- Email: [email protected]
Sloshing in microgravity
Cryogenics 39 (1999) 1047±1055
Sloshing in microgravity H.A. Snyder a,b,* a
Campus box 429, University of Colorado at Boulder, Boulder, CO 80309-0429, USA b Ball Aerospace and Technologies Corporation, Boulder, CO 80306-1062, USA
1
Received 20 September 1999; accepted 20 September 1999
Abstract The International Space Station provides a low gravity environment for experiments that require very low acceleration. The steady component of acceleration due to the gravity gradient is in the microgravity range. It is possible to achieve microgravity levels for the variable component by using isolation racks. For experiments cooled by liquid cryogens sloshing may increase the variable acceleration at the experiment beyond acceptable levels. Sloshing of cryogens in microgravity can be predicted using a surface wave model. The model should include: a calculation of the shape of the unperturbed liquid±gas interface; a listing of the normal modes and resonant frequencies for the container; a prediction of the amplitude of the modes in response to the motion of the container; and a test to detect the breakdown of linear theory. A model is presented that contains these components. The shape of the interface is calculated and it is found that for most anticipated applications the interface is nearly cylindrical or spherical. Since gravity is not aligned with the symmetry of the container, the depth of the liquid is variable. Examples are presented to show how to estimate the extent of variable depth and curved interface on the normal modes and resonant frequencies. Equations are derived for the dynamic interaction of the isolation rack, the dewar and the sloshing motion. Damping is introduced by using boundary layer theory. Random vibration theory is applied to the incoherent component of the driving spectrum while standard resonance formalism is used for the coherent component. The model cannot be used if the wave amplitude becomes so large that linear theory does not apply. A procedure is developed to check for nonlinear diculties. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Super¯uid helium (b); Fluid dynamics (c); Two-phase ¯ow (c); Space cryogenics (f)
1. Introduction One use of the International Space Station (ISS) is to provide a low gravity environment for experiments that require low acceleration. Many experiments need both an acceleration that is as low as possible and low temperatures. When a cryogen is used to cool a low acceleration experiment, the apparatus is usually placed in a dewar ®lled with the cryogen. Sloshing of the cryogen contributes to the acceleration level of the experiment. The purpose of this research is to describe methods to calculate sloshing of cryogens in dewars when both the constant and variable components of the acceleration are in the range experienced on the ISS and to calculate the acceleration level of the experiment when the sloshing is known.
*
Tel.: +1-303-492-7635; fax: +1-303-492-7881. E-mail address: [email protected] (H.A. Snyder). 1 Correspondence address.
The structure of the ISS is expected to have a variable component of acceleration that is strongly dependent on frequency. It ranges roughly from about 0.5 lg at 10ÿ2 Hz to about 103 lg at 50 Hz. The variable component of the acceleration can be reduced by several orders of magnitude with passive and active isolation racks. For experiments that are sensitive to acceleration, an upper limit is usually 10 lg. Frequently, these conditions must be maintained for long periods of time ± up to several months. The motion of the dewar is quite small at these acceleration levels and sloshing consists of surface waves. The literature on the theory of surface waves is extensive. However, a low acceleration environment leads to conditions not usually encountered in 1-g. Surface tension dominates over gravity and the gas±liquid interface is curved. Since the constant component of acceleration is comparable to the variable component, the interface can become unstable. On Earth an apparatus is usually positioned so that a symmetry axis is aligned with gravity. On ISS the constant component of gravity is
0011-2275/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 9 9 ) 0 0 1 2 0 - 4
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
rarely in the direction of the axis of the dewar. This leads to a problem with variable depth. If sloshing increases the acceleration level of the experiment appreciably, it may be reduced by installing a system of baes or aluminum foam. The change in wave amplitude due to this type of mitigation is analyzed. Helium II presents a special problem. Khalatnikov [1] shows that if the normal component of the super¯uid is not constrained, such as by aluminum foam, the dynamics of surface waves at a helium interface is the same as that of an ordinary ¯uid. Even when the normal ¯uid is immobilized by viscous drag, the super¯uid propagates surface waves with a slightly modi®ed dispersion relation. Thus, it is not possible to use conventional methods to reduce sloshing of super¯uid helium. The approach presented below is analytic. It appears that computational techniques presently available are inadequate for several reasons. The problem is threedimensional and lacks symmetry because the axis of the apparatus is not aligned with gravity and the interface is not ¯at. Most computational methods are restricted to two dimensions or to situations where there are symmetry elements. The most important results of this type of calculation are the resonant frequencies and amplitudes at resonance. Damping is low and the tuning curves are very narrow. A computational method that searches for the resonances would be very inecient. The problem is resolved into several topics. First, a standard method is used to calculate the shape of the gas±liquid interface. Then, the stability of the surface is analyzed for the case when the constant and variable components of acceleration have the same order of magnitude. The coupling between the motion of the wall of the container and the sloshing is considered next. Damping is introduced into the governing equations using methods common in acoustics. The tuning curve for resonance is derived from the damping term. The interaction of the isolation rack, the dewar and sloshing is analyzed in terms of transfer functions. When resonance occurs, the waves may attain large amplitudes. The conditions when the normal mode analysis lacks accuracy and a wave may break are described. Most of these calculations use the spectrum of the normal modes for surface waves of the container. The eect of a curved interface and variable depth on the normal modes is outlined in the next three sections. Finally, two methods to reduce sloshing are considered: baes and aluminum foam. Most of the work on sloshing for space applications is concerned with fuel management in rockets. Here, the gravity level is high and the amplitude can be large enough to require a nonlinear analysis. This research was published mainly in the 1960s. Abramson [2] edited an extensive review of this work. Many of the methods used below are based on material presented in that review and the classic text by Lamb [3]. The aim is to distill
from the previous work that which applies to the present problem and to synthesize a systems level procedure for calculating the acceleration of the experimental chamber given the acceleration of the ISS. Whenever possible an attempt is made to simplify the previous methods. Thus, analytic methods are preferred over variational and computational calculations. Since most of the earlier work assumes that surface tension is negligible, the results are reworked for the present situation. The normal modes for gravity and capillary waves are the same but the interface condition diers and the new dynamical equation has to be applied to the normal modes to get the dispersion relation. The earlier work also assumes that the motion of the container wall is known and that the sloshing does not react directly on the container. Equations are derived that couple the motion of the dewar and the sloshing. Also, the previous work relied on variational calculations to ®nd the eect of a curved interface and variable depth on the normal modes. In order to simplify the calculation, a method is developed to use analytic solutions to approximate these corrections. The presentation is kept suciently general so that it is helpful in solving most problems in the design of the ISS that involve sloshing.
2. Surface contour The shape of the quiescent gas±liquid interface is determined by a balance of two forces: surface tension and the constant component of acceleration or gravity. On the ISS the axis of the container is usually at an angle with the gravity vector. There are two coordinate systems that are convenient to describe the force balance: one with the z-axis along g and the other aligned with the z-axis of the container. In the latter case, the g vector can be resolved into a component along the z-axis gz and a component along the other axis gx . When the z is aligned with g, the equation to be solved is: z00 2z=a2 1 z02
3=2
;
where a is the capillary constant a2 2r=qg with r the surface tension, g the acceleration and q the density of the liquid. For all cryogens the contact angle, h when in contact with most materials used for the walls of dewars, is near zero. Since h 0 the boundary conditions on this equation are that z0 cot / at z d=2, where / is the angle of inclination of the container. The most general procedure to ®nd the shape of the interface appears to be due to Hastings and Rutherford [4] who derived a general set of dierential equations. They used a Runge±Kutta shooting method to solve them. We found the shooting method frequenstly to be unstable for these equations. The method can be
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
simpli®ed using a method by Landau and Lifshitz [5] that has always succeded for our calculations. The equation for z00 is integrated once and the undetermined integration constant is called A. 1 1=2
1 z02
Aÿ
z2 : a2
1 0
Then apply the conditions z cot / at x d=2 in 1=2 Eq. (1) to ®nd zÿd=2 zd=2 a A ÿ sin / . Then, 0 solve Eq. (1) for z and integrate again: Z z A ÿ z2 =a2 dz h i: 2 x 2 zd =2 1 ÿ A ÿ Z 2 =a2 Before Eq. (2) can be integrated we need A. In Eq. (2) set 1=2 x d=2 and z a A ÿ sin / and use an equation solver to ®nd A. Then, integrate Eq. (2) numerically to get x versus z and plot z versus x. This problem diers from the rise of liquid in a capillary tube because the amount of liquid is constant. The reference line z 0 has to be chosen so that there is an equal volume of liquid above and below z 0. For the other approach x is aligned with the axis of the dewar. In the new coordinates the equation to be solved is z00 2z=a2 1 z02
3=2
gx =gx gz =gz:
Now the boundary conditions are
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the length of the interface is suciently small relative to a so that no calculation is necessary. 3. Stability An interface can become unstable with plumes of gas plunging into the body of the liquid. It is known that a cylinder that is accelerated along the axis with the acceleration vector pointing from the gas to liquid is unstable if h 0 and the Bond number exceeds 0.85. The Bond number is qg0 R2 =r, where R is the radius of the cylinder and g0 is the additional acceleration beyond that which determines the shape of the static meniscus. Here, g0 is a constant vector. This experimental value of Masica and Salzman [6] agrees with the stability calculation of Reynolds and Satterlee [2]. The most sensitive cryogen for instability is liquid helium. For a cylindrical dewar of helium with R 1 m, instability occurs when g0 70 lg. It is unlikely that instability is a problem for ISS experiments for several reasons. First, the isolation racks are designed to reduce g0 to less than about 10 lg at the dewar. Secondly, the disturbance is oscillatory and is destabalizing during half the cycle and stabilizing during the other half. Also, the frequency of the disturbance is high compared to the rate of growth of the instability. Furthermore, the disturbance is not concentrated along the axis of the dewar but is random in direction.
