H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 7
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Indeed, this tank geometry is perhaps the most important of all, and, fortunately,the analysis of lateral liquid sloshing is notespecially difficult. However, because of thefrequent necessity to reduce sloshing massesor to shift the liquid natural frequencies, subdivision of such containers into compartmentsby either radial or concentric walls is relatively%])-A~b'(2n+l)'+a~(2n+l)'](2.16b)pabh(a2+ b3common practice, so that the basic simplegeometry transforms into other more complexforms. For purposes of our presentation here,it is somewhat advantageous to begin with themost general compartmented tank, the ringsector, and then present results for other configurations as special cases.
It is interesting tonote that the solution to the shallow water(tidal) oscillations of a liquid in a ring sectortank was suggested by Lamb (ref. 2.13, art.191) and that the natural frequencies for tidaloscillations of a fluid in a sector tank wereobtained by Rayleigh (ref. 2.30, art. 339).21LATERAL SLOSHING IN MOVING CONTAINERSRing Sector Compartmented TankThis configuration is not only of practicalimportance, but from the analysis of liquidsloshing in this type tank, the behavior ofliquids in all cylindrical tanks composed ofradial and circular walls may be obtained byappropriate selection of the geometric variables.For purposes of review, the procedures employedto obtain the natural frequencies and to solvethe problem of lateral sloshing of a liquid in aring sector tank induced by translational motionof the container will be outlined first, followingclosely the discussion of reference 2.3.
Resultsfor other types of container motion (refs. 2.3,2.19, and 2.31) are also then given. It shouldbe noted that the theoretical results presentedare d i d for a single sector tank. To use theresults for compartmented cylindrical tanks,one must superimpose the effects of all thecompartments. For example, the contributionto the total force in the xdirection of all eightcompartments has been accounted for in thetheoretical curves presented in figure 2.8.Fme OIcillationrand a flat bottom is obtained from the solutionto the Laplace equation V2q1=0 with theboundary conditions*l--0a2-dr -0at the tank bottom z= -h(2.17a)at the circular cylindrical tank wallsr=a, b(2.17b)r-0I.a t the sector walls 4=O, 2mw(2.17~)a*+, a@,~ $ 1 z 9= Oat the free surface z=0Assuming a product solution of the formand substituting into the Laplace equation, thefollowing expression for ql is obtained*l=eiU'(01cos 4+Ca sin 4}[4(cosh XzThe geometry of the tank and coordinatesare shown in figure 2.3.
The flow field of aliquid with free surface in a cylindrical containerof ring sector cross section with vertex angle 2rawhere v and X are constants and J,(b) andY,(AT) are Bessel functions of the first andsecond kinds of order v. Care must be takenin the choice of the "separation" constants vand A, and their signs, to insure that the mathematical solutions actually describe the physicalproblem. The velocity potential which satisfiesthe boundary conditions a t the container walls(eqs. (2.17b) and (2.17~))iswhereFIG-2.3.-Gordinate system and tank geometry forM g nectar tank (ref. 2.3).22THE DYNAMIC BEHAVIOR OF LIQUIDSThe values h, are the positive roots of theequation I,A=J&, (€>YL,, (kt)and k=b/a is the diameter ratio of the innerfind outer tank walls.
The unknown constantsA,, can be obtained from the initial conditions.The equation for the eigenvalues of the liquid isobtained from the free surface condition,equation (2.17d)It is seen that the frequencies of the liquiddecrease with liquid depth and with increasingtank radius. For a given tank size, the possibility of shifting the frequencies is expressed by706050$4030The roots of this determinant are plotted in fig. 2.4,and tabulated in table 2.1, as a function of k for thespecial case of m/2a=1.
Further discussion on theroots of this determinant may be found in ref. 2.2,while extensive tables and graphs of A r l r ( [ ) =0 arepresented in ref. 2.32.FIGURE2.5.-Natural"00.2F'xcUBE 2'4.RootP0.40.6kgabof A1(E)=O(ref. 2.2).a81.0FID1-frequency parameter for cmtainerr of circular, annular, and quartercircular crou m i o n (ref. 2.3).LATERAL SLOSHING IN MOVING CONTAINERSTABLE2.1.-R00t~of the Determinant Amn,(€) =0, Eq. (2.18d)[ m k =11,For-large values of h/a (i.e., h/a>l), theapproximationW29=; Ensis accurate.
Figure 2.5 gives the natural frequencies for fluids. in containers of circular,annular, and quarter-circular (90' sector) crosssection.For forced excitation x=x,,etncnormal to thecontainer wall (see fig. 2.3), the boundary conditions are:a@ i ~ z o e t ncos-' 4a t the circular.cylindrical tank walls r=a, bdcp1 a@---a t the sector wall 4=0r*-0*----I3-inz0emLsink--..+Amuvra5Da@-+g ==Oa t the--I1IMl-1 --WoLlIa$-ogbz-for 4=0, kn%l=ins~,pcos 4for z=0m2o=&,r;and K=,[.(2.20)h-athe disturbance potential isyr-uru---.Icos 4)et~'boundary conditions for the disturbance poten229-648 -67--3forz=-Ilbz-By extracting the container motion+*M1--04 = ~ 4 f=L,,at the free surfacez=o@={@Ifor r=a, band therefore the disturbance potential @,(r, 4, 2) which satisfies the Laplace equation hasthe same form as equation (2.18b).
