H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 91
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Reynolds and Hugh &I. Satterlee11.I HYDROSTATICS AND HYDRODYNAMICS ATLOW gIntroductionA full range of problems associated with thesloshing motions of liquids have been consideredin previous chapters. The sloshing was presumed to arise as a result of a body force (gravitational or. equivalent accelerational) acting onthe liquid as a whole. When the magnitude ofthis body force becomes very small, other forcescome into lay and these must be consideredin analyses of the sloshing motions. I n thischapter, we shall consider the influence of themost important of these additional forces,capillary forces, on the sloshing motions andassociated phenomena.The term "zero g" is a misnomer. Even inthe most advanced solar system mission onecan contemplate, gravitational forces are neverabsent.
I n fact, the strength of the Earth'sgravitational field 1000 miles from its surfaceis about 64 percent of the ground-level value.When we refer to a low-gravity environmentwe really mean that the statics or dynamicsof a system relative to its traveling vehiclecan be treated as though it were in fact ina :ow-gi-avitj; field. To i!!ustrate this pci2t,consider Newton's law as applied to a particlemoving with acceleration a in some inertialframe:F,,, denotes the forces which the vehicle exertson the particle, and W is the force exerted bythe gravitational field. The acceleration maybe represented as the sum of the vehicle acceleration, a,, plus the particle accelerationrelative to the vehicle, a,,,.
The weight isgiven by W= mgL, where gL is the local acceleration of a free particle in t,he gravitational field.Equat,ion (11.1) then becomeswhich may be writtenUnder the special condition where the vehicleis accelerating a t the exact acceleration that theparticle would experience if i t were free in thegravitational field; that is, when gL=a,, thedynamical equation for the particle relative tothe vehicle would be identical with the equationfor the particle in a truly zero-g environment.This condition is very nearly met in a freelyfalling vehicle, be it falling toward the Earth,around the Earth, or around the Sun.The condition gL=a, is met exactly only ifthere is no external drag on the vehicle, andthen only if the particle is exactly a t t,he centerof gravity of the vehicle.
Orbital systemsexperience drag decelerations of the order ofgo.' Aerodynamic drag is less significantin deeper space, but the existence of a gradientin the gravitational field means that the acceleration of a free particle a t one point in thevehicle differs from that of a free particle elsewhere in the vehicle. Near Earth the gravitygradient is of the order of loM9g,,;olcm.
I twould seem, then, that in the missions of mostimmediate concern, the effective residualgravity will never be much smaller than1o-8 go.The term "zero g" must be interpreted asmeaning t,hat the difference gL-a, is sufficiently small that the effective body forcesgo is here used throughout to denote 980 cm/secz.38 7Preceding page blank Z6388THE DYNAMIC BEHAVIOR OF LIQUIDSseen b y an observer moving in the referenceframe of the vehicle are very small compared toother forces, and consequently do not influencethe behavior of fluid particles. I n this chapter,we are concerned primarily with the relativeimportance of body forces and capillary forces,and with the dynamics of liquid rnotion underthe combined influence of both kinds of forces.Hydrostatic Regimes: The Bond NumberWe can make some sirnple estimates of thecondition under which capillary and body forcesare important.
Consider the rise of a liquid ina tube as shown in figure 11. l . The height, h,of the liquid may be estimated from a forcebalance on the column of liquitl between point.;1 and 2, which givesBssuming thnt the liquid wets the wall completely, a force balance on the meniscus give*(see eq.
(11.12))FIGUREll.l.-Capiliarytt~tionnlforce predorninntes, and the column ofliclr~id \vould barely rise up the tube. Forvalues of Bo1, cnpillary forces predominate,und the liquid mould rise high in the tube. wecan apply this criterion to the case of liquid in apttrtially filled conttiiner. At vely high valuesof the Bond number, gravity dominates, and wewould expect that the interface would be nearlyhorizontal. Conversely, a t very low Bondnumbers the capillary forces predominate, andthe hydrostatic configuration of the containedliquid would involve a strongly curved interface(nleniscus).
This behavior is in fact observed,us urlyone who has ever compared the behaviorof mercury in u marlometer to that in a thermometer can testify.Regimes of hydrosttitic behavior are qualitatively separated by the condition Bo= 1.These regimes are indicated for a number ofliquids in figure 11.2. The straight lines correspond t o constant values of the physicalproperty "kinernatic surface tension," 8= alp,which is the relevant property in capillaryfluid mechanics. Values for typic81 liquids areindicated in figure 11.2.
