On liquid motion in a circular cylinder with horizontal axis.pdf

8/10/2019 On liquid motion in a circular cylinder with horizontal axis.pdf

1/14


2/14

20g K. Baxtkowiak, B. Gamper , and J. Siekma, n:

reference is made to Lamb [1], Stoker [2], Wehausen and Laitone [3], Wehausen

[4], Moiseev [5] and Moiseev and Petrov [6]. A comprehensive view of the general

subject with emphasis on space technology applications is given in a monograph,

edited by Abramson [7].

In recent years several papers have been published which investigate the

behavior of a liquid in a circular canal, both theoret ically and experimentally.

Budiansky [8] employed a rigorous mathematical theory (integral equation

technique) which allows to determine the antisymmetric natural frequencies.

Experimental work has been reported by McCarthy and Stevens [9]. k com-

parison between theory and experiment shows that good agreement is obtained

for transverse oscillations (see [7, p. 48, Fig. 2.23]). Petrov et al. [10] calculated

the free oscillations of a liquid in immovable containers by a variational method.

A numerical procedure, based on a method (potential of the simple layer) proposed

by Siekmann and Chang [11], has been further developed by Chang and Wu

[12]. Their numerical data agree very well with Budiansky's results.

In the present paper the flow pattern of the liquid motion in a partially

filled circular cylinder with a horizontal axis is studied analytically and experi-

mentally in some detail. A simple device, designed by the junior author (K.B.),

allows the determination of the flow. Streamline patterns, surface profiles

(-motions) and frequencies showed no contradiction to theory, which could

not be explained by the shortcomings of the experiments.

2. Analysis

Consider a homogeneous, inviscid and incompressible liquid contained in

a canal of circular cross section with horizontal axis and having a free surface.

The cylindrical cavity is assumed to be inf initely long and subject to a transverse

excitation. We denote the two-dimensional domain occupied by the liquid by

P- and its boundary curve by ~2, i.e. ~/2 = So u S, where So is the free surface

of the liquid and S is the (wetted) rigid wall of the vessel. To study the lateral

sloshing of the liquid in the tank, we introduce a cartesian coordinate system

(0; x , y fixed to the top of the cylinder as shown in Fig. 1. With the above

assumptions the governing equations of the liquid motion are the Euler equations

and the cont inuity equation:

~ u ~ u au ap ~ 1 )

o W + u ~ + v ~ ) _ a x

e ~ + u ~ x + v - ~y

~ , 2 )

+ va-- = o. (a)


3/14

On Liquid Motion in a Circular Cylinder

Y

2 0 9

g r v i t y

S o ~ . ~ . . . . . .. . .. . .. .

0

c o n t a i n e r

[ c i r c ul a r c a n a l

Fig. 1 . Circular canal, ge om etry and n ota t io ns

I n t h e se e q u a t i o n s w e d e n o t e t h e v e l o c i t y c o m p o n e n t s p a r a l le l t o r e c t il i n e ar

a x e s x a n d y b y u a n d v , r e s p e c t i v e l y , ~ i s t h e c o n s t a n t ) d e n s i t y o f t h e li q u i d ,

g i s t h e a c c e l e r a t i o n d u e t o g r a v i t y , a n d t i s t h e t i m e . T h e d i r e c t i o n o f g r a v i t y

i s p e r p e n d i c u l a r t o t h e a x i s o f t h e c a n a l . T o s a ti s f y t h e m a s s - c o n s e r v a t i o n E q . 3 ),

w e e m p l o y t h e s c a l a r s t r e a m f r u n c t i o n ~b x, y , t) , w h i c h is r e l a t e d t o u u n d v b y

u - ~ Y , v = - - -- . ~ x 4 )

W i t h t h a t w e o b t a i n f r o m t h e E u l e r e q u a t i o n s t h e f o ll o w i ng e q u a t i o n f o r ~b

a l o n e

A ~ - - A ~ A ~ 0 , 5 )

a t ~y ~x ~x ~y

w h e r e A i s t h e t w o - d i m e n s i o n a l L a p l a c i a n . F r o m o b s e r v a t i o n c f. t h e s e c ti o n

o n e x p e r i m e n t s ) w e n o t e t h a t a l l l i q u i d p a r ti c l e s o s ci l la t e s y n c h r o n o u s l y a n d

a b o u t a s t a t i o n a r y m e a n p o s i t i o n . M o r e o v e r , t h e a m p l i t u d e o f t h e o s c i l l a t i o n s

i s a f u n c t i o n o f p o s i t i o n o n l y . P u t t i n g

~ b x , y , t ) = T x , y )

s in wt , 6)

w h e r e eo d e n o t e s t h e c i r c u la r f r e q u e n c y , w e g e t f r o m E q . 5 ),


