LOW-GRAVITWf SLOSHING IN RECTANGULAR TANKS

by Franklin T. Dodge

L v I s R. Garza

TECHNICAL REPORT NO. 1 Contract NAS8-24022

Control No. DCN 1-9-75-10061 ( lF j , ( E l ) ( l F ) SwRI Project 02-2578

Prepared for

National Aeronautics and Space Administration George C. Marshall Space Flight Center

Marshall Space Flight Center, Alabama 35812

January 1970

~70-35590 n (THRU) $ (ACCESSION NUMBER)

5 '2-? -- \ =:%. " 3

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(NASA CR OR TMX OR bP N U ~ ' JER) (CATEGORY)

SOUTHWEST RESEARCH INSTITUTE $AN ANTONIO HOUSTON

https://ntrs.nasa.gov/search.jsp?R=19700026274 2019-01-25T23:00:02+00:00Z

S O U T H W E S T R E S E A R C H I N S T I T U T E

Post Of f ice Drawer 28510, 8 5 0 0 Culebra Road

Son Antonio, Texas 7 8 2 2 8

LOW-GRAVITY SLOSHING I N RECTANGULAR TANKS

b r Frankl in T. Dodge

Luis R. G a r z a

TECHNICAL REPORT NO. 1 Contract BAS8-24022

Control No. DCN 1-9-75-10061 (IF), ( S l ) ( l F ) SwRI Project 02-2578

P r e p a r e d for

National Aeronautics and Space Administration George C. Marshall Space Flight Center

Marshall Space Flight Center. Alabama 35812

January 1970

Approvt d:

+

H. N c r m a n Abramson, Director

Department o f Mechanica l Sciences

FOREWORD

This is the seventh in a series of Technical Repr t r dealing with fuel sloshing under low-gravity condi- tions. The previous reports. all issued under Contract NAS8-20290. were: Technical Report No. 2, October 1966; Technical Report No. 4. March 1967; Technical Report No. 5, Decemkr 1967; Technical Report No. 6, February 1968; Technical Report No. 7, February 1969: and Technical Report No. 8, April I M1). A generalized digital cmmputer program for computing the resonance parameters and equivalent mechanical model of low-gravity sloshing in an arbitrary axisymmetric tank has also been documented.

ABSTRACT

Liquid sloshing in rectangular tanks is studied theoretically and experimentally under low Bond number conditions. The static fret. surface stirpe is computed accurately by an approximate technique. and the results are used in the equations of motion for the fluid to determine the sloshing parameters; these equations are solved by Galerkin's method. The natural frequency parameter is found to increase and the equivalent slosh mass to decrease under low Bond number conditions. Nonlinearities in the experimental results prevented a close comparison of theory and test, but the trends of both are similar. Exploratory tests with square tanks show that nonlinear effects prevail also ior reasonably large Bond numben,

iii

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS

PRINCIPAL NOMENCLATURE

I. lNTR@DU(TION

11. ANALYSIS

Static F m Surha Shape Sloshing Allrlysir

111. EXPERIMENTAL RESULTS

IV. CONCLUSIONS

V. REFERENCES

LIST OF ILLUSTRATIONS

Figure

1 Coordinate System and Nomenclature

2 Static Free Surface Parameters

3 Shape of Free Surface for Various Bond Numbers

4 Natural Frequency vs NBo for Depth Ratio of 1 .SO

5 Natural Frequency vs Nso for Depth Ratio of 0.75

6 Natural Frequency vs NBo for Depth Ratio of 0.50

7 Variation of Slosh Mass with Nso and Depth Ratio

8 Experimental Apparatus

PRINCIPAL NOMENCLATURE

Nondimensional quantities are shown in parentheses at the end of the definition of the corresponding diaensiona! quantity

d liquid depth

f static free surface height (F = f/w)

g gravity, or equivalent linear acceleraticn

h dosli wave height (H = h a l t s )

K1 spring constant ir! equivalent mechanical model

KO nondimensional free surface curvature at x = 0

N~~ Bond number, pgw2 l o

m 1 slosh mass in equivalent mechanical model

m~ total liquid mass per unit length of tank

nondimensional variable, see Equation (Sa)

one-half tank width

amplitude of lateral excit~tion

Cartesian coordinates, see Figure 1 (.Y = x/w. Z = z/w)

nondimensional parameter, see Equation (5b)

contact angle

liquid density and surface tension

velocity potential amplitude (@ = el@)

natural frequency (a = mg)

I. INTRODUCTION

Certain present and contemplated space missions rtquire that a large mass of liquid propellant remain in the tanks throughout long durations of reduced gravity flight. It has therefore become necessary to under- stand the dynamics of a liquid having a free surface in reduced gravity environments. As part of a continuing research program, fuel sloshing in low gravity tias been analytically and experimentally investigated for cylin- drical tanks with flat and inverted ellipsoidal bottoms [1,2,3], spherical tanks 131, and oblate ellipsoidal tanks [4]. Several digital cornputer rcutines also have been formulated to alialyze low-gravity sloshing in axisymmetric tanks [5,6,7,8]. The u!ihty of these researches in the nationel spilce effort is apparent.

