General Disclaimer. One or more of the Following Statements may affect this Document

(1)

General Disclaimer

One or more of the Following Statements may affect this Document

This document has been reproduced from the best copy furnished by the

organizational source. It is being released in the interest of making available as

much information as possible.

This document may contain data, which exceeds the sheet parameters. It was

furnished in this condition by the organizational source and is the best copy

available.

This document may contain tone-on-tone or color graphs, charts and/or pictures,

which have been reproduced in black and white.

This document is paginated as submitted by the original source.

Portions of this document are not fully legible due to the historical nature of some

of the material. However, it is the best reproduction available from the original

submission.

Produced by the NASA Center for Aerospace Information (CASI)

(2)

J _G

JPL PUBLICATION 79-99

Homogeneous Vortex Model for

Liquid Slosh in Spinning

Saherical Tanks

!

Michael fl-Raheb

Pat l Wagner

(NASA-CR-162725) HOMOGENEOUS VORTEX MODEL N80- 16299 .

FOR LIQUID _{SLOSH IN SPINNING SPHERICAL TANK}

S

(jet Propulsion Lab,) 46 P IIC A03/NF AO 1 ,}

CSCL 20n

IInclas

61/34 46971

I

C bo ' _{^N ^a oN}A Ma r ^ •^ a^ w r

November 15, 1979

National Aeronautics, and

Space Administration

Jet Propulsion -Laboratory

California institute of Technology.

(3)

u TECHNICAL REPORT STANDARD TITLE PAGE

i.

Report

No.

J

pr

,

M. 79-99

2. Government Accession No.

3. Recipient

'

s Catalog

No.

4,;

Title and Subtitle

5. Report Date

I[onogoneoua Vortex Model for Liquid Slooh ITovenber 15, 1979

6. Performing Organization Code

in

SpinMng Spherical Tz nIm

7. Author(s)

B. Performing Organization Report No.

Michel El-ltaheb and

Paul 11amor

9. Performing Organization Name and Address

10. Work Unit No.

JET PROPULSION LABORATORY

11. Contract or Grant. No.

California In; ti tttto of Technology

4800 Oak Grove Drive NAS 7-100

13. Type of Report and Period Covered

Pasadena,

California

91103

JPL Publication

12. Sponsoring Agency Name and Address

NATIONAL AERONAUTICS AND SPACE AD14INISTPLATI-ON

14. Sponsoring Agency Code

Washington, D.C. 20546

15. Supplementary Notes

16. Abstract

The prodem

of forced fluid sloshIn" in

a '*yartially filled spinning spher

•

i.ct.l tank is

solved numerically

ucsin

the finite clowent nethod.

The governing equations include

Coriolis accelermti.on

t

em_o:i.ri.ca.l fluid camping and spatia:11v howog

,

Yueous

vor

ti

c

ity

first introduced by Pfeiffer.

An e.-pun.enti al instability similar to flutter is detecte

to

the

present ,inulation for fill ra

•

i,ios below

5ni'o.

This instability ap pears in the

t oriel as a result of the h.or.;ogeneous vortex ansumntion since -the free :slosh equations

4xo neutrally stable in the ld.apunov sensee

4

L11^J,; r J}}t C1G^r

3 [ r.!, 1

l

A

T

O ORRlJ I

NTATl L"

A 7 .{

'`

7tti _{!fit ^^ 1:^}

i

11. t:ey Words (Selected by Author(s))

18.

Di. r%ution Statement

utic;; (General)

,r=

^ *'"'"nica and Heat Transfer

Unclassified - Unlimited

445..

yy // ,_jr cs

•

_{s^vs^ty Cl^rsif.}

(of this report)

20. Security Classif. (of this page)

21. No. of Pages

22. Price

Unclassified

r

(4)

JPL PUBLICATION 79-99

Homogeneous Vortex Model for

Liquid Slosh in Spinning

Spherical Tanks

Michael EI-Raheb

Paul Wagner

November 15, 1979

t

National Aeronautics and

Space Administration

Jet Propulsion Laboratory

California Institute of Technology

4..

Pasadiena, California

(5)

The research described in this publication was carried out by the Jet Propulsion

Laboratory, California Institute of Technology, under NASA Contract No, NAS7-100.

k

(6)

ACKNOWLEDGEMENT

The major portion of this work was performed for and was sponsored

by the Galileo Flight

Project.

The

effort was partially supported by A. Amos and R. Goetz, Office of Aeronautics and Space Technology at NASA.

We wish to express our appreciation to Professor T. K. Caughey of

the

Applied Mechanics Department, California Institute of Technology, for his constructive suggestions regarding

the

stability analysis and

his

help in assessing the initial

slosh

analysis.

Appreciation is also extended to Dr. A.

Pohl

of Ilesserschmitt-Bolkow-Blohm in Munich for his preliminary work.

e

11

(7)

r

F ?

ABSTRACT

The problem of forced fluid sloshing in a partially filled spinning spherical tank is solved numerically using the finite element met;lod. The governing equations include Coriolis acceleration, empirical fluid damping and spatially homogeneous vorticity first introduced

by

Pfeiffer. An exponential instability similar to flutter is detected in the present

simulation for fill ratios below 50%. This instability appears in the model as a result of the homogeneous vortex assumption since the free slosh equations are neutrally stable in the Liapunov sense.

