H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 22
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(ref. 4.13). Theeffects on damping of the liquid depth, effluxrate, liquid amplitude, kinematic viscosity,and tank size were investigated, and a dampingfactor was employed which is very similar (butnot equivalent) to that given by equation(4.8a).''This equation was obtained independently by others(rcfs. 4.3 and 4.4).111,DAMPING OF LIQUID MOTIQN'S AND LATERAL SLOSHINGData Tank radius,pointcm[ reference]Test liquid,123456715.23.815.27.615.238.138.10.8360.009290.02230.009290.00929SAE 1 0 - W oil [413]Water [4 41Keroseie y 4 ]Water 4 4Water 4 131915250.00929Unpublished Convair IAstronautics dataaFIGURE4.5.-Viscous damping coefficient for liquid in acylindrical tank with a spherical bottom (ref.
4.15).Kinematicviscosity,stokesThe theoretical value of K given by Miles(ref. 4.3) of K = 0 . 5 6 X 2 ~differs from the valuedetermined by these experiments of K=5.23.For large depths (h/R> I ) , equation (4.10a)reduces tog,5.23v1/2R-3/4g-1/4(4.10b)This equation, as well as equation (4.Sb) areplotted in figure 4.6, from which it is seen thateither equations (4.8) or (4.10) may be used toobtain results which contain discrepancies ofno more than the same order as the experimental scatter.Some investigation was also made (ref. 4.13)to det,ermine the effect of efflux rate on dampingof the free surface displacements; no significanteffect was observed for a range of efflux rateswhich caused surface velocities between 9.27cm/sec and 0.266 cmlsec.
I t is of interest tonote that two theories have been presented topredict the relationship between "damping"and tank drainage (refs. 4.1 and 4.5) and havebeen thoroughly discussed in reference 4.25.Except.for a single term, which can generallybe neglected as being small, the two theoriesare equivalent.
I t was concluded that thedamping of the amplitude of the free surfacedisplacements of a liquid during tank drainingis a small positive quantity.Oblate Spheroidal TankA limited experimental investigation has beenconducted (ref. 4.12) to determine the dampingof the fundamental antisymmetric mode ofoscillation of water in an oblat,e spheroid. Thedamping factor was defined in the formSY-1a 01,l l , , . , /0.05 a 100.50 1.00R-'*<"'Damping parameter,I, .5.0FIGURE4.6.-Damphg-viscosity-tank size relationshipsfor liquid in a cylindrical tank (after ref.
4.13).where Mo is the amplitude of a selected initialmoment and M, is the amplitude of the moment after n cycles of fluid oscillation. Thisdamping factor, as a function of liquid depth,is given in figure 4.7. The increase in dampingat the low liquid depths and the increase atlarge depths may be attributed to the shapeof the container with respect to the shape ofthe free surface.Spherical TankThe damping characteristics of liquids inspherical tanks have also been investigatedexperimentally (refs. 4.14 and 4.15).
The effects of kinematic viscosity, tank size, andexcitation amplitude on the damping of thefirst-mode slosh forces were reported in refer-112THE DYNAMIC BEHAVIOR OF LIQUIDSence 4.14. The liquid depth was held constantat h/R= 1.0, since the maximum slosh forcesin a spherical tank occur a t the first naturalfrequency for this depth (ref. 4.26), and anempirical relationship was obtained in the formwhere C3is given as a function of h/R in figure4.9 and B is defined in equation (4.11a). Thefollowing empirical relationships were alsoobtained :whereThe damping factor was defined bvu-here F is slosh force and n indicates thecycle number.
I t is interesting to note thatliquid swirl was never observed for thoseliquids ha\-ing kinematic viscosities greaterthan approximately 0.929 stokes. The experitnental dutn and the faired curve given byequation (4.11u) are shown in figure 4.8.111 an independent study (ref. 4.15), dampingcoefficients were also obtained ILS n functionof liquid kinematic viscosity, the gruvitationalucceleru tion, tank radius, and liquid depth.T h e dti~npiripratio was defined as the lognrithmic decrernrnt of thc amplitt~deof the freesurfnce clisplncements.
leuding t o the empiricalreln t ionshipFor h/R<O.l, equat'ion (4.9) should be employed.I t is of interest to compare the dampingfactors given by equations (4.11) and (4.12) forthe special case of the half-full tank. For thiscase, C3= 1.0 and the relationship between6R and bN is found to beI n general, these damping factors are notequivalent; the discrepancies may be seen intable 4.2 where the damping factors are givenfor water in sphericill tanks of radius 15.25centimeters and 30.5 centimeters (both referencesemployed water to est ablish their equationsnnd both employed small test tanks with radiivarying from nboiit 12.2 centimeters to 40.6centimeters).