z0 cot h 1 at x d=2: This equation is more dicult to solve than the previous equation for z00 because it does not have a ®rst integral that can be represented by simple functions. However, it should be solvable by numerical methods. But when conventional numerical schemes are applied, it shows the same instabilities as the described above. The technique of Landau and Lifshitz using the z-axis along g seems to be the best choice for ®nding the shape of the interface. All of this may be an academic exercise in many situations. The capillary constant a has dimensions of length. If the distance across an interface is of the order of a or less, surface tension dominates the force balance and the surface is either cylindrical or spherical. If the diameter of a cylinder is d and d a, calculations show a deviation from a spherical surface of less than 1%. When d 2a, the dierence is about 5%. Thus, if the distance across the interface is less than about 2a, a calculation of the shape is not necessary. Liquid helium has the smallest value of a of all the cryogens. For a gravity level of 5 lg, a 0:3 m and it is weakly dependent on temperature. Interfaces up to 0.6 m can be approximated as cylindrical or spherical for helium. The next smallest value of a for a common cryogen is nitrogen with a 0.55 m. For most applications on ISS
4. The driving force When the driving disturbance is small, as here, the motion of the interface can be represented by a sum of harmonic surface waves. This theory is linear in the velocity and cannot predict the distortion and breaking of the waves. Also, it cannot represent the motion after a wave breaks. We will use surface wave theory as the basis of analysis. The normal mode amplitudes will be calculated. Then, we will use nonlinear wave theory to check for breaking. Large amplitudes only occur at resonance. Damping is small compared to the inertial terms of the governing equations. The analysis can be simpli®ed considerably by calculating the normal modes using inviscid equations and adding damping when the frequency response is calculated. That is the approach used here. For small amplitude inviscid surface waves the velocity ®eld can be derived from a potential: v r/ and r2 / 0. This equation is independent of the size and shape of the container. At all solid boundaries the velocity normal to the boundary vanishes, o/=onjwall 0. The free surface boundary condition will be discussed below. The geometry of the container determines the
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
normal modes. There are no body forces for this application and the ¯ow is driven by motion of the container. Thus, the velocity of the ¯uid at the wall of the container equals the velocity of the container. We will assume that the dewar is suciently sti so that the motion of the wall is the same everywhere ± the vibrational modes of the dewar do not have to be considered. The driving forces of the walls are included in the theory by writing / as a sum: /total / /wall , where /wall Vx x Vy y Vz z and Vwall (t) is the vector velocity of the wall. The boundary condition on / is as above o/=onjwall 0 at the mean location of the wall. Then, o/total =onjwall Vwall t and the normal modes are calculated as if the container were stationary. When the normal modes are found, /wall is expanded in a series of normal modes. The amplitude of the normal mode component of /wall determines the amplitude of that normal mode in /. Many normal modes do not get excited by motion of the wall because they do not occur in the series expansion of /wall . The most general motion of a dewar consists of the three translations included in Vwall (t) and three rotations. For rotation the components of Vwall (t) are not the same at all locations on the wall as they are assumed above. The distinction between translation and rotation is only signi®cant when the dewar is close to an axis of rotation. The ISS does not rotate at frequencies above 10ÿ2 Hz and an axis of rotation is unlikely to be near the dewars being considered. It appears that rotation of the container can be neglected. 5. Damping and resonant amplitude Inviscid theory predicts an in®nite amplitude when Vwall (t) excites a resonance. Damping reduces the frequency response near resonance. Consider a normal mode with wave number k k 2p=k, where k is the wavelength) and circular frequency x excited to an amplitude A0 by the motion of the wall. When the excitation is turned o, the amplitude decays as A t A0 exp ixt ÿ ct, where c is the damping constant. Damping is caused by viscous shear in the wave and in boundary layers at the wall. There are three additive contributions to c. There is shear in: the wave in the body of the ¯uid cw ; the side wall boundary layers perpendicular to the propagation of the wave cs ; and the bottom boundary layer cb . Cane and Parkinson [7] published a method in which the linearized boundary layer equations are solved exactly for gravity waves. When this method is applied to our case, the damping cs is proportional to the area of the sidewalls and cb is proportional to the bottom area, both scaled by the volume. The method was adapted to capillary waves and the approximation of wall areas is used.
The wave contributes cw 2mk 2 to c, where m is the kinematic viscosity of the liquid. This eect is usually small except in large tanks with small surface to volume ratio. The boundary layer at the wall has thickness 1=2 m=x . At a frequency of 10ÿ2 Hz the thickness is about 3:8 10ÿ4 m for liquid helium and 15 10ÿ4 m for liquid hydrogen. For other cryogens it is smaller. The side wall and bottom boundary layers have dierent eects on c. This is due to the decay of the wave amplitude with depth; in most cases this is exp kz. For the side walls cs L=S mx=81=2 , where S is the area of the cross-section perpendicular to the propagation of a normal mode and L is the perimeter of the side wall in this cross-section. The bottom boundary layer adds 1=2 2 cb w=S mx=8 1 ÿ exp ÿkh=kh , where w is the width of the bottom and h is the depth. When h > k=3 the bottom boundary layer can be neglected. Thus, c cw cs cb . Each excited normal mode has a c associated with it. The decay constant c for a tank without baes 0:5 m 0:5 m 0:5 m at a driving frequency of 1 Hz is about 1 10ÿ3 =s for helium and 5 10ÿ3 =s for hydrogen. The decay constant c determines how long it takes a mode to grow to equilibrium amplitude and to decay. The time constant is 1/c. Damping causes the resonant frequency of a mode xres to be lower than the undamped frequency x0 by x2res x20 ÿ 2c2 . It also modi®es the frequency response and prevents a large amplitude at resonance. It is customary to relate c to the quality factor of a normal mode: Q xres =2c. For the tank of the last paragraph the Q factors are about 3000 for helium and 700 for hydrogen. We may expect Q factors of about 1000 for most applications without baes. The tuning curve for each normal mode driven by amplitude Am is A x
x20
Am x20 ; ÿ x2 ix0 x=Q
3
where Am is the amplitude at a frequency far below resonance. At resonance the amplitude is Am Q. The width of a resonance at half height is Dx x0 =2Q. The displacement of the dewar that drives the waves consists of a spectrum of frequencies. Each small interval of the driving spectrum drives all the normal modes. If the driving frequency is about x0 /Q below resonance, the response is Am . When it is about x0 /Q above resonance the response is negligible. The response spectrum is a series of narrow tuning curves for each normal mode. They do not overlap because the values of Q are of the order of 1000. The output spectrum is a product of the driving spectrum and the response spectrum and represents steady state conditions. Since the driving force is made up of three components Vx , Vy and Vz , each spectrum consists of three plots. The relation between the driving spectrum of the dewar and the spectrum of the ISS will be discussed in
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
the next section. The driving motion of the ISS is not a continuous harmonic signal. Part is random and part is coherent. The steady state analysis only applies to the coherent part of the signal. The time to build up a resonance is about 4/c. This is called the settling time. For helium this ranges from 500 to 5000 s and for hydrogen 100±1000 s. If the coherence time is less than this, a resonant driving signal does not produce the resonant amplitude. Random signal theory [8] provides a method to calculate the output when the coherence time is less than the settling time. The response and output spectra have the same shape for coherent and random signals but the amplitude for random signals is much smaller. If the ISS spectrum is a root mean square power spectrum and is smooth, the data refers to incoherent signals. Then the amplitude of the acceleration driving the system can be read directly from the spectrum and used as described below. If it is known that there are coherent sources, they will appear in the power spectrum as large spikes. Usually the coherent part of the spectrum is treated separately and the power spectrum does not contain these spikes. Then, it is necessary to calculate the response at these frequencies separately. We cannot calculate the wave amplitudes from the ISS spectrum until we consider the motion of the dewar.