(8)

CONTENTS

I. SUMMARY . . . . 1

II. INTRODUCTION . . . . 2

III. EQUATIONS OF MOTION . . . . 3

IV, SLOSH FORCES AND MOMENTS . . . . 14

V. PROCESS OF SOLUTION . . . . 17

VI. SLOSH DAMPING SIMULATION . . . . 19

VII. NUMERICAL, RESULTS . . . . 22

VIII. STABILITY ANALYSIS OF FREE SLOSHING . . . , . 24

REFERFMCES . . . . 27

Figures 1. Reference and local rotating coordinate systems . . . . . 28

2, , Accelerations on fluid volume . . . . 28

3. Comparison for translational slosh . . . . 29

4, Nondimensional tangential slosh force for case A . . 29 5. Nondimensional tangential slosh force for case B . . 30 6. Nondimensional tangential slosh force for case C . . . . 30

7. Stability boundary . . , . . . , , , , 31

' 8a. Characteristic frequencies for = 0.03 . . . . 32

8b. Characteristic damping for V = 0.03 . . . . 32

9a. Characteristic frequencies for

V

= 0.10 . . . . . . , . . 33

10a. Characteristic frequencies for

V

= 0,22 . . . . . . . 34

lla. Characteristic frequencies for V = 0.35 . . . . 35

(9)

F

_ i gures

11b. Characteristic damping for V - 0,35 . . , . , . . . . 35 12a. Characteristic frequencies for V a 0.43 . . . . 36 12b. Characteristic damping for V - 0.43 , . . . . . 36 13a. Characteristic frequencies for

V -

0,5C . . . . . 37 13b. Characteristic damping for V = 0.50 , , , . . , 37 14a. Characteristic frequencies for

V

= 0.57 . . . . . 38 14b Characteristic damping for V = 0.57 . . . . 38 15. Variation of nondimensional negative damping with

₀

.

39

Table

1. Representative test cases . . . . . . . 23

i

vi

(10)

I. SUMMARY

The Galileo dual-spin Jupiter orbiter spacecraft carries an amount of fluid that represents about half of the total weight. The fluid is contained in four tanks symmetrically located about the spin axis of the spacecraft. In order to evaluate the behavior of the fluid slosh in the spinning configuration as it affects the attitude control, an analysis was initiated early iri the project. An analytical slosh model that accounts for the steady mcecraft spin was performed by Messerschmitt-Bo1kow-B1ohm Company (MB at Munich, W. Germany, as part of the effort of manufacturing the retro-propulsion module for the Galileo spacecraft. The MBB analysis had several simplifying assumptions:

(a) In order to arrive at a practically solvable set of equations, the vorticity was taken to be independent of the spatial

coordinates while it is a function of time only. This model was first introduced by Pfeiffer at MBB in 1974.

(b) The steady-state spin as well as angular perturbation was felt instantaneously in the fluid. This assumption is inconsistent with the mechanism of vortex transfer from the tank wall into the fluid.

(c) The Coriolis acceleration was retained inside the fluid but was neglected in the free surface pressure boundary conditions in order to arrive at an exact differential form.

The assumption of homogeneous vorticity was found to lead to fic-titious instabilities for certain fluid fill ratios. The inconsistency of this model was further substantiated by a stability analysis per-formed by T K. Caughey, who showed that a Liapunov function of the free vibration equations had a vanishing first time derivative leading to a neutrally stable system. It was concluded that the MBB analysis is only valid for fluid fills above 55%.

An equivalent simplified dynamic pendulum model to represent the fuel slosh was proposed for quick assessment of the fluid effect on the attitude control system.. Further investigation by JPL concluded that it is impossible to achieve such a simplified equivalence for rotating tanks where Coriolis acceleration and vorticity prevail. Therefore, a pendulum model can only give an incomplete fluid behavior representation.

A complete analytical investigation of the fluid behavior is now believed to be a very complex undertaking that for cost and schedule reasons is not recommended. Instead, a testing program of the complete spacecraft with its appropriate inertia, fluid fill, and spin rates is suggested to obtain an understanding of the attitude control/fluid interaction.

(11)

II. INTRODUCTION

The problem of liquid slosh in spinning containers has lately received substantial attention as it is an essential step in analyti-cally assessing the destabilizing energy dissipation in dual-spin space-craft, The solution of the governing equations is involved due to the occurrence of the Coriolis term which destroys the self-adjointness of the Euler equations. An alternate set of equations that excludes the pressure gradient can be obtained by applying the curl operator on the Euler equations, thus obtaining the Helmholtz equation of vorticity.

In 1974, Pfeiffer (Ref. 1) introduced the concept of homogenous vorticity; he argued that the spatially independent vorticity assumes the value of its average over the fluid volume. Although the assumption is valid for completely filled ellipsodial cavities, it is only an approximation for a partially filled cavity.

The fluid perturbation velocity relative to rotating coordinates fixed in the center of the tank is expressed by:

u V^ - s" -

^+ ilx

r

where ^ is a velocity potential, ^i is the homogenous vortex vector, and T is the Stokes-zhukovsky potential, The process of solution starts with the determination of trial functions ^ n satisfying the problem;

V 2^ n O in V

za^n'n

2

'5n - _{q ^h}n on SF (free surface)

Hin

Ean = 0 on S w (wetted surface)

where _{On are the eigenvalues of the potential slosh problem. The} poten-tial i satisfies the problem;

V2rp 0 in V 1 " rxn =onS 4.S Fy n F w 2

(12)

Substituting the set {fin, ") and ^ intro ho linearized Helmholz

equation and free surface boundary conditions and using the orthogonal-ity of the 0nl set, we arrive at a system of linear ordinary differen» tial equations in the dependent variables r^( t) and Fn(t) (the general-ized coordinates in the trial function expansion).

The steady spin of the spacecraft is assumed to be felt all over the fluid instantaneously, although the periodic angular motions from control and nutation are only transmitted to a very thinboundary layer of fluid adjacent to the wetted surface of the tack with a radial

thick-ness on the order of7 7

os where

v

is the kinematic viscosity and ^,; the

frequency.

The method of finite elements was adopted in determining Ian, inl as well as the coefficients in the resulting ordinary differential equations. Evaluation of the characteristic eigenvalues indicated that the equations possess an instability for a range o f the angle v (angle formed by the resultant gravitational acceleration and the spin axis). The phenomenon is very similar to flutter in the fact that two eigen-values coalesce in the range of instability. As in the case of flutter where bending and torsion frequencies approach each other, the present motion involves the free ^, , rface and vortex motions. The range of

instability was found to O crease as the fluid volume ratio Nfluid/ stank) increased, up to P. 55 16 volume ratio, beyond which the motion was

stable

for all o.

A stability analysis wa;o then performed on the original governing equations. A Liapunr y functional was established and shown to hkve a vanishing first time derivative. This proves that the motion is neu-trally stable. It would, therefore, appear that the assumption of homogenous vorticity may be true for filling volumes more than 5514, while for lesser volumes this assumption cannot possibly be true, even approximately.

