singular Perturbation Problems in Ship Hydrodynamics 



The first of these three equations means only that there is no net 

 source strength in the 2-D problem. The second relates the 2-D 

 vortex strength, tiq, to the dipole density, Y|, in the far field. 

 The latter can obviously be interpreted also as a vortex strength. 

 The third quality relates the "constant" term, Aqo* in the near-field 

 solution to the far-field solution's dependence on z, the spanwise 

 coordinate. It is important to include such a term as this in the 

 near-field solution, because it provides a three-dimensional effect 

 in the otherwise two-dimensional problems. 



Presumably, the near- field problem caji be solved somehow. 

 If the body is simple enough, an analytic solution may be obtainable; 

 with the available powerful methods of the theory of functions of a 

 complex variable, it is even reasonable to hope to find such solu- 

 tions. However, even if numerical methods must be used, the 

 solution can be found. Then all of the constants in (2-51) except 

 Aqq are known. The constant of most interest at this naoment is 

 r\^ it will be non-zero only if some nnechanism has been included 

 tnat can generate and determine a circulation around the body. I 

 shall assume that a Kutta condition is available for this purpose, 

 since the present section is concerned with wings. Then, with r\Q 

 known, we can find the first approximation to the vorticity (and 

 dipole density) in the far field, by means of (2-52). At the same 

 time, Aqo is determined. 



Nothing more can be done now unless we find a higher-order 

 term in either the near- or far-field expansion. It is interesting 

 to pursue the near-field solution further first. 



When we substitute the expansion, (2-48), into the Laplace 

 equation, (2-46), and keep only leading-order terms, we obtain the 

 partial differential equation for ^^: 



^'xx "*" ^'yy ~ " ^Ozr» ^^ ^^® fluid domain. 



Now ^0 was found to be 0(c) , and we might reasonable expect that 

 $1 would be 0(e ). In fact, this turns out to be quite correct. In 

 the equation just above, this means that the left-hand side is 0(1) 

 and the right-hand side is 0(e). Asymptotically, then, we have 

 that: 



$, + $1 =0 in the fluid domain. 

 'xx 'yy 



We again have a purely two-dimensional boundary- value problem 

 to solve, if we can state the boundary conditions appropriately. 

 From (2-47'), we find by the same arguments that: 



Q^i =0 on the body, 



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