singular Pevtuvhation Problems in Ship Hydro dynamics 



Fig. (5-5). Bluff Body in the Free Surface 



sketched in Fig. (5-5). Regardle^ of whether their description of the 

 flow at very low speed is correct , this jet model appears to be 

 entirely reasonable physically; a barge-like body usually causes a 

 region of froth just ahead of the bow, and this froth is probably 

 caused by such a jet being thrown upward and forward, then dropping 

 downward (which the theory overlooks). Thus it seems appropriate 

 to study the formation of such a free-surface jet by the use of free- 

 streajmline theory, and one may expect that the details of the formation 

 of the jet are not terribly sensitive to the effect of gravity. 



The body, as shown in Fig. (5-5), extends downstream to 

 infinity. (In a sense, the whole problem is part of the inner expan- 

 sion of a much larger problem, in which the stern of the body would 

 be visible and in which waves would follow the body.) Thus, there 

 is no Kutta condition or equivalent which can effectively cause a 

 circulation type of flow in the fluid region. In Green's problem, for 

 example, the flow at great distances appears to have been caused by 

 a vortex. It is this property that causes the apparent logarithmic 

 deflection of the free surface far away from the body, and it is this 

 property that requires the far-field description (as in Rispin's 

 problem) to contain a logarithmic singularity at the origin. Dagan 

 and Tulin have no such logarithmic solutions. 



They find that the jet appears, from far away, to be caused 

 by a singularity of algebraic type. Specifically, the outer expansion 

 of their inner expansion shows the complex velocity behaving like 

 Z" , where Z is the complex variable defined in the physical 

 plane, shown in Fig. (5-5). Thus, their far-field expansion must 

 exhibit a singularity at the origin of this same type. 



This result, if correct, is most interesting, for, as Dagan 

 and Tulin point out, it means that the far-field expression for 



Their Section III. 2 has some questionable aspects. 



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