Sec. 42.6 



POTENTIAL-FLOW PATTERNS 



41 



elocity in all tubes 



Fig. 42.A Schematic Diagram of Uniform Three- 



DlMENSIONAL FlOW AeOUND A STRAIGHT CIRCULAR 



Rod With Equidifferent Stream Functions 



velocity is ?/„ . This situation, pictured in Fig. 

 42.A, is derived immediately from the 3-diml 

 stream-function lA(psi) in diagram 2 of Fig. 2.M 

 by freezing the hquid within the central rod, an 

 operation which does not change the flow pattern 

 around it. The immovable quantity of frozen 

 liquid, extending out to radius Rq , is then 

 subtracted from the whole. It is convenient in 

 this process to retain the rod axis as the reference 

 axis for measuring the 3-diml stream functions. 



If the thickness of the first, second, or any other 

 tube of hquid is represented by An, and its mean 

 radius from the axis by R, then the increment of 

 hquid volume AF passing tlurough it in the 

 time At is represented approximately by 



A¥ = -U„{2ivR)An (42. i) 



where the velocity [/„ is constant throughout all 

 the tubes. If Ra is the radius to any selected 

 cyhndrical stream surface, the 3-diml stream 

 function of that surface is 



= \~){-u:)-k{ri-r^:) 



(42. ii) 



{Ra — Ro) 



At the soUd cylindrical surface of radius Ro the 

 3-diml stream function is zero. 



The subdivision of the uniform liquid flow into 

 stream tubes of equal area means that the normal 

 spacing between the tube-wall traces in any 

 longitudinal plane through the axis is no longer a 

 direct measure of the velocity in each tube, or 

 even of the relative velocity, as it was in the 

 2-diml case. For equidifferent values of the 

 volume increment per unit time or of the stream 

 function, the tube thickness diminishes inversely 



in proportion to the increase in its mean radius, 

 to maintain the relationship of Eq. (42.i). 



When the flow takes place around some body 

 of revolution which has a varied section along the 

 axis of flow, such as the sphere in Fig. 42.B, the 



Varying liquid velocity 

 at different rad 



Fig. 42.B Longitudinal Section Through Three- 



DlMBNSIONAL FlOW ArOUND A SPHERE, WiTH 



Equidifferent Stream Functions 



velocity in each stream tube is no longer constant 

 but changes with distance along the x-axis. In 

 fact, there are three variables in the stream- 

 function expression which vary with x, namely R, 

 An, and U, where R is measured normal to the 

 axis of the uniform stream flow. It then becomes 

 necessary to make use of some rather involved 

 procedures to dehneate the stream surfaces. For 

 two 3-diml bodies of revolution, including the 

 sphere, the stream functions in spherical coordi- 

 nates are worked out in Sec. 41.9. 



A 3-diml stream form, especially one derived 

 from a single source-sink pair, lends itself to 

 graphic construction of the flow pattern around it, 

 as well as to calculation of the elements of this 

 pattern. When so constructed, the radii -Ri , R2 , ■ ■ ■ 

 and the stream-tube thicknesses An^ , AUi , . . . are 

 measured, the local velocities U are determined, 

 and the accompanying pressures p derived from 

 Eq. (2.xvia). The graphic construction is de- 

 scribed in Sec. 43.8 and illustrated in Figs. 43.L, 

 43.M, 43.N, and 43.0. The formulas for a 3-diml 

 sphere are set down in Sec. 41.9 and in diagram 2 

 of Fig. 41. G. ■ 



Fortunately for the physicist, marine architect, 

 and others, certain staff members of the Aero- 

 dynamische Versuchsanstalt (AVA) at Gottmgen, 

 Germany, have worked out the pressure distri- 

 butions over 12 forebodies and 59 fuU bodies of 

 revolution, of a great variety of shapes and pro- 

 portions. These bodies were created by placing 

 various combinations of point and hne sources 

 and sinks along an axis, and then superposing 

 upon this combination a uniform flow of ideal 