4/14

210 K. Bartkowiak, B. Gampert, and J. Siekmann:

Since this equation holds for all times, we postulate

A ~ - - - - + - - O . 8 )

~x 2 ~y2

Now the bounding curve S of the canal (circular arc) is a streamline. In addition

we anticipa te th at the streamlines within ~9 are circular. Thus we pose the solution

of Eq. (8) in the form

M y

: - - 2 - - ~ x 2~- y~" (9)

This function corresponds to a doublet of strength M located at the origin of

the coordinate system. The equations of the streamlines are therefore obtained

from

M y

~(x, y , t ) = - - 2 --~ x 2 - [ - y~ sin tot = const. (10)

by giving arbitrary values to the constant. Since the motion is not steady, the

streamline p atte rn changes in general from instant to instant 9 However, in the

present case it can be shown that the streamlines are independent of time, since

the time depend~nt factor cancels out. In order to pro ve this we notice that

the differential equat ion for the streamlines can be writte n as

with

d y v ( z , y , t )

d x u ( x , y , t ) '

(11)

M y 2 _ x ~ - - M 2 x y

u --~ 2z~ (x 2 _ _ y2)~ sin t o t , v - - -- 2 ~ r ( x 2 q - y ~ ) ~ sin tot. (12)

Unfortunately, the resulting differential equation

2 x y d x - k ( y 2 _ x 2 ) d y = 0

is not exact. There exists, however, an integrating factor #(y)= c / y ~ , where

c ~-~ const. Hence

2x y~ -- x 2

- - d x - 4 - ~

d y - - - -

O 1 3 )

y y 2

from which follows

or

~

~ y ~ c ~ const ., c < 0, (14)

Y

1 5 )


5/14

O n L i q u i d M o t i o n i n a C i r c u l a r C y l i n d e r

Y

2 1 1

0 X

/.',- //,/[ \ 9 I ~\x.x-,~.\

Z A / // t \ - - , , 1 ~ \ > t ' , \

/ ' t r 1 6 2 " ' - - J " I I i l ~7\\

i l " t

IL

sire rni lne

F i g . 2 . S t r e a m l in e p a t t e r n o f p l a n e d o u b l e t

T h e c u r v e s ~ ~ c o n s t , d o n o t c h a n g e w i t h t i m e a n d a r e c i r c l e s (~ 'ig . 2 ) w i t h

c e n t e r ( 0 , + c [ 2 ) a n d r a d i u s fc /2 I . F u r t h e r m o r e , i t c a n b e s h o w n t h a t t h e p a t h

l in e s a r e c i rc u l a r a r c s a n d a l s o t i m e - i n d e p e n d e n t .

T o r e c o v e r t h e p r e s s u re , w e re c a l l E q . ( 2) . T o g e t h e r w i t h E q . (1 2 ), t h e s e c o n d

E u l e r e q u a t i o n c a n b e r e c a s t t o r e a d

~ p 1 o M [ 2 M 2 y ]

~ - y = - 2 ~ (x 2 + Y 2 ) ~ x y c o co s cot + 2"--~ " x ~ + y~ s in2 cot - - r (16 )

I n t e g r a t i o n w i t h r e s p e c t t o y g i v e s

p x , y , t ) = - - -

~ o x M 2 ~ s i n 2 c ot 1

9 c o s co t . . . . . . ~ g y + [ ( x , t ) .

2~ x 2 y ~ 8 ,n2 (x 2 + y2 )2

( 1 7 )

T o c a l c u l a t e

] ( x , t ) ,

w e a p p l y E q . (1 ), a g a i n w i t h E q . ( 12 ). W i t h E q . (1 7) t h e

f i r s t E u l e r e q u a t i o n y i e ld s ] ( x , t) = C = c o n s t . T h i s c o n s t a n t c a n b e e v a l u a t e d

b y c o n s i d e r i n g t h e l iq u i d a t a n i n s t a n t w h e r e i t o s c i l la t e s a b o u t i t s m e a n p o s i t io n .

