H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 33
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I n such cases it is oftenpossible to express significant deformntionalcharacteristics in terms of overall sectionproperties, as is done above for thin plate, andestablish similitude requirements for the overallstiffness properties. Thus by introducing anacceptable approsimntion, the requirement ofstrict geometric similarity in a small-scalemodel is al-oided. For the present, hen-ever,deflections of only the external thin-wall tankstructure \\-ill be considered.The significant additional parameters forthis model of the tank structure may besummarized as follows:Quantit,yILoad in tank n-all---------hioment in tank wall - - - - - - Local displacement, tankan11. . .
. . . . . . . . . . . . . . . . .Extensionnl stiffness- - - - - - - In-plnne shear stiffness-- - - Transverse shear stiffness- - Bending stiffness- - - - - - - - - - Torsionnl stiffness ---------Poisson's ratio--- - - - - - - - - - Bendlng Poisson-type ratio- Density of tank wall - - - - - - -SymbolIThe corresponding r-terms are denoted witha prime to denote that they pertain t o the abovespecial list.
Using the same repeating variablesas in previous terms:The first three r-terms of the preceding listare essentially response parameters. Holdingthe remaining eight parameters the same inmodel and prototype provides structural similarity insofar as dynamic deflections andfrequencies of the tank wall are concerned. Aspreviously noted, stress similarity is notattained, but is assumed to be unimportantwith respect to fluid-structure interaction. Itdoes not appear possible to simplify greatly theabove parameters for all types of tank-wallconstruction other than to minimize the appearance of liquid density, p, and tank diameter, D lin the eight important structural parameters.One result of such a manipulation is as follows(double primes denote results of the manipulation) :DimensionNhl2AB,,B,DbB5f'p.1,and r;,, r;,,T;, areunchanged:157SIMTJLATION AND EXPERIMENTAL TECHNIQUESThis form of the simulation parameters isvirtually that arrived at in reference 5.9.These simulation parameters are much simplified in the case that the tank wall in both modeland prototype can be represented by an equivalent homogeneous plate.By substituting the homogeneous plateequivalents for A, Bzv, BP, Db, B, and I' intoequations (5.46) and (5.47)vibration to be important with respect to fluidinteraction and, similarly, buckling not to bea problem.The foregoing illustrations perhaps point outthat simulation of fluid structure interac tionrequires some very considerable jud,ament onthe part of the designer of the experiment.
Ingeneral, the replica modeling approach is espensive and suffers from lack of a wide rangeof suitable materials. The success of the"adequate" structural model approach is highlydependent on a prior understanding of theinteraction phenomena being investigated.Dynamic Properties of Fluids: GeneralNeglecting interface phenomena and density,which have been considered from the outset,the common dynamic engineering properties offluids which are of interest may be listed asf 0110ws :For the liquid:SymbolI t can be seen that the assumption of homogeneous plate tank walls has the effect of simplifying the form of the results but not helpingthe simulation problem appreciably.
I n equaltion (5.48), r3,requires that the tank walls bescaled down in accordance with the geometricscale ratio, D,. If T: is held the same, thenit follows that pd/p must be the same in modeland prototype. Additionally, if ri, is the 3amein model and prototype rjl implies that ElpaDmust be the same in model and prototype.Thus, equation (5.48) for homogeneous platetank walls is exactly the same as the specification for replica modeling shown in equations(5.36), (5.37), (5.28), and (5.19). If the prototype tank wall can be replaced theoreticallyby a n equivalent plate of simpler geometry andgreater relative thickness, this approach maybe iiS6fii!.- , ~ .
~ =fc cthe %bsved;Icu!tywith the homogeneous plate case is in ?rj; whichresulted from the plate bending stiffness parameter, D,. If tank-wall bending stiffness can beneglected, considerable simplification of thefabrication may be effected by holding thegrouped parameters pstw/pD,Et,lpaD2, and ';the same in model and prototype. Neglectof bending stiffness is roughly equivalent toconsidering only extensional modes of shellIDynamic viscosity - - - Vapor pressure- - - - - - Bulk modulus--- - - - - -pP.ELDitnensionML-lT-'ML-IT-~.%fL-'T-ZFor the gas:SymbolDynamic viscosity - - - Bulk modulus - - - - - - - -p,E,DimensionML-IT-IML-lT+Five *-terms result when the previously notedrepeating variables are used :AU previous T-terms are assumed to hold, asbefore.0to theThe parameters ?r,, and ~ 4 correspondReynolds number, as can be seen more clearlyby performing a replacement operation :where the kinematic viscosity, pip, is signifiedby v.
