singular Perturbation Problems in Ship Hydrodynamics 



N 

 <Kx,y,z)~ ) $n(x,y,z), $„♦! = o(*n) as e -* 0, 

 n=0 



with (x/c, y/e, z) fixed. (2-48) 



The first term in this expansion satisfies the conditions: 



4)q + $Q = in the fluid region, (2-49) 



A* yy 



-^ =0 on the body. (2-50) 



From (2-45a), it is clear that the one-term inner expansion, 

 <^(x,y,z) ~ ^Q(x,y,z), must match the one-term outer expansion, 

 ^(x,y,z) ~ Ux, Thus $Q(x,y,z) is the solution of a two-dinnensional 

 potential problem, and a rather conventional problem at that: In a 

 section through the body drawn perpendicular to the spanwise axis, 

 the potential satisfies the Laplace equation in two- dimensions , a 

 homogeneous Neumann condition on the body, and a uniform-flow 

 condition at infinity. The direction of the uniform flow is the sanne 

 as the direction of the actual incident stream as viewed in the far 

 field. 



Since $q does satisfy the Laplace equation in two dimensions, 

 the methods of complex- variable functions are available for deter- 

 mining its properties. In particular , if we assume that V^q is 

 bounded everywhere in the fluid region and single- valued too, then 

 $0 can be expressed as the real part of an analytic function of a 

 complex variable, the analytic function being such that Its derivative 

 can be expressed by a Laurent series. Thus, we can write for 

 4>o(x,y;z): 



$Q(x,y;z) = Ux + S^log r + tIq tan ' ^ + A^ + 



Aoi cos 9 



, Bqi sin 9 , Aq2 cos 29 , Bq2 sin 29 , /? ki \ 



r J.2 ^2 ... ,1 - ) 



where r = (x + y ) . The "constants" are all unknown functions of 

 z, the spanwise coordinate. The first term represents a uniform 

 stream at Infinity, and I have already performed one matching to 

 determine this term. The second and third terms represent a source 

 and a vortex, respectively; the fourth term, a constant, Is Included 

 for generality; the fifth and sixth terms represent a dlpole; etc. 

 Such an expansion as (2-51) Is valid outside any circle about the 

 origin which encompasses the body cross section. 



701 



