



where x = 2.5 and M is the mesh size of the grids producing the turbulence. The 

 symbol S in Eq. (9) represents the Kroenecker delta. 



As for the aerodynamic response function our analysis is restricted to two- 

 dimensional theory applied to stripwise element of semichord b with spanwise 

 width R. The appropriate form of this function has been given by Sears (7) as 



H.(a;) = 2npW.b.SR.{C(k.)[],ik.) - iJi(kj)] + iJi(k.)} . 



where 



V. = the resultant velocity at the jth propulsor element, 



Jq, Jj = Bessel functions, 



C(k) = Theodorsen's function, 



k. = coh-/\- = reduced frequency. 



The relationships established so far permit the calculation of the response 

 of a propeller to turbulence by numerical means. Clearly it is desirable to ob- 

 tain a relatively simple expression for the rms thrust coefficient and the spec- 

 trum tensor in terms of readily available propeller parameters. For this pur- 

 pose, we make some approximations and assumptions: 



1. The axis of rotation of the propeller is colinear with the free stream 

 velocity vector. 



2. The resultant velocity and chord of the various propulsor elements may 

 be represented by those of a single "typical section" located at some fraction of 



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