282 



(X.r.Q) 



► Yp 



I|j„|(|k|p)K, |(|k|r) p < r < 



I, |(|k|r)K, ,(|k|p) 0<r<p 



(18) 



Ijn and K^ being the modified Bessel functions of 

 the second kind of order m. 



To secure the pressure field for an N-bladed 

 propeller, the blade position angle 9 is replaced 

 by 6 + 2irn/N and the sum over n from n = to N - 1 

 carried out. This sum yields a factor N and the 

 constraints on the frequencies A and m, given by 

 X - m = JIN with a = 0,±1,±2,±3. . .i.e. , products of 

 terms for which X - m ^^ JlN will sum to zero. The 

 total induced pressure at any field is secured by 

 summing over H from -"> to +". 



Upon use of (15) , (16) and (17) and looking after 

 the shifted time variable, using 9 = -ojt which shifts 

 to -ut + ui/U (x-x'), one obtains the velocity 

 potential in the form 



FIGURE 2. Propeller coordinate system-projected 

 view looking upstream. 



wake which is presumed to be known from wake survey 

 measurements . 



The pressure induced at a field point by a single 

 blade is given by the following distribution of 

 pressure dipoles 



,(x,r,f,t) = 



N 

 p'U 



iS,Nu)t 



Ap;^ (?,p)Pj„(x,r,f;C,P) dS 



A^^^ 



(19) 



in which the propagation function, P_, is given by 



p(x,rr,t) = — 



M 



\ ApA{C,p)e - dS 



) 8np R 



X=0 



(16) 



where Ap;^ is the complex amplitude of the pressure 

 loading on the blade arising from the wake harmonic 

 order X and, as illustrated in Figure 3, 



Sp is the surface of the blade, represented ap- 

 proximately by the helicoidal surface 5 = U/u a 



n is the distance directed normal to the surface 



R = [ (x-C) 2 + r^ + p2 _ 2rp cos (9 + a - Y ) ] is 

 the distance from a point (S,p,9 + a) in the 

 surface Sp to the field point (x,r,-f) 



9 = - Mt is the angular position of the blade 



We note that the representation of 

 only approximate for a wake adapted 

 being correct for a constant pitch 

 uniform flow. Here we also assume 

 jumps on the blades, hpx, have been 

 calculated by the unsteady lifting 

 such as developed and programmed by 

 (1973) . 



To place the harmonic content of 

 the following identity can be used 



the blade is 



propeller, 

 propeller in 

 that the pressure 



previously 

 surface theory 



Tsakonas et al. 



1/R in evidence. 



cp at 



1-1 ^ r 7, / li Iv ik(x-C),, im(9+a-f) 



R-^ L. J A, |(r,p,|k|)e dk e 



ni=-o5 -co 



-iMix 

 U 



4tt^ 



m 3n 



-ik? 



«.Nto A 

 + x' 



IT ' 



dx' 



im(a - f ) 



dk 



(20) 



where for each ^, m = A - J,n, and M is a practical 

 upper bound of the wake harmonic order number beyond 

 which the amplitudes of the wake harmonics are so 

 small as to render negligible values of Ap;^ for all 

 A > M. (A value of M = 8 is reasonable) . Details 

 of further reductions of the integrals involved in 

 (19) and (20) may be found in Jacobs and Tsakonas 

 (1975). 



BLADE 



REFERENCE 



LINE 



(17) 



■HELIX: ? = r[« 



where the amplitude Ai i is given by 

 m 



FIGURE 3. Propeller coordinate system-expanded view 

 of blade section at radius p. 



