H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 68
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Then, according to figure8.25, bubbles (or clusters of bubbles) no longercan remain on the bottom, and the cluster risesto an appropriate a. At this level, the cluster298THE DYNAMIC BEHAVIOR OF LIQUIDS-Water at atmospheric pressureand room temperature--7-Fritz et all. (ref.
8.37) have shown experimentally that tahesame sort of behavior occurswhen the excitation is random if the inputacceleration is large enough. Their experiments demonstrate, in a.ddition, that the largebubble cluster will eventually vent after asufficiently long time; the entire sequence ofevents then begins again.CavitationThe processes of cavitation and bubble f ormation are not well understood at the present time.- Existing theories tend to support the belief thatsmall nucleation sites, such as microscopic gaso 2 54 cm I.D. columnbubbles, must exist before a bubble may formin a fluid. The forces required to form a bubblenucleus solely from rupturing a totally degassed@ 10.16- fluid by low pressure (or high temperature) are* 20.32so large that theories which do not presupposeLine of equal values nucleation sites are generally not applicable.However, in most practical cases, includingcryogenic fuels, there is a sufficient amount ofdissolved gas or vapor to insure the formationof bubbles whenever the local fluid pressure issignificantly below the vapor pressure.No systematic investigation of bubble motionina vertically vibrating, boiling (or nearFIGURE8.27.-Theoretical and experi~nental value8 o fvibrational arr~leration required for Lrrhble ~ t a l ~ i l i z n - hoiling) liquid seems to have been reported,tion (ref.
8.31).although a review of the work that is availablehas been given (ref. 8.32). For the purposes ofthis chapter, however, it is sufficient to realizeis in stable equilibrium so long as + doe5 notthat in a near-boiling fluid, bubbles will alwayschange further.be present whenever the instantaneous pressureQunntitntive predictio~i of the large-scnledrops below the vapor pressure, provided thatcluster motion obviously cannot be obtninedthe bubbles formed in this way grow to awith t l ~ esinlple tlieory o~itliuedabove, t\ltl~ot~gl~sufficient size so that they do not collapse duringBleicli i~~dicntes(ref. 5.34) that t i some\vliutthe succeeding, positive part of the pressuresimiltlr tlieory would probably be npplicttble.cycle.
Once the bubbles are formed it isHowever, tlle sirnple theory is sufficient to preapparent that the vibration of the fluid will havedict the poilit nt which the process begins, andthe same effect as that discussed in the preto give a11 o\.ersll qualitati\-e picttire.ceding sections. In other words, the bubbleAlthougli the preceding discussioli ttppliesmotion is more or less independent of theirstrictly o~rl?-t o sirnplc harmonic excittltion,formation.-REFERENCES8.1. FARADAY,MICHAEL:On the Forms and StatesAssumed by Fluids in Contact With VibratingElastic Surfaces.
Phil. Trans. Roy. Soc.(London), vol. 121, 1831, pp. 319-340.8.2. MATHIES~EN,L.: Akustische Versuche, die Kleinsten Transversalwellen der Fliissigkeiten betreffend. Annalen der Physik, vol. 1341 1868,pp. 107-117.299VERTICAL EXCITATION OF PROPELLANT TANKS8.3. MATHIESSEN,L. : Uber die Transversalschwingungen tonender tropfharer und elastischerFliissigkeiten. Annalen der Physik, vol. 141,1870, pp. 375-393.8.4. LORD RAYLEIGH:On the Crispations of FluidResting Upon a Vibrating Support. Phil.Mag., vol. 16, 1883, pp. 50-58.8.5. LORD R A Y L E I ~ UOn: Maintained Vibrations.Phil. Mag., vol.
15, Apr. 1883, pp. 229-235.8.6. LORDRAYLEIGH:On the Maintenance of Vibrations by Forces of Double Frequency, and onthe Propagation of Waves Through a MediumEndowed With a Periodic Structure. Phil.Mag., vol. 24, Aug. 1887, pp. 145-159.8.7. TAYLOR,SIR G. I.: The Instability of LiquidSurfaces When Accelerated in a DirectionPerpendicular to Their Planes, Part I.
Proc.Roy. SDC.(London), A201, 1950, pp. 192-196.K.S . : The Problem of the Motion of a8.8. MOISEYEV,Rigid Body Containing Liquid Masses WithFree Surfaces. Matem. Sbornik (Math. Review), vol. 32, no. 1, 1953, p. 32.S . X. : Introduction to the Theory of8.9. MOISEYEV,Oscillation of Liquid Containing Bodies. Advances in Applied Mechanics, vol. 8, AcademicPress, 1964.8.10. BOLOTIN,V. V . . On Liqrlid Motion in a VibratingContainer. Prikladnaia Mathematika in Mekhanika, vol.
