IIVDRODNX VMICs 1\ Mill' 1)1 S1(;N 



, 10.: 



40.7 Double or Multiple Solutions to the 

 Equations of Motion. It is pos.-jiMc for i\ .ship 

 to travel from one port to luiothor l)j' two or 

 more difTerent routes, each of which may lie the 

 easiest and Kest under its own particular comiiina- 

 tion of circumstances. It is found that nature, in 

 her role as a canscr\er of enerRj-, cau.ses water 

 and other liquids to flow b}- difTerent paths from 

 one point to another whenever there is a good 

 reason for doing so. Speaking in terms of mathe- 

 matics this means that there may he two or more 

 solutions to the ecjuations of motion which govern 

 the action of the liquids or the bodies in (juestion, 

 just as there are two solutions to the ordinary 

 quadratic equations in algebra. 



Nature does not select these solutions at 

 random but chooses them in strict accordance 

 witii some secondary' or lesser cause, which maj' 

 appear only after the most careful examination 

 or extended study. A case in point, although 

 admittedly not pertaining to liquid flow, is the 

 multiple paths taken by high-voltage discharges 

 and by lightning in darting from one fixed point 

 to another. A shifting flow pattern frequently 

 encountered is that which takes place around 

 and behind a cylindrical stick which is drawn 

 rapidly through the water at right angles to its 

 axis. Fig. 40.A depicts the manner in which a 

 double row of vortexes known as the B(5nard or 

 KirmAn vortex trail or vortex street is formed in 

 the wake of a 2-diml rod, when drawn through 

 a liquid in a direction normal to its axis. Here 

 large vortexes roll up, first behind one quarter 

 and then behind the other quarter of the rod as 

 it moves along. 



The lowest diagram in the figure shows the 

 manner in which a jet of water i.ssuing from a 

 nozzle with a flared exit cone clings to one side 

 or the other of the cone, as the nozzle is flicked 

 quickly from side to side. 



IjOss familiar cases are those encountered by 

 mf)dcl Ijasins in the transition region between 

 laminar and turbulent flow, where the nature of 

 the flow may and does change with position and 

 time. Similar situations often occur when investi- 

 gating the flow around ship models. The water 

 passing through a given region on the model at 

 timcfl follows one |)ath anrl then, ai)parently 

 without reason, suddeiily changes to another 

 path, finly to return to the first as unexpectedly 

 BH it cleparl<-d from it. This phenomenon has often 

 been l(Mike<l upon as a matter of instability in the 

 flow but it can ju»t aa well be regarded ob a uhifling 



Double lightning 



paths to qroun* 



Position 1 



Time A 



■^f- 



Flow (^ ^ 



pattern 

 changes to 

 form large 

 vortexes on 

 alternate sides 



---«-~-^ '^////////yy,,.. 



Ce&el 



^T7^7777777777777777777777 

 separation mey occur 

 ^- on alternote sides 



Fin. 40.A Sevbrai. Exampi.ks Taken from Xaturb 



iM-f.STRATlNG DoCBLE SOLUTIONS OK THE Egt'ATIONS 



OF Flow 



In the top diagriim clectririty flows through the two 

 paths simultaneously. In the middle iiiid bottom diagrnms 

 the schematic flow patterns change with time or other 

 circumstances; these illustrations are intended only to 

 indicate that the flow patterns can and do change while the 

 surroiMulings remain the same. 



flow condition causcil li,\' two suiulions to the 

 equations of motion. 



As a consequence of this situation it may be 

 expected that any of the real solutions to a set 

 of equations of motion may be encountere<l in 

 actual practice. Considerable study may be 

 nece.s.sarj' to determine which of these solutions 

 will apply under conditions which arc apparently 

 identical in every respect. There is on record the 

 case of the fast man-of-war which, at a given 

 speed, rudder setting, and trim, would !ip|)iireiitly 

 make its own on-the-spot decision as to whether 

 it Would turn in ;i loose circle or a tight ipmc 