Great care should then be taken in using Pfeiffer's model in flow problems involving intrinsic resonances.

III. EQUATIONS OF MOTION

Let XYZ be a rotating orthogonal coordinate system with origin at the CG of the spacecraft such that the Z corresponds to the spin axis. Let xyz be a local rotating orthogonal coordinate system with its origin at the center of the spinning spherical tank and parallel to XYZ. The angular velocity vector of XYZ with respect to inertial space is:

w R = Q t w (1)

(13)

r T

where tl =

tij

is the steady-state spin about Z,

and

t

-

, is the

perturba-tion angular velocity vector applied to t he spacecraft from control, rigid body, and ela ,^tic appendage motions.

Let R be the position vector from the spacecraft cente ►, of mass C to any point P in the volume V or boundary S of the sphere. Then,

R X R

o

+

`r (2)

R4

is the radius vector from C to the sphere center o, and r is the radius vector from o to any point in V or on S (Figure 1).

The il otal velocity vector at P(x,y

)

z) is:

"v R 3 d (R

o

+r) X

'+u+

h

X

(3)

where vo - dRo/dt, and a is the liquid particle velocity relative to (xy

z)-At this point it is necessary to understand the mechanisms of transfer of vorticity from the boundary of the tank to the contained fluid. Assume that the tank is motionless for t , to and that the spin about Z reaches its steady-state value of no by accelerating smoothly in order not to incur any ripples on the free surface. Vorticity can only be transmitted to the liquid by stresses tangential to the boundary as those generated by viscosity. It can be shown that vortex motion in a viscous fluid is governed by the diffusion equation. It is evident from the solution that the time it takes to achieve a uniform fluid vortex is of the order:

2(^a)

t .. 0(

i4v

where a is the radius of the sphere, and v is the kinematic viscosity of the liquid. Furthermore, enforcing a steady periodic excitation to the same wort • city equation shows that only a thin boundary layer of fluid

is entrained, having a radial thickness of

r

L

aP

(4b)

r

= p

(14)

I

(7)

(81) The results

(aa,

b) demonstrate that although a steady-state spin equilibrium may be reached throughout the viscous liquid after some transient time, small periodic rotations will never be transmitted in the core of the liquid and can, therefore, be neglected.

The above argument explains the use of (aio x i) rather than (aR x `r) in equation (3), since the latter fomi is inconsistent with the proper mechanism of vortex transmission discussed above.

The total acceleration vector at P(x,y,z) is:

_ 6 _ WxAo

+

s'to x (sto X A) +^+M x u ( )

The centrifugal acceleration term so x (sa x R) does not include the perturbation angular velocity u, since th?s latter motion is not felt by the fluid. Using the definition of the substantial derivative d/dt and the identity:

u '

vu

^Vu u x

(v x u) in the momentum equation, wq get:

at

+Tu2 u x(vxu)+Mo x u

(6)

+w gip+fx Ao +std x(stox ^t)

-where p and p are the fluid density and pressure, respectively, and f is the constant body force per unit mass, Superscript ('" denotes the par-tial derivative with respect to time. For an incompressible liquid, the

continuity equation has the form:

v

> V

= 0=> v-

u = 0

We distinguish two sets of kinematic boundary conditions: A Neuman condition on the wetted surface Sw enforcing no flow through the boundary

v n

0==> is . n =

0 on Sw

5

(15)

Let F(R,t) - 0 be th4 equation of the free surface, Then,

dF( ,t) _{aF(R,t) + u}

vF(R,t) = 0 (8b)

dt at

In addition to the kinematic relations, the dynamic condition of constant pressure at the free surface SF is imposed:

p(R,t) = constant on SF (8c)

In general, the relative velocity vector u can be expressed as a combination of an irrotational potential and a rotational term to allow for vorticity:

u

v'P +

x r

(9)

where

2v x v = S1

= st

o

+

is the total vorticity. Using the definition in (10) and relation (3):

2(n + 5 0 = v x v = v x u + v x (50 x "r )

^vxu

= 2(s +5

vx(5 xr)

= 25

f

Therefore, -u x (vxu) = 2Sa x u

Substituting (9) in (7) leads to:

(16)

4

Substituting (9) in (8a), we arrive at the boundary condition i"or (p:

vq n+ (s

x

F) . n = 0

an

-(r x n)

n

on

S

The form of the boundary condition (11) suggests representing S p in the

form

We now introduce the concept of homogenous vortex moticn (Ref. 1). Assume that the vorticity si is independent of the spatial coordinates and is only a function of time. This assumption is an approximation for the partially filled tank, while it is an exact description for rotating motions of ideal fluids totally filling spheroidal containers (Ref. 2). Based on this assumption, (11) can be rewritten as:

Tn an all w

Since the operator on Tj and associated boundary condition can be prescribed arbitrarily, let:

v

2

^

= 0 in V;a =

r

x n

on

Sw

+ SF

(T3)

v2 ^ = 0 in V; a = 0 on Sw

The Stokes-Zhukovsky potentialvector is used to satisfy the inhomogenous boundary condition (11). The scalar potential 1, satisfies a homogenous Neuman condition on S W and, as will be shown later, the linearized free surface condition. Substituting (13) in (10) with the assumption that n 5(t): h

(12)

(14)

u = -vO - s v^ +5 _x r

7

I

(17)

The linearized free surface condition is now derived. At first we define the steady-state equilibrium free surface equation by estimating all time-dependent quantities in the momentum equation (6):

5

o

x (5 x R) = -1 P

o

+ f

where subscript "o" denotes quantities at equilibrium. Rewriting the cross product in the form of a gradient and using the dynamic condition on the free surface (8c):

Po

1(5 x R)z - f R = const. (15)

If the body force is a thrust -go along the Z axis, then

f•R=

-goZk, where k is a unit vector along z. The steady-state free surface equa-tion becomes:

F`o(X^Y^Z) = 2°2(X2 + Y2) goZ - Co = 0

Fo(X,Y,Z) = 0 describes a paraboloid of revolution, where C o is a con-stant that can be determined from the known fluid volume (Figure 2.)

