Pien and Strom-Tejsen 



This concludes our discussion on kernel functions for the case of steady 

 blade loading such as in the open water condition. The case of periodic blade 

 loading such as in the behind condition is discussed in the next subsection. 



Kernel Functions for Periodic Blade Loading 



Whenever a ship changes speed or course, the velocity or wake field be- 

 comes time dependent. Although it is feasible to analyze such problems within 

 the framework of our basic approach, we have chosen here a much simpler 

 example of the unsteady case, that of a periodic blade loading. 



When a ship is maintained on a steady course, we assume that the wake field 

 behind is time independent. Since wake strength varies spatially, a rotating pro- 

 peller blade experiences a periodic inflow variation. As a result the blade load- 

 ing also becomes periodic. Since the induced velocity is proportional to the 

 blade loading if everything else is kept the same, we can use the principle of 

 superposition. A periodic loading is first broken into its harmonic contents. 

 By summing up induced velocity due to each loading harmonic, we obtain the 

 total induced velocity due to the total periodic loading. Therefore, our problem 

 is essentially to obtain new kernel functions k^, k^, and k^, similar to those 

 defined by Eqs. (32), due to a pressure dipole with a periodic varying strength 

 giqQt g^|. ^ point Q(A,p,6'-nt) on the moving blade, where q is the order of the 

 loading harmonic. Now k^, k^, and k^ are complex functions with a real and an 

 imaginary part, which can be expressed as 



Ki = Ki, + iKib . (49a) 



(49b) 



(49c) 



Since the load variation over the blade area may not be in phase, we write 

 the load distribution as 



L(^,p,d) = L(^,p,0)^ + iL(^,p,e)^ . (50) 



Now we may write 



u(x,r,0)= r Tlc^./o,^) k^ dpd0 , (51a) 



s 



v{x,r,4>)^ j [L(^,p,e) k^dpdd, (51b) 



s 



w(x,r,g^)= i fL(^,p,0) K^ dpd0 , (51c) 



104 



