211 



HYDRODYNAMICS IN SHIP DI:SIC;N 



Sec. 50.5 



roflotttHi in Clmp. I(> of this part of tlio hook, l>tit 

 it is certainly a step in tlu- rinht liiicttion. 



Curve A in iliagrnm 3 of Fig. 50. B gives values 

 of the expre.'vsion /^h ((0.5B)J]''J, on a basis of 

 Frouiie ninnl>er /•'. = \'/\/gL, for the eah-iihited 

 waveinaking re.sistanec of the 2-din)l model in 

 diagram 1 of the figure, when moving ahead in 

 an iileal li<iuiii, with no boundary layer. Curve B 

 in diagram '.i repre.sents the calculated wave- 

 making resistance Ii,r divided by [{0.oB)'V'\, for 

 a symmetrical stern appendage corresponding to 

 Modification B in diagram 2 of Fig. 50. B. Curve 

 C gives similar data for Modification C. The 

 effect of easing the waterline slopes in the run 

 appears to be very large comparofl to the small 

 size of the appendages adiled. 



Allowing for the boundary-layer elTect i.s 

 discussed brieflv bv J. K. Lunde [SXAME, 1951, 

 p. 83). 



50.5 Formulation of the Velocity-Potential 

 Expression. Given the a-ssumjitioiis listed in Sec. 

 .50.4 preceding and accepting the limitations 

 stated there, one analytic procedure may be 

 outlined brief!}' as follows. Granting that the shape 

 of the underwater ship hull, as well as the motion 

 of the water around it, is defined by a given 

 combination of sources and sinks and a superposed 

 uniform flow, the internal sources and sinks, 

 balancing each other in strength, can determine 

 the hull shape but they have no effect upon the 

 external resistance. This is caused solely by the 

 image source(s) used to produce the .surface 

 boundary condition; that is, to keep the free 

 surface sensibly flat. By starting with the force 

 produced on a typical internal source by the 

 fluid velocity at that point resulting from the 

 image source, it is possible to arrive at an expres- 

 sion for the surface wave resistance. 



Expre.s.sed in more specific terms, one may start 

 with a .3-diml .source placed at some point in.side 

 the bow of the schematic ship, below the waterline. 

 Thi.s source is so placed that, in (Mimbination with 

 others to be added later, and when superpo.sed on 

 a uniform-stream flow, it produces an entrance 

 for the ship of the desired size and shai)e. 



Moving by it.self at a steady speed clo.se under 

 the free liquid surface, the low source, called A 

 for convenience, not only diverts the liquid 

 flowing toward it at its own level but also produces 

 a surface; wave above it. Following the procedure 

 dcHcribed at the beginning of Sec. .50.4, step (1) 

 is to flatten out the bow wav«' above source A, 

 bringing the free Hurfmc back to its original al-rest 



condition. For this purpo.se a .second 3-diml or 

 image source B, with a velocity potential of 

 opposite sign, is added at a distance above the 

 free-surface plane equal to the submergence of 

 source A below it. 



The velocity potentials of the two 3-(linil 

 sources, taken from Eq. (3.xiii) of Sec. 3.9 of 

 Volume I, are expressed as 



'i>* = JT ^^ *o 



/<"« 



They arc added to form the velocity potential of 

 the source mo\Ting under the flat free surface. A 

 general term 



*.s 





is then added to produce the elTect of all other 

 supplementary sources, whether they are "body" 

 sources below the liquid surface or image sources 

 in the air above it. Hence the velocity potential 

 for a moving source may, to the first approxima- 

 tion, be written 



= *.+ .^« +«.,= - ^ - ^ + 2.< 



Rs 



In many of the references of Sec. 50. K! it will be 

 foiuul that these expressions are modilied by 

 having 47r in the denominator. This is solely 

 because the 3-diml source strength ?h is defined as 

 equal to the quantity rate of flow Q, instead of 

 as Q/47r. The latter corresponds to the notation 

 in this book. 



Making use of the Bernoulli Theorem, ami 

 passing through a long series of mathematic 

 transformations, much too complicated and 

 involved to be given here, an expression is derived 

 for the velocity potential which permits almost 

 any finite number of sources and sinks to be used 

 to rejjrcsent the imderwaler form of the ship. 

 'J'liis velocity potential may, in fact, be expre.ssetl 

 ill a number of dilTerent forms, depeiuling upon 

 the plienomcna which are to be predicted or 

 calculated from it. In any of its forms, however, 

 it must first .satisfy the continuity conditions. 

 Second, it nuist sivtisfy the various boundary 

 conditions corresponding to tlie .shape of the 

 tmderwater form and the shape of the free surface 

 around the nioviiig shii) (this surface need not 

 nece.ssjirily be fiat in the final form of the velocity 

 pot.<'ntial; indeed it. is not fl.-il). 



In thewoi'ds of M. M Mmik, when spe;ikiiig of 