It seems worthwhile to roundout the research by a brief study of low-gravity fuel sloshing in tanks of square and rectangular planform. Although such tanks are not used in any existing booster, it does appear likely that they will be needed in space transportation systems (STS) such as shuttle vehicles. This report presents the experimental and theoretical results of the study.

11. ANALYSIS

As is usual in theoretical fuel-sloshing research, the assumption of inzompressible and inviscid fluid flow is made. The flow is further assumed to be two-dimensional; the coordinate system and symbol definitions are shown in Figure 1. The geometry of the static free surface, which is symmetrical about x = 0, is described by the function f(x); it is a function of the surface tension a, the density p, gravity g, the tank width 2w, and the contact angle 8 . The slosh wave is described by s(x, t ) and is determined by the fluid dynamics equations; the height of the wave above the static surface is h(x, t ) = s(x, r ) - f(x).

Static F m Surface Shape

A force bdlance on sn element of the f r e ~ surface shows that f(x) must satisfy:

For a zero degree contact angle, the boundary conditions are

Equations (I) through (3) can be solved numerically in a variety of ways but an approximate analytic integration is more appropriate here. Using the substitutions y = df/dx, y dyldf = d2 f/dx2, Equation (1) can be integrated once to give

where boundary condition Equation (2) has been used to evaluate the integration constant. KO is the curva- ture (d2f/dx2) at the center of the tank, and NBO = pgw2 In is the Bond number. Since df/dx = at x = w, it is clear that the wave height at the wall, f,,, , satisfies 1 - KO(f,,,,,Iw) - (112) NBo(f,,,,,/~)2 = 0. Equation (4) can be put in a convenient nondimensional form by the substitutions

whereupon it takes the form

The variablz u runs from u = (1 Cf= 0) at x = 0 to u = 1 (f = f,,,) at x = w . The positive square root has been taken in Equation (6) so that it is restricted to x d 0; the negat.ve square root corresponds to x > 0.

Equation (6) can be integrated approximately whenever 6' 4 1, which will be seen to be valid whenever NBo > 6.* Near the center of the tank (0 d u2 < 10 62 4 1) u2 is much smaller than one; thus, near the centerline, Equation (6) simplifies to:

*Equation (4) can be integrated exactly for the special case of zero gravity, NB0 = 0; the results an f = 1 - J IX~~ KO = 1.

- .-

Figure I . Coordinate System and Nomenclature

This can be integrated to give

after evaluating the integration constant by using u = 0 at x = 0. For the rest of the surface (10 S Z G u2 < I), Equation (6) reduces to

since here 62 is negligibly small compared to u2. Consequently,

where the integration constant hay been evaluated by using u = 1 at x = 1. Equations (8) and (I) both hold for u2 = 10 S2 , which thus gives zn equation defining the unknown S2 in terms ofNBo :

The solution of Equation (1 1) for 15 as a function of NBo is shown in Figure 2, as is the resulting v a l ~ e of KO = 6 \-. It can be seen that h2 < for NBO > 6, so that NBO > 6 can be taken as the range of validity of the proposed solutions. Curves nf the static free surface shape, computed from the above equa- tions, are shown in Figure 3 for typical values of NBo = 10,?0,40, and 75.

Sloshing Analysis

The characteristics of the fundamental vibration mode (slo;hing mode) of the liquid are determined by a potential flow analysis. In nondimensional form, the basic equations governing the amplitudes of the wave motion, H. and the potential. 9, are 111 :

a2cp a2@ V2@ = - -p -= 0 everywhere in the fluid

aP az2

Figw

e 2 S

tat c Free Surface Pa~am

e-ers

HEIGHT OF FREE SURFACE, y l w

The nondimensional quantities are defined in the Nomenclatilre section. In short, the equations are the inco~npressibility condition, the nonpenetration of liquid through the walls and the bottom, the constancy of tile contact angle.* the equivalence of the wave motii,;i and the liquid velocity at the free surface. and Bernoulli's equation writ ten for the free surface.