This discrepancy may be partlyexplained on the basis that the relationshipbetween the decay of the amplitude of thefree surface displacements and the decny ofthe slosh forces is apparently nonlinear. Aswas discussed in chtlpter 3, nonlinear beba\-ioris typical of liquid dynamics in sphericnl tanks.TABLE4.2.-Comparison of Damping Factors Qiiwnby Equations (4.11~)and (4.11b)--R, cm010JI0.20.4Tank fullness,FIGURE4.7.-Variation0.60.81.0h12bof ciatnping factor with tank fullness for ohlate ~pheroidaltank (ref4.12).6 ~ 1 6 ~(eq. ( 4 . 1 2 ~ ) )DAMPING OF LIQUIDAAcetylene tetrabromide9Water - glycerineAcetylene tetrabromidev0.01FIGURE4.8.-Average113MOTIONS AND LATERAL SLOSHING0. 1Viscosity parameter.1B=(v/m)10*10first-mode damping ratio as a function of viscosity parameter for liquid in a spherical tank(ref.
4.14).vestigated experimentally in reference 4.15.The empirical viscous damping relationshipFIGURE4.9.-Viscous damping coefficient for liquid in aspherical tank (ref. 4.15).I n reference 4.14 the damping coefficient 6Nwas observed to be independent or' the excitation amplitude; in reference 4.15 the dampingcoefficient 6~ was observed to be independentof liquid free surface amplitude up to amplitudes of O.lR measured a t the tank wall.Conical TankDamping of liquid motions in conical cavitiesnarrowing upward and downward was also in-was obt,ained, where C4 and C5 are given infigure 4.10 as functions of the cone semivertexangle, a, for cones narrowing upward anddownward, and r, the radius of the free surface. This equation was found to be valid forh/ro>l.O, and for liquid free surface displacements a t the tank wall less than about O.O1ro.For amplitudes, measured a t the wall, greaterthan O.O1ro and cone tingles c r > l O O , the damping coefficient depends upon the ampiitude oithe free surface displacements.
For example,it was found that in a tank with a semivertexangle of 17O, and an amplitude of O.lr,, thedamping coefficient increases by a factor of 2.Again, damping was defined as the logarithmicdecrement of the amplitude of the free surfacedisplacements.114THE DYNAMIC B E ~ V I O ROF LIQUIDSTank minor radius r = 6.35 cmTest liquid, acetylene tetrabromidep am0-F, 0.20Ega loama06a040.02FIGURE4.1O.-Viscoufi damping coefficients for liquid ina conical tank (ref. 4.15).0a260.4-Liquid depth ratio,I-R(L 81.0h 12 r1Toroidal TankA limited aliiount of data was obtained inreference 4.27 on viscous damping of acetylenetetrnbromide in horizontally oriented toroidaltanks, with typical results showl~in figure 4.1 1.The damping ratio tended: (1) to be independent of the tank's rnnjor radius, and (2) to decrease t o n minimum value nt the liquid depthratio for which the specific slosh forces are ni n a s i n l ~ ~ m The.danlpirlg ratio was defined bywhere j' is the slosh force and n indicates thecycle of liquid oscillation.
Since liquid dynamic behavior in toroidal tanks is extremelycomplicated (see ch. 2), the data shown infigure 4.11 should be applied to other ~ondit~ionsor tank orientations very cautiously.4.3DAMPING BY MOVABLE OR LIQUID SURFACEDEVICESThe necessity of danlping the forces andnloments produced by the sloshing liquid leadsFIGURE4.11.-Effect of liquid depth ratio on the dampingratio for acetylene tetrabromide in a toroidal tank(ref.
4.27).one immediately to the concept of suppressingthe motions of the liquid free surface. I nprinciple, this can be accomplished by sometype of rigid lid or cover which adjusts itselfto the continual reduction in liquid height, orby some type of floating device that can absorba t least a portion of the energy of the movingliquid. Both of these types of devices will bediscussed in the present section of this chapter,while various types of fixed slosh suppressiondevices (baffles) will be discussed in followingsections.Floating Lids or MatsThe floating-lid type of liquid suppressiondevice has been studied by Abramson andRansleben (ref.
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