Sec. 43.4 



DELINEATION OF SOURCE-SINK DIAGRAMS 



57 



at the intersectioii_B of the circular stream-func- 

 tion Una ypc = +4 and the uniform-flow stream 

 function Hne ^v = —4, the resultant flow is zero. 

 Therefore B is a point on the boundary of the 

 Rankine body to be formed. There is a second, 

 and corresponding point B for iZ-^ = at the left 

 of the sink. At the intersections of the circles 

 \pc = +8, +12, and 4-16 with the parallel-flow 

 lines ^f/ = —8, —12, and —16, respectively, the 

 pairs of points C, D, and E are found, all on the 

 stream-form boundary. 



A streamline through the lettered points where 

 yps = 0, passing through the stagnation points 

 Q and embracing the lower or mirror image as 

 well as the upper half of the flow diagram, forms 

 an oval-shaped stream form or Rankine body. 

 The flow pattern around this body is constructed 

 by drawing the streamlines for the functions 

 ^s = —2, —4, —6, and so on, using the method 

 described for the single source. The resulting 

 body shape and flow patterns are illustrated by 

 the heavy lines of Fig. 43. D. Points near amidships 

 along the body boundary are determined by 

 drawing intermediate streamlines which are 

 radial, circular, and parallel, or they may be 

 determined by calculation, whichever may be 

 found most convenient. The necessary formulas 

 are listed in Sec. 41.8. 



Flow from the source to the sink also takes 

 place within the boundary of the Rankine body, 

 depicted by the stream-function line ^/ = +2 

 in the figure. This inside liquid can be considered 

 as solidified and the flow neglected. 



The length of the oval-shaped form is partly 

 adjusted but not wholly controlled by the spacing 

 2s between the source and the sink. Its length- 

 beam or fineness ratio is a function of the relative 

 strength of the source-sink flow and the uniform- 

 stream flow. The absolute size of the form is 

 largely a function of the scale upon which the 

 diagram is laid out. 



With relatively simple radial- and uniform-flow 

 diagrams made up beforehand on translucent or 

 transparent material, the delineation of a stream 

 form and its surrounding 2-diml flow pattern is 

 a much shorter and easier task than might appear 

 from the detailed description just given. In fact, 

 with a httle practice, the freehand sketching of 

 the form and the flow pattern is the work of 

 only an hour or two. Figs. 3.0 of Sec. 3.11 in 

 Volume I and 43. G of Sec. 43.5 are examples of 

 this procedure. If the stream-form shape and 

 proportions are not suitable for the work in 



hand, a new form is as quickly sketched, using 

 different relative strengths for the several flows. 

 The shape of the stream form is preserved while 

 changing its velocity relative to the uniform 

 stream by the simple expedient of increasing or 

 decreasing all the velocities and stream functions 

 by the same factor. The oval "ship" thus retains 

 its form while changing its speed through the 

 liquid. 



43.4 Graphic Determination of Velocity Around 

 Two-Dimensional Stream Forms. The magni- 

 tude and direction of the resultant velocity in the 

 streamUne field ^s is determined in a simple 

 graphic manner, originating with W. J. M. 

 Rankine. By this method the velocity vectors of 

 the separate flows are combined to give a resultant 

 velocity vector of the streamline flow. The inter- 

 relation between the composition of stream 

 functions and Uquid velocities for 2-diml flow is 

 explained in Sees. 2.11 and 2.14 and illustrated in 

 Figs. 2.L and 2.Q of Volume I. Fig. 43. E diagrams 



Fig. 43.E Diagram Illustrating Graphic Method 



oj' Determining Resultant Velocities for 



2-DiML Flow 



the graphic method for the example of Fig. 43. D, 

 as applied to several points around the body. 



Take first the body point B in Fig. 43.E. A 

 uniform-flow velocity vector — (7„ is laid off at 

 OiB, parallel to the reference axis. From B a line 

 is drawn toward A, tangent to the circular or 

 source-sink streamline of the ^c system passing 

 through B. The latter is shown by the light circular 

 arc in the diagram. From B a hne BC is drawn, 

 tangent to the body surface at B. From Oj a 

 line OiA is drawn parallel to the body-surface 

 tangent BC, cutting the hne from B which is 

 tangent to the circular-arc streamline. The 

 velocity Ub at the point B is then defined by the 

 vector OiA. For any point on a streamline in the 

 field away from the body, the procedure is exactly 

 the same, considering the streamline as the surface 

 of a new body. 



