H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 66
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Such sinking bubbleshave been observed in n number of experimental programs, and, in fact, chemical engineers have long been interested in the effect ofvertical vibration on heat and mass transferand the role of bubble dynamics in improvingsuch mixing and transfer (refs. 8.30 and 8.31).Other investigations have centered around theeffects of bubble vibration in cavitation andsound propagation.
In more recent years,determination of the motion of vapor bubblesin vibrating fuel tanks of large missiles hasassumed an added importance, since clusters ofsinking or stationary bubbles could seriouslyaffect thc fuel flow through thc tank and pumping systems (refs. 8.32 through 8.35). Innddition, n ready supply of bubbles is availablehere because of the near-boiling condition ofcryogenic fuels. Nonetheless, one should note\\-it11 care that while such difficulties couldcbccur in actual vehicles, seemingly none hareyct materialized.In ordcr to set the stage for thc follor\-inpanalysis of bubble dynamics in vibrated tanks,the results of a typical series of tests (ref.
S.35)in a simulated rocket fuel tank, using water asthe test fuel, are summarized in the next fewparagraphs. In these experiments, the bubblesmere formed by free surface sloshing, since thewat,er temperature was far too low to formrnpor bubbles; however, the results appear tobe independent of the method of bubble formation as long as the bubbles somehow reachthe interior of the liquid.The entire process begins by exciting thetank a t a relatively high frequency with a totalinput vibrational ncceleration of about 7 g's."At f i s t , thc violent surfnce agitation (shown infig.
8.19) entrains air bubbles a t various depthsl o The resulting bubble motion seems to be practicallyindependent of the actual vibration frequency, and,therefore, thc resulting sequence of events might bccaused by n random input. The particular accelerationlevels given here refer, of course, to small laboratorymodels and may not be npplicablc to actual vehicles...1t : : .. :. :.. . .- . .I;;..::,.;'.) ... . . . . . . ..';a.!,*.-_.*:-k..:.....,;'.~*'4.$:.,:.~,-:.,>,:; .: :: ......:.c..
;.I . . ' ."L. . .:.:. -.-:; : .... . , .:., 1 .', .,!.. . .... : -',:"'. . ."... -.. .. . ... ... ..'I. "..,-, ,,..-+pF~CURE8.19.--Generation of bubble^ by violent aurfacemotion (ref. 8.35).in t,he liquid. These bubbles do not return totjhe liquid surface, but, instead, they streamdownward to the bottom of the tank. Thedeeper the bubbles initially are thrown fromt,he surface, the more easily they sink.As they reach the bottom, the bubbles beginto coalesce into n cluster that continuallygrows as time goes on; this is illustrated infigure S.20. Stroboscopic observation revealsthat the cluster as a whole pulsates a t the samefrequency as the container, and, as the clustergrows, the phase of its vibration begins to lagmore and more behind that of the container.In addition, the amplitude of the clustervibration, and the fluid pressure in the tank,.:..-I..'.-:';*,-..,-.
. .*,-?.:..4,VERTICAL EXCITATION OF PROPELLANT TANKSFIGURE8.20.-Bubble291cluster growing at tank bottom(ref. 8.35).continually increases. Suddenly, the clustermotion becomes extremely violent, as shownin figure 8.21. Bubbles rapidly shoot downward from the intensely agitated free surface.Fluid pressures are very high a t this time.After a relatively short period of violentmotion, the bubble cluster- leaves the tankbottom and rises to a new position in the bodyof the liquid, as shown in figure 8.22. Theexact level at which the cluster then settlesappears to vary with the amount of air originally entrained in the cluster, but as long as thevibrational input is not changed and as longas no more air bubbles are entrained, thecluster remains a t this level indefinitely.
The!in?kid e e e E U s Eat~t&~ l tke,&j undergoi~gvery large antisynlmetric sloshing motions.Increasing the level of vibrational accelerationnow causes the cluster to rise, and lowering itcauses the bubble to sink. If the input acceleration is decreased too much, however, thecluster no longer remains submerged but,instead, rises to tho surface and vents. Thisfinal phase of the cluster nlotion seems to bedependent upon the elasticity of the tankFIGURE8.21.-Bubblecluster during most violent phase(ref. 8.35).because, in another series of experiments witha more rigid tank (ref. 8.31), the clusteralways vented, and the entire process thenbegan anew, even with no change in the input.Motion pictures have been taken whichdepict very vividly all of these aspects of theL..Ll-1.uuuula mo'cions. It shouia be mentioned,again, that the pressures in the fluid caused bythe pulsating cluster are quite large.Theory of Bubble VibrationThe first step in understanding the startlingbubble motions outlined in the precedingsection is to realize that bubble and liquidtogether represent an elastic system thatpossesses a definite natural frequency.