Equations (12) through(l7),which donot have an "exact" analytic solution in terms of known func- tions, may be solved approximately by Calerkin's method. The approximating functions are taken as the normal modes for sloshing with a flat free surface (F (X) = O): these are

For numerical reasons. the series is truncated after M tenns and the cosh term in the denominator is used as a normalizing factor. Thea,,'~, thus, are the unknowns. Each an, and their sum, satisfes Equations(l2), (l3), and ( 1 4) iden t icali y.

M

The wave height, H = Z a,,Hn, might be computed by substittiting Equation (I 8) into Equation (1 6) n = 1

and collecting terms; but this calculation of H does not satisfy Equation (1 5). at least for each H, term by term. This is due to the asturned an's not beiig the true normal mode fuilctions, term by tenn, of the s l o 6

M .W

ing Presumably, if the sum C a n a n converges to the true as M + w, the sum C aA,, also will con- n = 1 n = l

verge t o the true H and dH/dX. For numerical work, however, it is much more desirzble that each Hn satisfy the correct boundary condition, Equation (1 5). This can be ammplished by expanding H,,, wmputed by substituting Equation ( I 8) into Equation (l6), into a Fourier series, thus

00

H, = 1 b,, sin (m - 112)lrX PI= I

I b,, = 2 J H,,sin(m-I/?)nXdX

0

where. from Equation ( I 6). the H, to be inserted in the integral is

n (n - I I2MF + dfw) H, = n(n - 112) G,

cosh n (n - I /2)(d/w) r is -- wsh n (n - 1 12WF + dlw)

m s (n - I 1 2 ) n ~ dX cosh n (n - 1 /2)(d/w)

Each H, and W,/dX. computed using Equation (l9), now satisfy the correct boundary conditions and. pre- sumably. both converge to the true limit.

The unknown an's and natural frequencies R2 are determined by Galerkin's method. After substituting Equations ! 1 8) and (I 9) into Equation (I 7)- it becomes

'Since dF/dX ==at the wall for a zerodegree static contact angk, the dynamic contact angle, arccot (dF /dX + W l d X ) , wal wmain at zero for any value of W l d Y at the wall. But tuis is 'rue only for d F / U = 00, so that unless 8 = O0 is a s m h point, it must be required that d H / U = 0 at the wall for I!' contact angles.

Multiplying this equation by both @,and a weigt~ti~~g factor [ 1 + ( c I F / ~ X ) ~ ] - ' I2 , and then integrating from 0 to 1 gives

This process yields a set of hl linear equations:

(The weighting function is required in order that A,, =A,, and B,, = B,,.)

The eigenvalues ' ' and eig?nvectorsa, may now be computed by matrix methods. The results presented here are for M = 5, whicn appears to be sufficiently large to assure convergence.

Figures 4 , s . and 6 give values of R2 = 30' w/g for various NBO and depth ratios d/w. It can be seen that the natural frequency is always larger for f i i t e NBO than for the limit asNBo -* -. This is a little different than the results with cylindrical tanks, for which the natural frequency in the range NBo = 40 to 200 or 300 is slightly less than the infiiite NBO limit. (The data points on Figures 4.5, and 6 will be discussed in the next section.)

Knowning R2 and the a,, equivalent mechanical model parameters can be computed. The xcomponent of the slosh force exerted on the tank by the liquid, per unit length of the wall, is

where a, is the x-acceleration of the liqu~d and dm is the element of mass. By using nondimensional variables, and several vector identities to convert a2@/athr into other forms, realizing that V' a = 0, and using the divergence theorem, it can be seen that the force amplitude also is

where a , is an arbitrary amplitude and anla, are the eigenvectors of Equation (23) for the first m d e . Like- wise. the amplitude of the kinetic energy is

The corresponding quantities for a first-mode mechanical model are

where yo is the amplitude of vibration of the "slosh mass" mi, and kl is the spring constant. Comparing Equations (27) and (28) to Equations (25) and (26) gives:

61;nn

~ ' A3N3nb3&I 1WIIlV

N 1VH

)ISNH

IIO - NON

I I

I I

I I

8

2

a

**

*.

wu

u

dd

dd

5 *

****

ZZ

ZZ

aaa#,,,,

0 u

uu

uu

++

+

- uu

u0

ug

gg

gt:E

EE

-

a

--=aYY

YY

a

a5

3k

2Z

gk

=g

gS

Zg

H

O~

~O

0

0

00

-5 2

66

&&

&&

66

66

6&

d

ac -a

a

~e

*e

*o

ma

mm

.a

ao

*

- V

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)

10

8

I -

>

I -ca

I

I C

I

- I

- !2

- I

-a

t -

YI

e

I I

<V

0

0

00

'a

Q

4

cri cr;

4

4

4

c\i

61:nW

' A3N

nb3Id 1WIIlV

N lm

01SN3;N

IO - NON

I I I ?

