'N"

II cqil HE

Z 44 0

IPILICATION OF StMiUT17DFS THEOtY. TO llff. PROBLEM

or .SLOSHING AbramDTANK

R.' Martin,Oulo E. Ranilcben. Jr.

Cont ract DA -23.-0-72 OD12514SwR I ProjeCt,43-768.2'1

4

.- 3 May, 1958 2

D00

SOTHES RSAROUTSITT

SAN ANTONIO. TE XAS

N. DO C I,-b,,.s!-

SOUT4WEST PESEARCH INSTITUTE8500 Culebra Road. San Antonio 6. Texas

Department of Engineering Mechanics

APPLICATION OF .SIMILITUDE THEORY TO TIlE PROBLEMOF FUEL SLOSHING IN RIGID TANKS

by

HI. Norman AbramsonR,J. Martin

Guido E. Ransleben. Jr.

Tf :haiical Report No. ICont kc. DA-23-07Z-,RD-ZSl

"wRI Project 43-768-Z

Prepared for

U.S. Army Ordnance Miss:1r r.ommaridA.my Ballitic Missile Ap-ncy

Re.dstone Arqenal. Alabam;.

23 May 1V58

APPROVED:

E6iw.rd Wenk , Chir. rm nD.,partment of EngineeringMch.ranilc s

DISTRID UT!ON LIST

Sci.'ntific Ass,..tant to the DirectorArrobailliqtics LaboratoryU.S. Army Ordti.tam. Msiile CommandArmy BDel1iic LMissilt AgencyRedstone Arsenal. AlabamaAttn. ORDAB-}iS! 14 copies

U.S. Arny Ordnance District, St. Louis4 i00 Goodfellow Blvd.St. Louis 2O. MissouriAttn: Contracting Officer

Lt. Cot. Norman C. Pardue I copy

Southwest Rosearch InstituteR500 Culebra RoadSan Antonio 6, Texas 3 copies

Best Available Copy

ii

ABSTRACT

Similitude theory is applied to the problevn of fuel sloshingt in

accelerated tanks to establish criteria for the design of model experiments.

it is found that dynamic modeling is possible even if liquid viscositY is

considered. The ranges of significant parameters and the selection. of

model liquids are discussed.

TABLE OF CONTENTS

1. INTRO)UCTION I

It. SMNIt.L1UDF iRELATIONS FOR FUEL SLOSHING 3

A. Gem.ra" r3

B. Sun ,h|tud" Relationq - Translational

:x¢ ctt ation 4

C. Sitm .tucl Rolations - Rotational

Excitation S

III. 1M{ODELIXG CONSIDERATIONS BASED ON THE

SIMILITUDE RELATIONS 10

A. Gerne-.d 10

It. Modeing Considerations - ViscosityNelce

10N,£ttcte'd

C. Modeling Considerations -Viscosity

Included

LIST OF REFERENCES 14

Iv

I. INTRODUCTION

Sloshing motions of fuel in partially filled tanks may be established

by essentially lateral accelerations of the vehicle. Such fuel motions are

of impurtanLe for ther, is the possibility of extreme osc.1latione if the

frequency of the excitation is in 'he neighborhood of one of the natural

frequencies of the liquid fuel. The forces exerted by the fuel on the tank

walls may then lead tc. pr- ur,ations in the flight trajectory, or ma) even

impose severely high stresses on structural components.

Suppression of sloshing modes has been attempted by meant of

various mechanical devices, andthereforeit ia of some importance to

evaltate the effectiveness of iuch devices. Purely mathematical approaches

to thi, problem are exceedingly difficult because of the boundary conditions

introdurced by such suppreision devices, and are.thereforerrestricted

,.asenti.lly to very simple cases. On the other h.and, it may be possible

tI- evalhate suppression devices by means nf suitably designed and con-

ducted model tests.

The problem of conern in the .repent stucy is the spec ification oi

criteria for moeel tests which will give useful quantitative 3nformation

cecerning aloshing ard slosh uppressor's in full-scale fuel tanks. This

is to be accomplished by the application of Similitude Theory to obtain

appropriae expressions for the model - prototype relationships. These

maty in turn be e,.nioyed to yield a ratio"l design of model tests. This is

in contrast to previous experimental studies of the fuel sloshing problem

(Re(*. 1-4). which have apparently been conducted without regard U~r simi.

