i:j^?i.^»t*vw«!>(rtwwRw^. 



9. HALF-PLANE T^JBLEMS WITH MEAN FLOW: WAKES AND THE KUTTA CONDITION 



Consider again the half-plane y ■ 0, x < with prescribed velocity 

 v(x) » v exp(iqx), but suppose now that there is uniform parallel flow 

 at the same speed U on both sides of the plate, the plate edge being a 

 trailing edge . The potential A . satisfies the convected wave equation 



[(-!*> u£)' -cj(& + fi,) W -0 (9.1) 



and the boundary condition on the plate is that of continuity of 

 displacement (not of normal velocity) , giving 



where n. • v/(-im) is the surface displacement. Thus 



*total 



3A 

 -iu> 3T t0tal - (-iu + iUq) v o e lqx on y - 0, x < 0. (9.2) 



Because of the presence of mean flow there is a new possibility on the 

 extension of the plate. A wake can exist there, across which there may 

 be a discontinuity in tangential velocity 3A ./3x as long as the 

 normal displacement (and hence here also the normal velocity) and the 



pressure are continuous across the wake. Take a single Fourier com- 



h 

 *total 



iAx 

 ponent Ae of A for y - 0+, x > 0; then for y - 0-, x > 0, 



iAx 

 below the wake, the corresponding component is -Ae , because $ . must 



be an odd function of y. The pressure jump across the wake associated 



with this particular component of potential is 



P(*,0+) - p(x,0-)-(-p)(-lw + U j^) (2Ae lXx ) 



62 



F aa tuwaiift-rffinf , ■., /,? »nrt . ^.r. ■, ■■„,—..■,-,■■»■■. „>.,.. .•if^fai 