This form of 7r3~can be immediatelyrecognized as the Reynolds number as used in158THE DYNAMIC BEHAVIOR OF LIQUIDSfluid flow problems. Sandorff (ref. 5.9) derives a viscosity parameter identical to r,,,while in reference 5.10 the form chosen was1. The characteristic damping of thesloshing of the free surface was correlated witha "damping parameter" equivalent to (r3,)'14by Stephens (ref. 5.14). A similar but numerically different "damping parameter" was usedby Sumner (ref. 5.15) for correlation andextrapolation of first-mode slosh damping in aroughly ellipsoidal tank. In this case, the"damping parameter" used corresponds to(T,,)"~. The material contained in chapter 4points up the importance of the viscosityparameter to the fuel-sloshing problem. Theviscosit,y parameter for the gas, ~ 4 0 is, most oftenneglected, indeed it may only have been mentioned once or twice in the sloshing literature(refs.
5.11 and 5.12).I t seems to be generally doubted that anygas properties will have measurable influenceon normal lateral sloshing problems because ofthe much larger relative density of the liquid.In special problems involving liquid surfaceinstability under sudden reversals of an acceleration field normal to the liquid surface, theproperties of the gas may be important.
Similarly, the properties of the gas may assumegreater importance in the low gravity case.The gas v i s c ~ s i t ,parameter,~s,o, and theelasticity parameter, r 4 2 , will be neglected inthe remainder of the present treatment exceptto comment that A42, the elastic parameter forthe gas, may be quite easily transformed intoMach number utilizing r-terms previouslydetermined.The term T,, in equation (5.49) (the formderived in ref. 5.10) may also be transformedinto a Mach number for the liquid.
(Ananalogous relationship is used in eq. (9.10) ofch. 9.) Replacing r4, as follows, lettingImp=co, - T g - wD(5.51)All-&Cowhich may be seen to be of the form of k inequation (9.10),. . which has to do with cou~led-pressure resonances in an elastic tank (in thatcase some of the similarity terms of the preceding sections are also evident).The last r-term of equation (5.49) to be commented upon is r4,,which is a parameter relatedto cavitation in the liquid.
The parameter Pois analogous to an ultimate strength. If thepressure at a point in the liquid isP=Po+Pd+pah(5.52)whereP=Ambient pressurePo=Pressure a t free surface (ullage pressure)P,=Dynamic pressure in the fluid inducedby sloshingh=Depth from free surface to point inquestion (measured in the directionof a)This pressure must be scaled in accordance withI n order that cavitation occurP-P,<O(5.54)This criterion may be written as the differenceof equation (5.53) andI t is to be noted that each term in equation(5.55) has been previously specified as a similarity parameter. The h/D term is the same inmodel and prototype if geometrical similarity ismaintained, as is the dynamic pressure term (aresponse) if both geometrical similarity and thetime scale implied by equation (5.20) are maintained.
If for some reason the ullage pressuremust be scaled in accordance with PolpaD,then r,, fixes the required vapor pressure of themodel fluid for similarity of cavitation effects.If strict similarity of the ullage pressure may beneglected, the cavitation effects in the fluid maybe considered to be governed byP-P0paD0--APpaDAThis parameter may be transformed with theaid of *,, into the form of the "cavitation number" usually employed in external flow problems159SIMIJLATION AND EXPERILMENTAL TECHNIQUESin high-speed hydrodynamics.
Because muchof the detail of the basic mechanism of cavitation is imperfectly known, the "cavity pressure" (P,, a measurable experimental quantity)is often substituted for the fluid vapor pressureP, in such investigations.Under the assumptions that inertial scalingis required, gas properties and absolute ullagepressure are unimportant, and that the tank isrigid, the three liquid s-terms allow roughsimulation charts to be made.
Transformingragand rhland equation (5.56) to ratio notation,(~(1)'(5.57a)*'g=(pr)2a,(~r)3=u:, =--I ( E L ) r -pra,Drs ; 3 (AP)=d=lp,~lrDr-D y n a m i c P r o p e r t i e s of F l u i d s : V i s c o s i t yEquation (5.58a) summarizes the basic problem in simulation of the viscous dampingeffects discussed in chapter 4. The data onfluid properties presented in the appendix tothis monograph allow a rather crude idea of therangeof kinematic viscosities to be anticipated(5.57b)for the fluids used as propellants in boosterrockets and spacecraft.
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