-tude requirreneats and consequently are believed to.be of liited useful.

nest.

Tho purpose of the present report is to discuss the basic require -

7-

"-metrit for model tudis of fuel #1oah!i by apphL~ation of sinittde theory,

23- .

VZll

,Ak

-RP

11. SIMILITUDE RELATIONS FOR FUEL SLOSING

A. General

We consider the vehicle in vertiral flight with constant Acceleration

directed along the flight path (Fig. I). Sloshir.K I" * ssutmerl to be excited

by time-dependent accelerations acting essentially normal to the flight

path. For purposes of the present study, the exciting accelerations will

be considered either as purely translational. or purely rotational about .in

axis normal to the night path.

The essential features of the sloshing motion are further governed

by the physical properties of the a3quid and the pressurizing gas. The

importance of the liquid density and total mass are obvious; hnwev,,r.

viscosity and surface teneicn require tiditional commer. If sappression

devices are incorporated it is clear t!:t their effc tiveness mirout, to.'

very substantial degree, depend upon the damping f,rces they can provide

as the result of visco.,s ac.ion of the liquid. If mods-Is of very small size

are employed, it is conczvable that the effects of sloshing may be sub.

stantially modified by surface tension forces acting on the suppression

dvvices and at the tank walls. As will he discus sd later, it does not seem

possible to provide rigorous modeling if surface tension forces are included.

and further it is believed that the total contr bttion of such force,, would he

smail compared to the inertial and viscous forces. Consequently. surface

tension forces will be omitted it the analysis.

4

Simirly. it is~ tuppotied thait the forces produced by the pirceouriamng

gas are rieIkgiLk comipared iuith th other force*. Mind walL thereforebe

omitted. The gai pressure may be accounted for imnplicitly, however. as

it to Included vt the' total prss~ure' at the tank wall by direct .-uperpou.,tiori

otyWt' V~ie gais vodutnt' rrainp i vsetial!V constant during sloshinig.

F~urther rvti.ti~m tu thec analysi .rt provided by the Asstimption

of stwit r ig-d t-ina, ,n~d thej v4tvace of geomeicial similarity

4a 411 ri ;pects betvver miodcl itsid pru:typ. The' suppression dev.-C-s

Iitsv1iv'es will hie co -ide rid geoiaitric~itll iiinar in all respect* betweetsIE

m~odel arid prototype. ai,. the correspondinig inertial anid app~lrent mass

foc - scat" with them will be ngettd kn comparison with the via-

r cjai forces.

Al1~ Ssri iide. i ela ons~ Trtii--l Excitaition

Tkt, r 'ue.aat pairameters. bsd on th prcedir~ _disuosjo -w-il

a io-tu~di *ccelrtiori acting on tank (L~T"

- d tank d&,inoter (L)

rvaiutant tiqu~d farcer on tank wall (MLT

h depth of liquid in tanic (L)

X eciatioun amaplitude JLI

i hcjuid vi-&coglty %t'T

licjlutddUimsit IN"Lt s

An equation relating the%e eight parameters analytically can be writtet, in

the general form (Ref. 5)

*'(ni. nz" n3* ... , nn )= 0 13t)

where n is the n'imber of physical parameters invoiled and m is the itiln-

ber of fundamental dimensions (in this case n-rn t.1 S). The n' are

dimensionless combinations cf the parameters listed. Th general rex uuiesion

11 = a dP F-f h' Xrog P nrv

or

il = (LT") (MO1 (MLT' " ( ) (L ) ( ' T ' I) (M- L )r JT) v (3)

is first formed. In order for eachf Igroup to be dimensionless. the final

exponent on each group trut be zero. Conridering the fundamental dame-n.

sions, we find

o + + - + 6 + c- -31 = 0 (condition on L)

-'Q-2-t IV z 0 (condition on T) 14)

,Y + +1 n 0 (condition on M) j

From th'se three equ.ttions in right unknowns, we are to obtain

five dimensionleas groupa. For h --' these g. .rwisp f.ivr o( the ,,nknowna

must he set arbitrarily. Since the resultant liquid force on the tank wall

is the d.pcndeit *ar:ali,, of the problem. we :,,:iy assign -Y the -,dlue of

unity in one solution and zero in all otlh.rs, thur. ;nsurin that F accitri

only once and only to the firs' powc-r.

