TABLE 2. ASYMPTOTIC CHARACTERISTICS OF BLADE FREQUENCY COMPONENTS 



OF THE TORQUE-ASSOCIATED POTENTIAL 0q FOR A 5-BLADED PROPELLER 



AT LARGE AXIAL DISTANCES 



285 



Wake Propagation 



Order Order 



X m 



Influence 

 Coef. 



'"'^1 m I 



1 +2|m| 



Relative 

 Loading* 



2P«h 



Dependence on x,i^ and vP; 



Without With 



Free Surface Free Surface 



With 

 Free Surface 



(y,, = 0) 



N-l =4 



N = 5 



N + 1 =6 



-3.86 



-1.57 



1.57 



26.7 



4.8 



2.1 



Ixl" 



Ixj^ 



_l_ 



Ixl 



e'^f. 



15^, 



Ixl 



IxP 



*These are relative values as obtained from calculations of a 5-bladed propeller 

 using the wake of the SS Michigan. 



2p«b 





Ap^(p,C[)e"^"dc< = Ap,(p) 



for Ap^ independent of a 



expressions for the moduli of the blade-frequency 

 forces , viz. , 



|Zt(o)| = " ^ ° 



C„ W(z2+b2)^+l 



4TT2p 'n^Rp 



sin [(N+1)Y„] 



1.0 



0,2 



N+1. , > 

 p Apo (p) sin NUj^ 



(29) 



where 



(1.3. 5.. ■ (2N-1)) 



N 

 2 (N+1) ! 



and y^ = tan 



•1 /b 



for the contribution from the mean loading, (A = 

 0) , and 



(N) 



Sirp'n^R^ 



1.0 



0.2 



p I Ap^ I in 



(b+ Js2+22+p2+b2) 

 ab+ /f2+z2+p2+b2) 



f2+z2+p2 



? ? ? 

 s^+zj+p-^ 



(30) 



dp 



for the contribution from the blade-frequency loading 

 on the blades. 



Evaluations of (29) and (30) were carried on a 

 hand calculator for various integration lengths f 

 forward of a propeller using assumed radial distri- 

 butions of Apq and Apj^ and representative values 

 from computer calculations for a 5-bladed 22.5 ft 

 propeller in a single-screw ship (model) wake. The 

 calculations were made for a flat-bottomed hull of 



half-breadth b 



2 Rn at z,- 



1.5 Rq (25 percent 



tip clearance) and a stern overhang s = 1. Results 

 shown on Figure 4 show dramatically that the force 

 arising from the blade-frequency (b-f) loading is 

 (asymptotically) 65 times larger than that from the 

 mean blade loading when the free-surface effects 

 are omitted (note that Apg = 40 Ap^) . Furthermore, 

 the total force due to b-f blade loading rises very 

 slowly to its asymptotic value as the integration 

 length is increased and even the force from mean 

 blade loading requires integration of the pressure 

 to three radii forward of the propeller. 



To allow approximately for the effect of the 

 free surface, one can subtract terms of the same 

 form as (29) and (30) with z^^ replaced by Zg^ + 

 4dh with d being the depth of submergence of the 

 propeller axis and h the hull draft in way of the 

 propeller. The reduction in force for d = 3.5 and 

 h = 2 is significant for Zip'^) j^ut is found negli- 

 gible for the smaller force. As expected, the 

 asymptotic value Zr]i(N) (f-Ko) is more quickly achieved 

 due to the presence of the free surface, but, never- 

 theless, requiring that one integrate to some 8 

 diameters to achieve the final value. 



These results tell us that the current practice 

 in European model basins (in which b-f pressures 

 are measured on models in the vicinity of the 

 propeller and these are integrated in an attempt to 

 secure the b-f hull force) is highly suspect because 

 the slowly decaying pressures from b-f blade loadings 

 contribute large sectional force densities far from 

 the propeller . This effect is exacerbated by the 

 "growing" cross-sectional shape as one integrates 

 forward which is not accounted for in the constant 

 beam "ship" used in the foregoing analysis . 



