Bow Waves and other Non-Linear Ship Wave Problems 



position beneath the unperturbed level is practically independent of 

 Fr|_. Nonlinear inertial effects may be important nevertheless 

 near a blunt bow. 



A systematical experimental confirmation of the role played 

 by the bow bluntness has been provided recently by Baba [ 1969] . 

 From towing-tank tests with three geosims of a tanker (C^ = 0. 77) 

 it was found that in ballast conditions at a Fr-~ 1.2 a breaking wave 

 appears before the bow. At the maximum Fr|_ tested (Fr. = 0.24, 

 Fr J = 1.7, Fig. la) the energy dissipated in the breaking wave con- 

 tributed 18 per cent of the total resistance, while the energy radiated 

 by waves gave only 6 per cent. Baba has suggested a two-dimensional 

 representation of the breaking wave of this experiment, as If it were 

 uniform and normal to the bow (Fig. lb), and has estimated 

 equivalent length as half the beam. The drag coefficient per unit 

 length, corresponding to a two-dimensional flow across the breaking 

 wave is Cd= D'/O.S pU'^T' = 0.08 for Fr^=1.7. Sharma [1969] 

 has indicated a larger breaking wave resistance for a higher block 

 coefficient tanker and has suggested that the bow bulbs main effect 

 is to reduce the breaking wave resistance. With the development of 

 large tankers, as well as large and rapid cargo ships, the study of 

 the bow free-surface nonlinear effect becomes particularly important. 



We present here some of the results of our last year's 

 studies, which are reported in detail in two reports (Dagan and 

 Tulin [1969, 1970]). 



In this first stage we have attacked the two-dimensional prob- 

 lem of free-surface flow past a blunt body of semi-infinite length. 

 The two-dimensional study is a necessary step in the development 

 of a theory for three-dimensional bows since it provides a valuable 

 gain in insight at the expense of relatively simple computations. 

 Moreover, it gives an estimate of the bow drag of flat ships and opens 

 the way to more realistic computations by further approximations. 



Taking the length as semi- infinite is very useful from a 

 mathematical point of view and it is equivalent to the limit Fr^ "*" 0. 

 This assumption is entirely justified for the small Fr^ considered 

 here and for determining the bow flow, which is not sensibly in- 

 fluenced by the trailing edge condition. 



II. THE FREE SURFACE STABILITY (SMALL Fr^ EXPANSION) 



We consider the two-dimensional gravity flow past the body of 

 Fig. 2. The box-like shape has been adopted for the sake of com- 

 putational simplicity, but the method can be easily extended to any 

 other shape. 



When Fr- is small the free-surface is smooth. We assume 

 that breaking wave inception is related to the instability of the free- 

 surface. According to Taylor's criterion (Taylor [1950]), the 



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