singular Perturbation Problems in Ship Hydrodynamics 



Salves en expands the complex potential In a series which he groups 

 in two alternate ways: 



F{z) = [ Uz + FbJ + [ Ff, + Fb,] + . . . (5-3) 



= Uz +[FbQ + Ff|] +[Fb, +Ff2] +... (5-4) 



These terms are defined in terms of the iteration scheme already- 

 mentioned. The grouping in (5-3) is to be used near the body, and 

 the grouping In (5-4) applies far away from the body; in particular, 

 the latter applies on and near the free surface. Salvesen points out 

 that this distinction means that: a) near the body, we are consider- 

 ing the zero -order flow to be that flow which would occur in the 

 presence of the body and the absence of the free surface, and b) near 

 the free- surface , the basic flow is just the uniform incident stream. 

 Thus, in (5-3), we must determine FbQ so that [ Uz + Fbg] satisfies 

 the kinematic boundary condition on the body and so that | VFbQ| ~*" 

 as |z| — *■ oo (in any direction). 



Next, Salvesen assumes that Fbg is 0(e) far away from the 

 body. The two terms so far obtained do not satisfy a free- surface 

 condition, and so Ff| must be determined so that, when it is added 

 to the first two ternas , the sum satisfies the appropriate free-surface 

 condition, which is: 



R 



e {F^Q + F/| + i/cFbQ + i/cFf, } = on y = b (5-5) 



where K = g/U . Since F^q is assumed to be 0(e) near the sur- 

 face, then the same should be true for Ff . . 



Now the three terms in the series do not satisfy the body 

 condition, and so Fb, is determined so that, when it is added to 

 the first three terms, the sum satisfies the condition properly. 

 Then Fbi is assumed to be 0(€ ) near the free surface, and a new 

 function Ff^ is found to provide a further correction needed near 

 the free surface. 



It is in this last step that the Tuck-Salvesen approach differs 

 from the previous treatments of such problems . If Fbi is really 

 O(e^) , then the free-surface condition ought to be satisfied to that 

 order of magnitude. It can be shown that this implies the following 

 condition on F, : 



Re {f;^ + F^^ + i/cFb, + iKFf^} 



= ^, Im (F^^ + F;- + i/cF^^ + i/cF/^ } - (1/2U) | F'^ + F/^ f. (5-6) 



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