APPENDIX C 



Calculation of Hydrodynamic 

 Forces by Strip Theory 



Evaluation of Hydrodynamic Forces 



Thits Appendix is arranged to l'oll(j\v as closely as 

 possible Appendix 1 of an earlier paper (Korvin-Kroukov- 

 sky, 19555) so that the changes made can he easily seen. 

 Where sufficient discussion was given in that reference, 

 the details will be omitted here. 

 Formulation of Problem 



Consider a ship mox'ing with a constant forward \'e- 

 locity V (i.e., neglecting surging motion) with a train of 

 regular waves of celerity c. Assume the set of co-ordinate 

 axes fixed in the undisturbed water surface, with the ori- 

 gin instantaneously located at the wave nodal point 

 preceding the wave rise, as shown in Fig. 14.- With in- 

 crease in time t the axes remain fixed in space, so that the 

 water surface rises and falls in relation to them. This 

 vertical displacement at any instant and at any distance 

 X is designated -q. Imagine two control planes spaced d.v 

 apart at a distance x from the origin, and assume that the 

 ship and water with orbital velocities of wave motion 

 penetrate these control surfaces. Assume that the per- 

 turbation velocities due to the presence of the body are 

 confined to the two-dimensional flow between control 

 planes; i.e., neglect the fore-and-aft components of the 

 perturbation velocities due to the body, as in the "slender 

 body theory" of aerodynamics. This form of analysis, 

 also known as the "strip method" or "cross-flow hy- 

 pothesis," is ihus an approximate one in the sense that 

 a certain degree of interaction between adjacent sections 

 is neglected. 



The cross section of the ship at .r will now be taken as 

 semi-circular; the correction necessary to repre,sent other 

 ship sections will be introduced later. Following F. M. 

 Lewis (1929) and Weinblum and StDenis (1950), the 

 flow about the semi-submerged body used in the basic 

 derivation will be assumed to be identical with that 

 about the lower half of a fully submerged body. Correc- 

 tions to account for the presence of the free water surface 

 will be brought in later. 



In considering the pitching and heaving motions 

 of the body it is necessary to introduce a second co-ordi- 

 nate system moving with the ship with its origin at the 

 center of gravity of the ship. The longitudinal distance 

 of any section of the ship from the origin is designated ? 



' Appendi.v from Korvin-Krovikovsky and Jacobs (3-1957). Ref- 

 erences in this Apix'udix will be found at the end of Chapter i. 

 ' Figure and ecuiation numl)ers ot tlie original paper are retained 

 in this reproduction. 



(positive forward). Vertical displacement of the CG (i.e., 

 the heave) is designated by z (positi\'e upwards) and an- 

 gular displacement or pitching motion is designated by 

 9 (po.sitive for bow displaced upwards). The vertical 

 displacement of the section at .v due to pitching is then 

 ^9, with 9 in radians, for the relatively small angles en- 

 countered. (It is also assumed that cos 9=1.) 



The two-dimensional flow pattern between the control 

 planes at .r results from three imposed motions: 



1 Vertical \'elocity of the center of the circle 



v = :■ + ^e - ve (8) 



2 A^ertical component of wave oribital velocity 



rje 



2ir»/X 



-2-!rhc 



7ry/\ 



COS 



, - --^ , (.<• - ct) (9) 



A A 



where h is wave amplitude, X is wa\-e length, ij = —R cos 

 a, the depth below the still-water level to any point in the 

 fluid, and r; = h sin 27r (x — ct)/\. 



3 Apparent variation of the radius ;■ of the ship sec- 

 tion at the control planes with time; r = r(t). 



All motions are assumed t(; be sufficiently small so that 

 the derivatives of the potential can be taken on the sur- 

 face of the circle as if its center were at its initial position 

 y = 0, and the knowii expression for the potential in a 

 uniform fluid stream can be applied, despite the slight 

 nonuniformity induced by the waves which are assumed 

 to be small. 



The vertical hydrodynamic force acting on the length 

 dx of the submerged semi-cylinder is given by 



dF T""/- 



— = 2r I p cos ad a (10) 



dx Jo 



where F denotes the vertical force, a is the polar angle as 

 defined in Fig. 14, and the time-dependent part of the 

 pressiu'e p is obtained from Bernoulli's ecjuation. Ne- 

 glecting the squares of small perturbation velocities 



P - 



dt 



(11) 



where 4> is the velocity potential and p the mass density. 

 The velocity potential of the flow about a cylinder due 

 to the relative vertical velocity (v — v,t) is given by 



</>6 = -('' 



'- 

 i\.) - cos a 

 a 



(12) 



The first term of equation (12) may be considered as the 

 potential due to the body motion in smooth water, desig- 



338 



