H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 65
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That is, the line @ = 0 iil. ;,:.>-,equation (8.30) is dependent upon slightimperfections in the symmetry of ally actualsystem; thus, it is not prescribed by theidealized theory given in the preceding sections.Instead, it must be determined experimentallyfor each particular case.I t should be emphasized that the modelshown in figure 8.17 will not give a goodanalog for synlmetric sloshing, although thependulum parameters could probably be adjusted to duplicate the stability properties ofthe liquid free surface for this type of sloshing.The problem is that the equivalent sloshingmass, ml, inherently gives forces of the sameform as antisymmetrical sloshing. Some sortof pendulum which vibrates up and down.rather than to and fro, woiild be needed toduplicate symmetrical sloshing forces, but itis not clear how this could be arranged andstill retain the stability properties of the freesurface.The problems associated with equivalentmodels for subharmonic vertical sloshing canbe summarized, then, as: What kind of pendulum or spring mass has the same sort of stability288THE DYNAMIC BERAMORas the liquid free surface, and, at the same time,is able to duplicate the sloshing forces? Atthe present time this question has not beenanswered satisfactorily.
Furthermore, the nextfew sections show that for higher excitationfrequencies, the free surface responds in waysthat are entirely different from the responsesconsidered up to now. Other kinds ofmechanical models would be needed to simulatethese responses.8.3LIQUID SURFACE RESPONSE TO HIGH-FREQUENCY EXCITATIONWhile performing experiments a t relativelyhigh excitation frequencies, Yarymovych (ref.8.18) observed large amplitude surface waveswhose frequencies were of the order 1/25 to 1/50of the excitation; a typical wave is shown infigure 8.2. (Similar observations have beenmade by Kana in tanks of various geometries(refs. 8.25, 8.26, and 8.29).) Such low subharmonic responses cannot be explained on thebasis of any of tho large amplitude wave theoriesdiscussed in previous sections, and so Yarymovych advanced the hypothesis that thesewaves are generated by a complex interactionof the free surface and the spray droplets formedby the high-frequency free surface motion.
I tis also apparent that for such high-frequencyexcitation, tank flexibility may play an important role.The free surface waves that first form whenthe excitation is of high frequency and lowamplitude are short-wavelength capillary waves.Even though surface tension has a dominantinfluence on these waves, their frequency is stillexactly one-half that of the excitation, as discussed previously. The amplitudes of thesecapillary wares build up until the waves becomeunstable, and then they disintegrate by formingdroplets that separate from the surface; withincreasing escitation amplitude, the dropletsdescribe higher trajectories. A dense spray isthus created, since each of the multitude of theoriginal waves releases a droplet. For certainexcitation conditions, a first, second, or higherorder low-frequency wave is formed, with theformation usually requiring a considerableperiod of time.OF LIQUIDSNo satisfactory theory for this sort of surfacemotion exists at the present time.
An exactquantitative analysis of the origin and perpetuation of the low-frequency waves is a formidabletask because of the statistical nature of theresponse; that is, there are many different sprayparticles thrown off a t different instants andwith different masses and different initialvelocities. Also, the transition from high-frequency, short-wavelength ripples to low-frequency, long-wavelength sloshing is probablycaused by some kind of low-order instability, asevidenced by the relatively long period neededto accompli~hthe transition.
Such instabilitiesare generally difficult to predict analytically.However, the steady-state motion which eventually occurs should be easier to analyze if oneaccepts Yarymovych's hypothesis that the impacts of the spray droplets on the free surfaceare the primary agents in sustaining the lowfrequency sloshing.It has been observed that the portion of thewave near its peak produces a higher anddenser spray than the depressed positions.There is also a periodic spray-density distribution with respect to the time of the cycle whenthe droplets are released.
In order to producesuch forcing as to help maintain an existingwave, rather than retarding it, the dropletsmust return to the wave surface when it ismoving downward. Only droplets having certain trajectory times and released during certainintervals will produce positive forcing.A crude attempt to translate the foregoingobservations into a quantitative theory isgiven in the following paragraphs. I11 the firstplace, the spray action is much more importantover certain critical areas of the surface than itis over the rest.
(See fig. 8.18.) If the wavecrests are excited properly, the rest of thesurface will naturally follow wit11 tlie properfrequency and wave shnpe. In the secondplace, the accelerations, and the velocities, ofevery part of these critical portiorls of the freesurface are nearly equd at every instant; thus,the spray droplets in the critical areas arerelensed a t practically the same time and withpractically the same initial velocities. If thetrajectory time of these droplets is about equalto one-half the period of the low-frequencyIIIi1..- *1:.If.'-.tc;1-'-*289VERTICAL EXCITATION OF PROPELLANT TANKS1 g downward, which, because xowZis large, isjust after cos ot passes through zero and thenincreases.
Because cos (Mwt 9,) and cos(w,t+ 9,) both vary much more slowly thancos wt, the phase angles may be neglected, sothatsin wt=sin %wt=sin w,t=l+at the critical time when the drops are released.This is tantamount to assuming that the dropsare released at the instant when the total\relocity of the surface is a maximum. Thus,the initial velocity of the drops released nearthe wave crest isVd= xoo+%amw+anw,FrCURE8.18.-"criticalarea" for symmetrical surfacewave.motion, the droplets will impact when the wavecrest is moving downward, and the low-frequency motion will be reinforced.In analytical terms, the total displacement ofthe free surface is approximatelyS,(x, y) is the normalized mode shape (eigenfunction), with m a large integer, and a, thecorresponding amplitude of the higher frequencyripples (the frequency of these capillary wavesis one-half that of the excitation) ; S,(x, y) is themode shape of the low-frequency wave (n asmall integer), a, its amplitude, and w, itsnatural frequency; 9, and 4, are phase angles;and z, cos wt is the forcing motion.
Now, a,may be determined from known results whichaixre- thee-.--- nmnlit.iirl~---=------ of R_ ~~plll-y wave jilst as itbecomes unstable and begins to disintegrate(refs. 8.12 and 8.17); however, a, cannot, ingeneral, be calculated using the crude theoryoutlined here, except for the case of themaximum amplitude wave when a, is alsodeterminate.The spray droplets are released wheneverthe surface acceleration is a little greater than'In terms of the drop velocity, the trajectorytime is 2Vd/g. Since for positive forcing thistime must be approximately nlw,, the finalresult isThis equation includes, in u general way, mostof the phenomena observed in experiments.Analysis of certain preliminary data indicatesthat a satisfactory correlation for the firstsymmetrical mode in a circular cylindricaltank may be obtained fromwhere d is the tank diameter.
By tl liberalinterpretation, it can be seen that this empiricalequation and the previous theoretical equationare approximately of the same form. Consequently, although there are obvious differences between the two equations, it appearsthat the theoretical approach taken above isprobably fundamentally sound.While it is obvious that more experimentaland theoretical development work need to bedone in this area, especially as concerns theeffect of tank elasticity on the liquid motions,the significance of the spray-formed waves inactual applications remains to be determined.THE DYNAMIC BEHAVIOR OF LIQUIDS8.4 BUBBLE BEHAVIOR AND CAVITATIONBubble Motion in Vertically Vibrated LiquidsThe possible generation of bubbles with anapparent negative buoyancy was mentionedbriefly in section 8.1.
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