Let n be the perturbation wave height in the direction normal to the free surface and (^,p) its components in the Z and R directions. Incrementing R b p and Z by ^ and retaining terms to the same order of smallness in (15^,

P' = s^o2Rx p - 9 °4

(16)

r

where p' is the perturbation free surface pressure. Let nR and nZ be

the direction cosines of the unit normal to the free surface. Then (Figure 2), n - - aF° 2R f2° x R gR a R gR 1 8F° go (17) nZ _ - 9 R a Z 9R 9R = 92 + (^o2Rx)2

(18)

Substituting (17) in (16) and noting that

n =-nRp+nz4

we obtain the perturbed free surface boundary condition

I 's 90(18)

By

definition,

n = n u on S F (19)

Substituting (14) in (19) using (13),

n = -n , vO - -an (20)

(20) states that the perturbation velocity normal to the free surface results only from the potential ^, while is purely geometric. Substi-tuting (14) in (6) using (3) and (5),

p

{_^_ 8t

^'+p - f

R+

2u2

+ (wx R

o )

r --z(

x R)2}

(21)

+at x

r

+2stF x

u

- o

• The physical meaning of each of the terms in (21) is elaborated in what follows:

t

8t unsteady translational potential satisfying the

free surface boundary condition

PI

-P f R pressure and body forces

1 u 2 convective acceleration

9

(19)

perturbation tangential acceleration caused by (W x Ro) r

rotation about the spin axes assuming that only translational oscillations are transmitted to the

liquid

1(5o x R)2

steady-state centrifugal acceleration

3t

x r tangential acceleration from vortex oscillations

255 x u total Coriolis acceleration from spin and vortex motions

In order to express the perturbation pressure at the free surface

p' (equation 18) explicitly in terms of the

-

,^, and si

o,

the terms

sex ^+25

R

x

u

(22)

are omitted in (21) so that the momentum equation may be expressed as an exact differential. That is,

PI ==^+0 • ^- (WxR

o

)

r

(23)

The neglect of the acceleration terms (22) is consistent with the linearized free surface condition (18). Eliminating p'/n from (18) and (23), we obtain the perturbed free surface boundary condition:

9Rn = at + 0 + (r x Ro ) w _(24a) i

Differentiating (24a) with respect to time and eliminating n using (20),

r2

^

..

^

u

gR •+

8_

=-

sz

^,45

0

xr)xR

o

1

w 1 (24b) - (r x R o ) w

An approximate solution to the slosh problem can be found by adopt-ing the Galerkin technique. A set of simple trial functions is sought

10

(20)

that satisy the boundary conditions of the irrotational problem (equa-tion (24b) with si M n, 0). A candidate set is composed of the eigen-functions of the problem:

v 2 $0 = 0 in V an = 0 on Sw (25) a0 0 a2^o 0° + 8tz 0 on SF an 9R

The eigenfunctions of problem (25) satisfy the orthogonality condition:

9R ¢m OndS = Sinn

SF

where

amn is the Kronecker delta,

The functions 0 and n in (24a) are expressed in terms of a trunca-ted series in on:

N an(t)^n(R) R nil ' N 1 - (26) n( R ) _ 9R9nMO (R) n=1

Combining (20) and (25) using (26), we obtain:

-!On

_ cn

2

PA an n n an .n g R n ^n R n 9

^n

On (27)

I where {6n } are the eigenfrequencies of problem (25).

(21)

Equation ('27) yields to relations between

P' ►1 and an

gn = °n2xn (28)

Substituting (26) and (28) in (24a), N

^idn to + 4nIOn •' +(rxRo) w (29)

n-1

Multiplying both sides of (29) by ^_n/gR , integrating over the free surface, and using the orthogonality of the

{fin} set, N n + °n 2 n = (Ainw + Ci-nai) i=1 {Ain) = rn ^Ro^ bin} ( C inI = _{Yn fain)} Y n f

^n

/g RdS (30) SF ' fy, a^

ain axn nk dS = fin an dS

5 k S f(r x n) i ^n dS S fOn Ln ax b in xiaxn nk dS axe n k dS S k 5k fOn ni dS _ S

[ R oJ is the skew symmetric tensor form of Ro.

(22)

1

The total vort'c'ty n F is governed by the Nelmholz equation in rotating cordinates: dn* ^+ np X _ ^^

q)

y

11 F =

2

X V (31)

(31) can be derived by applying the curl operator to both sides of the momentum equation. In order to remove the spatial dependence, we define an average vorticity over the volume:

nF - f 5F dV = n + 50 Averaging equation (31), dstF ₁ "v_ dt + 5o x 5F = V1 v)dV (32) V

The expression for

v

is found by combining (3) and (14):

V = - x Ro + n x F - st Vii vO

(33)

The gradient of the first two terms in (33) vanishes identically. The contribution of the third terms v(3) is:

3

(5F ' v)vi(3) _ -

n

j

ax (5 F_' V^_,7) j=1

Averaging throughout the volume and linearizing, 3 -1 V

f

^^F v)v (3) dV = -

L

_JR 13 V 3 = 1 R i j = V f d i (5 F • v0 j ) dV R V (34) 1 _Qo a f (5F

voj

) n i dS = V

r :xni dS

J 3

S _S 13, i •mss i

(23)

(37)

(38) The linearized average of the fourth term v (4) can be expressed as;

V f

(

F

p )

^ 4) dV

n

V _{n - 1 -n}

U

(35) a^

hn .. V f-

L

(Vo

n

)

'V = ^^ az n d V S

The linearized vector equation for homogeneous vorticity is then:

h

st 4. (R + _n_o_)st

n

(36)

tn-- n

where R is a tensor whose components are given by (34),'F is the vector defined in (35), and 7.r

0 is the skew symmetric tensor of vector ^o;

The coupled equations given by (30) and (36) define the approximate rotational unsteady slosh motion of an ideal fluid with homogeneous vorticity in a spinning tank. The time history can be calculated once the truncated sets

K

and (an )are determined.