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similarly. Whenever a procedure is described below in one dimension, it is assumed that the other two dimensions can be treated similarly. In the early work on sloshing the dewar is assumed to be attached to the spacecraft rigidly. The motion of the spacecraft is estimated, the sloshing is calculated from this, the net pressure force resulting from the sloshing is applied to the dewar and the motion of the spacecraft is recalculated. There is no direct coupling between the sloshing and the motion of the dewar. Peterson, Crawley and Hansman [9] showed that under some conditions the results of a coupled and uncoupled model can deviate considerably. These authors developed a nonlinear coupled model for the sloshing of fuel. We do not need a complicated nonlinear model because the driving displacement is very low. A simple linear model can be made by summing the forces on the dewar in each coordinate direction. The model consists of the solid dewar Ms connected through the transfer function Tr to the displacement of the ISS structure and with the forces Fx and fx applied to Ms as in Fig. 1. If Tr consists of a simple spring jr and damping Cr , then Ms Md Xd00 ÿCr Xd0 ÿ jr Xd fx Cr X 0 jr X : The spectrum of ISS w(x) provides the data to ®nd X : X x ÿw x=x2 :
6. Acceleration of the dewar due to sloshing The purpose of this model is to ®nd the displacement of the dewar Xd , Yd and Zd in terms of X, Y and Z, the displacement of the mounting point to ISS. The dewar is attached to the structure of the ISS through an isolation rack. The motion of ISS is transmitted through the rack according to a transfer function Tr (x). The liquid has a component of solid body motion equal to that of the walls. It is represented in /total by /wall . The force on the dewar needed to maintain Vwall (t) is equal to the pressure produced by /wall integrated over the surface of the wall in the three coordinate directions. Since p ÿqo/=ot ÿiqx/, this force in the x direction is Z Fx ÿixq /wall dy dz exp ixt:
If active control is used by sensing and feeding back Xd , an additional term in Xd occurs in the above equation. If a more complicated Tr is used, it is easy to modify the above equation. Now we have to ®nd / and /wall and eventually fx in terms of Xd The analysis follows the procedure described above. The normal modes are found and represented as /. Then, /wall is expanded in the normal modes. Each term in the expansion is proportional to Vx . At this point the amplitudes of the normal modes of /wall are known and / are unknown. The interface boundary condition connects these amplitudes. The interface boundary condition is o2 /total =ot2 ÿ r=qo=ozr22 /total 0
at z 0:
Here, z is perpendicular to the interface and positive toward the gas while r22 refers to the two coordinates parallel to the interface. This equation can be rewritten
Furthermore, /wall Vx x and the integration results in Fx ÿixMd Vx x2 Xd Md , where Md is the mass of liquid in the dewar. The surface waves also produce a force on the walls of the dewar Z fx ÿixq / dy dz exp ixt #; where # is the phase dierence between the motion of the wall and each sloshing mode. This is a one-dimensional model. The other two dimensions are treated
Fig. 1. Dewar suspended by isolation rack and driven by liquid pressure.
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
as x2 ÿ r=qk 3 / ÿx2 /wall at z 0. Here we assume the time dependence is exp ixt and a space dependence of exp ikx kz. The wall term is the driver and the resonant frequency of the normal mode of the surface wave is evident. The resonant term is modi®ed by the addition of damping as explained above. Since / and /wall are expanded in the same normal modes, we can match terms in the equation. If wn is a normal mode and An Vx is its amplitude in / and an Vx the amplitude in /wall , then, with x20 r=qk 3 , jAn j ÿan x2 = x2 ÿ x20 2 xx0 =Q2 1=2
4
with a phase dierence p ÿ arctan xx0 =Q x20 ÿ x2 . It is easy to show that Z fx x2 Xd Rn An qWn dy dz from its de®nition. When fx is replaced in the dynamic equation, we ®nd xx 2 2 Xd x xr ÿ x i Qr R P 2 Wn dy dz q 2 n an x xd x ÿx 2 2 Ms Md xn ÿ x ixn x=Qn x2r ixr x=Qr X x:
5
Here, xr and Qr refer to the resonant frequency and quality factors of Tr and xn and Qn to the normal modes. This is an algebraic equation for Xd (x) in terms of X(x). At a chosen frequency only the closest resonance xn need be included in the summation because the resonance are very narrow Dx xn =2Q. Eq. (5) appears to be complicated but it has a simpli®ed interpretation. If surface waves are suppressed, the dewar oscillates with the mass Ms Md on the suspension. When the surface waves are present, the second term changes the apparent mass and thus the resonant frequency and the damping of the suspension. At resonance the fractional change in mass is Z qan Qn
Wn dy dz= Ms Md :
The apparent mass changes rapidly across the tuning curve and is positive below resonance and negative above resonance. If the apparent mass becomes as large as Ms Md , the system is unstable. After Xd is found by the above procedure for each signi®cant normal mode, the acceleration spectrum of the dewar is ÿx2 Xd x. This is usually the desired result. But the problem was treated as linear and this implies that superposition of normal modes holds. We have to verify that the amplitudes do not exceed this assumption.
7. Waves of large amplitude For linear theory to hold, some nonlinear terms are omitted in the governing equations and the boundary conditions. [3] It is assumed in deriving linear theory that fmax < k, where fmax is the maximum amplitude of the interface. Experience with other nonlinear systems and experimental results leads us to believe that the nonlinearity does not become noticeable until this limit is exceeded by a factor of two or three. The ultimate nonlinear eect is the breaking of waves. It has been shown [3] that when kh P 2 a wave does not break until fmax 140k. For waves on a sloping beach, breaking occurs when fmax h. For a particular dewar we have to estimate fmax and compare with these limits. The relation of=ot o/=oz at z 0 implies ixfmax o/=ozjmax;z0 . First, we have to use the ISS spectrum to ®nd the an , then the An at resonance and then o/=ozjmax;z0 . As an example consider a rectangular container of depth h with h > k. The length in the x direction is lx and the coordinate origin is midway between the ends; the z axis is upward. The container is oscillating in the x direction at a frequency x1 and at the point of mounting to ISS w x1 ÿx21 x1 . The lowest normal mode is the easiest to excite and has the lowest damping. Therefore, it is probably the mode with the largest amplitude. Let x1 be the resonant frequency of that mode. Since Vx x is an odd function only sine terms appear in /wall and these are the only modes excited in /. Then, / A1 Vx W1 A1 Vx sin px=lx exp pz=lx exp ix1 t # and /wall a1 sin px=lx exp ix1 t with a1 4lx =p2 : At resonance A1 a1 Q1 . Applying the above equation for f; jfmax j a1 Q1 p=ix1 lx Vx 4QR1 =pXd . We have to ®nd Xd from Eq. (5). Next we need W1 dy dz 2lx ly =p. The Tr should have a resonance far from the most dangerous normal mode resonance. Thus, we can neglect damping in Eq. (4) for the suspension. Substituting in Eq. (5) we ®nd Eq. (6): Xd x1
h
x2r
8l2 l Q
q x2r ÿ x21 1 xpy3 1 Ms M L 3 r p 4 jfmax j Q1 Xd x1 : x21 q lx p
i X x1 ; 6
The term with Q changes sign from positive for slightly below resonance to negative just above resonance. We now have a formula for fmax in terms of known parameters. The next step is to compare fmax with k and
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
make a judgement if linear analysis is adequate. This procedure has to be repeated for each normal mode. 8. Normal modes: general considerations The normal modes of surface waves are solutions of r2 / 0 that satisfy the boundary conditions. The equation is separable in all the common coordinate systems and the normal modes are products of function of one coordinate. One of the product functions is always oscillatory, like trigonometric or Bessel functions, and the other is always a monotonic function, like exponential and hyperbolic Bessel functions. The side wall and bottom boundary conditions determine the shape of the normal modes while the interface condition provides a relation between the frequency and the wave number. When the depth h is large compared with the wave length k, h > k=2, the velocity ®eld decays downward from the surface as exp kz (z positive upward) and the bottom boundary shape has negligible eect on the normal modes. When all the walls of the dewar and the interface coincide with a common coordinate system such as cylindrical or spherical, the normal modes are easy to calculate. For the present application the side walls usually conform to this requirement but the interface and bottom boundary do not. The interface is either cylindrical or spherical but with a dierent axis than the walls. When the axis of the dewar is not aligned with gravity, the depth h varies ± frequently as cos nh if the dewar is cylindrical. The normal modes for a curved interface and/or variable depth generally cannot be found analytically. In this case the problem is three-dimensional and numerical methods, with present computing capability, are also inadequate. Variational methods are the best approach but are frequently time consuming. However, many problems are suciently close to an analytic solution that an approximation is in order. We will illustrate the eect of a curved interface and variable depth in Sections 9 and 10.