T h i s e v e n t r e c u r s p e r i o d i c a l l y , n a m e l y i f s i n cot = =j= 1 , i .e . ,

6

co t = -~ - -4- n~r , n -~ O; 1 ; 2 ; . . . ( 18)

:N o w o n t h e f r e e l iq u i d s u r f a c e w e h a v e t o fu l fi l t h e c o n d i t i o n o f c o n s t a n t f lu i d

( a i r , g a s ) p r e s s u r e P 0 . L e t u s d e n o t e t h e r a d i i o f t h e c i r c u l a r s t r e a m l i n e a r c s

b y R , w i t h R o a s t h e m i n i m a l ( f r e e s u r fa c e ) a n d R 1 a s t h e m a x i m a l ( .c an a l w a l l )

v a l u e ( F ig . 3 ). A p p l y i n g t h e c o n d i t i o n o f c o n s t a n t f l u i d p r e s s u r e a t t h e p o i n t


6/14

2 1 2

K . B a r t k o w i a k , B . G a m p e r t , a n d J . S i e k m a n n :

Y

F i g . 3 . S k e t c h o f f r e e s u r f a c e p r o f i l e

x - -- - O , y = Y o = - - 2 R o ,

w e g e t f r o m E q . ( 1 7) , t o g e t h e r w i t h E q . ( 18 ),

M S ~

C = P o ~ - 1 2 8 u ~ R o a - -

2 g ~ R o .

( 1 9 )

T h e s h a p e o f t h e f r e e s u r f a c e y :

y o X , t )

f o l l o w s r e a d i l y f r o m E q s . ( 1 7 ) a n d ( 19 )

b y t a k i n g i n t o a c c o u n t a g a i n t h e c o n d i t i o n o f c o n s t a n t f l u i d p r e s s u r e P o. H e n c e

M o o x M ~ 1

- - - - c o s cot ~ - - - 9 s in s cot - -

g( 2R o + Yo) + 27~ x ~ + y0 ~ 8~ ~ ' (x ~ + y0~)~

s

1 2 8 u ~ R o ~ "

( 2 0 )

A l t h o u g h t h is e q u a t i o n g i v e s t h e f r ee s u r f a c e s h a p e i n i m p l i c i t f o r m o n l y , p o i n t -

w i s e c o n s t r u c t i o n o f t h e f r ee s u r f a c e p r o f i l e is e a s i l y a c c o m p l i s h e d . S e t t i n g

r 2 = x ~ ~- you , ( 2 1 )

E q . (2 0) c a n b e r e w r i t t e n i n t h e f o r m

W i t h

M s M e) co s cot x M 2 s i n 2 cot 1

Y o = 1 2 8 n S g R o 4 2 R ~ - - 2 ~ g 9 r-- - - 8 n ~ g 9 - ~ . ( 2 2 )

M c o

co s cot 1

9 - - ( 2 3 )

= o~( r , t ) = 2~ g r 2 '

w e

f i n d

t ~ = t ~ ~ , t ) -

M 2 M 2 s i n 2 co t 1

2 R o - - 9 - - ( 2 4)

1 2 8 ~ S g R o a 8 z ~ g r a ,

Yo = ~ x ~ f t . (25 )


7/14

w h e n c e

r~ = x ~ + ~2x2 + 2 ~ x + 5 ,

Y

l0

f o ll o w s. H e n c e

( 2 6 )

{t)

x 2 + ~ x + ~ = 0 (27)~

1 -I- ~ 1 + c~2

1

x l . ~ = 1 + ~ ~ ~ z i 1 / 1 + ~ / r ~ - - Z ~ ] 2 S /

a n d w i t h t h a t ( e l . E q . ( 2 5 ) )

Yol,2 = c~xl,2 +

.

( 2 9 )

T h e c o n s t r u c t i o n o f p o i n t s (x ~, Y0~) a n d ( x 2 , Y02) o f t h e s u r f a c e p r o f i l e , i s d e m o n -

s t r a t e d i n F i g . 4 .