IV. SLOSH FORCES AND MOMENTS

The slosh force and moment vectors (F, T), acting at the spacecraft reference point, are given by:

F_ fpfidS = fVpdV

S _V

T _ f R x p5 dS= r R x vp dV_J

S V

(24)

Using the expression for Vp in (6) and the definitions of a in (19),

vp

_PBt+1 52 +SEX u+ 2Sto x U+wa x Ro +Ito X (no

x R)

f}

(39)

U -VO + rl x r

:t

n

0n n=1 an 1 .. ₁

9R

4

n

On Substituting (39) in (37), F - M R - $1o x (50

x

0 x w) a bn ^ n + 2s0 x bn n + an2dn4n (40) n-1 n - _{V fRdV} = Ro + s (fluid CG) V n

bin

axi dU - S ^pn ni dS V d. -

f

n. dS 3F n ^ F

The terms M(T - Ti x (5o x S)) represent the equilibrium constant thrust and centrifugal o body forces. The term M(Ro x,,,) is the trans lational force (constant throughout the fluid) generated b the per-turbation angular acceleration 6. The last bracket of (40) involves the slosh unsteady forces as it depends on the generalized coordinates rn and their derivatives:

6 J ---.► translation inertia of slosh

250 x 6n^n Coriolis force due to relative velocity

d

n g n a n 2 restoring force frc.<n potential energy }

15

(25)

Note that the Coriolis effect is equivalent to a velocity proportional damping farce, Substituting (39) in (38),

T M(3? +

S

OO

S5q)

+ MOSw + (Q + A)ca + 2(U - U)5

r .A, 1 a +Rb +2OW + 2 (41)

n- 1 Q

n

a n

n

rr o n

4n

On

is the skew symmetric tensor of vector a in (40)

A pfrrdV V B T p

r

n _W dS I S

pJ r r^ dV

V

ft D = f -pQ0 6 _^3 f1 (V^j)^ dV V F n = J_"r(V^n)_'dV V f r n dS = rr dS f-a = n S ^n e n _SF n b f dS - f dS = n S n

n

g n SF n n fn = in + Ro9n S - RoR 0 + "s Ro [(MR

o sT +C) coo - MRo sDoT

V =

p - M R

o

6

0 "s

41 n - Fn - Ro bn

(26)

where M* is the total number o Within each element, the field its values at the nodal points Ni(J)(x,y,z) (k)(x^Yz) El , r _j

Quanti ti acs superscri pted by (")

are tensors,

whil

e

(`) denotes the

skew symmetric matrix form of a vector. The first, three moment terms in (41) result from the first three force terms in

(40),

while the Cn dependent contributions are born from the slosh unsteady forces, The

and ri dependent terms correspond to the couples produced by the vortex tangential acceleration and Coriolis acceleration, respectively.

V. PROCESS OF SOLUTION

The eignevalue problem (25) and the inhomogeneous potential problem ►^ are solved using finite elements, Essentially, the fluid domain

bounded by the sphere wall and the equilibrium free surface is sub -divided into isoparametric finite elements, The Laplace equation and free surface boundary condition are written in a quadratic integral

form:

I (? J (vO)-2dV _{f - OdS}^' _42a

V

_F

gR

(

)

14 1 )

l

f(V^ i

)

2

0

- J (r x n)*. ds

(42b)

i VS

is

The condition that the first variation of the functional in (42a) be stationary leads to the Laplace equation and free surface boundary condition in (25). A similar process on (42b) leads to the Laplace equation and inhomogeneous Neumann conditions of the potential j,.

The functional I(^) can be discretized as;

(43) kEl

t

F finite elements in the fluid volume V. variable ^ can be expressed in terms of using the interpolation functions

n*

= _Ni(k)(x_Y)Z)_{^ (k)} ₍₄₄₎

i-1

. 17

(27)

n* denotes the number of nodes for some element and

is the field

variable at node

I.

The condition that the functional

I

be stationary

is

U*_rl

614)

r, 61

j x I

x 0 i

x

1,...,n

Using (42a) and (44),

ON

I

i

!ON

i + M + - OZ dV ] j rlx

ON aN

O y By

6 F

I N dS 0 ₄₁ S F R

which can be written In matrix form for each element as: n* E (A j () _

B

(

i k

)

)Oj .

0

Vi j i i j=1

A

(k)

4

(!

N

i

!N

j

+ !

Ni

over V

(45)

O

x ax

ay

lay +

!

a

T

Z )k B(k) N on S I i F 9LR I it

where

^j

denotes the column

—

vector of_the nodal values `ri. Furthermore,

by Splitting ^ -into ^j and ^2, where ^2 contains the free surface nodes

only, (45) becomes;

I A

2

01 ;l

0

B

I

B 2 B

¢21

(46) a 2 - B,

Al

I

Ad

BI ;

2

0

;l

=

„A11 A

_{2 ;2} I

18

(28)

9 1

t

►

Applying tnu above procedure to the functiona l.

1f;.

0)

)

with the same

interpolation functions Ni(x l ypz) leads to a set

of

inhomogeneous

simultaneous equations of the form:

n A ii (k)^J , b

*

i

--I-

A^ X (47)

A(

k

) is identical to the matrix in (45) and

U

b

f

(i x h)

i

N

i

dS S

VI.

SLOSH DAMPING

SIMULATION

The energy dissipation associated with a sloshing liquid is

governed

by

many complex processes that depend mainly on:

(a) Fluid kinematic viscosity

W

Frequency and amplitude of excitation

(c)

Tank geometry and fluid volume

(d)

Ratio between centrifugal and thrust accelerations

To date, no theoretical investi ation exists that models even

qualitaLively the damping mechanism ?Ref.3-11).

It is thought in this

work that the problem is tractable only through empiricism that relies

largely on experiment,

Summer and Stofan (Ref.

5)

measured slosh forces on spherical tanks

subjected to translational periodic motions of vario,,

,

The

liquid kinematic viscosity rangedfrTfl 2.29 x 10-7

,

111

^

sec (10- 9 f

sec)

1

for water to a maximum of 9.29 x 0- m /sec

0

ft /sec) for glycerine.