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interface. The boundaries of the last three must be chosen so that they coincide with constant coordinate surfaces and thus are solvable by analytic methods. There must also be an obvious mapping of the normal modes for each case with the others. This is best illustrated by an example. For an annulus the surface is nearly cylindrical. An annulus with ¯at interface is solvable analytically. The annulus can be straightened to a channel with the width of the annulus and the length of the mean circumference. If the bottom has the same curvature as the interface, the depth will be constant. If the interface is ¯at, the bottom is parallel to the surface. Three of these cases are solvable analytically and there is an obvious mapping of normal modes. Solve the cases of a channel with ¯at and with curved interfaces and make a table of the ratio of resonant frequencies of the curved interface and the ¯at interface. Then solve the annulus with ¯at interface and apply the correction for curved surface to each normal mode found for the channel. The eect of surface curvature is illustrated with the example shown in Fig. 2. The shape of the containers are chosen so that the boundaries coincide with constant coordinate surfaces, the left in cylindrical and the right in rectangular coordinates. Both cases can be solved analytically. Both have the same constant depth h R2 R1 . They also have the same width at the interface pR1 and length p along the y-axis. On the left the interface is cylindrical while on the right it is ¯at. We will compare the spectrum of undamped frequencies xres . The velocity potential takes two forms. When waves stand across the channel, there is no dependence in the y direction and for a curved interface / cos nh arn brÿn with n an integer and a and b determined by the bottom boundary. For a ¯at interface / cos kx a sinh kz b cosh kz; where k np=lx n=R1 . When waves stand along and across the channel, for a curved interface
9. Normal modes: eect of surface shape When the capillary length a is less than about twice the longest span of the surface, the surface is either nearly spherical or cylindrical. This is the usual case for the present application. The former occurs in an open container, the latter in an annulus or torus. The method proposed for an approximate correction for the surface curvature uses a comparison of the normal modes of four geometric shapes: the actual boundaries with curved interface, the actual boundaries with ¯at interface, a simpli®ed boundary with the same shape as the actual problem and a simpli®ed boundary with a ¯at
Fig. 2. Eect on normal modes of a curved interface ± a comparison with a ¯at interface.
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/ cos nh cos ky aIn kr bKn kr where k mp=p, with m an integer and In and Kn are the hyperbolic Bessel functions, while for a ¯at interface / cos kx x cos ky y a sinh kz b cosh kz; where kx n=R1 ky mp=p and k 2 kx2 ky2 . The constant b is determined as a multiple of a by the bottom condition. These functions become fairly complicated when the bottom boundary condition is applied. The surface condition provides the relation between the resonant frequencies xres and the dimensions of the container. For the curved and ¯at surfaces these conditions are respectively: r o 1 o2 / o2 / 2 / 2 2 ; xres / q or R21 oy rR1 oh 7 r o o2 / o2 / 2 : xres / q oz ox2 oy 2 z0 For the ¯at interface: x2res r=qk 3 tanh kh. The corresponding equation for the curved interface is very lengthy and will be discussed in a separate report. The modes that are most aected by curvature are those that stand across the channel. They also have the simplest dispersion relation. After the bottom boundary conditions are satis®ed and the appropriate / are inserted into Eq. (7): 2n r n n2 ÿ 1 R2n 2 2 ÿ R1 ; xres q R2n R2n R3 1 8 3 1 2 r n R2 ÿ R1 2 tanh n xres R1 q R31 for the curved and ¯at interfaces, respectively. The curved surface mode n 1 is not allowed. This is also the case when waves propagate on a circular jet, a similar problem. For n 2 and R2 2R1 the resonant frequency of the ¯at surface is 20% higher than the curved surface. As R2 and n increase, the resonant frequencies get closer together. The eect of the curvature in this worst case situation is small. A few calculations on waves standing along the channel show a much smaller eect of the curvature.
of the examples can be transformed easily into a sliced container. However, a rough estimate of the eect of variable depth is useful. Recall that this eect is negligible if h > k=2. Most reported studies on the eect of variable depth deal with waves in V-shaped channels. Perhaps this is the worst case situation for our problems. In Fig. 3, we compare the spectrum of a V-shaped container with walls sloping at 45° with that of a rectangular tank with the same interface length and depth. There are two forms of the velocity potential, one for waves standing across the channel and the other for waves standing both across and along the channel. The former is most sensitive to the bottom conditions and is selected to illustrate the eect. Several calculated data points using the latter indicate considerably less dependence on variable depth for waves traveling along the channel. The velocity potential for the V-shaped container is / A cosh ky cos kz cos ky cosh kz B sinh ky sin kz sin ky sinh kz:
9
The ®rst term is associated with symmetrical waves while the second is related to antisymmetrical modes. There are no boundary conditions that select k. When Eq. (9) is inserted into Eq. (7) evaluated at z h, the requirement on k is tanh kh tan kh. The negative sign is used for symmetrical and the positive for antisymmetrical modes. After k is determined, another requirement is that x2res r=qk 3 tanh kh for symmetrical modes and x2res r=qk 3 cotanh kh for unsymmetrical modes. For the comparison rectangular channel x2res r=qk 3 tanh kh. The wave numbers for the rectangular channel are k np=2h with n an integer. The V-channel has normal modes with kh 0:79 np=2. Thus, the resonant frequency of the lowest mode for the V-channel is about twice that for a rectangular channel with the same interface. As n increases, the resonant frequencies converge. For this case, at least, the eect of variable depth is large. Here, the ratio of variable depth to depth is one. This is not likely to occur in actual dewars until they are nearly empty. The eect is probably much smaller in actual dewars but this is not
10. Normal modes: eect of variable depth The depth correction can be approximated using the same method used for a curved interface. The surface correction and the depth correction are additive. For the present problem we need solutions in a sliced cylinder or annulus. In LambÕs review [3] of the literature he ®nds solutions for sloping beaches (edge waves) and for Vshaped channels. These can be translated into capillary wave problems by applying the interface condition for capillary waves to the / found for gravity waves. None
Fig. 3. Eect on normal modes of variable depth ± comparison of a Vshaped on a rectangular channel.
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
obvious. It may be necessary to resort to variational methods for most dewars because comparable analytic solutions are not available and the eect is not negligible. 11. Amplitude reduction methods There are two methods commonly used to reduce sloshing: baes and foam. Baes have two eects. They increase the surface area of the boundary layers and thereby increase c. The also move the spectrum of resonance's to higher frequencies. Since the amplitude of a wave at resonance is An x0 =2c and c is proportional to the area of the side wall boundary layers, the amplitude is inversely proportional to the side wall area. End walls do not have boundary layers because the velocity there is zero due to the boundary conditions. Adding end walls increases the resonant frequencies. When the depth h is larger than about k/2 the dispersion relation is x2 r=qk 3 and the boundary conditions require k 2w=n, where w is the width between end walls and n is an integer. Thus, if an end wall reduces w by two, the resonant frequency increases by 23=2 . Since the driving spectrum of ISS increases rapidly with x, a detail calculation is needed to determine if the end wall baes reduce or increase the acceleration of the dewar. Bottom baes are impractical because of varying ®ll level. Baes do not reduce the amplitude of resonances in He II because they only reduce the amplitude of the normal component. However, end wall baes can be used to increase the resonant frequency. Foam creates boundary layers of thickness m=x1=2 around the ®bers of foam. Damping is proportional to the volume of the boundary regions and thus to the total length of the ®bers. Foam produces a uniform distribution of shear throughout the liquid. We expect that the formula for side wall layers will apply with l equal to the perimeter of all ®bers in a cross section. 1=2 Thus the additional cf is proportional to both m=x and the length of all the foam ®bers in the dewar. It will be necessary to determine cf experimentally. Again, this method will not reduce waves in He II because only the normal component of the super¯uid is damped and the super¯uid component will oscillate unaected.
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12. Summary A procedure is presented to calculate the eect of cryogens sloshing in a dewar on the variable component of acceleration experienced by an experiment situated in the dewar. The various steps in the process are discussed. The model is specialized to conditions that may apply on the ISS: surface tension forces are much stronger than gravity and the vibration of ISS is suciently low so that sloshing may be treated as linear surface waves. The aim is to use these conditions to simplify previous results whenever possible. Previous published results are reworked to meet these constraints. It is assumed that the liquids are cryogens. A simple dynamic model is developed that directly couples the sloshing and motion of the dewar through the transfer function of the suspension to the motion of ISS. A simple method is discussed to ®nd the shape of the free surface. Simple methods of estimating the eect on the normal modes of the surface waves of variable depth and a curved surface are presented.
Acknowledgements This work was partially funded by the Jet Propulsion Laboratory on project LTMPF contract 960464. References [1] Khalatnikov IM. Introduction to the theory of super¯uidity. New York: WA Benjamin, 1965. [2] Abramson HN. The Dynamic behavior of liquids in moving containers. NASA SP-106 1966. [3] Lamb H. Hydrodynamics. New York: Dover, 1945. [4] Hastings LJ, Rutherford R III. Low gravity liquid±vapor interface shapes in axisymmetric containers and a computer simulation. NASA TM X-53790 1968. [5] Landau LD, Lifshitz EM. Fluid Mech. Reading, MA: AddisonWesley, 1959. [6] Masica WJ, Salzman JA. Lateral sloshing in cylinders under lowgravity conditions, NASA TN D-5058 1969. [7] Case KM, Parkinson. Damping of surface waves in an incompressible liquid. JFM 1957;2:172±84. [8] Rice SO. Mathematical analysis of random noise. Bell Tel. J. 194445;23:282 and 25:46. [9] Peterson LD, Crawley RJ, Hansman RJ. Nonlinear ¯uid slosh coupled to the dynamics of a spacecraft. AISS Journal 1989;27:1230±40.