T h u s f r o m E q . ( 21 )

Fig. 4. C onstru ction o f free surface profile Fig. 5. Pen du lum ana logy of free surface.

m o t i o n

On Liq uid Motion in a Circular Cylinder 2 1 3

T o d e t e r m i n e t h e c i r c u la r f r e q u e n c y ro a p p r o x i m a t e l y , w e c o n s i d e r a l i q u i d

p a r t i c l e o s c i l l a ti n g o n a c i r c u l a r a r c o f r a d i u s R 0 o f t h e f r e e s u r f a c e ( s tr e a m l i n e ,

p a t h l i n e ) a b o u t t h e e q u i l i b r i u m p o s i t i o n ( F ig . 5 ). F o r s m a l l o s c i l la t i o n s w e c a n

a p p l y t h e f a m i l i a r fo r m u l a f r o m e l e m e n t a r y p h y s i c s f o r t h e fr e q u e n c y c o f a

s i m p l e p e n d u l u m , v i z .

- ~ o ~ . ] . / 3 0 ),

A t t h e s a m e f i ll h e i g h t ( a t r e st ) , t h e c i r c u la r f r e q u e n c y d e p e n d s - - v i a R o

y e t o n t h e s t r e n g t h o f t h e o s c i l l a t i o n ( s t r e n g t h o f t h e d o u b l e t , e x c i t a t i o n , a m p l i -

t u d e ) . O n p r i n c i p l e t h i s d e p e n d e n c e w a s c o n f i r m e d b y t h e e x p e r i m e n t s . I t i s:

o f s e in e i n te r e s t t o c o m p a r e o u r r e s u lt w i t h L a m b ' s [1 ] a n d R a y l e i g h ' s d e t e r -


8/14

2 1 4 K . B a r t k o w i a k , B . G a m p e r t , a n d J . S i e k m a n n :

r u i n a ti o n o f th e f r e q u e n c y o f t h e s l o w e s t m o d e i n t h e c a s e w h e r e t h e f r e e s u r f a c e

i s a t t h e le v e l o f t h e a x i s. L a m b f i n d s a p p r o x i m a t e l y ~ ---- 1 , 1 6 9 g R 1 ) l l L t ~ a y l e i g h

( s ee [ 1 , p . 4 4 5 ]) o b t a i n e d , a s a c l o s e r a p p r o x i m a t i o n , co = 1 . 1 6 4 4 g / R 1 ) t / L

A t a f il l h e i g h t ( a t r e st ) u p t o t h e m i d d l e o f th e c o n t a i n e r , s u c h a s i n t h e c a s e o f

L a m b a n d i ~ a y l e i g h , w e h a v e /~o => 0 .5 R 1 . T h e r e f o r e , w i t h R 0 = 0 . 5 R 1 a n d

o s c i l la t io n s o f i n f i n it e s i m a l a m p l i t u d e , o u r e s t i m a t e f o r th e f u n d a m e n t a l e i ge n -

f r e q u e n c y o f a h a l f - f u l l c a n a l y i e l d s

1 . 4 1 4 2 g I R l ) 1 I ~ .

T h e n o t i c e a b l e d i f f e r e n c e

b e t w e e n o u r r e s u lt a n d t h a t o f L a m b ( an d R a y l e ig h ) m a y b e a t t r ib u t e d t o t h e

f a c t t h a t L a m b a s s u m e s t h a t t h e fr e e s u r f a c e r e m a i n s a l w a y s p la n e , m a k i n g a

s m a l l a n g l e w i t h t h e h o r i z o n t a l . W i t h R 0 = 0 . 7 3 1 8 171 a n d t h e a s s u m p t i o n o f

f i n it e (l ar g e) a m p l i t u d e s , w e f in d a g r e e m e n t w i t h L a m b , f o r R 0 = 0 . 7 3 7 6 R 1

a n d t h e a s s u m p t i o n o f f in i te ( la rg e ) a m p l i tu d e s , w e o b s e r v e a g r e e m e n t w i t h

R a y l e i g h . H o w e v e r , t h e v a l i d i t y o f t h e t h e o r y i n th e d o m a i n o f l a rg e a m p l i t u d e s

is s o m e w h a t q u e s t i o n a b l e . F o r o u r a n a l y s i s , t h e v a l u e s g i v e n b y E q . (3 0) a r e

a c c u r a t e e n o u g h .