For some fixed geometry and liquid properties, the nondimensional first

mode slosh force Fs/ogD 3 was found to vary linearly with the excitation

amplitude parameter X

O

/D for small X

O

/D, where

F

s

=

slosh force

p = liquid density

g g

ravitational acceleration (or thrust)

D = tank diameter

X0

=

amplitude of periodic excitation

0

(29)

A smooth transition region follows, beyond which the slosh force

asymptotes to a constant value depending on liquid kinematic viscosity. The observed trend can be explained as follows. For small amplitude vibrations,

FS

is proportional to Xo as expected for a linear oscillator with constant proportional damping:

F

s

= C

F

Pg

D3

D

_(Wz_{- W}z

r

₎z_{+ (}1 _{2; WWr)2} $ D S 0(1)D

C

F

= force coefficient function of geometry and fluid filling W,W

r

= excitation and resonant frequencies, respectively

b = equivalent viscous damping

4 depenae only on kinematic viscosity fur linear motions. As the ampli-tude increases, free surface motions become nonlinear while the observed asymptotic behavior suggests that

4

is proportional to X

0

1D kNC^Xo/D)

X 2

F

s

= C

F

P9D3 w_u

r

_C^ p > 0(1)

It appears from the above limiting cases that the ^ variance with X

0

/D is a hyperbola:

F+

4

21 ( X

D)

4

F An empirically derived expression for c

0

is given in (Ref. 5) in the form: C 0.131 v x 104 0.359 f (h ) 0 2^r F9

M

1 s h _a (49)

f (

h) =

1 + 0

.46 2 - ha

for h =

h

> 1

h a

1.46(2- ha

a

fh(ha ) = haa 1

`

h is the depth of the quiescent liquid in the tank.

(48)

(30)

V —

--In order to determine the dependence of 5 1 on the liquid and

geometric parameters, the steady-state slosh was computed for a periodic excitation corresponding to the fundamental slosh resonance in transla-tion. A viscous damping term, proportional to the free surface normal velocity, was introduced in equation (30);

n + 2C _{E n + vn 2}_i:_n_-_{f(w, n,)} (50)

The damping factor K wai varied until a numerical value of the

slosh force Fs coincided with the experimental value. A hyperbola was then

fitted to the resulting {c, X o /D} set in the form:

X Y=

Z02 _{+ 10(1 + 27rco)(}

D

(51)

The frequency response spectrum was computed for a periodically excited spherical tank based on the damping factor expression (51). The geometric and liquid characteristics were:

k a = 18.4 cm (7.25 in.) P = 1000 kg/m 3 (62.2 lb/ft 3) 0.5 < n* < 2.0

D

0.5 D —o = 0.0083 g 9.81 m/sec t (32.2 ft/sect) n* is the nondimensiona'l frequency parameter

n* = W

9

i Figure 3 compares the pres

The correlation is satisfactory mental slosh resonance. Beyond estimate the slosh force, while quency. A possible explanation follows.

°nt analysis with test data from Ref. 12. up to n* = 1.3 just after the funda-this point, the analysis tends to under-the err-or increases uniformly with fre for this discrepancy is given in what

Liquid slosh in spherical tanks falls in one of four different regimes:

(1) At low frequencies

(Ti*

< 1.1) the small amplitude free surface

motion is in phase with the excitation. The liquid is mainly

acted upon by slug forces (externally applied translational accelerations). The response is very similar to that of a

21

. 3 L

(31)

linear damped single degree of freedom oscillator having a mass equal to the total fluid mass.

(2) As the frequency increases beyond q* = 1.1, the fundamental resonance is approached. This results in an increase of the liquid mobility (degrees of freedom) and greater free surface amplitude. The slosh wave is broken down into small drops as a result of splashing against the tank wall. In addition, it froths due to surface cavitation and release of entrained vapor. These phenomena are associated with an appreciable increase in energy loss that is considerably higher than that in the linear regime.

(3) Beyond the first resonance (n* ? 1.25), the excitation and response are out of phase. This leads to an increase in relative velocity between the slosh wave and the solid boundary. Breaking of the slosh wave as it follows the adverse slope of the wall occurs whenever the tank is more than 40% full and the wave amplitude is sufficiently large. It is believed that the dissipation mechanism is far more complex in the case of a spinning tank. Experiments were conducted at Hughes Aircraft in 1972 (Ref. 13) on a spinning table supported by an air bearing. The measured rate of energy dissipation from slosh was found orders of magnitude higher than that predicted analytically. In fact, the energy dissipation from slosh has destabilizing effo cts for a certain range of the ratio between spin and transverse inertias of the spacecraft.

VII. NUMERICAL RESULTS

Since the resultant gravitational acceleration g R enters as a parameter in th^ definition of the nondimensional frequencies, a value gR = 9.81 m/sec was assumed throughout the computations while the angle 't t tan-1 ( Rox + s )st02 1 B = g

was varied by changing both thrust g and spin rate 0 0

for a mixed geom etery and fluid volume. The "'Galileo" tank was assued with the follow-ing parameters: a = 0.37 m Rox 0.64 m 22 k 3"1

(32)

P = 872 kg/m 3 (light-fluid) v = 0.277 x 10 -6 m2/sec

V liquid Vtank - 0.5

The tank was spun to a constant rate (9 0 ) on a rigid platform with a superimposed periodic excitation about the spin axis such that the equivalent maximum oscillatory amplitude at the center of the tank was always one percent of the tank diameter (X o /D = 0.01). Thus,

w = wo s i n cot R

wo = 0.O1 D

ox

where wo and w are the excitation amplitude and frequency, respectively. Table 1 shows three test cases in the ^ range of interest.

The variation of steady-state nondimensional circumferential force Fy/(pg D 3Xo/D) with frequency parameter n* for the three cases in

Table ^ are shown Figures 4,5, and 6. As expected, case A (no spin) exhibits one resonance at n* = 1.25. A similar behavior is noticed for case C (no throst) except that the resonance occurs at a lower n* = 1.04. The response for case B is different in that three resonances are

encountered. The first resonance is associated with a vortex dominant motion. The next two resonance correspond to mostly potential irrota

tional modes about two orthogonal axes on the free surface. A weak fourth resonance can be detected at n* = 1.6 and relates to the slosh mode with two circumferential nodes at the free surface.