Omittingthe double summation and indices and introducing the abbreviationsat the bottom of thecontainer z= -h&=OM-a,.1-0rTmnrlotionol Fomd OscillationsbT-tid, which are homogeneous a t the containerwalls, are obtained asTo determine the unknown coefficients A,,,from the condition of the free fluid surface therighbhand side of this boundary condition hasto be expanded into a series where cos 4 isrepresented as the Fourier seriesmcos 4 = C a,cosm-oz(2.22424TEIE DYNAMIC BEHAVIOR OF LIQUIDSwithsin G Ja o sin= ii~ &=J 2Z((m9-Z2)a(Z= h a )and the pressure at depth (-2)from equation (2.8) asis obtainedThe function r is represented as a Bessel serieswhere$C(u)dubmn=tmnJt>c2(U)du-w2(tmn)P C 2 ( k t m n ) ] -e ~€[ m~ - nC 29 (Ckm~ nm n ) ](2.22b)The coefficients A,, becomeA""=iQa,,,bm,laq2(1_q2)(2.22c)where q=Q/u is the ratio of the exciting frequency to the natural frequency.
The velocity potential ch for translational container excitation in direction is then@(r,+,2,t)=iS2xoefn'&bmnC(c)q2 ~ o s h( ~ +cosl ) S)(1-$)coshx+(2.23)The &st term (potential of the rigid body)satisfies the boundary conditions at the tankwalls, while the second part (disturbance potential) vanishes at the tank walls. The freesurface condition is satisfied by both parts ofthe formula.
The free surface displacement,the pressure and velocity distribution, as wellas the forces and moments of the liquid, canbe determined from the potential by d8erentiations and integrations with respect to thetime and spatial coordinates.The surface displacement of the propellant,which is measured from the undisturbed position of the liquid, may be obtained fromequation (2.10), with po equal zeroAt the outer container wall, r=a, the functionC(u) = (ZJrb,), while at the inner containerwall, r=b, the function C(u) has a valueC(k&,,,). At the sector walls +=O, +=E, thecosine assumes the value 1 and (-I)", respectively.
The pressure distribution at the tankbottom is obtained from equation (2.25) withz=-h(t=-K).By integration of the appropriate componentsof the pressure distribution, the liquid forcesand moments can be obtained. The resultingforce in the x-ctionis thereforeFz=L'S:h(upa-bpb) cos &4&-1sin ZdrdzHere the first integral represents a contribution of the pressure distribution a t the circularcontainer walls, and the second integral stemsfrom the pressure distribution a t the sectorwalls.
With the mass of the liquid Mr=pra2k(l-P), the fluid force becomesF*=MrQ2wf0'[1+2(-l)m+la,,,bmnsinZq2tanh KEa(1(1-q2)The force component in the y-direction isgoverned by25LATERAL SLOSHING IN MOVING CONTAINERSwhich then leads toand-fJpz--h?(!+ z ) P*-lJ'h+(; r ) p1-Here (see appendix A)The first term, MrVa,e'Ot, in equation (2.26a)represents the inertial force of the liquid (thatis, the force that would be produced by anequal volume of solidified liquid). The fluidmoments M, and Mz with respect to thepoint (0, 0, -h/2) are given by-l -.J:hp+sin a (i+z)d~drsin2Z 1 ambmu[l-(-1)"M+=-M'2Crzoe'O'2(9,~-2)point (0, 0, -h/2) parallel to the y-axis whileM, is the moment about an axis parallel tothe x-axis through the same point.
In theseformulas, the first integral again represents thecontribution of the pressure distribution at thecircular cylindrical tank walls. The secondintegral is the contribution of the bottompressure, while the remaining integrals represent the pressure contribution a t the sectorwalls to the moment. After the integrationhas been performed, the moments of the liquidare2Mpga (I-k3) sin ZwhereN,9 2 t%muA.2 AT.a \$mu//r \Lzn-kC(kUJJ+=FsM1 r~C(U)&-cos Z]q2{[tanh-f-2eos a d zMuis the moment about an axis passing through2=2GnN2(Emu)+(#m2-2)~ C O S ~x+sin W T(im2--21%cash r3 1)1K<(L cosh xI)] [NO(L~)&Xvfps ( 1 - p ) (;-cos-3(2.26~)ii%(2.26d)Since the reference axis does not pass throughthe center of gravity of the undisturbed liquid,the last term in the moment formula representsthe static moment of the liquid.THE DYNAMIC BEHAVIOR OF LIQUIDSThe velocity distribution in the radial, angular, and axial directions areIn these equations for pressure, forces,moments, and velocity distributions of theliquid, the term for the solidified liquid isrepresented as a single term and not as aseries.
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