Ap-292THE DYNAMIC BEHAVIOR OF LIQUIDSIThus, the work done in compressing the bubblefrom pressure pl to p isBy assuming t,hat L\/a<<l, t.he work done,that! r m of the bubble,the potturnt to beP.E.=6.rrayplA2(8.47)The kinetic energy of the fluid, when thebubble radius is changing a t a ratre A, isAccording to the 1a.w of conservat,ion ofenergy, the sum of equations (S.47) and (8.48)must be constant; consequentlyThus, the natural frequency of the bubbleliquid combination is-.FIGURE8.22.-Bubblecluster after moat violent phase(ref. 8.35).parently, this wns first noticed by RIinnnert(ref.
S.36) in his study of the origin of the"burblingJ' sounds of running water.Suppose that a bubble with arernge radius,a, is completely imnlersed in an infinite spaceof incompressible liquid. The bubble is assumed to esecute spherically symmetric pulsations, so thnt the bubble rndius is a + A ( t ) , as nfunction of time. As the bubble volumechnnges, the pressure inside the bubble followsthe polytropic law: p,asy=p(a+A)sy, where p,is the fluid pressure in the ricinity of thebubble (surfnce tension forces are neglected).The exact value of 7 in equation (8.50) appears to varv from about 1.4 (adiabatic pulsaiions) to nearly 1.0 (isotherdaldepending on the experimental conditions andthe bubble size.Although equation (S.50) is not sufficient topredict sinking bubbles, it does help explain theformation of the large clusters shown in figure8.20. According to the theory of vibrations, aharmonic oscillator (in this case, the bubble)will vibrate in phase with an exciting motion ift,he exciting frequenc.j- is less than the oscillator's nntural frequency, but otherwise it willoscillate out of phase.
(If there is damping,the change in phase occurs over a band offrequencies instead of instantaneously.) Thus,bubbles smaller than the resonant size will allbe in phase with each other and in phase withVERTICAL EXCITATION OF PROPELLANT TANRSthe exciting motion; the small bubbles shown infigures 8.19 through 8.22 are all smaller than theresonant size for the excitation frequency usedin the reported experiments.
But if two nearbybubbles oscillate in phase with one another, themotion of the fluid around them is such that anattractive force is set up between them, andconsequently they tend to coalesce or orbitabout a common center (ref. 8.32).The mechanical strain of the fluid in thevicinity of a pulsating bubble can be increasedas much as 10 000 times over that caused solelyby the exciting pressure. In support of thisobservation, Kana (ref. 8.29) noted that thewall motion of an elastic tank underwent largeerratic vibrations whenever a bubble wastrapped in the vertically vibrating fluid.Dynamics ofBubble Motion293buoyant force) is larger during the downwardacceleration part of the cycle. If the differenceof these buoyancies is great enough, the staticbuoyancy caused by gravity may be exceeded,and thus the bubble will sink.In order to make numerical calculations, amore explicit formulation of the theory thanis given in the preceding paragraph is needed.For simplicity, the necessary theory is workedout for n rigid tank; in most practical cases,however, the tank is sufficiently elastic a t theforcing frequencies used in experiments thatsome account must be taken of its flexibility;hence, the theory will be modified immediatelyto include tank elasticity in an approxi~natefashion.The most significant part of bubble behaviorobserved in experiments is that part in whichan individual bubble initially begins to sink.Once this occurs, all of the rest of the phenomena described earlier ritt turally follow.Thus, it is this situation for which the theoryis developed, but even here there Inay be otherbubbles prevent in the liquid, ttnd their effecton the fluid compressibility must, be included.To obtain the equations of motion for thebubble-tank-liquid system shown in figure8.23, it is convenient to use Lngrange's method.The cause of stationary and sinking bubblesis a coupling between the bubble's pulsations,which are described above, and its overall motion through the body of the vibrating fluid.This has been shown by Bleich (refs.
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