f.2 



IIYDRODYNAMICS IN SIIIl' DKSK.N 



Src. -f^.S 



pxplainod presently, to he able to delineate 

 aicuratcly tiie potential flow of an ideal liquid 

 around this circular form. This is accomplishetl 

 by combining the circular doublet streamlines of 

 the ^o-system with the parallel streamlines of 

 llie ^^systcm, as described for other graphic 

 constructions in the sections preceding. However, 

 the ^o-system is constructed in quite different 

 fashion. 



Selecting the desired radius R of the circular 

 stream form, the stream functions of the uniform 

 flow arc laid off at intervals of say tenths of the 

 radius R on each side of the a.\is. This is illustrated 

 in Fig. 43.J, drawn for one side only. The inner- 

 most circular streamline of the doublet system, 

 not shown in the figure, has a radius of R/2. Its 

 geometric center, as for all other centers of the 

 system, lies on a line normal to the doublet a.\is 

 at the doublet position 0. If this innermost circle 

 were drawn from the center P it would be tangent 

 to the doublet a.\is at and to the uniform-flow 

 streamhne for iu = —10 at M. Since M is a 

 point on the stream form where \f/s = 0, this 

 circle has a stream-function value ^o = -f-lO. 

 The next circle representing a doublet stream 

 function ^o is tangent to the a.xis at and has a 

 radius (7?/2)(10/9). The remaining circles of 

 the i^c-system have the radii (/?,'2)(10/8), 

 (/<:/2)(10/7), and .so on to (ft/2)(10/l) for the 

 outermost circle, ^o = +1- Expressed in another 

 way, at the top of Fig. 43.J, the radii of the 

 \^/)-systcm maj' be taken as A-/1, fc/2, k/3, . . . k/n, 

 where n has the numerical value of ^,; at the 

 point M, and k = nR/2. 



The algebraic addition of the 4'd and the ^u 

 stream functions produces points in the resultant 

 ^s-isystem. The stream function for ^., = is a 

 circle of radius R. Other stream functions such as 

 ^fl = 1, 2, and so on define the streamlines 

 depicted in Fig. 43..J for the ^.,-systcm, covering 

 as large a field as may be desired for analysis. 



Both by this graphic construction and b^' pure 

 calculation, using the formulas of Sec. 41.8 for 

 flow about a 2-diml rod of circular section, it is 

 possible to determine easily the stream function, 

 the velocity potential, the flow pattern, the pre.s- 

 »ures, the velocities, and other flow characteristics 

 around the circular stream form. Bj' the proccjvs 

 of confornial transformation, mentioned in Sec. 

 41.11, the circular boundary of Xhi' 2-diml doublet 

 Htrcam form is transformed into almost any 

 dceired Hha()e, taking the flow pattern and the 

 variuu.H flow characteristics along with it, so to 



speak, to suit the new contour. Sections of hydro- 

 foils, propeller blades, rudders and control 

 surfaces, even transverse sections and watcrlines 

 of ships, are the end result of this procedure, 

 complete with data for the flow around them. 

 However, because of the discontinuity at tlie 

 trailing edge of a hydrofoil which is definitely 

 sharp, the method breaks down for the region in 

 this vicinity. 



43.8 Graphic Construction of Three-Dimen- 

 sional Stream Forms and Flow Patterns. The 

 manner in which an oval 2-diml body is formed by 

 inserting a 2-diml source-sink pair into a uniform 

 stream, flowing in a direction parallel to the source- 

 sink axis, is described in Sec. 43.3. It is possible, 

 in much the same way, to form an axisymmetric 

 3-diml body by inserting a 3-diml source-sink pair 

 into a uniform stream moving parallel to the 

 source-sink axis. 



For the mathematical formulations of Sec. 41.8 

 and the spherical coordinates employed there, it 

 was convenient to use a zero stream-function 

 reference for the 3-diml source (and sink) in the 

 form of a transverse plane through the source 

 center, normal to the source-sink axis. For the 

 graphic construction, however, it is much simpler 

 and more straightforward to use as a zero reference 

 for both source-and-sink flow and uniform-stream 

 flow the source-sink axis itself. The representation 

 of the stream function \pv for the 3-diml uniform- 

 stream flow is the same in both cases. It is depicted 

 in diagram 2 of Fig. 2.M of Sec. 2.12, on page 31 

 of Volume I, where the 3-diml stream function 

 ^[/ corresponds to — U^y'/'l, with y measured 

 radially from the source-sink axis. 



Using the same zero reference for the 3-diml 

 flow out of the source and into the sink, the 

 radial flow is split up into a cone-and-funnel 

 pattern symmetrical about the source-sink axis 

 rather than into a "pyramidal" pattern uniformly 

 distributed in all directions around the source 

 (or sink) as a center. This means that the inner- 

 most subdivision of the radial 3-diml flow t-akes 

 the form of a cone whose axis is coincident with 

 the source-sink axis, as at B in Fig. 43. K. The 

 radial flow is assumed to pass through the circular- 

 contour spherical base of this cone, notwithstand- 

 ing that it is shown closed at B in the figure. A 

 similar cone lies diametrically opposite the source 

 or sink, also shown at B. 



The next subdivision of the 3-diml radial flow, 

 reckoned outward from the source-sink axis and 

 out of (or into) which a unit (|uiinlity rate of 



