H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 5
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For liquid depth todiameter hld>l, the liquid n a t u . frequenciesmay be considered to be independent of theliquid depth, as will be shown later. For containers of symmetric cross section, the frequencies of primary interest are the lowest few corresponding to the first few antisymmetric modeshapes. - I t is intuitively clear that the symmetric liquid mode shapes do not cause anyresultant lateral forces and moments to acton the container; they can, however, couplewith the elastic container and lead to other interesting aspects of the overall problem. (Thisgeneral subject will be discussed in detail inch. 8.) In the case of a circular cylindrical tank,the orthogonality condition for trigonometricfunctions insures that antisymmetric motion isthe only class of motion which contributes tothe net horizontal force and moment exertedby the liquid on the container (refs.
2.1 and 2.2).Several methods have been employed tominimize the effects of liquid motion. Bafflesof various configurations have been devisedwhich add a small amount of damping to thesystem and mainly affect the magnitude of theslosh forces and the amplitude of the liquidmotion. (See ch. 4.) If a shift in the range ofthe fluid natural frequencies is desired, a moreeffective method is to divide the tank intosubtanks by means of radial or concentric walls,which has led to interest in the ring sector tankand its variations (sec.
2.3). Bauer (refs. 2.3to 2.5) has shown theoretically that compartmentation of a cylindrical tank into sectors hasthe effect of raising the first natural frequencyanci iowering the seconci so that these i w o $9quencies are not widely separated, as has beenc o h d experimentally (refs. 2.6 to 2.8).These experiments have also indicated that:(a) radial compartmentation with solid wallsdoes not reduce the slosh forces as much as doordinary ring baffles, and (b) there is a marked13Preceding page blank-14THE DYNAMIC BEHAVIOR OF LIQUIDSdependence of liquid natural frequency onexcitation amplitude. (Such nonlinear effectswill be discussed in more detail in ch. 3.)The cylindrical tank with an annular crosssection has also been investigated theoreticallyin reference 2.2, but experimental verification ofthis theory is not yet available.
The attraceiveness of this tank configuration resides in thewide separation between frequencies of the innerand outer compartments and in the fact thatphasing of the liquid motions between the twocompartments results in a lower total force.Eulitz (refs. 2.9 and 2.10) has suggested thatsloshing be avoided by using an experimentallybased analytical method. Virtually countlessother schemes for the suppression of sloshinghave been proposed a t one time or another, butwe shall not recount them here.In general, very good agreement betweentheory and experimental observations has beenobtained for lateral sloshing.
Wall-pressuredistrib~t~ionshave been obtained from sloshingexperiments with rigid model cylindrical tankshaving flat bottoms (ref. 2.11), and it has beenfound that the total force and moment obtainedby integration of the pressure distributions arein good agreement with theory. Sloshingbehavior in tanks with conical bottoms can berepresented by sloshing behavior in a tankwith an equivalent flat bottom (based on equalliquid volumes); this conclusion appears to bevalid a t least through the second mode andpossibly through the third, and implies that asimiiar technique may be applied (with caution)to tanks of other bottom geometries (ref.
2.12).A few general theoretical analyses for predicting natural frequencies of fluids in containeraof various shapes are available (refs. 2.13 and2.14). Lamb (ref. 2.13) discusses the ellipticcanal in section 291, the parabolic container insection 193 (tidal wave theory), and the 45"conical canal in section 258, Troesch (ref.2.14) discusses the conical tank and thehyperboloid, and also presents graphs of container cross sections which were obtained usingan inverse technique.
I t is interesting to notethat Troesch found that the fundamentalfrequency for a container which looks verymuch like a cone differs from that of a cone by25 percent, indicating that the eigenvalue canbe quite sensitive to changes in containershapes. Somewhat different results are indicated in reference 2.15 for a cylindrical container with elliptical cross section, in which itwas determined that the effect of small tankellipticity on the natural frequencies is small.It is seen from this discussion that it is di6cuItto draw general conclusions about the behaviorof liquids in containers.