6

Setting y " 1,a 6 0q " 0. Eq. (4) yields 1 - -4, v a 2. q -I

so that

and for a 1, 6 0. we find 1: -1, : Z, n " 0, so that

n1z (6)

Proceeding in a similar manner, the remaining dimensionless groups

may be found to be

n I (7)

x0 (8)

115 o (1 19)

A get-ral solution of the form of Eq.(l1) may now be written in

terms of the [I' as

{" d Pd (d/)]

We, note from 113 and 114 that all linear dimensions are scaled in the same

ratit, apt *he dtameter. Frtha'r, should other shape parameters be con-

uldlered in this analysis, we note that the effect would be only to add corre-

sponding shape ration (with the diameter) in Eq.( 10). Also, we note that

I in equivalent to Froude's Number and that f15 is equivalent to Reynold's

Number.

I

Should interest reside not in the resuiant liquid force on the t1.nk

w'all, but rather in the resultatit lqu:tdpre-ure, a smtilar anal as ioisad

yield

P f(- ) (I I

so that

pdz p (d/r)2

which is in the form of a pressure coefficient (Euler Number).

The preceeding analysis has considered ths excitation to 'ie a dis.

placement, in anticipation of actual model experiments. The exctlatann

could, however, have been taken equally %ell as a force " (or een an

,cceleration), in which case the dpprupsiate dime sionlese group wou~d

become|P

(pd3) d/)

Considerattun of the surface tendiun in the foregoing analysis

would have resulted -n a new TI group of the form pd /e r, which corre-

sponds to a Weber Number. At will be seen from later consderattons.

the simultaneous solution of this new group and tI5 would be exceedingly

difficult, in view o! the model liquids readily available. Further. we

assume that surface tension forces are small t ompared with the inertial

. S

and viscous forces, and consequently we ,hall omit any additional considera-

lion of suarface tenston.

C. Skmilitude RelatltLnx - Rotational Excitation

In thip a c the pertinent paramcters remain the sante asin the

tr.an4latlonal cate except that the excitation is 'now defined by two para

nit.te-'.. the ang!ular rotaticit 00 and the location of the rotational axis b

(Fig. 2). Since we con-idter only -small excitation amplitudes, it may be

noted that 1 0 -Y X V and. therefore.the general functional Equation (10)

applies to both ksnds of excitation. The use of a moment M a Pb as the

excitation would result in a group formed by fl4 multiplied by the ratio

b/d.

FIG. I

Ia

ot wt

FIG. d!

10

111. MODELING CONS!DEr.ATIONS BASED ON THESIMILITUDE RELATIONS

A. Crn% r.

Equation ( 10). and the associated alternate expressions for special

ca sesa governs the design of models to sirmulate a given prototype. If all

dimensionless groups in this equation have the same numerical value for

both model and prototype. then the forces and prc. sure. m~easured on the

model are directly applicable to the prototype.

Denoting the ratio of model to prototype parameters by the sub-

sc ript r, we notc that fluid depth and excitation displacement amplitude

are scaled in the same proportion ar the geometrical scale. Thus

dr dm/(lp (14)

and,

hr !Xor 'br rdr * r 2 (15

The remaining model parameters are to be estpblished by further

atudy of Eq. ( 10).

B. Modelint! Considerations - Viscosity Neglected

It ts instructive to cons-Wer the modeling parameters in the case

that viscous effects are neglected throughout. This means that 115 drops

out of the previous analypits, and the relation between model and prototype

depends an the geometric factors discussed above and the time scale factor

Z .dr/a. (16)

II

This result means that an' s1?e modt,-I an a !g' acccterto't field can he

used to simulate any size pr-Ptotype in ar.v acceleratton field by varying

the time scale, and would further permit the use o.f any model liquid since

the density appears only in the force (or press-ire) parameter.