20, no. 2. 1956, pp. 293-294.8.11. BENJAMIN,T. BROOKE;A N D URSELL,F.: TheStability of a Plane Free Surface of a Liquidin Vertical Periodic Motion. Proc. Roy. Soc.(London), A225, 1954, pp. 505-515.8.12. PENNEY,W. G.; AND PRICE, A. T.: FinitePeriodic Stationary Gravity Waves in a PerfectLiquid, Part 11. Phil.
Trans. Roy. Soc.(London), A244, 1952, pp. 254-284.8.13. TAYLOR,SIR G. I.: An Experimental Study ofStanding Waves. Proc. Roy. Soc. (London),A218, 1954, pp. 44-59.I.; AND KELLER,J. B.: Standing8.14. TADJBAKSH,Surface Waves of Finite Amplitude. J. FluidMech., vol. 8, no. 3, July 1960, pp. 442-451.8.15. FULTZ,D.: An Experimental Note on FiniteAmplitude Standing Gravity Waves. J.
FluidMech., vol. 13, no. 2, June 1962, pp. 19.3-212.8.16. VERMA, G. R.; A N D KELLER, J. B.: ThreeDimensional Standing Surface Waves ofFinite Amphtude. Phys. Fiuias, voi. 5, no. i,Jan. 1962, pp. 52-56.8.17. MACK,L. R. : Periodic, Finite-Amplitude, Axisymmetric Gravity Waves. J. Geophys. Res.,vol. 67, no. 2, Feb. 1962, pp.
829-843.8.18. YARYMOVYCH,M. I.: Forced Large AmplitudeSurface Waves. D. Eng. Sci. thesis, ColumbiaUniversity, Dec. 1959.8.19. SKALAK,R.; AND YARYMOVYCH,M.: ForcedLarge Amplitude Surface Waves. Proceedingsof the Fourth U.S. National Congress ofApplied Mechanics, 1962, pp. 1411-1418.820. DODGE,F. T.; KANA,D. D.; A N D ABRAMSON,H.
N.: Liquid Surface Oscillations in Longitudinally Excited Rigid Cylindrical Containers.AIAA J., vol. 3, no. 4, Apr. 1965, pp. 685-695.8.21. BHUTA,P. G.; A N D YEH, G. C. K.: LiquidSloshing Due to a Time Dependent Discontinuous Boundary. Int.
J. Mech. Sci., vol. 7,July 1965, pp. 475-488.8.22. BLEICH,H. H.: Longitudinal Forced Vibrationsof Cylindrical Fuel Tanks. Jet Propulsion,vol. 26, 1956, pp. 109-111.8.23. BHUTA,P. G.; A N D KOVAL,L. R.: CoupledOscillations of a Liquid With a Free Surfacein a Tank Having a Free Bottom. Zeitschriftfiir angewandte Mathematik und Physik, vol.15, 1964, pp. 466-480.8.24. BHUTA,P. G.; A N D KOVAL,L. R.: HydroelasticSolution of the Sloshing of n Liquid in aCylindrical Tank. J.
Acoust. Soc. Am., vol.36, no. 11, 1964, pp. 2071-2079.8.25. KANA,D. D.: Vertical Oscillation of PartiallyFull Spherical Tanks. Contract NASw-146,Southwest Research Institute, Apr. 1963.8.26. KANA,D. D.: An Experimental Study of LiquidSurface Oscillations in Longitudinally ExcitedCompartmented Cylindrical and SphericalTanks. NASA CR-545, 1966.8.27.
CHU, W. H.: Subharmonic Oscillations in a nArbitrary Axisymmetric Tank Resulting FromAxial Excitations. Tech. Rept. No. 5, Contract No. NAS8-11045, Southwest ResearchInstitute, Sept. 1965.8.28. HUTTON,R. E.: An Investigation of Resonant,Nonlinear, Nonplanar Free-Surface Oscillations of a Fluid. NASA T N D-1870, 1963.8.29. KANA,D. D.: Longitudinal Forced Vibration ofPartially Filled Tanks. Tech.
Rept. No. 6,Contract No. NASw-146, Southwest Research Institute, 1963.8.30. BAIRD,M. H. I.: Resonant Bubbles in a Vertically Vibrating Liquid Column. CanadianJournal of Chemical Engineering, vol. 41, Apr.1963, pp. 52-55.R. H.; JAMESON,G . ; A N D OEDJOE,D.:8.31. BUCHANAN,Cyclic Migration of Bubbles in VerticallyVibrating Liquid Columns. Industrial andEngineering Chemistry Fundamentals, vol. 1,no. 2, 1962, pp. 82-86.o n---n--0.01.