Sloshing in microgravity H.A. Snyder a,b,* a
Campus box 429, University of Colorado at Boulder, Boulder, CO 80309-0429, USA b Ball Aerospace and Technologies Corporation, Boulder, CO 80306-1062, USA
1
Received 20 September 1999; accepted 20 September 1999
Abstract The International Space Station provides a low gravity environment for experiments that require very low acceleration. The steady component of acceleration due to the gravity gradient is in the microgravity range. It is possible to achieve microgravity levels for the variable component by using isolation racks. For experiments cooled by liquid cryogens sloshing may increase the variable acceleration at the experiment beyond acceptable levels. Sloshing of cryogens in microgravity can be predicted using a surface wave model. The model should include: a calculation of the shape of the unperturbed liquid±gas interface; a listing of the normal modes and resonant frequencies for the container; a prediction of the amplitude of the modes in response to the motion of the container; and a test to detect the breakdown of linear theory. A model is presented that contains these components. The shape of the interface is calculated and it is found that for most anticipated applications the interface is nearly cylindrical or spherical. Since gravity is not aligned with the symmetry of the container, the depth of the liquid is variable. Examples are presented to show how to estimate the extent of variable depth and curved interface on the normal modes and resonant frequencies. Equations are derived for the dynamic interaction of the isolation rack, the dewar and the sloshing motion. Damping is introduced by using boundary layer theory. Random vibration theory is applied to the incoherent component of the driving spectrum while standard resonance formalism is used for the coherent component. The model cannot be used if the wave amplitude becomes so large that linear theory does not apply. A procedure is developed to check for nonlinear diculties. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Super¯uid helium (b); Fluid dynamics (c); Two-phase ¯ow (c); Space cryogenics (f)
1. Introduction One use of the International Space Station (ISS) is to provide a low gravity environment for experiments that require low acceleration. Many experiments need both an acceleration that is as low as possible and low temperatures. When a cryogen is used to cool a low acceleration experiment, the apparatus is usually placed in a dewar ®lled with the cryogen. Sloshing of the cryogen contributes to the acceleration level of the experiment. The purpose of this research is to describe methods to calculate sloshing of cryogens in dewars when both the constant and variable components of the acceleration are in the range experienced on the ISS and to calculate the acceleration level of the experiment when the sloshing is known.
*
Tel.: +1-303-492-7635; fax: +1-303-492-7881. E-mail address: [email protected] (H.A. Snyder). 1 Correspondence address.
The structure of the ISS is expected to have a variable component of acceleration that is strongly dependent on frequency. It ranges roughly from about 0.5 lg at 10ÿ2 Hz to about 103 lg at 50 Hz. The variable component of the acceleration can be reduced by several orders of magnitude with passive and active isolation racks. For experiments that are sensitive to acceleration, an upper limit is usually 10 lg. Frequently, these conditions must be maintained for long periods of time ± up to several months. The motion of the dewar is quite small at these acceleration levels and sloshing consists of surface waves. The literature on the theory of surface waves is extensive. However, a low acceleration environment leads to conditions not usually encountered in 1-g. Surface tension dominates over gravity and the gas±liquid interface is curved. Since the constant component of acceleration is comparable to the variable component, the interface can become unstable. On Earth an apparatus is usually positioned so that a symmetry axis is aligned with gravity. On ISS the constant component of gravity is
0011-2275/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 9 9 ) 0 0 1 2 0 - 4
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
rarely in the direction of the axis of the dewar. This leads to a problem with variable depth. If sloshing increases the acceleration level of the experiment appreciably, it may be reduced by installing a system of baes or aluminum foam. The change in wave amplitude due to this type of mitigation is analyzed. Helium II presents a special problem. Khalatnikov [1] shows that if the normal component of the super¯uid is not constrained, such as by aluminum foam, the dynamics of surface waves at a helium interface is the same as that of an ordinary ¯uid. Even when the normal ¯uid is immobilized by viscous drag, the super¯uid propagates surface waves with a slightly modi®ed dispersion relation. Thus, it is not possible to use conventional methods to reduce sloshing of super¯uid helium. The approach presented below is analytic. It appears that computational techniques presently available are inadequate for several reasons. The problem is threedimensional and lacks symmetry because the axis of the apparatus is not aligned with gravity and the interface is not ¯at. Most computational methods are restricted to two dimensions or to situations where there are symmetry elements. The most important results of this type of calculation are the resonant frequencies and amplitudes at resonance. Damping is low and the tuning curves are very narrow. A computational method that searches for the resonances would be very inecient. The problem is resolved into several topics. First, a standard method is used to calculate the shape of the gas±liquid interface. Then, the stability of the surface is analyzed for the case when the constant and variable components of acceleration have the same order of magnitude. The coupling between the motion of the wall of the container and the sloshing is considered next. Damping is introduced into the governing equations using methods common in acoustics. The tuning curve for resonance is derived from the damping term. The interaction of the isolation rack, the dewar and sloshing is analyzed in terms of transfer functions. When resonance occurs, the waves may attain large amplitudes. The conditions when the normal mode analysis lacks accuracy and a wave may break are described. Most of these calculations use the spectrum of the normal modes for surface waves of the container. The eect of a curved interface and variable depth on the normal modes is outlined in the next three sections. Finally, two methods to reduce sloshing are considered: baes and aluminum foam. Most of the work on sloshing for space applications is concerned with fuel management in rockets. Here, the gravity level is high and the amplitude can be large enough to require a nonlinear analysis. This research was published mainly in the 1960s. Abramson [2] edited an extensive review of this work. Many of the methods used below are based on material presented in that review and the classic text by Lamb [3]. The aim is to distill
from the previous work that which applies to the present problem and to synthesize a systems level procedure for calculating the acceleration of the experimental chamber given the acceleration of the ISS. Whenever possible an attempt is made to simplify the previous methods. Thus, analytic methods are preferred over variational and computational calculations. Since most of the earlier work assumes that surface tension is negligible, the results are reworked for the present situation. The normal modes for gravity and capillary waves are the same but the interface condition diers and the new dynamical equation has to be applied to the normal modes to get the dispersion relation. The earlier work also assumes that the motion of the container wall is known and that the sloshing does not react directly on the container. Equations are derived that couple the motion of the dewar and the sloshing. Also, the previous work relied on variational calculations to ®nd the eect of a curved interface and variable depth on the normal modes. In order to simplify the calculation, a method is developed to use analytic solutions to approximate these corrections. The presentation is kept suciently general so that it is helpful in solving most problems in the design of the ISS that involve sloshing.
2. Surface contour The shape of the quiescent gas±liquid interface is determined by a balance of two forces: surface tension and the constant component of acceleration or gravity. On the ISS the axis of the container is usually at an angle with the gravity vector. There are two coordinate systems that are convenient to describe the force balance: one with the z-axis along g and the other aligned with the z-axis of the container. In the latter case, the g vector can be resolved into a component along the z-axis gz and a component along the other axis gx . When the z is aligned with g, the equation to be solved is: z00 2z=a2 1 z02
3=2
;
where a is the capillary constant a2 2r=qg with r the surface tension, g the acceleration and q the density of the liquid. For all cryogens the contact angle, h when in contact with most materials used for the walls of dewars, is near zero. Since h 0 the boundary conditions on this equation are that z0 cot / at z d=2, where / is the angle of inclination of the container. The most general procedure to ®nd the shape of the interface appears to be due to Hastings and Rutherford [4] who derived a general set of dierential equations. They used a Runge±Kutta shooting method to solve them. We found the shooting method frequenstly to be unstable for these equations. The method can be
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
simpli®ed using a method by Landau and Lifshitz [5] that has always succeded for our calculations. The equation for z00 is integrated once and the undetermined integration constant is called A. 1 1=2
1 z02
Aÿ
z2 : a2
1 0
Then apply the conditions z cot / at x d=2 in 1=2 Eq. (1) to ®nd zÿd=2 zd=2 a A ÿ sin / . Then, 0 solve Eq. (1) for z and integrate again: Z z A ÿ z2 =a2 dz h i: 2 x 2 zd =2 1 ÿ A ÿ Z 2 =a2 Before Eq. (2) can be integrated we need A. In Eq. (2) set 1=2 x d=2 and z a A ÿ sin / and use an equation solver to ®nd A. Then, integrate Eq. (2) numerically to get x versus z and plot z versus x. This problem diers from the rise of liquid in a capillary tube because the amount of liquid is constant. The reference line z 0 has to be chosen so that there is an equal volume of liquid above and below z 0. For the other approach x is aligned with the axis of the dewar. In the new coordinates the equation to be solved is z00 2z=a2 1 z02
3=2
gx =gx gz =gz:
Now the boundary conditions are
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the length of the interface is suciently small relative to a so that no calculation is necessary. 3. Stability An interface can become unstable with plumes of gas plunging into the body of the liquid. It is known that a cylinder that is accelerated along the axis with the acceleration vector pointing from the gas to liquid is unstable if h 0 and the Bond number exceeds 0.85. The Bond number is qg0 R2 =r, where R is the radius of the cylinder and g0 is the additional acceleration beyond that which determines the shape of the static meniscus. Here, g0 is a constant vector. This experimental value of Masica and Salzman [6] agrees with the stability calculation of Reynolds and Satterlee [2]. The most sensitive cryogen for instability is liquid helium. For a cylindrical dewar of helium with R 1 m, instability occurs when g0 70 lg. It is unlikely that instability is a problem for ISS experiments for several reasons. First, the isolation racks are designed to reduce g0 to less than about 10 lg at the dewar. Secondly, the disturbance is oscillatory and is destabalizing during half the cycle and stabilizing during the other half. Also, the frequency of the disturbance is high compared to the rate of growth of the instability. Furthermore, the disturbance is not concentrated along the axis of the dewar but is random in direction.
