212 



11M)R()1)^\ AMICS IN Sllir DlSKiX 



Src. •in.4 



To l)o suri', ns is cxplninrtl shortly, the ciirroiit 

 (195o) mathoiimtioal theory Uoos not recognize 

 the fact tliat the actual sliip moves in a real 

 hqiiid, tliat it is surrounded by a boundary layer 

 of varj-ing thickness and velocity, that it is 

 generally accompanietl by a separation zone at 

 tlie stern, or that it is driven by propellers with 

 velocity and pressure fields of their own. Xo 

 more dcx-s the simple beam theory lake account 

 of the complexities in the stmcturc of a modern 

 ship, yet it is contiiuially employed to predict 

 the stresses and strains in this stmcturc. If cor- 

 responding complications do not prevent everyday 

 use of the simple beam theory, the jissumptions 

 implicit in the i)resenl-day applications of theo- 

 retical hydrodynamics to a prediction of ship 

 behavior should not hinder its use wherever 

 applicable. Continuation of the theoretical and 

 analytical development of the past 50 years for 

 another half-century into the future, correspond- 

 ing to the full century that the beam theory 

 lias been in use for ship structures, may well 

 bring to the hands of naval architects a flow 

 theory equally simple in application if not in 

 character or exprossion. 



50.4 Assumptions and Limitations Inherent 

 in Present-Day Calculations. The a.s.sumptioiis 

 which must be made to obtain a solution of the 

 theoretical wave resistance of a ship by the most 

 modern mathematical methods (1955) will, it is 

 believed, appear in a ch^arcr li^lit if the analytic 

 procedure is first explained. Paraplnasiiiff T. II. 

 Ilavelock, to whom we are principally indebted 

 for it, this procedure is described as follows 

 [Lunde, J. K., "On the Theory of Wave Resistance 

 and Wave Profile," Norwegian Model Basin 

 Rep. 10, Apr 1952, p. 2]: 



(1) The first step is to neglect any wave motion 

 produced on the free surface, as if the latter were 

 covered with a sheet of ice which moves aside to 

 pemiit pa.s.sage of the .ship, and to con.sider only 

 the lii|uid motion jjrotluccd by the ship 



(2) The second stop is to obtain the wave dis- 

 turbance produced l)y this motion while ignoring 

 the presence of the ship in its effect upon these 

 waves 



(3) The third step, not yet po.ssible with existing 

 theory, is to evaluate the influence of the ship 

 on the waves so calculated 



(4) Finally, by a stjries of HUcce8.Hive approxima- 

 tions whiili remain to be worked o\it, to determine 

 Ihc actual wave disturbance aroinid the ship. 



From th<? pressures developed on the ship surface, 

 or from the energ.v in the wave system, to calculate 

 the ship resistance due to wavemaking. 



The actual assumptions made, as embodied in 

 the Lunde 1952 reference, are quite definite and 

 straightforward: 



(a) The liquid is homogeneous, it retains its 

 continuity, and it is incompressible; that is, it is 

 not subjected to ehistic d(?formation 



(b) The li(|uid is ideal in that it is without 

 viscosity 



(c) The action has continued for a sufficiently 

 long time so that steady motion is established 

 everywhere 



(d) The wave height is small in compari-son to the 

 wave length, with a wave slope and a wave 

 steepness that are likewise relatively small 



(e) The velocities due to wave motion are small 

 compared to the ship spceil 



(f) Outside of the displacement thickness 5*(delta 

 star) of the boumlary layer, which is relatively 

 thin compared to the beam or draft of the ship, 

 the liquid motion is irrotational and can be 

 characterized by a velocity potential (/)(phi) 

 which, when dilTerentiatcd partially, produces the 

 three comi)oncnt velocities along the body axes: 



d(t> 



M = T^ V = — 



ax ay 



dz 



These velocities are as.sumed to be so small in 

 comparison with the ship's velocity that their 

 squares and higher powers can be neglected, 

 (g) The liquid motion around the ship can be 

 represented by the combination of a uniform- 

 stream flow parallel to the ship axis and a railial 

 flow associated with the desired or necessary 

 combination of sources, sinks, and doublets, 

 placed anywhere within or on the hull boundary 

 (h) For rea.sons largely nuithematical, to keep 

 certain integrals determinate, it is a.s.sunu>tl that 

 the ideal li(|uid ilocs exert a small friction force 

 proportional to the liquid velocity. The damping 

 coeflicieiit thus arbitrarily introduced is dimin- 

 i.shed to zero in the analysis as soon lus it has 

 .served its purpose. Actually, this procedure insures 

 that the surface waves always trail the ship, 

 (i) Other than as listed in (h) i)receding, the 

 effects of viscosity in the liiiuid are neglecteil and 

 the boundary layer as such is considere<l absent 

 (j) The pressure resistance due to wavemaking, 

 under the foregoing contlitions, can be consiilercti 