U U U b n ,...m rns. IAL%T)-..:--.-r Dno....-..hCt..rl;a.~ r ~ u u vur r. u r u - e r r~ \ l c v l ~ VAw---nnBubble Motion in Liquids Contained inVertically Vibrating Tanks. Tech. Rept. NO.1, Contract No. NAS8-11045, SouthwestResearch Institute, Dec. 1963.8.33. BLEICH, H. H.: Effect of Vibrations on theMotion of Small Gas Bubbles in a Liquid.Jet Propulsion, vol. 26, 1956, pp. 958-978.8.34.
BLEICH, H. H.: Motions of Clusters of GasBubbles in Vibrated Vessels. Rept. NO.GM-TR-27,Contract No. AF(600)-1190,Ramo-Wooldridge Corp., 1956.300TRE DYNAMIC BEHAVIOR OF LIQUIDS8.35. KANA,D. D.; AND DODGE,I?. T.: Bubble Behavior in Liquids Contained in VerticallyVibrated Tanks. J. Spacecraft Rockets, vol. 3,no. 5, May 1966, pp.
760-763.8.36. MINNAERT,M.: On Musical Air Bubbles and theSounds of Running Water. Phil. Mag., ser.7, vol. 16, no. 104, Aug. 1933, pp. 235-248.8.37. FRITZ,C. G.;PONDER,C. A., JR.; AND BLOUNT,D.,H.: Bubble Coalescence in a LongitudinallyVibrated Liquid Column. Proc. of the ASMESymposium on Cavitation in Fluid Machinery,1965 ASME Winter Annual Meeting, Nov.7-11, 1965.PRINCIPAL NOTATIONSSymbols i n parentheses are the nondimenswnal equivalents of the preceding quantitya= average bubble radius~ ~ ~ ( c ~ ~ ~ ) = ecoefficient~ p a n s ofi ~m,n nth component of velocity potentialb =tank wall thicknessbmn(Bmn)=expansion coefficient of m,nth component of free surface modec=water-hammer wave velocityd =cylindrical tank diameterE= modulus of elasticityg=acceleration due to gravity, or equiralent a~celerat~ionh ( H )=fluid depthJm=mth-order Bessel function of firstkindK=compressibility of gas-liquid-tanksystem!=critical dcpth for sinking bubblesL=rectanyular tank lengthhTu=escitation frequency (N is a positivenumber)p =fluid pressures=ratio of gas volume to total mixturevolumeS, =mth eigenfunction of V29--0t(r) =timexo(c)=excitation amplitude" y1 '}=tank-fixed ed~llrt~esimaxesr, 9,a =crit,ical depth ratio, l/hr=poly tropic constantA = time varying part of bubble radius{=fluid particle displncementsrl(t)=free snrface displacementA,= mth eigenvnluev = surface tension=fluid densityu=frequency parnnletel-, w/w,,@(4)=fluid \-elocsity potenti:ddimensiunles~wavelength, 2wll.l~w,,=nat~lrnl frequency of 7n,nth free surface rilodeQ=bubble pulsatioll niltural frequenc,~'+=..VERTICAL EXCITATION OF PROPELLANT TANRSAPPENDIXThe constants in equation (8.37) areK11=0.122515-kll=0.045199-0.043438Xol 0.010759Xllf 0.09500Xz1- 0.149793X:tanh XolH+tanh hlHXzl tanh XzlH0.045199Xfl 0.149793X:1 0.010759Xftanh holH -k Ll tanh k l ~ tanhhlHkol=0.165118+ 0.171812X11tanh XolH0.310343X: 1k21=0.198686+b1 tanh A.B-0.022291Xlltanh h .
3H is the nondimensional liquid depth; t,hat is, ~ = h &tanh~ ilIh;illis the first root of J : (~I,R)=o;-and terms Xkl(k=O, 1 , 2 ) are defined by Xkl=,Xi1All tanh illhChapter 9Interaction Between Liquid Propellants and the ElasticStructureDaniel D. Kana,!9.1GENERAL DISCUSSION OF COUPLED PROBLEMthe investigations of the influence of theliquidon breathing vibrations of the tankThe previous chapters have dealt primarilywalls.The solutions of these simplified probwith a description of the variety and complexitylemsstillgive valuable information that conof liquid behavior that can be encountered intributestothe description of the overalla rigid container in motion. The tanks andbehavior. As methods and ideas progress, inentire structure of liquid-fueled space vehiclestime, all of the solutions of the individualare, however, by no means rigid; hence, theproblems will be consolidated into the generalcoupling of the various possible liquid responsesdescription of the coupled liquid-elastic strucwith elastic deformations of the tanks andture system.
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