51 



IIVDRODVN \MK:.S IN SUIT DI-SIGN 



Srr. 43.3 



now flows across the radial sogincnls BF, CG, 

 DH, and so on. A fair line joining the points 

 B, C, D, and E becomes the streamline or bound- 

 ary where 4^3 = 0. Another line through F, G, H, 

 and corresponding points becomes the streamline 

 where 4's = —4. A line joining J, K, L becomes 

 the streamline yps = —8. 



If the same construction is followed inside the 

 line BODE, tlie radial flow preponderates at the 

 point M, where ^t; = — 4 and ^so = +8, where- 

 upon (^Ao = -fSplusi^y = —4 becomes^/ = -|-4. 

 Similarly, at N, ipso = +12 and fpu = —8, 

 whence rff, = +4. A streamline representing 

 ^/ = 4, for inlernal flow, is then drawn from the 

 source through M, N, P, and a corresponding 

 series of intersections. The four radial broken-line 

 vectors, each representing unitj' quantity rate of 

 Uquiil, originally lying below the line \pso = +4, 

 turn upward and to the left as they pass between 

 tlie boundary BCDE and the inside streamline 

 ^i = 4. 



If all the lii|uid issuing from the source SO and 

 doubling back inside the boundary BCDE 

 freezes into ice, or is otherwise suddenly solidified, 

 the flow outside that boundary remains exactly 

 the same as defined by the streamlines of the 

 ^s-system. 



Bj' following this extremely simple and straight- 

 forward construction in both quadrants above the 

 reference plane, and then adding its mirror image 

 below that plane, the ovoid-sliaped bounding 

 surface ABODE ... is extended to any desired 

 limit downstream. All the streamlines around it 

 can be sketched through the respective series of 

 intersections by the process of adding the stream 

 functions algebraically and joining the points 

 having equal ^s values, as in Fig. 43. B. As few 

 or as many radial and parallel stream function 

 Unes are drawn as may be desired. The more are 

 drawn, the more intersections arc found and the 

 more accurately is the body defined and the result- 

 ing stream pattern delineated. Filling in stream- 

 function lines at intervals of unit quantity rate in 

 Fig. 43. B instead of in increments of four units 

 would give sixteen time^ as many intersections and 

 four tim&s as many streamlines as are shown 

 there. 



The width of the ovoid-shaped boundary 

 opposite the source is equal to the uniform-stream- 

 function position for a value corresponding to the 

 radial-.streatn-function number at 'JO deg from the 

 Hlream direction. Therefore, to make the solid 

 boun<liiry half as wide as in Fig. 43. H, a source 



is chosen with a radial stream function \l/so = +8 

 at 90 deg, in.stoad of ^.^o = -|-1G as shown. 

 Similarly, the uniform-fiow stream function ^j; , 

 if given values twice as great, produces a body 

 only half as wide. It is to be noted particularly, 

 in this and other applications suh-socjuenlly 

 described, that the velocity of the uniform flow 

 can be increased without changing the shape of 

 the form represented by ^.s = if the radial-flow 

 velocitj' is increased in the same ratio. This is 

 equivalent to multiplying or dividing all stream 

 functions in Fig. 43. B by the same factor; the 

 form contour and the various streamlines in the 

 diagram remain the same. 



The solid boundarj' around a single source in 

 a uniform stream in 2-diml flow continues to 

 widen with distance downstream so that it is of 

 limited practical application to ship design. 

 However, Sees. 41.9 and 07. 7 tell how a 3-diml 

 single source in a uniform stream may be used 

 to delineate an underwater bulb shape for the 

 bow of a ship. The solid boundary' around a 

 source and adjacent sink in uniform flow is 

 closed and of oval shape, resembling some 

 parts of a ship. Because of the ea.se with which 

 2-diml streamlines are constructed around it, tliis 

 bodj' serves well as a simple ship form, by which 

 to explain and depict many kinds of flow phe- 

 nomena. 



43.3 Graphic Construction of a Two-Dimen- 

 sional Flow Pattern Around a Source and a Sink. 

 The construction of a solid boundary surface and 

 a streamline diagram for a 2-diml source and a 

 2-diml sink placed in a uniform stream involves 

 one more step than the procedure described for 

 a single source. It is convenient, as before, to 

 take up each of the steps separately. They are 

 simple graphic operations which, when once 

 learned, are easily and rapidly performed. 



Let the horizontal line from — oo to -|- <» in 

 Fig. 43. C, pa.ssing through the source and the 

 sink and coinciding with the x-axis, represent the 

 trace of the reference plane for the 2-iliml uniform- 

 stream flow. The y-ax'is is midway between the 

 source and the sink. Using separate sheets of 

 tracing cloth, or other suitable transparent 

 material, lay down orthogonal axes in the center 

 of each sheet. In each of the four ((uadrants draw 

 an equal number of uniformly sijaceil radial lines, 

 representing equidifl'erent radial stream functions. 

 Series of numbers difi"ering by 2 or 4 are used 

 generally in the present ch.'ipter but any series is 

 licrmissible. 



