Sec.41.R 



GENERAL LIQUID-FLOW FORMULAS 



*• yi whe 



y^-n of Doublet is '; ^ , 

 /■ » where p is the doublet strength ^ 

 ^. R'X^ I y-y for Uniform Flow is -Ua 



•^ H e nce l^s-- U<»V^^^^ 



Sphere 





Plane of Paper 15 ^r^ Plane 

 throuqh Reference Axis 



>!-Dfor 3-Biml Doublet 



/jsin^6 



V -U ' -0.5 U qo R^sin^e 



where yr for this Cose is 



Q 3-Diml Axisymetric Function 



Then 11 51 n ^6 

 -i/r' -0,5U„R^3in'^e+^^^-f^ 



/Licos9 

 i= -U„Rcos6--=S;? — 



-Reference 

 Axis 



Fig. 41. G Definition Sketch and Formulas for 



Stream Functions and Velocity Potentials 



for 2-Diml Rod and 3-Diml Sphere 



The velocity potential </>/, of a 2-diml doublet 

 whose strength ix = 2ms is found by taking the 

 limit of Eq. (4Lvui) as s approaches zero. This 

 gives 



^1 cos 6 . 



R 



IXX 



in polar coordinates 



(41.xiv) 



X- + y' 



in cartesian coordinates (41.xiva) 



The velocity potential of a uniform flow from 

 right to left in diagram 1 of Fig. 41. G is 



4>u = — U^x 



Combining the two algebraically gives, for the 

 resultant flow around the rod, 



(4Lxv) 

 (41.xva) 



(41.xvb) 



K' 



In polar coordinates, the tangential velocity at 

 any radius R, Ue = R dd, is equal to minus the 

 partial derivative of the stream function with 

 respect to R, or d\{/s/dR = —Ug . From Eq. 

 (41.xiu), 



dR 



= - C/„ sin e 



R^ 

 R' 



= (-[/„sin 6») 1 + 



[/„ sin d 



Rl 



R' 



19 



(41.xvia) 



At any point on the surface of the cylinder, 

 R = Ro and the radial velocity Ur = 0. Then the 

 resultant velocity at any point on the surface is 

 the vectorial combination of the tangential and 

 radial velocities, or 



U = 2t/„ sin 



(41.xvib) 



It is interesting to note the tangential velocities 

 encountered abreast the 2-diml cylinder when 

 9 = 90 deg, at a succession of radii R. Substituting 

 R = Ro , 2Ro , 3Ro , and so on in Eq. (41.xvia), 

 the local tangential velocities parallel to the 

 uniform stream are: 



R = R(s 2Ro 3Ro 4i?o 5Ro 



U = 2f7„ 1.25C/CO l.llC/co l.OGU^ 1.04C7„ 



The pressure p at any point on the surface of 

 the cylinder is, by Eqs. (2.xvi) and (41.xvib), 



P = P« + |pf/i(l - 4 sin' 0) (41.xvii) 





^ = 1 - 4sin' 9 (41.xviia) 



IV. For a 2-dimil cylinder around which circula- 

 tion is taking place, depicted in diagram 3 of 

 Fig. 14. E in Volume I and tabulated in (d) 

 preceding, the stream function xpH for the hydro- 

 foil flow is formed by combining the function ^s 

 for streamline flow with the function ipc for 

 circulation. From Sec. 14.3, the circulation 

 strength F (capital gamma) = 27rfc = 2vUeR, 

 where Ue is the tangential velocity at the cylinder 

 surface. Then ^Ac = / UrR dd - f Ug dR. Since 

 the radial velocity Un = Oat the cylinder surface, 



,„=-/l,,.K._i/f 



--^log,S 



(41.xviii) 



Adding the stream function for flow around the 

 2-dunl circular cylinder gives, for the hydrofoil 

 flow, 



^H 



■UJR 



r) 



sin 6 



2x 



logeK (41.xix) 