I -

IZ

I ;

and

where mT = 2pw d is the total mash of liquid in the tank per unit length. The "rigidly attached mass" is mo = mT - m, . The locations of m , and mo also can be computed but are not usually needed.

Figure 7 presents Equation (30) graphically for various NBO and depth ratios. As can be seen, the slosh mass for finite NBO is always less than it is for the limit NBO + -;this is also the case for tanks ot other geometries.

111. EXPERIMENTAL RESULTS

All the experimental data were acquired by using small scale model tanks to obtain an appropriate Bond number similarity to large prototypes. Three different rectangular tacks were used, all having a plan- view width-to-length ratio small enough to insure two-dimensional flow over most of the tank; the exact planview dimensions were 0.2 X 2.0 in., 0.35 X 3.50 in., and 0.5 X 5.0 in. Three different liquids-methanol, acetone and carbon tetrachloride (CCl4)-were used to obtain a wide range of Bond numbers with the three tanks. For comparison, a hrief series of tests also we:e run with three different tanks of square planview, 0.35 X 0.35 in., 0.70 X 0.70 in., and 1.5 X 1.5 in.

The experimental setup, which is shown in Figure 8, is a slightly modified and improved verslon of the basic apparatus used previously and described in References 3 and 4; the test procedures also were similar to the previous ones. With the rectangular tank, the direction of the lateral excitation was Frpen- dicular to the long sides; with the square tanks, the excitation was perpendicular to any two opposite sides. Enough information was taken during each test to plot the slosh force amplitude as a function of frequencv around the first mode resonance; in this way, the resonant (natural) frequency, damping, and peak force could be determined.

The Bond number range covered in the rectangular tank tests was 3.75 (CC14 in the 0.2 X 2.@in. tank) to 23.5 (CC4 in the 0.5 X 5.@in. tank). Ratios of the liquid depth-to-width varied from 0.5 to 1.5, a depth ratio of 1.5 being effectively infinite; the liquid depth is defined as the distance from the tank bottom to the lowest point of the static free surface. (Lowel Bond numbers could have been obtained by using acetonc. or methanol in the 0.2 X 2.041. tank, but the smaller densities of these two liquids would have resulted in slosh forces too small to measure accurately.)

The experimentally determined natural frequencies for the rectangular tanks are shown in Figures 4,5, and 6. It is evident that the data points all fall below the theoretical curves, and some points even fall below the limiting frequency for NBO = 00. On noting the spread of the data for equal Nso but different xo , it is clear that the sloshing was decidedly nonlinear. The extent of the nonlinearity can be realized by comparing the two data points for Nso = 3.75 in Figure 4. For xo/w = 0.030, R2 = 3.76, which gives a natural frequency for this tank of 13.52 cps; but when xo/w = 0.040, the natural frequency works out to be 12.82 cps, a change of 5.5 percent; the predicted natural frequency is 14.3 cps which is 6 percent above the highest experimentally determined frequency. The spread of the data for the higher NBO's is not quite so great, but the discrepancies between theory and test are still significant. In every case, however, the natural frequency increases as the excitation amplitude decreases. Since slosh forces smaller than 0.0001 Ib cannot be measured accurately by the test apparatus, it was not possible to use very small excitation amplitudes in the hopes that nonlinear effects might have been eliminated; consequently, a direct comparison of test and theoiy is not possible, although the qualitative trends of both are the same.

Nonlinearities also prevent a valid comparison of the equivalent mechanical model and the tests, and like- wise the damping values, which were computed on the basis of a linear response, are not accurate. Other test programs, at much higher Bond numbers, involving tank geometries possessing sharp corners also have revealed substantial nonlinearities [9,10,1 I]. In addition, the results of the tests with the square tanks were nonlinear, even out to Bond numbers of over 200. For example, when NBo = 210(CCL in !he 1.5 X 1.5-in. tank), SZ2 varied from 3.05 when xo/w = 0.0054 to 3.09 when xo/w = 0.0027, for a depth-to-width ratio of 1.5; the theoretical value for NBO = w is 3.14. Other Lases are equally nonlinear, and thus detailed results are not presented here.

