MATHEMATICAL FORMULATION 



The physical flow problem Is to compute transient two-dimensional flow 

 about a body moving in a free surface. It is formulated mathematically as a 

 potential flow problem in which the velocity potential along the free surface 

 and the position of the moving free surface are sought as the solution to a 

 nonlinear initial-boundary value problem. 



In particular, the physical problem is to determine the fluid motion 

 caused by the prescribed movement of a body partially submerged in a fluid and 

 the resulting hydrodynamic force on the body. The prescribed motions are 

 forced harmonic heave motions never so large that the body becomes completely 

 submerged in or rises out of the fluid. The body considered In this report is 

 a closed U-shaped cylinder. Gravity is the only body force acting on the 

 fluid which is inviscid, incompressible, and initially at rest. The fluid 

 motion is irrotational and thus a velocity potential <() ' is assumed to exist. 

 Surface tension is neglected. 



All variables are nondimenslonalized. Lengths are scaled by a length L 

 characterizing the size of the body. Time is scaled by l/o where a is the 

 frequency of the body motion in radians per second. Velocities are scaled by 

 oL; the velocity potential (}> ' , by oL^; pressure, by pa^L^; and force, by 

 pa^L-^ . Here p is the fluid density. Thus, for example: 



x' = L X, y' = L y, t' = t/a , 



<})• = aL2(t) , p' = pa2L2p, F' = pa2L3F, 

 where (x,y) are variables representing the coordinate system, t is time, p is 



