H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 31
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A similarity analysis fora practical fluid dynamic or structural problemwill always result in strict geometrical similarity.Thus, several trivial r-terms useful formanipulative purposes can be written as:AnyAnyAnyAnyAnyAnyGeometric variablelength, b- - - - - - - - - - - - - - - - - - - - - - - - - angle, 8 . . . . . . . . . . .
. . . . . . . . . . . . . . .area, A - - - - - - - - - - - - - - - - - - - - - - - - - - volume, V - - - - - - - - - - - - -- - - -- - - - - - - area moment of inertia, A2 - - - - - - - - - - volume moment of inertl, V 2 - - - - - - -z-termblDrz=8za= AIDZr,=VID3rs=A2/IT*a=V2/DSrl=(5.19)These results apply to any length variablespertaining to the configuration of the fluid,and in the strict sense to the thicknesses, andso forth, of structural elements.Deviations from strict geometric similarityto simplify the model must be justified ongrounds other than those of the similarityanalysis.
The only class of fluid dynamicproblems where strict geometric similarity ofthe fluid boundaries is not maintained is thatof the study of flow in rivers, harbors, anddams. There has vet been no justificationadvanced for deviaGons from strict geometricsimilarity in the fuel-sloshing problem, andnone are generally made as far as the fluidboundary conditions are concerned.Geometrical considerations pose no conceptual difficult,^ in any simulation, but oftenpose practical fabrication problems whichmay severely limit the experimental design.All the previously cited authors (refs.
5.9through 5.11) have employed rather strictgeometric similarity for the fluid boundaryconditions in their analyses and usually singleout one or more geometric variables of particularimportance (such as the static liquid depth)which are formally included in the similarityspecifications.Kinematic RequirementsIn general there is some characteristic timeassociated with the sloshing problem. This151SIMIJLATION AND EXPERIMENTAL TECHNIQUEScan be a natural period of some liquid freesurface mode, the duration of an excitationtransient, etc. If a particular characteristictime is denoted by T, and a *-term is formedwith the chosen repeating variablesprototype at the corresponding times definedby equation (5.20).
Similar results to equations(5.23) and (5.24) obtain for angular velocitiesand accelerations :(for any angular velocityThese are the forms derived and used in previous studies (refs. 5.9 through 5.12). If frequency, w, is conceptually more meaningfulThe latter form of equation (5.21) has beenextensively used as the independent frequencyparameter in reporting slosh-force responsedata. I n most dynamic problems, the numberof frequency or time parameters is nearly asgreat as that of the length parameters.
Including these other conceptual times or frequencies results in additional H-terms similarto equations (5.20) and 5.21), and these may bereduced to ratios of times or frequencies, as=*lo=7(st=anyother time parameter)wt/w(wi= any other frequencyparameter)(5.22)A great variety of kinematic variables are ofpotential interest in the fuel-sloshing problem.It can be seen that the magnitude of any otheracceleration, i,impressed upon the tank systemresults inrl1=i/a(5.23)hi^ impressed veiocity considered xi.,results inEquation (5.24) is a general form of the familiarFroude number.
Equations (5.20) through(5.24) are perfectly consistent with timedependent impressed motions which have geometrically similar magnitudes in model andj)(for any angular acceleration b). Basically,the simulation analysis implies a uniform distortion of the time scale in accordance withmaintaining equation (5.20) (or a variant) thesame in model and prototype.
Because thisanalysis presumes that a linear acceleration, a,is of prime importance, this parameter appearsin every kinematic H-term. So long as a bodyforce affects the fluid dynamics of the problem(in the laborat,ory or in the prototype), theacceleration parameter cannot be neglectedand \\-ill materially affect what type of simulation is possible.The analyses of references 5.9 through 5.11are of this type and each contains the sameessential specifications for kinematic similarity.Mass Density RequirementsIn general, the fuel-sloshing problem haselements which are solid, liquid, and gaseous.Each of these elements has a mass density.Consequently, consideration of the mass densityof the gas and the tank structure as additionalparameters of interest results in(where p is the previously chosen liquid density).In words, the ratios of the densities of liquid,soiid, and gas must be the same in the model asin the prototype, though nothing is revealedthus far about the relative magnitudes of massdensity between model and prototype.
Onesloshing simulation study (ref. 5.11) consideredthe gas density, where exactly the above relations were found. For the most part, thedensity of the gas is neglected as being of verysmall magnitude relative to the density of the152TH3 DYNAMIC BEHAVIOR OF LIQUIDSliquid (implicit in refs. 5.9 and 5.10) and thisassumption is apparently borne out by experiment, at least in the normal fuel-sloshing problem. Specification of mass density of the solidsin fixed ratio to the liquid density 1141 satisfythe requirements of rigid body dynamics. Forexample, if the mass density of floating-canslosh-suppression devices (see ch.
4) bears thesame ratio to that of the liquid in model and inprototype, geometrically similar cans \\ill floaton the surface of the liquid at scaled drafts.The mass density of the tank is quite properlyneglected in reference 5.10, since the tank wasassumed rigid and t,he interest mas \vholly inthe fluid dynamics involved, all experimentalresults being independent of or corrected forthe rigid body dynamics of the tank.For reasons which will be apparent in a latersection, SandodT (ref. 5.9) considered the tankwall mass per unit area or, in' other words,justified a departure from strict st'ructurd geometric similarity in combination with considerat,ion of solid mass density.Dependent Variables, Forces, Pressures, ResponseparametersThe most common quantities of interest insloshing simulation have been the net dynamicforces, F, and moments,exerted by theliquid on the tank:m,Similarly, for pressure PWhen fluid elevations, £, velocities, f , and accelerations, i', in response to tank motion areof interest', the result,ing r-terms have exactlyt,he same form as the corresponding kinematicparameters:~ 2=0 €ID(5.32)Actually, the force or pressure rariables mightunder some circumstances be the independentvariables and t,he tank motion the dependentvariables.
The form of the immediately preceding nondimensional parameters is the sameas that of SandorfT (ref. 5.9). In reference 5.10and elsewhere, slosh-force amplit,ude response,lFI, is sometimes presented as(where zo is the tank excitation amplitude).Essentially this is the replacement of s17byr 1 7 / r 1 and implies t,hat the slosh-force responseand t,he tank excitation are related by a lineardifferential equation.Summary of Inertial and Mechanical ScalingThe usual sloshing simulation problem involves measuring one or all of the dependentvariables when the tank is given an acceleration as a function of time.
It can be notedthat some of the parameters derived in thelast four subsections are redundant, and arerelated to one another in quite ordinary mechanical ways. Certain other sets of parameters are of potential use as consistent nondimensional ways of presenting experimental results.Yet other parameters are useful for the manipulation of the form of the remaining parameters.Though such lists of r-terms as have beendeveloped in the preceding paragraphs are notuncommon results of dimensional analysis, therestrictions on experimental design which h a r ebeen imposed thus far in the analysis are veryfew. The most important relation developedbetween model and prototype quantities to thispoint is equation (5.20), which can be mittenin ratio notation asThis result means that if all the importantrariables pertaining to the problem have beenoutlined in the preceding sections, a model ofany geometric scale, D,, can be used in a 1 gacceleration field by varying the time scale,7,.This relation fixes the time scale if thegeometric scale, D,, is det,ermined from otherconsiderations.SIMULATION AND EXPERIMENTAL TECHNIQUESNo restrictions have been placed upon thegeometric scale, D,,or upon the mass densityscale, p,, thus far.
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