284 



L (X) , . m -2 

 m (p)p' ' 



-imf 



-im f. 

 e 'i 





where 



(2|m|+l) r (1/2) r ( m +1/2) 



|m| 

 function 



and 



r( m +1) 



, r being the Gamma 

 (27) 



L (A) I , , , ima , , , , , 

 m = p I Apj^(p,a) e da, the load density 



(28) 



Here the effect of the free surface is included by 

 the last terms in each integrand. To exclude the 

 free surface, take the propeller depth of submergence 

 d = °°. 



Limiting our attention to blade rate (S. = 1, - 1) , 

 we see that, although the mean pressure jumps Apg 

 (X = 0) are much larger than those at all other 

 wake harmonics, the propagation functions for m = 

 X - S.N = ±N exhibit extremely rapid decay with 

 increasing x. In addition, we observe that the 

 radial loading for m = ±N obtained from Apg is 

 weighted by the oscillatory function giNa which 

 has the effect of producing an Lj^{o) which is 

 inversely proportional to N. In contrast, the 

 contribution for X = N, i.e., m = 0, is of the 

 form „ 



T (N) , ^ 



which has a "non-destructive" weighting function of 

 unity. Another feature which reduces the mean 

 loading contribution to the generation of forces 

 on the hull (wherein integration over the athwart- 



ship variable yp is involved) is the presence of 

 the space angular function 



iN 



f = 



-iN(tan 



yp/-p) 



yielding pressures and velocities at different yp 

 which are not in phase. In strong contrast in the 

 propagation mode for the blade frequency loading 

 APf] (for which m = 0) , (()rp has no dependence on )o 

 or f ^, and all yp locations receive velocities and 

 pressures which are in phase with each other. On 

 the other hand, the coefficient C|n,| is large for 

 X = {being 5.5 for a 5-bladed propeller), whereas 

 C^ = 11 , the multiplier for the contribution from 

 the blade frequency Ap's. 



These observations are succinctly summarized in 

 Tables 1 and 2 for the case of a 5-bladed propeller, 

 displaying the rate of decay with x, the variation 



of the influence coefficients C I 



and mC L /l+2|m| 



and the dependence on the angular space coordinates 

 ■f and -f^, without and with the free surface effect 

 for the dominant terms at blade frequency arising 

 from the loading at wake harmonics X = 0, N - 1, 

 N and N + 1. 



One may observe in Tables 1 and 2 that the effect 

 of the free surface does not generally increase the 

 rate of attenuation of the potentials with x except 

 at or near all points in the vertical plane yp = 

 with the exception of the '^^.(^^ and 'if-^^^' arising 

 from blade frequency loading on the blades, i.e., 

 X =N and m = 0, which show a change from x~2 to x""^ 

 and x~^ to x~3 everywhere, respectively. 



A dramatic contrast in the force-generating 

 capabilities of the pressure field components arising 

 from the mean (the largest) and the blade-frequency 

 loadings on the blades can be found by integrating 

 the pressures 



-P' 



3*^(0) 



at 



and 



-P' 



3*^(N) 

 "Tt 



over a rectangular region of half-breadth b arranged 

 symmetrically Zq units above the propeller and 

 extending from -f radii forward to s radii downstream 

 of the propeller plane. Upon defining the coeffi- 

 cient of the vertical force on the rectangle as Z^ ' ' 



F^(X) 



^Zn"* 



/p'n^D^, we can arrive at the following 



TABLE 1. ASYMPTOTIC CHARACTERISTICS OF BLADE FREQUENCY COMPONENTS 

 OF THE THRUST-ASSOCIATED POTENTIAL ^t FOR A 5-BLADED PROPELLER 

 FOR LARGE AXIAL DISTANCES 



Wake Propagation Innuencc 



Order Order Coef. 



8.48 



N-l=4 



N=5 



N + 1 =6 



-1 



4.71 



3.14 



4.71 



Relative 

 Loading* 

 I (X)** 



2pau 



26.7 



4.8 



2.1 



Dependenec on x.v^ and Vj 



Without With With 



Free Snrlaee Free Surface Free Surface 



(yp = 0) 



x(ei5^-e''% ^6d(d-7.p)x 

 x'5 

 10d(d-Zp)x 



ur 



6xd(d-Zp) 



IxP 

 10d(d-Zp)x 

 jxl^ Ixl' 



IxP 



X 



xe-'' 

 |x|5 