k_

tY

I ~ U max

Fig. 6. Oscillating f luid pa rt icle on the free surface

N e x t w e h a v e t o e v a l u a t e M . M a k i n g u s e o f th e p e n d u l u m a n a l o g y , w e g et ,

i n t h e f i rs t p la c e , f r o m a n e n e r g y b a l a n c e ( F ig . 6 )

1 . d m "~ 3 1 )

d i n . ~7" H = ~ - 9 ureax

a n d t h e n c e f r o m E q s . (1 2) a n d ( 18 ), t o g e t h e r w i t h

x = 0 , y ~ - - 2 R o ,

= l / H . 3 2 )

W i t h t h e d e t e r m i n a t i o n o f R e ( m e a s u r e f o r t h e fi ll h e i g h t ), H ( m e a s u r e f o r t h e

a m p l i t u d e o f t h e o s c i l l a t i o n ) , ~ ( d e n s i t y o f t h e l i q u i d , t h e d e n s i t y o f t h e f l u i d


9/14

On Liquid Motion in a Circular Cylinder 215

Y0

- ~ l O r n m

RI O

f i n a l p o s i t i o n { l e f t )

t = 0 + n , 0 , 3 1 7 2 s

m e a n

p o s i t i o n

t = 0 , 0 7 9 3

+ n . 0 ,1 5 8 6

~ _ _ _ _

i n 2 e . . r m e d i a t e p o s i t i o n

_ ~~ , ~ ~ I t = 0 , 1 1 8 9 S 0 , 3 1 7 2 s

.... ~ 0 . 1 9 82 s + n . 0 . 3 1 7 2 s

Fig. 7. Numerica lly calculated free surface profile

above the free surface is assumed to be negligeable), g (local gravitational ac-

celeration) and P0 (ambient pressure, fluid pressure at the free surface) all flow

quant i t i e s u , v and p) a re known.

Fig. 7 i l lustrates results of numeri cal comp utat ions of the free surface profi le

y o x , t )

for different times. The radi us of the canal was R1 = 40 ram, further more,

the followin g va lue s were as sum ed : R 0 = 25 ram, H = 2 mm a nd g = 9 81 0m ms -'~.

Wi th th at co0 become s 19.8091 s -1. P art icu larl y, the init ial posi tion at time

t = 0 coincides with the final posi tion at the left ha nd side (~t = 0 :E n 9 2~).

The plot s show also the fina l pos iti on a t the rig ht ha nd side (o~t ---- :t :J: n 9 2x),

the me an posi tion (wt -- z/2 n 9 z) an d an int erm edi ate po sitio n (o~t = 3~/4

:E n 9 2:r; 5~/4 -t- n 9 2~) betw een the me an posit ion an d the right final positio n.

We no tic e th at for n ~- O; 1; 2; ... the profiles at ~t a nd ~ot~ = :J:o~t -t- n . 2~ are

identical . Moreover, to each profi le at ~t corresponds i ts reflection in the y-axis

(x = 0) at o~tll = z~ :E n 9 2~ (br oc ken line of in te rm ed ia te positi on).

15 Ac ta Me ch. 54/3--4


10/14

216 K. Bar tkow iak , B . Gam per t , und J . S i ekmann:

3 Experiments

T o s t u d y t h e s l o s h in g b e h a v i o r o f t h e l iq u i d e x p e r i m e n t M l y , a c i r c u la r c y l i n d e r

o f 80 m m d i a m e t e r a n d 2 0 m m l e n g h t w a s u t il iz e d . T h e f l a t f a c e - p la t e s o f t h e

c y l i n d e r w e r e m a d e o f t r a n s p a r e n t c e l lu lo i d. S m a l l p l a s t i c b a l ls o f 0 .4 m m d i a m -

e t e r w e r e d i s t r ib u t e d i n t h e l iq u i d . I n o r d e r t o g u a r a n t e e t h e s a m e d e n s i t y

f o r b o t h t h e p l a s t i c m a t e r i M a n d t h e t e s t l i q u i d , s a l t w a s d i s s o l v e d i n w a t e r

u n t i l t h e t i n y s p h e r e s r e m a i n e d i n a f l o a t i n g p o s i t i o n .

