H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 87
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As in the preceding paragraph, eachpoint reflects the highest pressure recorded onany of the pressure taps during a test firing.Figure 10.15 shows that, within the limitsdictated by the appreciable data scatter, the372THE DYNAMIC BEHAVIOR OF LIQUIDSis a practical value may be seen by noting, inaccordance with section 10.2, that Z/E is roughlyequivalent to rocket lift to drag ratio, and thatthis value of lift to drag might be achievedwith a rocket angle of attack between 5" a n d2X0.
According to figure 10.2, a11 initial fluidunzlc of 6" or 8" is possible for un tirrglr of attackof?yo.By noting that the pressures for the differinginitia! fluicl ;~11gle:,11r~ the b>Lnie within d a t a~ t ~ t t c t~l i or , tiypotlicsis that imptic% pressurestirt8 riot i~~flileric~edby the magnitudes of eitheri r l i tial Huitl ungle or lateral acceleration mayh~ nppr~uim:ltc.ly true. Perhap.: all that isirnpo~.t:irlt is to htivcb an initial angle itrid 1 islidit l ~ i l t ~ t~c~c*c~l~~~-titiont~lto set the rnodc ofFIGURE 10.15.-Maximum pressure coefiiiicient vers115motion of tht1 f'rctl surface.lateral acceleration ratio.Small relative normal accelerations arid firstmodesloshin: ttrr both to be tlxpected in upressures are not appreciably influcnc,c~clI)y rhclin the atmosphere and, in themagnitude of lateral acceleration U I I t il t l i ~ booster rehic~lt~rthsrncbc.
of btlttcr data, it must probably btllateral accbeleration ratio falls somt>wh~rc>holnwi t ~ d the prtbsbures nlc~tisured in tlie0.008. The transition in pressurcbs b ~ ~ t w t ~ c ~tr i s ~ ~ ~ n that( ~ 1 5of) 0.00s and 0.0 is unfort~~rlntt~lvnot 1 - o v inculincd firings of rcf(~reric~e10.3 are indicativerrrd by the data. T h a t a valuc~oi :C ( i f oof prtlrtic*al pohsibilitiea.Part 11. Liquid Rotation and Vortexing During DrainingFranklin T.Dodge10.5 INTRODUCTORY REMARKS ON ROTATIONALLIQUID MOTIONSOf the various types of liquid motion thatmay occur in rocket fuel tanks, those that involve rotational liquid motions are of specialinterest for several reasons. As one example,unexpected behavior during the flight of Transit2-A has been tentatively explained by postulating that rotational sloshing (see ch.
3) occurredin the fuel tank (unbaffled); the sloshingproduced a certain amount of fluid angularmomentum, which in turn caused a roll torqueof about 3 kilogram-meters to be exerted onthe missile (ref. 10.10). Furthermore, if a considerable amount of liquid angular momentumexists, it is relatively easy for a large vortex toform during draining; this can result in a hollowcore over the drain (see fig. 10.16) and a consequent decrease in the fuel flow rate.10.6 RECENT RESEARCH TRENDSProblems of liquid rotation and vortexing areintrinsically very complex.
Sufficiently accurate and detailed experimental data to giver-1 insight into the actual physical processesare extremely difficult to obtain, and theoreticalanalyses are hindered by this same lack of understanding. Available theories and experirnenta 1results ar&reviewed in this section; however, aswill be seen, they leave considerable room forimprovement.The central subject of this section is the vortex that forms whenever R.
t,nnlr dra_i~?sthrcugh ssmall orifice; however, in order to carry out areasonable discussion of this problem, i t is alsonecessary to discuss liquid rotation, since by itsvery nature a vortex is accompanied by a substantial amount of swirling liquid motion. Inthis context, "rotation" implies only that acertain amount of liquid angular momentumexists about some axis (usually the vortex core) ;FIGURE 10.16.-Vortexformed during steady gravitydraining from a cylindrical tank (ref. 10.11).it does not necessarily mean that the smallestliquid particles rotate abol~t their own axes,which is the more common definition of liquidrotational motions.As anyone who has ever seen a drainingvortex has noted, the centrifugal forces in therotating liquid are sometimes large enough toform a hollow core over the drain.
