singular Perturbation Problems in Ship Hydrodynamics 



5.3, Submerged Body at Finite Speed 



Since the principal difficulty in solving free-surface prob- 

 lems follows from the nonlinear conditions at the free surface, we 

 are always seeking new arguments to justify linearizing the condi- 

 tions. One possible basis for linearizing is that a body is deeply sub- 

 merged. Then its effect on the free surface will presumably be 

 small, even if it is not appropriate to linearize the problem in the 

 immediate neighborhood of the body itself. 



Such problems were discussed by Wehausen and Laitone 

 [ I960] , where the previous history may also be found. Tuck [ 1965b] 

 introduced a more systematic treatment for the case of a circular 

 cylinder, Salvesen [ 1969] solved the problem for a hydrofoil (with 

 Kutta condition and thus with circulation), and he compared his 

 results with the data from experiments which he conducted. In the 

 earlier studies of such problems , the approach was usually an itera- 

 tive one in which the body boundary condition was first satisfied, 

 then an additional term was added to the solution so that the free- 

 surface condition would be satisfied; the latter would cause the body 

 boundary condition to be violated, and so another term would have to 

 be added to correct that error, but then there would again be an error 

 in the free-surface condition. And so on. The free-surface con- 

 dition that was satisfied once during each cycle was generally the 

 conventional linearized condition. Thus, if the procedure converged, 

 one obtained a solution which exactly satisfied the body boundary 

 condition and the linearized free- surface condition. The contribution 

 of Tuck seems to have been in systematizing the procedure in terms 

 of a small parameter varying inversely with depth of the body and in 

 pointing out that a consistent iteration scheme involves using the 

 exact free-surface conditions as a starting point. Then, as the 

 boundary condition on the body is corrected at each stage, so also is 

 the free- surface condition made more and more nearly exact. 



Tuck concluded, in fact, that it was more important to include 

 nonlinear, free- surface effects than to improve the satisfaction of the 

 body boundary condition if one were nnost interested in certain free- 

 surface phenomena, e.g. , predicting wave resistance and near- 

 surface lift. Salvesen agreed with this conclusion only on the con- 

 dition that the body speed be not too large. At fairly high speed, 

 his results indicated that precision in satisfying the body boundary 

 condition was just as important as precision in satisfying the free- 

 surface condition. Figure (5-6) is taken from Salvesen's paper; it 

 shows the theoretical wave resistance of a particular body as a 

 function of (depth) Froude number, the resistance being calculated 

 by three different approximations: 1) linearized free -surface 

 theory, 2) theory in which the free-surface condition is satisfied to 

 second order, and 3) theory in which both the free- surface condition 

 and the body boundary condition are satisfied to second order. The 

 differences are quite apparent. 



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