20 



HYDRODYNAMICS IN Sllll' D1.S1(;N 



"irc.-il.O 



In a similar manner, the velocity potential is 

 ol)taine«l Ity adding; tlie velocity potentials of the 

 separate flows. For circulation alone, 



^^ = f i\ tUi + J r,}{ de = ~e (ii.x.x) 



For the comliinetl hydrofoil How, 



i„ = -{•.(/<■ + 1^') cas + ^e (U.xxi) 



From either the velocity potential or the stream 

 function, the velocity at the surface of the cylinder 

 is ol)tainetl from the relationships 





1^ 



Rdd 



= U, 



U, 



4+1). 



in e + .T 



Hence, at the surface, where /? = /?„ , 



U = 2r. sin e + 



27r/f„ 



(ll.xxii) 



The pressure at any point on the surface of the 

 2-diml cylinder is, by Ec]. (2.xvi), 



V = V" ■'<- 



1 - 



2r. sin d + 





VI 



(41.xxiii) 



Upon integration of this pressure over the sur- 

 face to obtain the resultant force in the //-direc- 

 tion, the lift // is obtained. 



= -fvR. 



Jo 



sin e (le = pr„r fii.xxiv) 



As a matter of iiif(jnn;ition, llio npproximalc 

 value for the stream function in the case of a 

 2-diml cylinder of radius R„ in the middle of a 

 2-diml water passage of width .1, as workcil out 

 by II. I^mb, is 



2irx 



^ = 17- 



.Rl "'" A 



A , 2tt/ 2x1 



cosh —^ — cos — r- 

 A A 



where the origin of cartesian coordinates is appar- 

 ently taken at the center of the cylinder [Ilclc- 

 Shaw, II. S., IXA, 1808, Vol. XI., p. 28). A plot 

 of the streamlines for this ca.se is given by Ilele- 

 Shaw in Fig. 20 on Plate XI of the reference'. 



Knowledge of the stream function or velocity 

 potential for the 2-<iinil circular s<'ction is of 

 great value in deriving corrcspondinn dat.'i for 



other section shajjcs by confoniial t ran<foriiiatioii, 

 dcscribeil in Sec 11.11. 



41.9 Stream-Function and Velocity-Potential 

 Formulas for Three-Dimensional Flows. Paral- 

 leling the expositicjn uf Sec. ll.S, where formulas 

 are derived for the stream function, velocity 

 potential, local velocity, and local pressure in 

 the streamline and hydrofoil flows around a 

 2-diml rod with its axis normal to a uniform 

 stream of licjuid, the present section carries out 

 the same derivation for the flow around: 



(a) A 3-diral sphere 



(b) A 3-diml bodj' with a head of ovoid shape, 

 formed by placing a single .3-diml source in a 

 uniform stream. A partial longitudinal section 

 of such a bod}' is illustrated in Fig. G7.II. Only 

 that flow is considered which is symmetrical 

 about an assumed x-axis through the center of 

 the sphere and of the body described. By using 

 spherical coordinates, diagrammed in Fig. 41. H, 



Dioqram os Droivn is for the 2-Dimensionol Cose It Serves Also for 

 k»^ 1^ the 3-DimensionQl Cose b>( Rototmq It to on^ 



* "^^slr^^" ~--- Ilesired Anqle About the 

 ! [* — s... |"~- — , Souae-Sinh Axis 



I ^I§S!i"=t«) ftint 



Source 

 Axis 



Fin. 41. H Definition Sketch for Cartesian and 

 Spiikuicai. Coordinates ok Source-Sink Pair 



only liie coordinates A' and need be used to 

 define the flow, although the cartesian coordinates 

 for a selected point in any one longitudinal plane 

 through the axis, parallel to the direction of 

 uniform flow, are shown for convenience. 



l''or I he |)iii|inscs ciT this sei-li(in, as for use in 

 most text and reference books, the 3-diinl stream 

 fimction ^ is l/(27r) times the Stokes stream 

 function described in the concluding paragraphs 

 of Sec. 2.12, on page 32 of \'oluine I. Hence the 

 (liuiiitity rate of flow in a "rod" of radius ij, 

 with a uniform velocity of — ('„ , is 



^Blok.. = -V^Ttf 



However, for the 3-diml stream function used here 



