Recent Progress in the Calculation of Potential Flows 



H. Nowacki - the interaction between a simplified shipform and the propeller 

 pictured by a sink disc (JSTG 1963). This latter method is based on an iter- 

 ation procedure; the results are valuable notwithstanding a minor slip in a 

 boundary condition. 



It appears that a large number of problems in ship hydrodynamics will be 

 solved by properly extending Dr. Smith's method, following the lines indicated 

 by himself - the flow around bodies in the presence of a bottom, tank walls etc; 

 i.e., the determination of shallow water and blockage effects, which so far have 

 been treated by approximate procedures only. 



Valuable fluid charts of the pressure field around general ellipsoids have 

 been treated earlier by Maruhn (Jahrbuch der Luft-fahrtforschung, 1941) before 

 computers had been developed. . . 



Of special interest are the author's remarks on two-body problems as a 

 foundation for determining the interaction forces of passing or overtaking ships. 



DISCUSSION 



Louis Landweber 



Institute of Hydraulic Research 



University of Iowa 



Iowa City, Iowa 



I wish to discuss the integral equation applied in the paper to obtain surface 

 distributions of sources, viz 



a(p) = f(p) +1 K(P,Q)o-(Q) dSQ , K(P,Q) 



27^ 3np VpO 



where f (p) is a given function and P and Q are points on the given surface S. 

 This integral equation, which formulates the Neumann problem for the surface, 

 has the well-known properties that the eigenvalues X of the kernel K(P,Q) are 

 real, that x = -i is the eigenvalue of smallest absolute value, and that \ = +i 

 is not an eigenvalue. Then, according to the fundamental theorem of Fredholm 

 integral equations, a solution of (1) exists. 



If we write, instead of (1) 



^i + i(P) = f(p) + ^ K(P,Q) a.{Q) dSp 



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