H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 98
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Satterlee and Reynolds(ref. 11.24) show that when and H are eigenfunctions satisfying equation ( 1 1 9 0 ) thefunctionis stationary with respect to arbitrary variations satisfyingis stationary with respect to arbitrary smallvariations in @ and H satisfyingThe variational problem, equation (11.97),can be attacked directly by the Rayleigh-Ritzmethod. We assume a solution of the formThe functions $('Imust be chosen so as tosatisfy equations (11.97b) through (11.97d).The expression for I then becomes of the formwhere the Ail and Btj are numbers which canbe calculated by integration. The requirementthat I be stationary then produces a chtlracteristic determinant, from which Q2 must beevaluated,10 This may bc derived physically from Hamilton'sprinciplc (ref. 11.24).v%=O inD(1l.lOlb)Now, if we look for eigenf~nct~ionsassociatedwith a particular azimuthal mode, we can use@=xa, cos m9 .
\kCi'(R,2 )Ni-I(11.102)and the required numericd integrationsare purely one-dimensional. The functions\kCf)(R,Z ) cos m9 can be taken as the normalmodes for the high g (flat interface) problem,if these are known in a convenient analyticalform .Satterlee and Reynolds (ref. 11.24) haveapplied this idea in the analysis of the fundamental sloshing mode for a liquid in a cylindricaltank with a flat bottom. Convergence difficulties were encountered with nonwetting contact angles, but the method proved quiteeffective for wetting liquids. Figure 11.33shows the results of these predictions and somemeasurements obtained in a companion experi-417LIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO GLiquid-SymbolWaterMercuryMercury under wateraCarbon-tetrachlorideMethanolWater against paraffin00VaeFIGURE11.33.-Lateralsloshing in a cylinder of infinite depth.ment.
The lines denote the theory; "free"refers t o a calculation with r=0; and "stuck"denotes r= -a, for which the contact lineremains fixed. Note that the effect of meniscuscurvatures becomes increasingly important atlow and negative Bond numbers. For esample, the natural frequency for a highlywetting liquid is more than two times lessthan t h a t predicted earlier for a flat interface,while nonn-etting liquids slosh much morerapidly than wetting liquids a t zero g.Figure 11.34 shows the predictions and experirnents for zero g. The solid line representsthe theory in the range where the convergenceof the variational calculation mas consideredadequate, and the dotted line our present bestestimates, based on the theory. An approximate equation, based on an empirical fit t o thetheory for the fundamental (lateral) mode (freeinterface case), isp$p2/a= tanh (1.84L)[6.26+1.84Bo-4.76 cos 81(11.103)where 0 is t,he contact angle. This approximation will be suitable for most design estimates.4020110Q641,'1.
. .,iIIMercuryO Carbon -Tetrachloride"I'0204080 100 120Contact angle, deg.60140 160 180FIGURE11.34.-~ero-g lateral sloshing in a cylinder ofinfinite depth.418THE DYNAMIC BEHAVIOR OF LIQUIDS(2) Determine the critical Bond number i nthe manner described in the previous section(note that there is a sign change due to thedefinition of g).(3) The Q2-Bo curves seem to be straightlines, so put25-1"5bQ20~ 2 = ~ $ w ~ / ~ = a . + b B o(11.104s)15The information in (1) and (2) abovesuffice to determine the constants a and bb=Q2(11.104b)10a=IBo,,i,llb/-5(11.104~)This should provide a reasonable estimate forthe natural frequency of low-g sloshing.Elastic InteractionsSmith (ref.
11.40) has studied the effect ofelastic interact,ions with the container onsloshing frequencies and meniscus instability-rin a two-dimensional channel where the contactFIGURE11.35.-Effect of contact angle hysteresis on zero-gis 900. He finds that the critical Bondlateral sloshing in a cylinder of infinite depth.number for antisymmetric disturbances is unaffected by the elasticity of the wall, but asurprising result was found when symmetricThe experiments of reference 11.24 showeddisturbances were considered. When the wallthat for some liquid-container combinations theis elastic, one possible mode of vibration hassurface forces are so strong that the interfacebecomes "stuck" along the contact lines.
Thethe meniscus (which is flat in his model)theoretical prediction for Q2, including hystermoving up and down, without changing itsshape. In such a situation, surface tensionesis, is shown in figure 11.35. Note that thiseffect is quite pronounced at zero g. In makingcannot exert a restoring influence, and henceSmit,h predicts that the critical Bond numbera sloshing frequency prediction for zero g, it iswill be zero if the tank is elastic. As yet,consequently quite important to learn whetherthis surprising prediction has not been conthe liquid will move freely up and down thecontainer surface, or whether it is so tenaciousfirmed experimentally, nor is it expected tohold for realistic containers and contact angles.as to become stuck.
Experience to date seemsto suggest that the free-interface model is mostAlthough Smith's work suggests that elasticapproximate for typical propellant systems.effects are very important at low Bond numbers,in practical situations the elastic frequencies areA n Estimating Methodmuch greater than the sloshing frequencies,and hence the interactions are relatively weak.At this vniting, a large body of zero-g sloshingHence it is suggested that the rigid- all modelanalyses are noticeably lacking.