In the process of numerical experiments, a slosh instability was detected fluid fills G = ( V fluid/Vtank) less than 0.55. The exponential instability was found to exist in a range of R angles that depends on

the fluid fill G. 'the range is largest at the lower

G

and decreases

Table 1. Representative test cases

Case A li liquid t o 00' g' 2 m/s gR'2 m/s R , RPM Deg A 0.5 2.0 9.81 9.81 0.01 B 0.5 31.5 5.50 9.81 52 C 0.5 34.0 0.0 9.81 90 23

(33)

uniformly as V increases to a V of 0.55, be ond which the instability

vanishes. The stability boundary (Figure 7^ is independent of the

resultant gr

.

;,ivitational acceleration 9R'

The unexpected occurrence of the instability suggested further

investigation into its nature and mechanism. The complex characteristic

eigenvalues of the governing system of linear differential equations

(30) and (36) were determined for different V in the range 0 < 0 <

90 deg. The variation of the nondimensional -imaginary roots "Fepr

;

^sent-ing the frequencies and their correspond^sent-ing nondimensional real parts

representing the damping is Plotted versus 0 in Figures 8 through 14

with V as a parameter. For low V (nearly empty tank) the first

fre-quency vanishes at ^ = 0, then rapidly increases to coalesce with the

second resonance in the range of instability. The two frequencies then

separate once more in the stable regime, while the lowest frequency

approaches a finite value as 0 reaches 90 deg. The damping associated

with the first frequency starts at zero for 0 = 0, while that

corre-sponding to the second frequency is finite. As the instability region

is approached, the first mode damping assumes larger negative values,

while the second mode becomes overly damped. This behavior ceases as

we exit the instability range. The frequency separation between the

first two modes increases uniformly with fluid fill V, an indication of

a weaker instability. The dashed lines represent the first two

reso-nance

r

p of the approximate eigenvalue problem (25). The different

negative damping parameters of the first mode are plotted against

with V as a parameter in Figure 15.

The phenomenon bears resemblance to aeroelastic flutter in which

the bending and torsional elastic wing frequencies coalesce (Ref. 14).

In the present model, the vortex and irrotational slosh frequencies

couple in a manner similar to the wing frequencies in flutter.

VIII. STABILITY ANALYSIS OF FREE SLOSHING

A separate stability analysis on the "free-slosh" equations was

performed for all cavity fill ratios and all values of 0 < $ < 90 deg.

Consider Euler's equation for fluid motion in a rotating system

(Eq. 6 with @ = W4 =_ 0) with hu'llogeneous boundary conditions given by

(8a,b,c). The steady-state solution is given by (15).

Let V(x,t) be the infinitesimal perturbational velocity of the

fluid, and ^1(x,t) be the infinitesimal perturbational pressure. Then,

using (6) and T15), the equation of perturbed motion of the fluid is:

av

at

+ 2P x v = - P Vp,

(52)

V -

^ = 0 in Vo

4

(34)

Outside of V0 there is a "gravitational field" _gegiven by: ge = w21XI + _Y11¢

sk (53)

As a result of the perturbed motion, the free surface is perturbed. Let the elevation of the perturbed free surface relative to the equili-brium free surface SF be denoted by g, measured normal to SF .

0

From (15) and (53) it is easily seen that ge is normal to the free surface, and is directed into the fluid,

As before,

v n = 0 on SW

0

i

The pressure on the equilibrium free surface SF is given by: o

pl = P 9n on SF (correct to OW)

i SFo

o Consider the Liapunov functional (Ref. 15):

L- 2

f

_pv v

dV

o

+ 2 P

_g

e

f

_{^ 2 d}SF (54) Vo SF o 0 r ge = ge . n

Consider the time derivative of L:

L _{fp v'} 8t

dV

o

+ f

P

ge

8t

d S

F

(55)

Vo SF o Using (52) in (55):

L=

-

f

P (2 Q x v - v+ 1 v v p) dV + J p g e a dSF V P 1 o SFo o V o f H 25

(35)

Using the divergence theorem on the remaining terms in the first integral note c4—V-

v

= 0),

L = -f

p,v

v dV

o

- f p

1

v . n dS

F

-

_,1_{p l}

v , n dS

w + f

pg

e

^: dSF

V o _SF _{o S} _{o S} o wo Fo Now

v•v ^ O

in

V

o

;

G•n ^ OanS

w

v•n = ^^onSF

o

0

pl = p ge 4 on SF

0

Hence, L = 0. Thus, the functional

L=

J pv

.

vdV +

₁

J

_g

n^2dS

2 V o ° 2 SF e Fa

o

is conserved in free sloshing. Therefore, free sloshing is stable (marginally stable).

This proves the assumption of homogeneous vorticity is invalid for problems of free sloshing and, therefore, by implication, for problems of forced sloshing.

(36)

REFERENCES

1. Pfeiffer, F., "Ein Naeheruncdsverfahren fuer Fluessigkeitsgefuellte Kreisel," Ingenieur, Archiv 43, 1974, pp. 306-316,

2. Lamb, H., "Hydrodynamics," Dover Publications, 6th Edition, 1932. 3. Stephens, David G., Leonard, H, Wayne, and Silveira, Milton A.,

"An Experimental Investigation of the Damping of Lquid Oscilla-tions in an Oblate Spheroidal Tank With and Without Raffles," NASA TN D-808, 1961.

4. Stephens, David G., Leonard, H. Wayne, and Perry, Tom W. "Investi-gation of the Damping of Liquids in Right-Circular Cylindrical

• Tanks Including the Effects of a Time-Variant liquid Depth," NASA TN D-1367, 1962.

5. Summer, Irving E., and Stofan, Andrew J., "An Experimental Investi-gation of Viscous Damping of Liquid Sloshing in Spherical Tanks,"

NASA TN D-1991, 1963.

6. Mikishev, G. N., and Dorozhkin, N. Ya, "An Experimental Investiâ,-tion of Free OscillaInvestiâ,-tions of a Liquid in Contai ners (in Russian ," Izv. Akad, Nauk SSSR, Otd. Ì°ekh. Nauk, Mekh. i Mashinostr, No. 4, July/Aug. 1961, pp. 48-83.

7. Abramson, H. Norman, Chu, Wen-Hwa, and Garza, Luis R,, "Liquid Sloshing in Spherical Tanks," AIAA J., Vol. 1, No. 2, Feb. 1963, pp. 384-389.

