H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 30
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(5.3jj cannot be ciimensionally homogeneous, the variable is either notimportant or some additional parameter ismissing from the original list.Since most engineers are accustomed toforming dimensionless numbers by inspectionand to manipulating ratios, a method ofperforming dimensional analyses by inspectionis attractive. Equation (5.11) suggests a148THE DYNAMIC BEHAVIOR OF LIQUIDSsystematic way in which valid sets of *-termsmay be easily formed by inspection.
Afterforming the initial list of rariables, and writingdo~vntheir dinlensional form, a subset of mvariables sutisfying the conditions abore ispicked for the set of repeating rariables. Theproduct of all of these m rariables with eachof the nonrepeating variables is formed, n - mproducts in all, and proper exponents assignedto the repeating variables by inspection sothat the product is dimensionless.5.3APPLICATION OF DIMENSIONAL ANALYSISTO SIMULATIONWhen simulation is the object of the analysis,the conceptual functional relationship betweenthe *-terms is used in the sense that a completefunctionapplies to a system of any size. Consequently,if all the n - m dimensionless *-terms are heldthe same in model and prototype, the simulation is assumed to be correct.
If one or moreof the implied equalities is not met in the simulation, the simulation is "out of scale" withrespect to the r-terms involved. If one ofthe particular rariables is a source of scalingdifficulty, the set of r-terms should probably bemanipulated so that this variable appears inonly one term.
In this respect, the ambiguityof the dimensional analysis is useful, since theexperimental designer can insure that thedependent variable, or a highly controllable orprimary independent rariable, is included inonly the most convenient T-term(s), thusautomatically insuring a consistent nondimensional method of presenting results.It is usually the case that the analyst cannottell a t the outset what final form the s - t e r mshould have. This, like the form of thefunct'ion under investigation, must be revealedeit,her by analysis or by experiment. I n anyevent, the analyst is closer to his goal afterdimensional analysis than before, even thoughthe final and best grouping of variables is tobe determined.It is often the case that simulation cannot beobtained unless the effects of one or morevariables of possible importance are neglected.I n this situation, it is common to conduct, aseries of special experiments wherein theoffending parameter is varied as much asfeasible to assess its importance or to developan extrapolation method.I n many cases in the literature, the *-termsresulting from a dimensional analysis aretransformed into a ratio notation for convenience in experimental design.
Under the modeling assumption in a previous paragraph, each*-term for the prototype may be formallyequated to the corresponding r-term for themodel( * i ) ~ =(*OF(5.12)where the subscripts M and F denote modeland prototype values, respectively. Lettingsubstitution yieldsThe dimensions of the variables a k do notchange between model and prototype, butonly their magnitudes; consequentlj-, the subscripts apply only to the magnitudes of theak and equation (5.13) may be transposed andsimplified toIntroducing the notationsandwe haveSIMULATION A N D EXPERIMENTAL TECHNIQUESSince the form of n; is exactly the same asthat of ni, a set of n - m equations, which mustbe satisfied in the simulation, may be formedfrom the set of n-m *-terms by substitutingfor each variable in the r-term the ratio between its magnitude in the model and itsmagnitude in the prototype, and equating theresult to unity.
I t is evident that these simulation equations may be manipulated in exactlythe same way as the original n-terms; that is,any one of the n-m equations may be raisedto any power or can be replaced by its productwith any one or all of the other simulationequations.5.4SIMULATION ANALYSIS FOR LIQUID SLOSHINGSimulation analyses are implicit or explicit invirtually every experimental report cited in thepresent volume, though there are few referencesdevoted wholly to this subject. Perhaps themost detailed exposition is that of Sandorff(ref. 5.9), who considers the simulation of acompressible, viscous fluid in an elastic tankand gives experimental design examples as wellas some properties of typical model and prototype fluids, and of some typical structuralmaterials.
