142 



U\l)Rt)l)\.\AMlC.,S IN Mlli' ULMGN 



eh 



Sec. -16.8 



Wohe Velocitv 

 of Vortex Troil 

 - 0.14 U„ 



Absolute Velocitv 

 of Vortex Troil 

 h»-0.66U, 



I I llnstont-' 



a -1. 3D "- I '' 



4.0*^ 1 Dro(j* 



3.0 o 



Outside of Vortex 



Moves in Some 



IQ f- Frequency of ^heddint^ Eddies Direction os 5treor 



3.0 4.0 5.0 6.6 



Ua,D 

 5cole of Lo^iQ iJ 



Fig. 46.G Giidance Sketch and EDDV-FREqiEsrv Relation.'? for Vorte.x-Tb.mi. Probi.e.ms 



If the vortex trail is being generated l)y a 

 l)lunt-ended body instead of a 2-diml lireiilar 

 cylinder normal to the flow, the dimen^5ion D 

 corresponds to the effective diameter or width of 

 the trailing edge. Unfortiniately, there is no 

 known rule by which this effective diameter may 

 he determined for a trailing edge of any shape. 



Horause of the basic theorem of hydrodynamics 

 which requires that the circulation around any 

 clo-scd curve within a fluid must remain constant 

 with time (FIIA, 1934, pp. 192-193], the circula- 

 tion r(capital gamma) at any instant about the 

 2-fliml cylinder in Fig. 4().G must be cfjual to, 

 and mu.st be of oppo.site sign to the nd circulation 

 of all the vortexes previously formed [Rouse, H., 

 EH, 1950, p. 129]. At time zero, assume that 

 there is no circulation anywhere in the system 

 and that the uniform-stream velocity is zero. 

 This velocity may then be a.ssumcd to rise from 

 zero to t'„ . At the same time, the vortex circula- 

 tion approaches a steadj' value of dzFy , likewise 

 increasing from zero. This increase in vortex 

 circulation takes place in such manner that the 

 ulgebraic sum of all the vortex circulation from 

 time zero e(|uals ±r,;/2 after the stream velocity 

 has reached a steady value of U„ . iSincc the cir- 

 culation around the cylinder must be c<|ual to 

 and of opposite sign to the net circulation of all 

 the vortexes, the circulation around IIk- cylinder 

 varies from ■\- F, /2 to — F(;/2. 



The approximate magnitude of the lift force 

 per unit span of the 2-4linil body, acting in either 

 direction, i.s, by I0(|. (44. i), /> = pf ,F for the 

 general ciusc. For this parti<ular case, L = 



pU^Tr/'2. The absolute downstream velocity of 

 the vortexes as a group is aliout 0.8Gt/„ . The 

 strength of the vortex circulation F,/ is about 

 2.86(0. 14 i7 J = 2.8(4.3/)) (0. Mr J or about 

 1.7C/„D. 



The graphs in the left-hand diafz;r:un of Fig. 

 46. G give values of the relation.ships fO'/v and 

 JD/U„ in terms of the Strouhal number {D/U„ 

 and the applicable Reynolds number f ' ,D v. The 

 example of Sec. 41.G explains the method of using 

 this graph to predict the eddy frequency for a 

 2-diml rod of diameter D or for a blunt-ended body 

 of effective diameter D. As another example, 

 assume that a cylindrical rod having a diameter 

 of 0.1 ft forms part of an experimental speed log 

 on the ABC ship of Part 4, projecting 2 ft vertic- 

 ally below the bottom of the ship. A.ssumc also 

 that the average velocity U in the boundary 

 layer in way of this rod is 0.951'' or 0.95(7„ . At 

 20.5 kt, f/„ is 34.02 ft per sec, whence 

 U = 0.95(34.02) = 32.89 ft per sec. 



The vortex circulation F,/ is about 1.7(t/)D = 

 1.7(32.89)(0.1) = 5.59 ft' per sec. The cylinder 

 circulation Tuf2 is 2.8 ft' per sec, whence the 

 maximum transverse lift force is pUr,i/'2 or 

 1 .9905(32.89) (2.8) = 183.3 lb per ft length of span. 



TluM/-Rcynolds number is ID 'i> or (32.S9)(0.1) 

 (10'')/1.2817 or about 0.257 million, whence from 

 the left-hand diagram of Fig. 4(t.G the Strouhal 

 number //)'r = about 0.215, from which / = 

 (0.215)(32.89) (0.1) = 71 hertz or 71 cycles per 

 .sec. If the virtual nuuss of the rod, including the 

 ad<ied ma.ss of the entrained wat«r, is such that it 

 vibrates transversely lus a cantilever at a fre«iuency 



