283 



To account for the presence of the free surface 

 which, at the frequencies of interest acts as a 

 zero potential surface (see Eq. (7) , we merely add 

 to (19) the potential (j)p. = - (t>p (x,r^,yj^, t) in 

 which 



+ (2d 



-^p' 



2 _ 



= r when z 



(21) 



fi 



-1 



2d-z 



■f when z 



d and 



Tr/2 <f. < TT/r, f j^ (y=0) 







(22) 



where d is the distance or depth of the propeller 

 axis below the free surface; Yp, Zp are the transve 

 and vertical coordinates of any field point (Figure 

 2). Thus, the total potential arising from the 

 loadings on an N-bladed propeller in the presence 

 of the free surface (neglecting the feed-back on 

 Ap;^ from the free surface) is 



N 

 p'U 



i2.Nut 



Asymptotics of the Loading Potential 



The fact that the disturbances induced by each of 

 the pressure jumps Ap^ are propagated by widely 

 different functions of the space variables x,r, f 

 must be emphasized as these behaviors have a most 

 significant impact on the pressure, velocities, and 

 the resultant forces generated on the hull. These 

 diverse characteristics can best be illustrated by 

 examining the asymptotics of the potential for 

 upstream locations which are large only with respect 

 to the x-wise extent of the blade surface. The 

 x-wise extent of the blades is given by the (chord) 

 . sin i(jp,ij; being the local pitch angle which, in 

 the radial region of heaviest loading, is normally 

 of the order of 25°. For merchant ships, the blade 

 chord in this region is of the order of one-half 

 the radius and, hence, the x-wise extent of the 

 significant position of a propeller is only about 

 0.2 radius. Thus, for axial distances of the order 

 of one diameter, the x-wise extent of the important 

 region of the blade can certainly be neglected in 

 an asymptotic analysis. 



Using the expansion of R~ given by 



E 



A=0 



Apx(5,P) [Pm(55'r,r;C,p) 



-1 _ 



■n" , /pr 



' 2m- 1/2 (2) e 



im{Q+a-f) 



Pin(x,ri,ri;5,P)] ds 



(23) 



and the spatial derivatives of this function yield 

 the velocities induced by the propeller and its 

 negative image in the free surface. Clearly (fip + 

 $„, =0 for all X and y_ for z = d. 



where Q is the associated Legendre function, and 

 (x-C)2 + r^ + p2 



2pr 



Blade Thickness Potential 



The potential, (J)^, induced by blade thickness may 

 be constructed from a distribution of dipoles (with 

 axes tangent to the helical arc along the blade at 

 any radius) whose strenths are given by V^, V being 

 the local relative resultant velocity and t the 

 local thickness provided by the expanded blade 

 section drawing. Using the helical geometry as 

 before, one can obtain 



|)T-(x,r,f,t) 



1 

 4tt 



+ (ti)p)2 t(p,ci) 



and retaining only the leading term in the expansion 

 of Q for large Z, one can arrive at the following 

 behaviors for the consituents of the loading poten- 

 tial, i.e., <i>-p = <i' + <!>(-,, i^T bsing the part 

 associated with thrust loading and ifiQ being that 

 arising from the torque-producting loading in the 

 forms: 



m -iJlNe 

 C| .rl 'e 

 m 



4ti^P 'u 



(mSA-«.N) 



L (A) , , 

 m (p)p' 



3 1^^ 



(24) 



where aj)(p) and a^-(p) are the angular coordinates 

 of the blade leading and trailing edges. 



To allow for the free surface, 1/R is replaced 

 by 1/R - 1/Rj^ with R- being the distance from the 

 reflection of the dummy point in the free surface 

 to the field point on or below the water surface, 

 making use of relations (21) and (22) . Again, to 

 place the harmonic content of 1/R and l/R-j^ in 

 evidence and to facilitate integrations over the 

 blade surface, the Fourier expansion (17) can be 

 applied. 



-im-/' 



-imy. 

 xe 1 



(x2+r2+p2) I'^l+^/S [x2+r2+p2+4d(d-Zp)] l""l^^/2 



m -i«,Ne 

 mC 1 I r ' ' e 

 m 



4tt2p '0)2 





(1+2 m)£ 



continued on page 284 