8. Williams, D. D., "Estimates of Energy Dissipation and Nutation Damping Due to Fuel Sloshing in a Spherical 'lank," Hughes IDC 2280.03/202, 1 June 1965.

9. Porter, W. W., "Fuel Slosh Damping Test with ATS Tanks and Test Fixture," Hughes IDC 223/2466, May 12, 1970.

10. Edwards, R. H., "Fuel Damping of Intelsat IV," Hughes IDC 2245.2/8, May 9, 1969.

11. Neer, J. T., "Intelsat IV Fuel Slosh Dissipation Rates and Time Constants," Hughes IDC HS312-2286-3-Fl6l, Feb. 13, 1970.

12. Abramson, N., "The Dynamic Behaviour of Liquids in Moving Containers," NASA SP-106, 1966.

13. Neer, J., "Fuel Slosh Energy Dissipation on a Spinning Body," Hughes Aircraft Co., E1 Segundo, Feb. 1972.

14. Roccard, Y., "Dynamique Generale des Vibrations," Deuxie"me Edition, Masson et Cie, Paris, 1949.

15. Caughey, T. K., "Review of Galileo Slosh," Interoffice Memorandum, June 15, 1979 (JPL internal document).

(37)

Y f x )X 112x ) 0 X Z k .A

Figure 1. Reference and local rotating

coordinate

systems z

Figure 2.

Accelerations

on fluid Volume 28

(38)

A B C

F:

D -.7_. A - LINEAR RESPONSE """' EXP' B -SPLASHING REGION , ^ , THEORY C - BREAKING FROM R - 18.4 cm CHANGE IN PHASE D - ANTI-RESONANCE P x 000 1 k0!m3 HIGHLY DAMPED _h = 0.5 x0 0 0.0083 D 4 ` 10 9 8 X°1° 7 M c 6 ^ 5

4

3

2 0 0.4 0,6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

wJ

J9

Figure 3. Comparison for translational slosh

P9RD3 p 34.72 NEWTONS w 10 9 1,8 7 Flo b u.

M

5

Ol

^4

3 2 0

SPIN = 2 R. P.M. THRUST 9.81 m

/sec2

77r = 1.25

E a i t r

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

CiOR

Figure 4. Nondimensional tangential slosh force for case A

(39)

10 9 8 7 ^a 6 4 3 2

SPIN s 31.5 R.P.M. THRUST 5.50 m/soa2

'lr 1.11

x s a F i7_r 0

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

nffiW a OR

Figure 5. Nondimensional tangential slosh force for case Q

K

SPIN= 34 R. P.M. THRUST = 0.00 ,fir , 1.04

k

i

F a f t 1r

6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

'7 = C

l 10

9

8

7

,^I0 6

U. ^ 0 5 ^e 4 3 2 0

Figure F. Nondimensional tangential slosh force for case C

(40)

Figure 7. Stability boundary

(41)

1.6 1.4 1.2 1.0 0.8 o^ 0.6 0.4 0.2 0.0

' _{Figure 8a. Characteristic frequencies for G = 0.03}

i 0.6 0.5 -

V(F)/V(T) -

0.03 0.4 0.3 0.2 0.1 _—! -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0 10 20 30 40 50 60 70 80 90 Q

Figure 8b. Characteristic damping for V 0.03 f.

t

i

(42)

1.E 1.A 1,0 o^M 0,6 0.6 0.4 0.2 0.0 i V(F)N(T) N 0.10 10 20 30 40 50 60 70 80 9

Figure 9a. Characteristic frequencies for V = 0.10

0.6 0.4 V(F)/V(T) ¢ 0.10 0.2 M 1^ ^0) 0.0 t M F -0.2 C . -0.4 -0.6 0 10 20 30 40 50 60 70 80 90 0

Figure 9b. Characteristic damping for V = 0.10

33

(43)

1.b 1.4 V(FVV(T) 0.22 1.2 1.0

3

0.6

0.

4 f 0.2 0.0 0 10 20 30 40 50 60 70 80 90

i Figure 10a. Characteristic frequencies for V = 0.22

i 0.6 --V(F)/V(T) = 0.22 0.4 f 0.2 ^.n 6 -0.2 s -0.4 lam. e 0 10 20 30 40 50 60 70 80 90 u k

(44)

1.6 1.4 1.2 1.0 0.8 3 0.6 0.4 0.2 00 0 10 20 30 40 50 60 70 84 90

0

Figure lla. Characteristic frequencies for

V

= 0.35

0.6 0.4 0.2 °Ion 0.0 -0.2 k -0.4 µ -0.6 0 10 20 30 40 50 60 70 80 90

Figure 11b.

Characteristic damping for V 0.35 i

i

(45)

V(F)/V(T) = 0.43 1.6 1.4 1.2 1.0 o ^^ 0.8 3 0.6 0.4 0.2

0.0

v l0 20 30 40 50 60 70 80 90 0

Figure 12a. Characteristic frequencies for V 0.43

0.6 V(F)/V(T) = 0.43 0.4 0.2 C, 0.0 -0.2 -0.4 F' -0.6 ₀ 10 20 30 40 50 60 70 80 90 1 i

Figure 12b. Characteristic damping for V 0.43 , f 36

t 1

(46)

F , Ito

1.4

V(F)/V(T) 0.50

1.2

""",^^^ ^^

1.0 . o

`^ 0.8

3

0.6

0.4

0.2

0.0

0 10 20 30 40 50 60 70 so 90

0.6

0.4

0.2

i^

0.0 +.n

-0.2

-0.4 -06 t i 0 10 20 30 40 50 60 70 80 90

Figure 13b. Characteristic damping for V 0.50

37

(47)

1.6 V(F)/V(T) = 0.57 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70 80 90

0.6 V(F)/V(T) = 0.57 o.4 0.2 ° I rnw ---0.0 -0.2 -0.4 -0.6 0 10 20 30 40 50 60 70 80 90 f;

r

s8

S

1

(48)

tn It Ch N O c; O c; afi C-4 04 O II O LM Q-3. 0 V O O O O 0 m GL Rf 0) O V) 0 4-0

r-0 Ir-!mod S-cl) S-=3 0)

U-NASA—JPL—Coml, L.A., Calif