Shallow Water Problems in Ship Hydro dynamics 



motion of the section T as if it were an in finit ely long rigid cylinder 

 moving in the y-direction with velocity Ayjg/h, the fluid at y = ± oo 

 being at rest. 



The problem specified by (6. 1) subject to (6.2), the bottom 

 condition (3.3), the hull condition (6.7) and the rest condition (6,5) 

 is a classical Neumann boundary value problem, and <\> is deter- 

 m.ined by these conditions apart from an additive constant. If in 

 addition we prescribe the natural symmetry condition that cj) is an 

 odd function of y, in conformity with (6,4), then <j) is uniquely 

 determined, and the actual limit of (f) as y — ^ ± oo must, via (6,4), 

 provide a determination of P. 



A number of techniques are available for solving this two-di- 

 mensional boundary- value problem. If the section is sufficiently sim- 

 ple (e.g. rectangular) a solution may be found by conformal mapping 

 methods (Flagg and Newman [ 1971]). For actual and quite general 

 ship sections we have (Taylor [1971]) developed a computer program 

 based on the methods used by Frank [ 1967] for a similar problem. 



In this method one represents the flow by a distribution of 

 sources around the section T, with variable but unknown density. 

 These sources individually satisfy the "free" surface and bottom 

 conditions (6,2) (3,3), but not, of course the hull boundary condition 

 (6,7) on r. We now attempt to choose the source density function 

 in order to satisfy (6,7), thereby obtaining an integral equation from 

 which the source density is in principle obtainable. 



Since analytic solutions for general F are out of the 

 question, we adopt a numerical approach in which T is first approxi- 

 mated by a set of straight line segments, on each of which we assume 

 the source density to be constant. The integral equation then reduces 

 to a set of linear algebraic equations for these unknown constant 

 source strengths, and this set of equations is solved by direct matrix 

 inversion. 



It Is convenient to define a blockage coefficient 



C(x) = p^j, (6.8) 



as the inverse of the porosity. With this definition, we see from (6,4) 

 that to ob tain C(x) we merely divide the potential <j> by minus the 

 speed AVg/h of the motion of the section T and take the limit as 

 Y -*- + oo. Thus, if our aim is solely to deterrnine C(x) or P(x) , 

 we nnay, without loss of generality, take A = — /h/g for the purpose 

 of the present section only, and identify C as the limit of <|> as 

 y -* + oo, a quantity which is readily evaluated from the numerical 

 solution for the generating source strengths. 



651 



