HI 



ll^ i)R()i)> \ \\ii( s i\ SI 11 1' ni si(.\ 



StT.H.lO 



ninRiiitiHlf i>i ilir rc.-iilliiiit vclocii y at llic|niiiii 1'. 

 For pxamplr. assume tliat it is roqniriMl to find 

 the volofity niaRiiitinlo and diroction for the 

 point P in Fig. 41.1. at a ship speed equal to 

 r_ . or 34.tV2 ft per sec. The spherical coordinates 

 of P are 9 = 85 deg and 7? = 0.0 ft. Then from 

 Eq. (41.xxvsi'> the radial velocity 



W 



cos e = ^^ - (34 .62) (0.0872) 



= S.SS ft per sec. 



The tangential velocity ( '♦ is, from Eq. (4 1 .x.xxvii), 



U, = 17- sin 6 = 34.02(0.9962) 



= 33.49 ft per sec. 



The re.'^ultant velocity is 1(3:^.49)' + (8.88)']" * = 

 34.7 ft per sec. The value of tan"' (8.88/33.49) is 

 about 14.9 deg. This means that the direction of 

 the resultant velocity makes an angle of 

 (90 - 14.9) = 7.5.1 deg with the radiu.s R to 

 the point P. 



In Fig. 67.H of Sec. 07.7 the 3-diml source 

 used as a means of constructing the ovoid shown 

 there has its 0-valued stream function coinciding 

 with the positive x-axis. If a 3-diml source 

 and sink are both involved, as shown in Fig. 43.J, 

 the stream function of each, and of the combina- 

 tion, has a value of zero at the source-sink axis. 



WTien representing 3-diml sources and sinks 

 mathematically, e.specially when developing the 

 forms of and the flow around axi.s^Ttimetric bodies, 

 it is much more convenient to take as the refer- 

 ence for ^ = a plane perpendicular to the x-axis 

 (or the source-sink axi.s) through the center of 

 the source or sitik. The sources and sinks are 

 axi.sj-mmet ric with respect to the j-axis for either 

 methcKi of representation but in the latter case 

 the characteristics of the flow can be represented 

 by the two spherical coordinates R and B. 



As far as the shape of the streamline pattern 

 and the evaluation of liquid velocities at any 

 point are concenied, it makes no difference where 

 the reference line or plane is chosen. Hut changing 

 the reference of a source or sink changes the 

 stream-function value of a given streamline. 

 Thi.s is the reas<^>n why the stream-function value 

 which represents the surface of the 3-(linil ovoid 

 in Fig. 07.11 is zero, while in the mathematical 

 representation of the same flow, K(|. (41.xxxvb), 

 the Mlreain function at the surface of I he ovoid 

 hft.s a value of —m. 



41.10 The Determination of Liquid Velocity 



Around Any Body. Many of the problems arising 

 from the flow of iicpiid around a IkmIv or ship 

 resolve themselves, directly or indirectly, into the 

 determination of the magnitude and direction of 

 the liquid velocity at any or all point.s, on the 

 surface and in the vicinity. Once the velocity is 

 known, the pressures, forces, moments, and other 

 factors are derived by relatively simple and 

 expeditious methods. Xaval Constructor David 

 W. Taylor, in his paper "On Ship-Shaped Stream 

 Forms," for which he was awarded a gold medal 

 by the Institution of Xaval .Architects in London 

 in 1894, prefaced his remarks i>y the following 

 [p. 38.51: 



"Doubtless tlic day will come when the naval architect, 

 given the lines and speed of a ship, will be able to ralculato 

 the pressure and velocity of the water at everj- point of the 

 immersed surface. That day is not yet, but the present 

 state of our knowledge of the mechanics of fluid motion 

 is such that we can determine completely, under certain 

 conditions, the pressure and velocity in a perfect fluid 

 flowing past botiies whose lines closely resemble tho.se of 

 actual ships." 



Analytic and design problems concerning ships 

 and their parts involve real liquids, whereas most 

 of the mathematical procedures and formulas 

 apply only to motion in ideal lifiuids. Fortunately, 

 some adjustment is possible by expanding the 

 body or ship form so that it includes the displace- 

 ment thickness 5* (delta star) of the boundary 

 layer around it, explained in Sec. 5.15. Potential 

 flow, as in an ideal liquid, is then a.ssumed to 

 exist around this expanded form, in the manner 

 depicted by Figs. 7.1, IS.A, and 18.0. 



The flow net for 2-diml bodies, described in 

 Sec. 2.20 and constructed by graphical, electrical, 

 or other convenient procedure, is one way of 

 finding the velocity. Another method is to .shape 

 the body by a combination of radial and uniform 

 flow, employing sources and sinks. Then by cal- 

 culation or graphic procedures the velocities in 

 the surrounding field are derived. Methods of 

 following the latter procedure arc described in 

 Sees. 41.8 and 41.9. The steps for obtaining the 

 desired data by the second method are described 

 in Chap. 43. Both methods give the velocity 

 throughout the field as well as at the body surface. 



If a velocity jjotential <t> for the field amund 

 any body or ship form is a.ssumed or can be set 

 up, by the methoils outlin<'d in Sees. 41.8 and 

 41.9, or by any other methods, expres-sions for 

 the comi)onent velocities u, v, and «• are derived 

 by partial difTcrentiation of 4> ^^ith respect to 