As has been noted, different analysts will often derive slightly different similarity parameters even when the initial assumptions are the same, and will almost certainlyarrive at different results when the initialassumptions differ. For their purposes, Abramson and Ransleben (ref. 5.10) consider an incompressible, viscid fluid in a rigid tank withsome different criteria resulting than those ofSandorff. Similarly, Epperson, Brown, andAbramson (ref. 5.11) considered a rigid tankcontaining an incompressible viscid fluid andan incompressible gas with the fluid-gas interface having the property of surface iension a d ,as would be expected, arrive a t some stilldifferent criteria.I n simulation analyses for the fuel sloshingproblem, thermodynamic effects, and so forth,are usually omitted from consideration.
Because fuel sloshing is a dynamic mechanicalproblem, the basic dimensions involved areeither mass, length, and time or force, length,149and time, and Newton's second law holds.Consequently the n-theorem says that there willbe three fewer r-terms than variables, whatever the nature of the assumptions regardingthe importance of particular fluid properties,etc., made by the analyst.For present purposes, it is convenient toattempt to unify the treatments of the authorscited above by a parallel analysis. For thesepurposes, the method outlined in section 5.2will be followed as it has the advantage that ifthe repeating variables are selected and consistently used to form *-terms, a valid set isautomatically insured.I n support of what may be criticized as anarbitrary selection of these repeat,ing variables,it may be remarked that the physical systemunder considerat on consists of a solid (thoughpossibly elastic) tank which is filled with twofluids of sufficiently different densities andsufficiently insoluble in each other that a welldefined free surface can exist.
This system issubjected to various accelerations, both rapidlyvarying and slowly varying, which cause fluidmotion and a deformation of the free surface,thus giving rise to forces or pressures on thetank which interest the rocket or spacecraftdesigner, and which in almost all of the sloshingsimulation studies to date have been the dependent variables of the problem. With theexception of very low gravity problems, thepresence of a gravitational field of some sort isthe most important constraint on the configuration of the fluids and plays a vital part inthe results of theoretical fluid dynamics, whichare covered at length elsewhere in this monograph.
Additionally, forces or pressures resulting from these theoretical analyses aredirectly proportional to the mass densities ofthe fluids.I t is not unnatural then to choose as a repeating variable some characteristic length describing the size of the tank, say the diameter,D. One of the slowly varying accelerationcomponents, a, which might correspond to theacceleration of gravity in the laboratory and togravitational plus a constant thrust-inducedacceleration in the prototype may be selectedas a second, and the mass density of the more150THE DYNAMIC BEHAVIOR OF LIQUIDSdense of the two fluids in the tank can beselect,ed as a t,hird repeating variable:Repealing variableSymbol DimensionCharacteristic diameter--- - DLAcceleration (linear) - - - - - _Liquid mass density - - - - - - - -aL T-2pML-3I t can be seen that no nondirnensional ratiocan be formed from these three variables andthat all basic dimensions are represented.The remaining variables of potential interestmay be classified as follows:(1) Geometrical(2) Kinematic requirements, primary independent variables(3) Mass density requirements(4) Dependent variables, forces, pressures,response parameters(5) Properties of structural materials(6) Dynamic properties of fluids(7) Interfacial effectsConsideration of variables in the first fourgroups \\-ill yield dimensionless parameters ofthe type which result naturally in treatmentsof the rigid body and incompressible inviscidhydrodynamic aspects of the fuel-sloshingproblem.
Ellisticity and/or plasticity of thetank is introduced in the fifth group, and theremaining two groups introduce fluid propertiesother than mass density.Geometrical ConsiderationsIf the three repenting variables chosen in thepreceding pnragraph are used to form a r-termwith uny additional variable having only thedimension of length, the exponents assignedmnss density and acceleration must be zero, andthere results the ratio of the new variable withthe characteristic diameter, D. Consequently,if n linear dimensions, bi, in addition to Dl arerequired to fully describe the tankIn ratio notationEquations (5.17) and (5.18) are simply formalrepresentations of the notion of strict geometrical similarity expressly implied in everyengineering drawing.
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