I n the region abovethe line for the pertinent 8, the body forcesdominate; this is the gruvity-dominated regime,<<Assuming thut P 2 = P,, we combine the eqruttions above, and express our esti~nateof hnondinlensi onally asThe dinlensionless grouping, Bo=pyr2/a, iscalled the Bond n umber;2 it compares the relutive magnitudes of gravitutional 1i11d ctrpillar?forces, and is the pertinent purunleter delineating capillary-dominated and gravity-donlinatedhydrostatics. For values of Bo> 1, the gruvi->2 The dimettsioliless tlrirnber, Bo=pgLzlu, has corlieto be know11 i t 1 the current literilturc nu the Boridnumber (refs.
1 1 . 1 , 11.2, arid 11.3). I~efere~icr11.3 isthe earliest work k11ow11to the authors in which thegroup Ro is speciticiilly called the Burid r~unlherafterW. N. Bo~ld'seniploynie~itof the pnranirter to irldicatethe irnportilnct. of surfncc~ter~siotiin thr corrc.lntion ofthe rise r ~ ~ tofe bubbles ill liquids (ref. 11.4). Thepnrnrneter nppenrs in differcltit forrns perhnpa it1 theliterature at a niuch earlier dnte, however; e.g.,refs.
11.5 arid 11.6.action in a tube.LIQUIDFIGURE11.2.-IIydrostaticiP R O P E L L A ~ T BEHAVIOR AT LOW AND ZERO G389regimes for typical liquids.the sttbjecbt of the previous chapters of this~nonogrr~ph.Below the line, capillary forcesdonlin>tte: h i s is tlie "zero g" regime. I n theregion near the line, both capillary and bodyf o r ~ c sn i i ~ s tbe considered, and this is seen tobe the case for many situntions of practicctlinterest. Tt should he understood that theregime dil-ision is only qualitative, and thatonly for Bond numbers much greater thanunity can one be sure that capillary forces arenegligible.
Similarly, only for very small Bondnumbers is gravity effectively zero. S o t e thatgravity effects can be important in large systems, even though the magnitude of g might bequite small.Some feeling for the Bond number influencecan be obtained from the photographs ofmethyl-alcohol menisci (under air) in Lucitetubes, shown in figure 11.3. There is someoptical distortion, but the Bond number effectshould be evident.Again we see some difficulty in the meaningof "zero g." Henceforth, by zero-g hyclrostaticswe shall mean situations with very low Bondnumbers, such that body forces do not affectthe fluid statics appreciably. In small-enoughsystems, '(zero g"- can be effectively obtaineda t 1 go (as in the mercury thermometer).
Inlarger systems, even lo-' go cannot properly beconsidered as zero g. T o emphasize furtherthe importance of body forces in large boosterBo=4.5FIGURE11.3.-Methyl-alcoholmenisci (under air).390THE DYNAMIC BEHAVIOR OF LIQUIDSsystems, the Bond numbers associated with theliquid in cylindrical tanks approximately 1.5,3, and 9 meters in diameter have been calculatedfor effective body forces arising as a result ofaerodynamic drag, the gravitational field gradient, and rotational motion required for geocentric orientation. The results are shown infigure 11.4, which also portrays the parametersof the calculation. These calculations showclearly the importance of considering both bodyand capillary forces in many liquid-propellanthandling problems, and the danger of oversimplifying an analysis by the idealization ofI(zero g."Hydrodynamic RegimesThe dynamic motion of a liquid-gas systemmay be influenced by capillary forces, bodyforces, and viscous forces.
In many instances,all but one of these forces can be neglected, andthe analysis of the motion or correlation ofexperimental data is thereby considerablysimplified. B number of dimensionless pnrameters can be defined, and these serve to diridethe hydrodynamic behavior into regimes.The Weber numberprovides an estimate of the ratio of "inertialforces" to capillary forces. Here L is a"characteristic length" of the system. ForR t > 1, cltpilli~r~forces influence the dynamicbehtlvior only slightly, while for FVe<<l theypltty tt tlominant role in determining the motion.The Froude number>provides an estimate of the ratio of inertialforce t o body forces. For F r > > l , the bodyforce.. tire stiff iciently weak that, they cannotbe expected to play an important role in thefluid dync~tnics,while for F r < < l they mustcertclinly be considered.
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