z0 cot h 1 at x d=2: This equation is more dicult to solve than the previous equation for z00 because it does not have a ®rst integral that can be represented by simple functions. However, it should be solvable by numerical methods. But when conventional numerical schemes are applied, it shows the same instabilities as the described above. The technique of Landau and Lifshitz using the z-axis along g seems to be the best choice for ®nding the shape of the interface. All of this may be an academic exercise in many situations. The capillary constant a has dimensions of length. If the distance across an interface is of the order of a or less, surface tension dominates the force balance and the surface is either cylindrical or spherical. If the diameter of a cylinder is d and d a, calculations show a deviation from a spherical surface of less than 1%. When d 2a, the dierence is about 5%. Thus, if the distance across the interface is less than about 2a, a calculation of the shape is not necessary. Liquid helium has the smallest value of a of all the cryogens. For a gravity level of 5 lg, a 0:3 m and it is weakly dependent on temperature. Interfaces up to 0.6 m can be approximated as cylindrical or spherical for helium. The next smallest value of a for a common cryogen is nitrogen with a 0.55 m. For most applications on ISS
4. The driving force When the driving disturbance is small, as here, the motion of the interface can be represented by a sum of harmonic surface waves. This theory is linear in the velocity and cannot predict the distortion and breaking of the waves. Also, it cannot represent the motion after a wave breaks. We will use surface wave theory as the basis of analysis. The normal mode amplitudes will be calculated. Then, we will use nonlinear wave theory to check for breaking. Large amplitudes only occur at resonance. Damping is small compared to the inertial terms of the governing equations. The analysis can be simpli®ed considerably by calculating the normal modes using inviscid equations and adding damping when the frequency response is calculated. That is the approach used here. For small amplitude inviscid surface waves the velocity ®eld can be derived from a potential: v r/ and r2 / 0. This equation is independent of the size and shape of the container. At all solid boundaries the velocity normal to the boundary vanishes, o/=onjwall 0. The free surface boundary condition will be discussed below. The geometry of the container determines the
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
normal modes. There are no body forces for this application and the ¯ow is driven by motion of the container. Thus, the velocity of the ¯uid at the wall of the container equals the velocity of the container. We will assume that the dewar is suciently sti so that the motion of the wall is the same everywhere ± the vibrational modes of the dewar do not have to be considered. The driving forces of the walls are included in the theory by writing / as a sum: /total / /wall , where /wall Vx x Vy y Vz z and Vwall (t) is the vector velocity of the wall. The boundary condition on / is as above o/=onjwall 0 at the mean location of the wall. Then, o/total =onjwall Vwall t and the normal modes are calculated as if the container were stationary. When the normal modes are found, /wall is expanded in a series of normal modes. The amplitude of the normal mode component of /wall determines the amplitude of that normal mode in /. Many normal modes do not get excited by motion of the wall because they do not occur in the series expansion of /wall . The most general motion of a dewar consists of the three translations included in Vwall (t) and three rotations. For rotation the components of Vwall (t) are not the same at all locations on the wall as they are assumed above. The distinction between translation and rotation is only signi®cant when the dewar is close to an axis of rotation. The ISS does not rotate at frequencies above 10ÿ2 Hz and an axis of rotation is unlikely to be near the dewars being considered. It appears that rotation of the container can be neglected. 5. Damping and resonant amplitude Inviscid theory predicts an in®nite amplitude when Vwall (t) excites a resonance. Damping reduces the frequency response near resonance. Consider a normal mode with wave number k k 2p=k, where k is the wavelength) and circular frequency x excited to an amplitude A0 by the motion of the wall. When the excitation is turned o, the amplitude decays as A t A0 exp ixt ÿ ct, where c is the damping constant. Damping is caused by viscous shear in the wave and in boundary layers at the wall. There are three additive contributions to c. There is shear in: the wave in the body of the ¯uid cw ; the side wall boundary layers perpendicular to the propagation of the wave cs ; and the bottom boundary layer cb . Cane and Parkinson [7] published a method in which the linearized boundary layer equations are solved exactly for gravity waves. When this method is applied to our case, the damping cs is proportional to the area of the sidewalls and cb is proportional to the bottom area, both scaled by the volume. The method was adapted to capillary waves and the approximation of wall areas is used.
The wave contributes cw 2mk 2 to c, where m is the kinematic viscosity of the liquid. This eect is usually small except in large tanks with small surface to volume ratio. The boundary layer at the wall has thickness 1=2 m=x . At a frequency of 10ÿ2 Hz the thickness is about 3:8 10ÿ4 m for liquid helium and 15 10ÿ4 m for liquid hydrogen. For other cryogens it is smaller. The side wall and bottom boundary layers have dierent eects on c. This is due to the decay of the wave amplitude with depth; in most cases this is exp kz. For the side walls cs L=S mx=81=2 , where S is the area of the cross-section perpendicular to the propagation of a normal mode and L is the perimeter of the side wall in this cross-section. The bottom boundary layer adds 1=2 2 cb w=S mx=8 1 ÿ exp ÿkh=kh , where w is the width of the bottom and h is the depth. When h > k=3 the bottom boundary layer can be neglected. Thus, c cw cs cb . Each excited normal mode has a c associated with it. The decay constant c for a tank without baes 0:5 m 0:5 m 0:5 m at a driving frequency of 1 Hz is about 1 10ÿ3 =s for helium and 5 10ÿ3 =s for hydrogen. The decay constant c determines how long it takes a mode to grow to equilibrium amplitude and to decay. The time constant is 1/c. Damping causes the resonant frequency of a mode xres to be lower than the undamped frequency x0 by x2res x20 ÿ 2c2 . It also modi®es the frequency response and prevents a large amplitude at resonance. It is customary to relate c to the quality factor of a normal mode: Q xres =2c. For the tank of the last paragraph the Q factors are about 3000 for helium and 700 for hydrogen. We may expect Q factors of about 1000 for most applications without baes. The tuning curve for each normal mode driven by amplitude Am is A x
x20
Am x20 ; ÿ x2 ix0 x=Q
3
where Am is the amplitude at a frequency far below resonance. At resonance the amplitude is Am Q. The width of a resonance at half height is Dx x0 =2Q. The displacement of the dewar that drives the waves consists of a spectrum of frequencies. Each small interval of the driving spectrum drives all the normal modes. If the driving frequency is about x0 /Q below resonance, the response is Am . When it is about x0 /Q above resonance the response is negligible. The response spectrum is a series of narrow tuning curves for each normal mode. They do not overlap because the values of Q are of the order of 1000. The output spectrum is a product of the driving spectrum and the response spectrum and represents steady state conditions. Since the driving force is made up of three components Vx , Vy and Vz , each spectrum consists of three plots. The relation between the driving spectrum of the dewar and the spectrum of the ISS will be discussed in
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
the next section. The driving motion of the ISS is not a continuous harmonic signal. Part is random and part is coherent. The steady state analysis only applies to the coherent part of the signal. The time to build up a resonance is about 4/c. This is called the settling time. For helium this ranges from 500 to 5000 s and for hydrogen 100±1000 s. If the coherence time is less than this, a resonant driving signal does not produce the resonant amplitude. Random signal theory [8] provides a method to calculate the output when the coherence time is less than the settling time. The response and output spectra have the same shape for coherent and random signals but the amplitude for random signals is much smaller. If the ISS spectrum is a root mean square power spectrum and is smooth, the data refers to incoherent signals. Then the amplitude of the acceleration driving the system can be read directly from the spectrum and used as described below. If it is known that there are coherent sources, they will appear in the power spectrum as large spikes. Usually the coherent part of the spectrum is treated separately and the power spectrum does not contain these spikes. Then, it is necessary to calculate the response at these frequencies separately. We cannot calculate the wave amplitudes from the ISS spectrum until we consider the motion of the dewar.