The designer,faced with problems of estimation,.is t,herefore will be entirely adequate for consideration ofthe~ meniscusstability in a realistic system.forced to make educated e ~ t r a ~ ~ lBased~ t i ~ ~.TongandFung(ref. 11.41) have studiedon the results at hand, we suggest the followingtheelasticinteractioneffects in a cylindricalscheme:container,andfindthatthe increased system(1) Determine the high-g sloshing frequencyflexibility tends to reduce the natural frequency,-2 2-w rolgns expected.01 0+11010'lo'lobLIQUID PROPELLANT BEHAVIORI t is interesting t o note that the pertinentelastic p a r m e t e r is a "Bond number" wherethe equivalent membrane-force-per-unit lengthreplaces the surface tension (ref.
11.40).Hence, when the membrane Bond number islarge compared to the surface tension Bondnumber, elastic effects will be important.Reorientation Time EstimatesA problem related closely to low-g sloshingis that of determining the time required for ameniscus t o form its zero-g shape following theremoval of a strong body force. Siegert,Petrash, and Otto (refs. 11.42 and 11.43)have made an experimental study of this problem, correlating their result,s in the formt =KLY/~/J&(11.105)419AT LOW AND ZERO Greasonably well. I n this case, i t is the firstsymmetric mode which is predominantly escited, whereas the first antisymmetric mode isnormally the fundamental sloshing frequency.For a cylindrical tank, Fung finds that thefrequency of the resulting oscillation, as givenby the first symmetric mode, is about 2.5 timesthe fundamental frequency (first antisymmetricmode).Oscillations on a mercury droplet have bccnstudied in a similar manner by Shuleikin (wf.11.47).Petrash and Nelson (ref.
11.45) have studiedcapillary rise rates in tanks of the form offigure 11.l5(a), and find that reasonable prec1ic.tion of these rates can be made using conventional momentum ttnalyses.Oscillation of Drops and Bubbles in Zero gas suggested b y dimensional arguments. Withwetting liquids, the constant K \%-as foundt o have the following values:Cylindrical tank - - - - - - - - - - - - - - - Spherical tank, 50 percent full-- - Annular tanks :r,/ro=%-- - - - - - - - - - - - - -1-- *r./ro=% - - - - - - - - - - - - - - - - - *r,/ro=%- - - - - - - - - - - - - - - - - -0.150.
160.150.100.042Lamb (ref. 11.34) shows that drops andbubbles of fixed size are stable configurationsin zero g. On the basis of an inviscid analysis,the fundamental frequencies of oscillation andtheir corresponding periods are found to be(1) for the drop in an infinite voidO I = ~~ =$ 2, . 2 2 (11.1OBa)~*With D replaced by the gap width in equation (11.105).When the body force field is suddenly removed, the capillary energy is considerably inexcess of its equilibrium value. If there weredamping, when the interface reached its equilibrium configuration the liquid would possessthis extra energy in the form of kinetic energy.Paynter (ref. 11.44) has used this idea toestimate the velocity of the meniscus and hencethe time required to reach the equilibriumconfiguration.
His analysis is perhaps oversimplified, b u t does seem to represent the dataof Petrash, Zappa, and Otto (ref. 11.45) ratherwell.Unless the energy released is quickly damped,the liquid will tend to oscillate about the zero-gequilibrium shape. A linearized analysis ofthis oscillation has been given by Fung (ref.11.46), which seems to check experiments(2) for a bubble in an infinite liquid~ = 2 . 2 2 ~ , m(11.1OBb)Reid (ref. 11.49) studied the effect of viscousdamping on the oscillations of a liquid sphere.He showed thatur(1) if 7>I.$ damped oscillations occurPv(2) ifur<1.7pv2(1 1.107a)the motion is critically damped(11.107b)For water, the radius below which the motionsare critically damped is 0.23 millimeter.
Hence,slowly decaying oscillations are expected iri420TRE D ~ A M I CBEHAVIOR OF LIQUIDSdrops or bubbles of larger size, as are usuallyfound in zero-g systems.11.5FLUID-HANDLING PROBLEMS AT LOW 9Propellant Positioning and Control at Low gDrainingThere is a t present a considerable need forknowledge of and prediction methods for thedraining of liquid from a tank under low g.Saad and Oliver (ref. 11.50) hnve made nlinearized analysis of the sloshing problemduring draining for nearly flat interfaces.
Theirresults indicate a slight reduction in the criticalBond number for draining of an "inverted"tank.A visual study by Xussel, Derdul, andPetrash (ref. 11.51) indicntes a marked dipoccurs in the interface tls the liquid is forcedout by a pressurizing gm. Blowthrough occursif the draining velocity is too great. I t isevident that considerable precaution is necessary in the design of n liquid-renting system forzero-g operation, and that model tests aredefinitely in order in the present absence ofreliable design criterin.
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