singular Perturbation Problems in Ship Hydrodynamics 



high aspect ratio is to simplify (2-37) by arguments that relate the 

 sizes of the terms involving (x-^) and (z-t,) . (Quite comparable 

 arguments are used in the conventional approach to the theory of 

 slender wings.) If the radical in (2-37) can be sinnplified, then the 

 4 integration can be performed, and one is left with just the integral 

 over t,. In this way, the 2-D integral equation is reduced to a one- 

 dimensional integral equation, which is of a standard form. 



Using the method of matched asymptotic expansions, we 

 return to the original formulation of the problem and derive a 

 sequence of simpler problenns , rather than try to work out approxi- 

 mate solutions of the integral equation. The large-aspect- ratio 

 wing is "slender" in the spanwise direction. This means that cross 

 sections parallel to the z = plane vary graducdly in size and shape 

 as z varies; in particular, the maximum dimension in the z 

 direction, say 2S (the span), is much greater than the maximum 

 dimension in the cross sections. We shall make whatever further 

 assumptions of this kind that we need in order to keep the solution 

 well-behaved. The small parameter can be defined as the inverse 

 of the aspect ratio , that is , 



c = 1/(AR) = (area of H )/4S^ 



where H is the projection of the wing onto the y = plane. As 

 before, it is not necessary to be so specific about the definition of 

 C, and in fact it may be misleading. A wing with aspect ratio equal 

 to 100 might be slender in the required sense if, for example, there 

 were discontinuities in chord length in the spanwise direction. In 

 any case, the wing shrinks down to a line, part of the z axis, as 

 €-* 0. 



Let the body be defined by the following relation: 



y = g(x,z) ± h(x,z), (2-38) 



for (x,0,z) in H . See Figure (2-1). It is not necessary that the 

 body be a thin one, in the sense of the previous section. I do, 

 however, specify that it should be symmetric with respect to z, 

 for the sake of simplicity in what follows. Both of the functions 

 g(x,z) and h(x,z) really depend on €, of course*, but we shall 

 generally onnit explicit mention of the fact. 



There is an incident flow which, at infinity. Is uniform In 

 the x direction. Let the far- field solution be represented by the 

 asymptotic expansion: 



In fact, g and h are both 0(e). 



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