Further references oncontainers of various shapes are given inreferences 2.1 and 2.16.Boric Theory: AssumptionsAn "exact" solution to the general problemof fluid oscillations in a moving container isextremely dScult. The simplifying assumptions that are generally employed in such atheoretical analysis and which are employed inthis chapter are as follows (some of these willbe relaxed in following chapters):,Rigid tankNonviecoua fluidInoompmaaible fluidSmall dispbcemente,velocities, and slope6of the liquid-free surfaoeIrrotational flow fieldHomogeneous fluidNo ainke or source6Since the question of interaction between theliquid and the elastic tank is an exceedinglycomplex one, we shall assume for present purposes that the tank is rigid.
(Interaction problems will be discussed in ch. 9.) For unbaffledtanks of relatively large proportions, the viscousforces of the fuel may be neglected as beingsmall compared to other forces, which is a veryaccurate assumption except in a small regionnear the boundary. The assumption that thefluid is incompressible seems to be generallyvalid for the fuels and liquids in current use.The liquid free surface displacements, slopes,and velocities are assumed small, which linearizes the boundary conditions a t the free surface. Some discussion of the significance of thelinearization of the boundary condition is givenin references 2.13 and 2.17. Briefly, two nonlinear boundary conditions must be satisfied a tthe free surface; by employing the abovementioned assumption, these two conditions arelinearized and combined to give only a singlecondition.LATERAL SLOSHING IN MOVING CONTAINERSThe assumption of irrotational flow is compatible with the assumption of zero viscositysince, as is well known, motion in a nonviscousfluid which is irrotational a t one instant alwaysremains irrotational (ref.
2.17). This assumption, along with the linearized boundary conditions, yields a theory which can be employedfor the analysis of several practical containergeometries.The assumption of no sinks or sources,keeping in mind the assumed incompressibilityof the liquid, requires that the liquid volumebe constant; therefore, for example, the theorymay not be valid for a rapidly draining tank.This is discussed further in chapter 10.Basic Theory: AnalysisIt will be found convenient in this chapter-to employ either rectangulas or circular cylindrical coordinates located a t various points inthe container. Unless otherwise stated, thecoordinates will be located on the undisturbedliquid surface, with the z and y axes in theplane of the surface and the z coordinatenormal to the surface.
The assumption ofirrotational flow insures the existence of asingle-valued velocity potential @(x, y, z, t) inany simply connected region, from which thevelocity field can be derived by taking thegradient (it should be noted that someauthors prefer to define the velocity as thenegative of V@)V=V@(2.1)-#or in terms of the components of the velocityvector, V (in rectangular coordinates)+The vector statement of Newton's second law ofmotion for a particle in a nonviscous fluid is-1P Vp+ FB=A+-9(2.3)where P is the mass density, p ia the intensity ofnormal pressures, Fg is the body force vector,-#A the acceleration vsnor'3fact t h a tnoting the15the above equations may be rewrittenwhere it has been assumed that the only bodyforce present is due to the acceleration field.The continuity of the fluid (mass conservationlaw) must be preserved.
Enforcing this requirement, and noting the assumptionof incompressibility, we haveEquations (2.5) and (2.6) are sufficient, onceappropriate initial and boundary conditions areimposed, to determine the velocity componentsu, v, and w and the pressure, p, uniquely.Substituting equation (2.1) into equation(2.6), it is seen that @ must satisfy the Laplaceequation:where V1 is given in the above equation in rectangular coordinates.By noting that the flow is irrotational andp=constant, the equations of motion (2.5) maybe integrated and then linearized to obtain~ernoulli'slaw (the integration function is absorbed in the definition of 9 with no loss ingenerality, cf.
ref. 2.17)r,ccGle;-.'cn~ , d h-d be-- =,+,.iS~ksdto that of gravitation. Thus, @ is determinedfrom equation (2.7) and appropriate boundaryconditions, the velocity components from equation (2.1), and the pressure distribution fromequation (2.8). The free surface displacementis obtained from equation (2.10) to be derivedbelow (the pressure at the free surface is usuallyconsidered to be zero),,, while the forces andmomenta acting on the container may be found,I..$I.~16THE DYNAMIC BEHAVIOR OF LIQUIDSby appropriate integrations of the pressure.(See, for example, sec.
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