In presenting measured data, the parameter avz/d may be used

as the primary independent variable. Since frequency is a more com-

monly used variable, it may be introduced for the period to give dwz a.

The force (or pressure) group may also be modified by introduction of

Eq. (16) to give

F or- 2 - (17)

apd 3 apd

the measured dita may then be presented with F/apd 3 and dw 2/a as the

primary variables and the remaining quantities as parameters.

C. Modeling Considerations - Viscosity JIcluded

With viscous forces ircluded. the data niz, stil be presented as

indicated above; however, the model design is now subject to certain

restrictions arising from 115 , the Reynold's Number. Introducing Eq. (16).

we find

2 3/2 1/2 -

IA ;from which

I

dr P a1/3 119)

r 7-

f

This iniport4.at re-lult shown that the geometrical scale is determined by

the acceleration, liquid density, and liquid viscosity ratios. It it is

desired to test models so as to cover a range of prototype accelerations,

we may either fix the model diameter and vary the liquid properties, or

fix the roode!l liquid and vary the model size. Since the range of liquid

proprrtif,- 1 verV restrict r!. the latter course is chosen.

In order to employ models of reasonably small size to study large

prototv;.,, i: v,4'ctton.of F. I IQ) indicates that the model liquids must have

high density or low viscos~ty. or both, compared to the prototype liquid.

Mercury. of course, has a very high density, but is undesirable for a

variety of reasons; some of the organic solvents have very low viscosity,

therefore.offer some promise.

Aq an example of model simulation, assume the following typical

case:

Parameter Prototype Model

Liquid Kerosene Methylene ChlorideSpecific gravity .83 1.336Viscosity 2. 5 c.p. .465 c.p.

Acceleration 1-5g Ig

Diameter 9 ft

For a prototype acceleration of Ig.

dr o (( L '65 . I

I;

dm2 ddr 4. 14

m p r 4.21

For any other prototype acceleration, the model size is Z. 14 divided by

the cube root of the acceleratioi ratio. 'thus

i2. 14)( g .( /66 ft

Therefore. a 9-ft diameter prototype with kerosene futI ant undergoing

acrelerationb from l-Sg may be simulated by a series of models of 2 14

to 3.66-ft diameter with methylene chloride as th- model liqid. The

mudel frequency rar.,e is determined from Eq. (16) as

1/2r/T -Z (ar/d / "2.052 to 0.701

It in clear from the foregoing that the inclusion of an additional

group to account for surface tension would have rendered the model design

virtually impossible because of the inability to find a model liquid of appro-

priate properties to satisfy .11 diniennio.nless piroups simultaneousl,. in a

small model.

It may be ncted in passing th. the exper:micnts reported -n Pef,.

1-3 were restricted to small models with water as the model liqid. Based

on the c-)nsiderations of thit present report, those ret;ults cannot adequately

sinulate sloshing with, suppresor-i because of the f;.tlure to model %iscous

forces. While liqu:ds other than water were also employed in th, expert.

ments of Ref. 4, the liquid properties were not properly fitted to the

mc.iel sires and hence the results cannot be *lstd to evaluate Auppression

devices.

14

LIST OF REFERENCES

1. Broun. K,., "Laboratorv Test of Fuel Sloshing", Rept. No.St-D,.v. 7841, Dotgi. s Aircraft Co. , Inc.

2 . Ca., K ard Parkiii-.on. W.C. , "The Damping of a Liquid ina RIight (ircular Cylindrical Tank", Ramo-Wooldridge Rept.GM 41-75. Sept. loshi.

3. Howell. E. and Ehler, F.G. "Experimental Investigation ofthe Influence of Mech.ani.lL Baffles on the Fundamental SloshxigMode of Water in a CVltndrical Tank". Ramo-Wooldr:dge' Rept.GM 45.3-87. CM-TII-69, July 1056.

4. Eulttz, W.R., Unpublished work, ABMA, 1q57.

5. Larighaar, H.S.. Dimenqicnal Analysis and Theory of Models,

John Wiley. 1953.