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similarly. Whenever a procedure is described below in one dimension, it is assumed that the other two dimensions can be treated similarly. In the early work on sloshing the dewar is assumed to be attached to the spacecraft rigidly. The motion of the spacecraft is estimated, the sloshing is calculated from this, the net pressure force resulting from the sloshing is applied to the dewar and the motion of the spacecraft is recalculated. There is no direct coupling between the sloshing and the motion of the dewar. Peterson, Crawley and Hansman [9] showed that under some conditions the results of a coupled and uncoupled model can deviate considerably. These authors developed a nonlinear coupled model for the sloshing of fuel. We do not need a complicated nonlinear model because the driving displacement is very low. A simple linear model can be made by summing the forces on the dewar in each coordinate direction. The model consists of the solid dewar Ms connected through the transfer function Tr to the displacement of the ISS structure and with the forces Fx and fx applied to Ms as in Fig. 1. If Tr consists of a simple spring jr and damping Cr , then Ms Md Xd00 ÿCr Xd0 ÿ jr Xd fx Cr X 0 jr X : The spectrum of ISS w(x) provides the data to ®nd X : X x ÿw x=x2 :
6. Acceleration of the dewar due to sloshing The purpose of this model is to ®nd the displacement of the dewar Xd , Yd and Zd in terms of X, Y and Z, the displacement of the mounting point to ISS. The dewar is attached to the structure of the ISS through an isolation rack. The motion of ISS is transmitted through the rack according to a transfer function Tr (x). The liquid has a component of solid body motion equal to that of the walls. It is represented in /total by /wall . The force on the dewar needed to maintain Vwall (t) is equal to the pressure produced by /wall integrated over the surface of the wall in the three coordinate directions. Since p ÿqo/=ot ÿiqx/, this force in the x direction is Z Fx ÿixq /wall dy dz exp ixt:
If active control is used by sensing and feeding back Xd , an additional term in Xd occurs in the above equation. If a more complicated Tr is used, it is easy to modify the above equation. Now we have to ®nd / and /wall and eventually fx in terms of Xd The analysis follows the procedure described above. The normal modes are found and represented as /. Then, /wall is expanded in the normal modes. Each term in the expansion is proportional to Vx . At this point the amplitudes of the normal modes of /wall are known and / are unknown. The interface boundary condition connects these amplitudes. The interface boundary condition is o2 /total =ot2 ÿ r=qo=ozr22 /total 0
at z 0:
Here, z is perpendicular to the interface and positive toward the gas while r22 refers to the two coordinates parallel to the interface. This equation can be rewritten
Furthermore, /wall Vx x and the integration results in Fx ÿixMd Vx x2 Xd Md , where Md is the mass of liquid in the dewar. The surface waves also produce a force on the walls of the dewar Z fx ÿixq / dy dz exp ixt #; where # is the phase dierence between the motion of the wall and each sloshing mode. This is a one-dimensional model. The other two dimensions are treated
Fig. 1. Dewar suspended by isolation rack and driven by liquid pressure.
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
as x2 ÿ r=qk 3 / ÿx2 /wall at z 0. Here we assume the time dependence is exp ixt and a space dependence of exp ikx kz. The wall term is the driver and the resonant frequency of the normal mode of the surface wave is evident. The resonant term is modi®ed by the addition of damping as explained above. Since / and /wall are expanded in the same normal modes, we can match terms in the equation. If wn is a normal mode and An Vx is its amplitude in / and an Vx the amplitude in /wall , then, with x20 r=qk 3 , jAn j ÿan x2 = x2 ÿ x20 2 xx0 =Q2 1=2
4
with a phase dierence p ÿ arctan xx0 =Q x20 ÿ x2 . It is easy to show that Z fx x2 Xd Rn An qWn dy dz from its de®nition. When fx is replaced in the dynamic equation, we ®nd xx 2 2 Xd x xr ÿ x i Qr R P 2 Wn dy dz q 2 n an x xd x ÿx 2 2 Ms Md xn ÿ x ixn x=Qn x2r ixr x=Qr X x:
5
Here, xr and Qr refer to the resonant frequency and quality factors of Tr and xn and Qn to the normal modes. This is an algebraic equation for Xd (x) in terms of X(x). At a chosen frequency only the closest resonance xn need be included in the summation because the resonance are very narrow Dx xn =2Q. Eq. (5) appears to be complicated but it has a simpli®ed interpretation. If surface waves are suppressed, the dewar oscillates with the mass Ms Md on the suspension. When the surface waves are present, the second term changes the apparent mass and thus the resonant frequency and the damping of the suspension. At resonance the fractional change in mass is Z qan Qn
Wn dy dz= Ms Md :
The apparent mass changes rapidly across the tuning curve and is positive below resonance and negative above resonance. If the apparent mass becomes as large as Ms Md , the system is unstable. After Xd is found by the above procedure for each signi®cant normal mode, the acceleration spectrum of the dewar is ÿx2 Xd x. This is usually the desired result. But the problem was treated as linear and this implies that superposition of normal modes holds. We have to verify that the amplitudes do not exceed this assumption.
7. Waves of large amplitude For linear theory to hold, some nonlinear terms are omitted in the governing equations and the boundary conditions. [3] It is assumed in deriving linear theory that fmax < k, where fmax is the maximum amplitude of the interface. Experience with other nonlinear systems and experimental results leads us to believe that the nonlinearity does not become noticeable until this limit is exceeded by a factor of two or three. The ultimate nonlinear eect is the breaking of waves. It has been shown [3] that when kh P 2 a wave does not break until fmax 140k. For waves on a sloping beach, breaking occurs when fmax h. For a particular dewar we have to estimate fmax and compare with these limits. The relation of=ot o/=oz at z 0 implies ixfmax o/=ozjmax;z0 . First, we have to use the ISS spectrum to ®nd the an , then the An at resonance and then o/=ozjmax;z0 . As an example consider a rectangular container of depth h with h > k. The length in the x direction is lx and the coordinate origin is midway between the ends; the z axis is upward. The container is oscillating in the x direction at a frequency x1 and at the point of mounting to ISS w x1 ÿx21 x1 . The lowest normal mode is the easiest to excite and has the lowest damping. Therefore, it is probably the mode with the largest amplitude. Let x1 be the resonant frequency of that mode. Since Vx x is an odd function only sine terms appear in /wall and these are the only modes excited in /. Then, / A1 Vx W1 A1 Vx sin px=lx exp pz=lx exp ix1 t # and /wall a1 sin px=lx exp ix1 t with a1 4lx =p2 : At resonance A1 a1 Q1 . Applying the above equation for f; jfmax j a1 Q1 p=ix1 lx Vx 4QR1 =pXd . We have to ®nd Xd from Eq. (5). Next we need W1 dy dz 2lx ly =p. The Tr should have a resonance far from the most dangerous normal mode resonance. Thus, we can neglect damping in Eq. (4) for the suspension. Substituting in Eq. (5) we ®nd Eq. (6): Xd x1
h
x2r
8l2 l Q
q x2r ÿ x21 1 xpy3 1 Ms M L 3 r p 4 jfmax j Q1 Xd x1 : x21 q lx p
i X x1 ; 6
The term with Q changes sign from positive for slightly below resonance to negative just above resonance. We now have a formula for fmax in terms of known parameters. The next step is to compare fmax with k and
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
make a judgement if linear analysis is adequate. This procedure has to be repeated for each normal mode. 8. Normal modes: general considerations The normal modes of surface waves are solutions of r2 / 0 that satisfy the boundary conditions. The equation is separable in all the common coordinate systems and the normal modes are products of function of one coordinate. One of the product functions is always oscillatory, like trigonometric or Bessel functions, and the other is always a monotonic function, like exponential and hyperbolic Bessel functions. The side wall and bottom boundary conditions determine the shape of the normal modes while the interface condition provides a relation between the frequency and the wave number. When the depth h is large compared with the wave length k, h > k=2, the velocity ®eld decays downward from the surface as exp kz (z positive upward) and the bottom boundary shape has negligible eect on the normal modes. When all the walls of the dewar and the interface coincide with a common coordinate system such as cylindrical or spherical, the normal modes are easy to calculate. For the present application the side walls usually conform to this requirement but the interface and bottom boundary do not. The interface is either cylindrical or spherical but with a dierent axis than the walls. When the axis of the dewar is not aligned with gravity, the depth h varies ± frequently as cos nh if the dewar is cylindrical. The normal modes for a curved interface and/or variable depth generally cannot be found analytically. In this case the problem is three-dimensional and numerical methods, with present computing capability, are also inadequate. Variational methods are the best approach but are frequently time consuming. However, many problems are suciently close to an analytic solution that an approximation is in order. We will illustrate the eect of a curved interface and variable depth in Sections 9 and 10.
