HVDROUVNAMiCs 1\ sll||> DISIGN 



Sec. 11.9 



To obtain the doublet, s — » 0. O^o — ♦ Osk —* 6. 

 niul /i',,A — /^,<, — • It. Tlicii. for llic :>-(liinl (i..ul)li't, 





1 - 





4> = 



— /I cos 



I -MX COS , M 



(ll.xxxa) 



(U.xxxb) 



= 1 



siir 



(ll.xxxiv) 



Coml)ii>iiig tlie cloul)let and the uniform- 

 stream veloeity potentials and stream functions 

 gives, for the streamline flow around a sphere in 

 a uniform stream of velocity —U„ , depicted in 

 diagram 2 of Fig. 41.G, 



II. For the case of the :5-diinl bodj' of ovoid 

 shape, formed by placing a 3-diml source of 

 strength rn in a uniform stream of velocity — f/. , 

 as in Fig. 07. H, the .3-diml velocity potential and 

 the 3-diml stream function are obtained as 

 previouslj' explained, by adding the values of 

 <f> and }f/, respectivcl.v, for the two flows. Then, 

 from Eqs. (41.xx'via) and (U.xxviia), 



«. 



- U^R cos 8 - 



n cos 8 

 R- 



4> 



(U.xxxia) 



-U^R cose - j^ (41.xx.xva) 



1 ,, ,-,o . -. „ , u sin' 8 , , , ... 

 ^3.d.=.i = -o f^/?"sm- 8 + ^ (41.xxxib) 



Setting ^ = at the .spherical .surface, 



I l\R' sin= 



/I sin^ 6 

 R 



From Iv|.-;. (41.xxvib) and (4l.xx\iib), 



i/- = — .Jf/„/2"sin" - m cos B (41.xxxvb) 

 whence 



t/« =f| =^- fLcosO (41.xxx-A-i) 



72' = 1^ = 7?^ or ii-o = (1^)"' 



where Ro is the radius of the sphere about which 

 the flow takes place. Stated in another way, the 

 flow around a sphere of radius Ro is obtained by 

 adding a uniform stream — 1/„ to a IB-diml 

 doublet of strength /x = \U,.,R\ . 



Substituting m = §t/„/?o into the expressions 

 for (f> and ^ gives: 



«a.din.. = - 1/. cos 8\R + ^j (41 .xxxiia) 



[/. sin' 8 



"'- RdB 



[/„sin 8 



(41.xxxvii) 



To find the coordinates of the nose of the 

 body, set U r and [/» equal to zero since the nose 

 is al.so the stagnation point. Then 



7^ - i/„ cos e = 



[/„ sin = 



Hence the nose is at 



= 0, Ro = ^ 



\^3-dllI>l — 



2ft 



{Rl - R') 



(li.xxxiil)) 



From Eqs. (41.xxA-a) and (41.xxvd), 

 1 



f/, = - 



R sin 8 dfi ' ft 08 



The 3-diml stream function value which passes 

 through the stagnation point is ^ = — ;». But 

 this stream function is also the surface of the 

 body. Hence, in axisymmetric spherical coordi- 

 nates, tlic (I Illation of the surface is 



On the surface of the .sphere, R = Ro , U k = 0, 

 whence the local velocity is 



R- 



2?/i(l — cas 6) 

 U„ sin' 8 



(11 .xxxviii) 



For the points abreast the source, where 

 8 = 90 dcg, cos = and sin 8=1, whence 



U = If/.sin e 



(41.xxxiii) 



Rio = 



2 m 



or Rt 



The prcKHure cocilicicnt at any point on l!ic 

 jnirfncc of the Hphere Im, by F'lq. (2.xvi), 



From this it appears that Ron = /?o V 2. 



Till' Inmsvcr.se radius of a 3-dinil ovoid such as 



