Sec. 43.6 



DELINEATION OF SOURCE-SINK DIAGRAMS 



59 



diagram representing the ^c2-systena flow between 

 the secondary source and sink is prepared, by 

 the same method. When completed, one-quarter 

 of it appears hke diagram 1 of Fig. 43. F. 



It is possible to use the same basic radial 

 diagrams for constructing tlie circular streamlines 

 associated with the two sets of sources and sinks 

 by assigning to them different radial stream- 

 function values. However, both the ^ci and \i/c2 

 diagrams must be drawn with lines sufficiently 

 heavy to make them visible through each other 

 and through the sheet carrying the ^cz diagram, 

 subsequently to be placed on top of them. When 

 the ^ci and i/'c2 diagrams of the primary and 

 secondary source-sink flows are finished and 

 superposed, with the sources and sinks at the 

 distances Si and Sa , respectively, from the origin, 

 a third flow diagram is constructed by adding 

 the two sets of "circular" stream functions to 

 form a composite diagram for the two pairs, 

 called for convenience the ^C3 stream-function 

 system. It is shown by the curved broken lines 

 of the freehand sketch of Fig. 43. G, radiating 

 from the two sources. 



-64 

 Stream function "^ 



-Sttejm functions Jiijj 



Fig. 43. G Freehand Sketch of Source-Sink 



Streamlines for Two 2-Diml Source-Sink 



Pairs, Form of Body When Inserted in 



Uniform Flow, and Resultant Streamlines 



As a fourth and final step, the i/'c3-system is 

 combined with the uniform flow i^crsystem to 

 produce a stream form yj/s = 0, indicated by the 

 heavy line of Fig. 43. G. This has definitely more 

 pointed ends than the 2-diml body which would 

 be produced by either primary pair alone. 



The exact shape of the ends of a body such as 

 that delineated in Fig. 43. G depends upon the 

 strengths of the secondary source and sink and 

 the relative locations selected for them. A great 

 variety of body shapes can be drawn in a sur- 

 prisingly short time, simply by assigning different 

 stream functions to the radial and parallel flows 

 and changing the source-sink spacing by tacking 



down the inked radial diagrams in different 

 positions along the source-sink axis. 



To make the bow and stern even more pointed 

 a tertiary source-sink system may be added near 

 the extreme ends. The graphic construction of 

 such a form and the streamline pattern around it, 

 involving as it does the six steps listed hereunder, 

 is admittedly tedious but it is readily done for 

 special studies if the results are worth while. 

 These six steps are: 



(1) Flow between primary source and sink, \(/ci 



(2) Flow between secondary source and sink, if/ci 



(3) Flow between tertiary source and sink, \j/c3 



(4) Combination of flows ^d and \pc2 = 4'ct 



(5) Combination of flow \]/c4 with flow xj/cs = 4'cs 



(6) Combination of flow rpcs with uniform flow 

 iu = iPs ■ 



Regardless of the combination of stream-func- 

 tion values used in this operation the ends retain 

 some bluntness for any reasonable number of sets 

 or pairs of point sources and sinks. A solution to 

 this problem, developed by D. W. Taylor, is 

 described in the section to follow. 



It is pointed out here, as W. J. M. Rankine 

 did in his classic treatise of 1866 on "Shipbuilding: 

 Theoretical and Practical," that it is not strictly 

 necessary to limit a ship curve, for analysis or 

 design, to the contour of the Rankuie body or 

 stream form defined by the stream function 

 rps = 0. Parts of the lines of ships may be repre- 

 sented by streamlines which lie at some distance 

 from what is considered as the solid boundary of 

 the stream form, described in Sec. 4.2 on page 72 

 of Volume I. This is in accordance with the 

 principle previously enunciated that any stream 

 surface in an ideal liquid can be replaced by a 

 solid surface without changing the flow pattern 

 on the other side of it. 



43.6 The Construction of Two-Dimensional 

 Stream Forms and Stream Patterns from Line 

 Sources and Sinks. It is possible to construct 

 stream forms with the sharp ends customary in 

 ship waterlines, and with practically any desired 

 shape and degree of fineness or fullness, by an 

 ingenious method devised by D. W. Taylor many 

 years ago ["On Ship-Shaped Stream Forms," 

 INA, 1894, pp. 385-406; "On Sohd Stream 

 Forms, and the Depth of Water Necessary to 

 Avoid Abnormal Resistance of Ships," INA, 1895, 

 pp. 234-247]. This involves the use of what are 

 called line sources and sinks. 



In its simplest form this procedure embodies the 