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interface. The boundaries of the last three must be chosen so that they coincide with constant coordinate surfaces and thus are solvable by analytic methods. There must also be an obvious mapping of the normal modes for each case with the others. This is best illustrated by an example. For an annulus the surface is nearly cylindrical. An annulus with ¯at interface is solvable analytically. The annulus can be straightened to a channel with the width of the annulus and the length of the mean circumference. If the bottom has the same curvature as the interface, the depth will be constant. If the interface is ¯at, the bottom is parallel to the surface. Three of these cases are solvable analytically and there is an obvious mapping of normal modes. Solve the cases of a channel with ¯at and with curved interfaces and make a table of the ratio of resonant frequencies of the curved interface and the ¯at interface. Then solve the annulus with ¯at interface and apply the correction for curved surface to each normal mode found for the channel. The eect of surface curvature is illustrated with the example shown in Fig. 2. The shape of the containers are chosen so that the boundaries coincide with constant coordinate surfaces, the left in cylindrical and the right in rectangular coordinates. Both cases can be solved analytically. Both have the same constant depth h R2 R1 . They also have the same width at the interface pR1 and length p along the y-axis. On the left the interface is cylindrical while on the right it is ¯at. We will compare the spectrum of undamped frequencies xres . The velocity potential takes two forms. When waves stand across the channel, there is no dependence in the y direction and for a curved interface / cos nh arn brÿn with n an integer and a and b determined by the bottom boundary. For a ¯at interface / cos kx a sinh kz b cosh kz; where k np=lx n=R1 . When waves stand along and across the channel, for a curved interface
9. Normal modes: eect of surface shape When the capillary length a is less than about twice the longest span of the surface, the surface is either nearly spherical or cylindrical. This is the usual case for the present application. The former occurs in an open container, the latter in an annulus or torus. The method proposed for an approximate correction for the surface curvature uses a comparison of the normal modes of four geometric shapes: the actual boundaries with curved interface, the actual boundaries with ¯at interface, a simpli®ed boundary with the same shape as the actual problem and a simpli®ed boundary with a ¯at
Fig. 2. Eect on normal modes of a curved interface ± a comparison with a ¯at interface.
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H.A. Snyder / Cryogenics 39 (1999) 1047±1055
/ cos nh cos ky aIn kr bKn kr where k mp=p, with m an integer and In and Kn are the hyperbolic Bessel functions, while for a ¯at interface / cos kx x cos ky y a sinh kz b cosh kz; where kx n=R1 ky mp=p and k 2 kx2 ky2 . The constant b is determined as a multiple of a by the bottom condition. These functions become fairly complicated when the bottom boundary condition is applied. The surface condition provides the relation between the resonant frequencies xres and the dimensions of the container. For the curved and ¯at surfaces these conditions are respectively: r o 1 o2 / o2 / 2 / 2 2 ; xres / q or R21 oy rR1 oh 7 r o o2 / o2 / 2 : xres / q oz ox2 oy 2 z0 For the ¯at interface: x2res r=qk 3 tanh kh. The corresponding equation for the curved interface is very lengthy and will be discussed in a separate report. The modes that are most aected by curvature are those that stand across the channel. They also have the simplest dispersion relation. After the bottom boundary conditions are satis®ed and the appropriate / are inserted into Eq. (7): 2n r n n2 ÿ 1 R2n 2 2 ÿ R1 ; xres q R2n R2n R3 1 8 3 1 2 r n R2 ÿ R1 2 tanh n xres R1 q R31 for the curved and ¯at interfaces, respectively. The curved surface mode n 1 is not allowed. This is also the case when waves propagate on a circular jet, a similar problem. For n 2 and R2 2R1 the resonant frequency of the ¯at surface is 20% higher than the curved surface. As R2 and n increase, the resonant frequencies get closer together. The eect of the curvature in this worst case situation is small. A few calculations on waves standing along the channel show a much smaller eect of the curvature.
of the examples can be transformed easily into a sliced container. However, a rough estimate of the eect of variable depth is useful. Recall that this eect is negligible if h > k=2. Most reported studies on the eect of variable depth deal with waves in V-shaped channels. Perhaps this is the worst case situation for our problems. In Fig. 3, we compare the spectrum of a V-shaped container with walls sloping at 45° with that of a rectangular tank with the same interface length and depth. There are two forms of the velocity potential, one for waves standing across the channel and the other for waves standing both across and along the channel. The former is most sensitive to the bottom conditions and is selected to illustrate the eect. Several calculated data points using the latter indicate considerably less dependence on variable depth for waves traveling along the channel. The velocity potential for the V-shaped container is / A cosh ky cos kz cos ky cosh kz B sinh ky sin kz sin ky sinh kz:
9
The ®rst term is associated with symmetrical waves while the second is related to antisymmetrical modes. There are no boundary conditions that select k. When Eq. (9) is inserted into Eq. (7) evaluated at z h, the requirement on k is tanh kh tan kh. The negative sign is used for symmetrical and the positive for antisymmetrical modes. After k is determined, another requirement is that x2res r=qk 3 tanh kh for symmetrical modes and x2res r=qk 3 cotanh kh for unsymmetrical modes. For the comparison rectangular channel x2res r=qk 3 tanh kh. The wave numbers for the rectangular channel are k np=2h with n an integer. The V-channel has normal modes with kh 0:79 np=2. Thus, the resonant frequency of the lowest mode for the V-channel is about twice that for a rectangular channel with the same interface. As n increases, the resonant frequencies converge. For this case, at least, the eect of variable depth is large. Here, the ratio of variable depth to depth is one. This is not likely to occur in actual dewars until they are nearly empty. The eect is probably much smaller in actual dewars but this is not
10. Normal modes: eect of variable depth The depth correction can be approximated using the same method used for a curved interface. The surface correction and the depth correction are additive. For the present problem we need solutions in a sliced cylinder or annulus. In LambÕs review [3] of the literature he ®nds solutions for sloping beaches (edge waves) and for Vshaped channels. These can be translated into capillary wave problems by applying the interface condition for capillary waves to the / found for gravity waves. None
Fig. 3. Eect on normal modes of variable depth ± comparison of a Vshaped on a rectangular channel.
H.A. Snyder / Cryogenics 39 (1999) 1047±1055
obvious. It may be necessary to resort to variational methods for most dewars because comparable analytic solutions are not available and the eect is not negligible. 11. Amplitude reduction methods There are two methods commonly used to reduce sloshing: baes and foam. Baes have two eects. They increase the surface area of the boundary layers and thereby increase c. The also move the spectrum of resonance's to higher frequencies. Since the amplitude of a wave at resonance is An x0 =2c and c is proportional to the area of the side wall boundary layers, the amplitude is inversely proportional to the side wall area. End walls do not have boundary layers because the velocity there is zero due to the boundary conditions. Adding end walls increases the resonant frequencies. When the depth h is larger than about k/2 the dispersion relation is x2 r=qk 3 and the boundary conditions require k 2w=n, where w is the width between end walls and n is an integer. Thus, if an end wall reduces w by two, the resonant frequency increases by 23=2 . Since the driving spectrum of ISS increases rapidly with x, a detail calculation is needed to determine if the end wall baes reduce or increase the acceleration of the dewar. Bottom baes are impractical because of varying ®ll level. Baes do not reduce the amplitude of resonances in He II because they only reduce the amplitude of the normal component. However, end wall baes can be used to increase the resonant frequency. Foam creates boundary layers of thickness m=x1=2 around the ®bers of foam. Damping is proportional to the volume of the boundary regions and thus to the total length of the ®bers. Foam produces a uniform distribution of shear throughout the liquid. We expect that the formula for side wall layers will apply with l equal to the perimeter of all ®bers in a cross section. 1=2 Thus the additional cf is proportional to both m=x and the length of all the foam ®bers in the dewar. It will be necessary to determine cf experimentally. Again, this method will not reduce waves in He II because only the normal component of the super¯uid is damped and the super¯uid component will oscillate unaected.
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12. Summary A procedure is presented to calculate the eect of cryogens sloshing in a dewar on the variable component of acceleration experienced by an experiment situated in the dewar. The various steps in the process are discussed. The model is specialized to conditions that may apply on the ISS: surface tension forces are much stronger than gravity and the vibration of ISS is suciently low so that sloshing may be treated as linear surface waves. The aim is to use these conditions to simplify previous results whenever possible. Previous published results are reworked to meet these constraints. It is assumed that the liquids are cryogens. A simple dynamic model is developed that directly couples the sloshing and motion of the dewar through the transfer function of the suspension to the motion of ISS. A simple method is discussed to ®nd the shape of the free surface. Simple methods of estimating the eect on the normal modes of the surface waves of variable depth and a curved surface are presented.
Acknowledgements This work was partially funded by the Jet Propulsion Laboratory on project LTMPF contract 960464. References [1] Khalatnikov IM. Introduction to the theory of super¯uidity. New York: WA Benjamin, 1965. [2] Abramson HN. The Dynamic behavior of liquids in moving containers. NASA SP-106 1966. [3] Lamb H. Hydrodynamics. New York: Dover, 1945. [4] Hastings LJ, Rutherford R III. Low gravity liquid±vapor interface shapes in axisymmetric containers and a computer simulation. NASA TM X-53790 1968. [5] Landau LD, Lifshitz EM. Fluid Mech. Reading, MA: AddisonWesley, 1959. [6] Masica WJ, Salzman JA. Lateral sloshing in cylinders under lowgravity conditions, NASA TN D-5058 1969. [7] Case KM, Parkinson. Damping of surface waves in an incompressible liquid. JFM 1957;2:172±84. [8] Rice SO. Mathematical analysis of random noise. Bell Tel. J. 194445;23:282 and 25:46. [9] Peterson LD, Crawley RJ, Hansman RJ. Nonlinear ¯uid slosh coupled to the dynamics of a spacecraft. AISS Journal 1989;27:1230±40.