Ogilvie 



formulated and can be solved. 3) Frona the solution of the $i 

 problem, the function A|o(x) can be determined, which, through the 

 matching, gives a|(x), and the far-field two-term expansion is known. 

 4) From the matching relation for C|(x), along with formula (2-6la"), 

 the near-field potential is known completely to two ternas , and the 

 C|{x) term includes the most important effects of interaction among 

 sections. This sequence of steps shows what an intimate relation- 

 ship exists between near- and far-field expansions. 



The source strength, a, (x) = A|q(x), can be computed without 

 the necessity of solving the flow problem. In the near-field picture, 

 draw a circle which encloses the body section. The net flux rate 

 across this circle is just A|q. From the body boundary condition, 

 (2-63), one can show that there is a net flux rate across the body 

 surface, and it is given by Us'(x), where: 



s(x) =T ) d9 rQ(x,6) = cross section area at x, (2-67a) 



The two fluxes naust be equal, and so we find that: 



(r,(x) = A,q(x) = Us'(x). (2-67b) 



Thus, the source strength is proportional to the rate of change with 

 X of the body cross sectionsd area. 



I shall not pursue the solution to higher order of magnitude, 

 although there is no insuperable difficulty in doing so. Rather, I 

 prefer to point out several Interesting facts about the solution and 

 then close this section. 



In the far field, the solution to two terms Is cixlally sym- 

 metric, although the body is not a body of revolution. The near- 

 field two-term expansion is not symmetric In this way unless the 

 body Is circular and is aligned with the incident flow. However, 

 the near-field solution can be represented by the series, 



x/ \ ~ TT J. o'l(x) , ^ ^ / \ X -All cos G + B|| sin , 

 <|>(x,y,z) ~ Ux +-11;^ log r - -^i^W ^ -^ 2^^: + .., , 



and, at large r, the axially symmetric terms dominate this series. 



If the far-field expansion Is carried to three terms, It will be 

 found that the third term can be interpreted in terms of a line of 

 dlpoles, both vertically and horizontally oriented. Such terms will 

 be of the form given in {2-6lb), with n = 1; they contain unknown 

 functions si^^ (k) and b2i (k) , which miust be determined through 

 matching. These unknown functions will depend entirely on the 



720 



