58 



HYDROOVNAMICS IN SHIP DESIGN 



Sec. 0.5 



When the resultant velocity Ub is known at a 

 8olcct«l point, the corresponding Ap is found by 

 the rclationsliips Ap = 0.bp{l'\ — Ul) and 

 ^p/q c= (1 — (i'g/U.Y], described previously in 

 Sec. 41.12. Values of the pressure coefficient cor- 

 responding to various combinations of velocities 

 squared, in English units, for liquids of any mass 

 density p(rho), arc given in Table 41.c of Sec. 

 41.13, on page 27. 



The forward neutral point Nb occurs at a 

 point in the surface where the streamline velocity 

 U,v , represented by the vector OjN in Fig. 43. E, 

 equals in magnitude the uniform velocity — U. , 

 represented by the vector OjH. By striking an 

 arc on 0, as a center, using the radius 0,H = f/. , 

 it is known that the extremity of the vector OjN 

 lies somewhere on the arc HM. It is also known 

 that: 



(1) The streamline-velocity vector OjN is parallel 

 to the tangent NbL at the body surface, at the 

 proper location of the neutral point Nb 



(2) The vector HN is parallel to a tangent to the 

 arc NflK, which represents a portion of the 

 circular-arc source-sink streamline of the 

 ^c-system through Nb , where that arc crosses 

 the body boundary. 



By a process of trial and error a point Nb on 

 the body surface is found where the necessary 

 conditions are met [Rankine, W. J. M., Phil. 

 Trans. Roy. Soc, 1871, Vol. IGl, p. 305, where 

 Rankine calls the neutral point "the point of no 

 disturbance of pressure"]. 



The U.SC of Rankinc's graphic method gives a 

 neat solution for the conditions at the stagnation 

 points Qb and Qs , where it is known that the 

 resultant velocity is directed normal to the body 

 surface. At Qb , for example, the circular-stream- 

 function lines of the ^c-sj'stem are directed 

 ahead, opposite to the uniform-flow line rpa = 0. 

 Therefore, at this point the circular-stream 

 velocity vector Qb03 is equal to the vector U„ in 

 magnitude but is of opposite sign, hence the 

 resultant velocity at Qn is zero. 



At the midsection of the stream form or Ran- 

 kine bfxiy, and abreast it, the grapliic method 

 describo<l in the foregoing breaks down. The 

 cirrular-flow velocity vectors are parallel to the 

 uniform-flow vectors and the intersection cor- 

 n-Hpoiifiing to .\ in Fig. 13. E i.s indetfrminate. 



43.5 Laying Out the Two-Dimensional Flow 

 Pattern Around Two Pairs of Sources and Sinks 

 in a Uniform Stream. The 2-diml Rankine 



stream form resulting from the combination of 

 uniform flow with the "circular" flow between 

 one source-.sink pair is often too blunt to represent 

 the no.se of a body or the bow of a ship, even 

 schematically. This bluntness is modified by the 

 addition of a second source-sink pair on the same 

 axis, placed farther apart than the main pair, and 

 having somewhat less strength (Rankine, W. J. M., 

 INA, 1870, p. 178]. The construction of the body 

 shape and the flow pattern involves four steps as 

 compared to the two of the previous section, but 

 the procedure for each is equallj- straightforward. 

 An example is worked out here to illustrate the 

 method, using a main source-sink pair, each 

 having a total quantity rate of flow of 128 units, 

 and an auxiliary pair, each having a flow of 32 

 units. 



First, a stream-flow diagram representing the 

 "circular" ^ci-systcm flow between the primary 

 source and sink is constructed. When drawn by 

 the method described in Sec. 43.3 and illustrated 

 in Fig. 43. C, one-quarter of it has the appearance 

 of diagram 2 of Fig. 43.F. Second, a stream-flow 



— S2 — H 



Half of source-alnK distance 



V'C2" System 



Half of aourca-»ink di»tsnca M:ircular »tr«A<n 

 h 5, *] funcLOn 



Fin. -13. F-" Soi'iir-K-SiNK SriiKAMi.iNr..s t-t>n Two 



SoUHCE-SlSK PaIIW OK DlF«:RKNT OuTPlT AND 



DiF>xRt.sT .VxiAL Spacing 