This causesa decrease in the draining rate for two reasons:the effective area of the drain is decreased bythe air core, and the liquid potential head ispartially converted into rotational velocityinstead of axial (drain) velocity. A physicalpicture of these phenomena can be constructedin a relatively simple way. To start with, afree vortex in an ideal liquid carries along withit a rotational or swirling fluid velocity ofmagnituderVe=%(10.34)373374THE DYNAMIC BEHAVIOR OF LIQUIDSwhere I' is the circulation and r is the distancefrom the vortex.
r measures the vortexstrength; its exact value may be computed b ytaking the line integral of the velocity aroundany curve enclosing the vortex, since the integralis equal to r. I t is clear from equation (10.34)that the vortex flow field possesses a definiteamount of angular momentum (the momentumis theoretically infinitely large if the fieldextends to r = a); however, no physical significance can be attached to this flow fieldnear r=O, since infinite velocities are predictedhere. For this reason, Rankine hypothesizedthat a line vortex must consist of a filament offinite radius, a, and constant vorticity, w,surrounded b y an irrotational vortex field ofthe same circulation, r=soa2, as the peripheryof the filament.' The liquid velocity a t r=a,from equation (10.34), is equal to 7rwa2/27ra= Kua.
Hence, since the velocity varies linearlywith radius in the forced vortex, the flow fieldof Rankine's combined free and forced vortex is-rFIGURE10.17.-Characterietics of the Rankine combinedvortex (ref. 10.12).1Ve=- WT for O l r l a2oa2Vs=- for r 2 a2r.(10.35)The free surface shape can be computed fromEuler's equation by inserting in it the velocityprofiles of equation (10.35). Typical resultsare shown in figure 10.17 for two differentvalues of the filament radius, u.
I t can he seenthat increttsirlg the vorticity, w, or decreasingthe filament size, a, increases the depth of thedepression a t the center; ultimately an air corewill form. (Fig. 10.12 and no st of the precedingdiscussion are taken from an informative articleby Rouse (ref. 10.12). A rather completebibliography is included in this article; morerecent developments.are reported in ref.
10.13.)The preceding explanation of vortexing is byno means complete, since, in 'fact, radial andaxial (drain) velocities also exist; moreover,the size of the core and the assumed vorticity1 The flow field of equation (10.34) is irrotationalexcept a t the origin. This can be seen by computingthe actual particle rotation, which isdistribution within it are only approximations.However, more serious questions' than thesewise. Motions of the type described byequation (10.34) or (10.35) cannot arise in aperfect fluid, which a t some instant was free ofvortices, unless a t least a part of the forcesacting on the liquid itre not conservative; thatis, unless they are not derivable from a scalarpotential.
I n fact, according to Prandtl (ref.10.14), application of the principle of angularmomentum to an inviscid fluid leads to theconclusion that in all cases the moving fluidmust previously have possessed circulation.Consequently, one may ask: How does theangular momentum (or the vortex) come intobeing? One answer is that when the flow fieldexists on a large-enough scale, such as in hurricanes or tornadoes, the radial velocity towardthe center can generate a rotational flow fieldthrough the Coriolis effect of the Earth's rotation. B u t Coriolis forces are not sufficient toexplain the vortices of even sizable drains (ref.10.12), so that other effects are primarilyresponsible here.Dergarabedian (ref.
10.15) has given an approximate theory of vortex formation duringROTATION AND VORTEXING DURING DRAININGtank draining. According to his discussion, thecore formation is a consequence of the unsteadyfree surface boundary conditions. He hasshown that during draining, any small initialswirling velocity increases in magnitude withtime and asymptotically approaches the distribution given by equation (10.34), exceptnear the center r=O. However, his analysis isnot completely satisfactory because it is necessary to assume an initial rotational flow field,even though the velocity of this field may beindefinitely small.
A recent series of experiments (ref. 10.16) have shown that under someconditions the combination of a rectilinearboundary layer flow with a symmetrical sinkflow is unstable in thc sense that amplific*titionsof perturbations of the secondary vorticityassociated with the curved streamline patternlead to the creation of circulation around thesink outlet. These observations suggest thatthe circulation in a "bathtub vortex" might becreated in a similar manner.To circumvent the foregoing difficulties, it iscustomary in both theoretical and experimentalinvestigations of draining vortices to produceinitially a large swirling motion by artificialmeans.
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