IV. CONCLUSIONS

The analytical and experimental rcsults of this study of twodimensioral, low-gravity sloshing in rec- tangular tanks s!low, as expected, that the sloshing is qualitatively similar t o that occurring for high Bond numbers(i.e., flat free surfaces). The increase in the natural frequency and the decrease in the amount of liquid participating in the sloshing in low-gravity conditions are perhaps the most interesting analytic results. The severe nonlinearities, or dependence of the slosh parameters upon amplitude, are the most significant test results. Although the trend o,'both test and theory is the same, the nonlinearities encountered in the tests precluded a direct comparison of the tests and the theory.

If rectangular or square tanks are to be employed in STS, the effect of the nonlincar sloshing on guidance and control should be more thoroughly explored, since a discrepancy of over 10 perceiit between linear theory and actual nonlinear behavior, such as was encountered in these tests, might prove to cause trouble during flight. It would be valuable to know if a "size effect" exists between these small-model tests and full-scale prototype behavior such that significant nonlinearities would not occur in large tanks, or, if, alternatively, low Bond member behavior accentuates the nonlinear behavior that always exists (even though usl?diy negligible) regardless of : h ~ size of the tank; the present tests seem to support the latter supposition.

V. REFERENCES

Dodge. F.T. and Gar~a. L.R.. "Esperinle~ttul and Tl~eoreticral Studies of Liquid Sloshing at Simulated Low Gravities." h ~ s . .l.S:llt.'. J. :tpplicd:~ftsc/w~its. 34. No. 3. :September 1967. pp 555-562. (Also. Tecb. Rept. No. 2 . Contract N ASX-202%. Southwest Research Inst itute. Octaber iC)hh.)

Dodge, F.T. and Garza, L.R.. "Simulated LowCravity Sloshing in Cy!indrical Tanks Including Effects of Damping and Small 1.iquid Depth." Prcweedi~~.q oj'rl~e I Hear Tmrrsfi.ru~d Fluid Alec./ut~its Institute. pp b7-79. Stanford University Press. (Also. Tech. Rept. No. 5. Ccntract NAS8-20230, Southwest Research Institute. December 1967.)

Dodge. F.T. and Garza. L.R.. "Simulated LowCravity Slosri;ng in Spherical Tartks and Cylindrical Tanks with Inverted Ellipsoidal Bottoms." Tech. Rept. No. 6. Contract NAS8-20290. Southwest Research Institute. February 1968.

Dodge. F.T. and Garza. L.R.. "S!osh Force. Natural Frequen~y. and Damping of LowCrdvity Slohing in Oblate Eilipwidsl Tanks." Tech. Rept. No. 7. Contract NAS-20-30. Southwest Research Institute, February 1969. (Also. .41AA J. Spcec-tuft & Rockers. to appear.)

Chu, W.H.. "Low Gravity Liquid Sloshing in an Arbitrary Axisymmerric Tank Performing Translational k-llations." Tech. Rep. No. 4. Contract hiASs20-30. Sjuthwest Research Institute. March 1967.

Chu. W.H.. ' l ~ w Gravity Fuel Sloshing in an Arbitrary Axisymmetric Rigid Tank," Tech. Rept. No. 8. Contract XASs20290. Southwest Research Institute. April !969. (Also. Trans. ASIME. J. Applied dfec hanks. to appear.)

Concus. P.. Crane. G.E.. and Satteriee. HY.. 'Small Amplitude Lateral Sloshing in a Cylindrical Tank with a Hemispherical Bottom Under Low Gravitational Conditions," NASA CR-54700. Contract KAS3- 7 1 19. Lockheed Missiles and Space Co.. January 1967.

Concus. P.. Crane. G.E.. and Satt'rlee. H.M.. -Small Amplitude Lateral Sloshing in Spheroidal Yon- tainers Under Low Gravitational Conditions," NASA CR-72500, Contract NAS39704, Lockheed Missiles and Space Co.. February 1969.

Abrarnson. H.N.. Garza. L.R.. and Kana. D.D.. 'Some Notes on Liquid Sloshing in Compartmented Cylindrical Tanh.'*ARS J., 32, Nc. 6. June 1962. pp 978-980.

Ab.amson. H.N.. Chu. W.H.. and Garza, L.R.. "Liquid Sloshing in 45' Sector Compartme~ited Tanks." Tech. Rept. No. 3. Conrract N.9,SS-1555. Southwest Research Institute. November 1961.

Chu. W.H.. Dalzell. J.F.. and Mcdisette. J.E.. '4h:oretical and Experimental Study of Ship-Roll Stabilizatior, Tanks," Joumal of Ship Research. 12. September 1968. pp 165-1 80.