18 



Satnt-ond- Sink k« 



in I1R()|)^ \ AMKs IN Mill' 1)1 sK.N 



<t>.i = — ['-X + J m log. 



Sec. 11. S 



l(x + ay + j/'J 



(11. ix) 



Reference 

 Axis 



." Sourte. JJjo" inlfln'-jf?j- 



Adding th% SUtotn Function ¥-y • -U^y for the Uniform Row, 



ftirSinh, ^5K--mlogeRsK - -05mloq« jjx.s)'*/] 

 fcrSooree-SmW Riir, j^c " T'^'^^e 7 — (t i 

 A<Jdinq th« Velocity Moitiol ^u'-Ua>^ 'e"" the Uniform Flow, 



*3--U»>.*-Z-"'l' 



>[l^] 



Fifi. 41. F Definition Sketcfi and Formulas voh 

 Stream Function and Velocity Potential ok 



CoMBrNATION OF UNIFOR.M FlOW AND SoORCE-SiNK 



Pair 



Ailding the stream function ^,/ = —U„y of the 

 uniform flow in the direction of the sourcc-sinlv 

 a.xis gives for the stream function of the entire 

 2-diml combination 



^. = -U.y+ wtim- r , J'-'! ,1 (41.vii) 



Lx + 2/ - s J ' 



The ova! body shape for a value of ^.,- = 0, 

 and the streamlines for ^., = —2, —4, and so on, 

 are delineated in Fig. 43. D. 



To determine the velocity potential for the 

 2-diml source-sink pair of Fig. 41.F, that for a 

 source is 



(t>so = m log. Rso = \m log. [{x - sY + if] 



That for a sink is 



<t>iiK = -m log. /?.,K = -\m log. [(x + sY + ;/] 



where the relationship of the coordinates is as 

 shown in Fig. 41. H of Sec. 41.9. Adding the 

 two Rvalues, the velocity potential for the field 

 set up by the source-sink pair is 



^c = \m log 



" L(x + sY + v'j 



(41.viii) 



Adding to thi.M the velocity potential <f>i, = — f '...x 

 for a uiiifurin utrcam, the velocity potential for 

 a 2-dinil miiirce-siiik pair lying in a unifonn stream 

 parallel to the source-sink axis is 



III. For the 2-diml circular cj'linder diagrammed 

 at 1 in Fig. 41.0 and tabulated in (c) preceding, 

 it is shown in Fig. '.i.M that this form can be 

 represented by the resultant flow from a 2-<liml 

 doublet and from a uniform stream. It is explained 

 in Sec. 3.10 that a doublet is formed by moving 

 the source and sink so close together that they 

 almost coincide, but never do. At the same time 

 the source and sink strength is increased so that 

 the product of the distance 2s separating the 

 source and sink times the source strength m 

 remains finite [Glauert, H., EAAT, 1948, p. 29). 

 Stated mathematically, Ai(mu) = 2ms as s 

 approaches zero, where n is finite and is called 

 the doublet strength. The stream function of a 

 doublet is the limit as s diminishes to an infini- 

 tesimally small distance in Eq. (41.vi), the stream 

 function of a source-and-sink pair. Expressed in 

 .sjTiibols, 



J/D = lim HI tan ' ■■ , "•-■ 5 = 



Lx- + r - s J 



Sim 



^n = 



M.V 



X- + y- 



2msy 



(n.x) 



Adding to this the uiiiform-flow stream function 

 \l/u — —U^y, the function of the coinliiacd flow 

 becomes 



-V~y + -i 



ny 



x' + y' 



(41.xi) 



Setting \l's = 0, which is its value at the reference 

 surface, in this case that of the soliij 'J-dinil rod, 



U.y = -T^ 

 X + y 



(41.xii) 



Hence the flow takes place around a cylinder of 

 radius /?„ where /?,, = 'Vti^l\, . 



The .stream function yf/s of the flow around a 

 J-diml circular cylinder in a uniform stream 

 nomial to it-s axis can be written in several 

 alternative forms: 



^. = -r..,/ + f =-f.«(i-^;) 



= -I'Ju - S'')^*" « (41.xiii) 