C e r t a i n l y , t h e c o n c e p t o f a n i n v i s e id l iq u i d c o u l d n o t b e r e a li z e d in t h e e x -

p e r i m e n t s ; h o w e v e r , t o r e d u c e f r ic t i o n a l l o ss e s t o a m i n i m u m , t h e c y l i n d e r

w a s s u p p o r t e d h o r i z o n t a l l y in b e a ri n g s s u c h t h a t b y v i r t u e o f a n a c t u a t o r t h e

b o d y c o u l d o s c i l la t e a b o u t i ts g e o m e t r i c a x i s . T h e o r e t i c a l l y t h e ta n g e n t i a l v e l o c -

i t y o f t h e f l u id p a r t i c le s a t t h e c a n a l w a l l a n d t h e t a n g e n t i a l v e l o c i t y o f t h e

o s c i l l a ti n g v e s s e l s h o u l d b e t h e s a m e . T h i s , o f c o u r se , i s p o s s i b l e f o r c e r t a i n

p a r t s o f t h e m o v i n g w a l l o n l y . M o r e o v e r , l i k e t h e o s c i l l a t i o n o f t h e l i q u i d , t h e

a n g u l a r s p e e d o f th e v e s s e l m u s t b e a s i n u s o id a l f u n c t i o n o f t i m e . F i n a l ly , t h e

c i r c u l a r f r e q u e n c y o f t h e p e n d u l u m m o t i o n o f t h e c y l i n d e r s h o u l d b e e q u a l

t o t h e c a l c u l a t e d c o - va lu e . T o a c h i e v e t h i s , t h e a c t u a t o r w a s a d j u s t e d u n t i l l i q u i d

a n d v e s se l w e re i n p h a s e . T h e e x a c t a d j u s t m e n t o f t h e a c t u a t o r w a s ac c o m p l i s h e d

b y t h e i n f i n i te l y v a r i a b l e s e r ie s r e s i s ta n c e o f a d i r e c t c u r r e n t d r i v i n g m o t o r .

F l u i d o s c i l l a t i o n s w e r e b u i l t - u p a f t e r s w i t c h i n g - o n o f t h e m a c h i n e r y , o w i n g

t o t h e s t r o n g e f f e c t o f th e w a l l s h e a r s t r e s s o n t h e l iq u i d a t t h e b e g i n n i n g o f

t h e l i q u i d m o t i o n .

F ig . 8 sh o w s a f l a sh - l i g h t p h o t o g r a p h o f t h e e x p e r i m e n t M a r r a n g e m e n t in

o p e r a t i o n . T h e i n c a n d e s c e n t b u l b a b o v e t h e l i q u id a n d t h e t r a n s p a r e n t s c re e n

w i t h t e a r e d - u p d o u b l e t - li n e s s t re a m l i n e s ) a r e g o o d fo r o b s e r v a t i o n w i t h t h e

n a k e d e y e o n l y i n o r d e r t o e x a m i n e t h e c o i n c id e n c e b e t w e e n p a t h l in e s a n d

d o u b l e t - li n e s , t ~ e f le x e s f r o m t h e e d g e s , w h i c h t h e l i q u i d f o r m s w i t h t h e f a c e -

Fig. 8. Photography of the osci l lat ing cyl inder


11/14

On Liqu id Motion in a Circular Cylinder 21 7

p l a t e s o f t h e v e s s e l , e x h i b i t i n t h e i r m a i n c o u r s e a g r e e m e n t w i t h t h e n u m e r i c a l l y

o b t a i n e d s u r f a c e p r o f il e s.

F i g. 9 d e p i c t s t h e p o w e r s u p p l y u n i t 2 2 0 V ~ - + 4 . 5 ; 6 ; 7.5 ; 9 V = ) , s e ri es

r e s is t a n c e , d i r e c t c u r r e n t d r i v i n g ~ m o t o r a n d c o n t r o l li n g m e c h a n i s m o f t h e t e s t

a p p a r a t u s , a r r a n g e d b e h i n d t h e c o n t ai n e r . T h e c y l in d e r i s r o t a t e d w i t h a c r a n k

a n d a d r iv i n g ro d . T h e r a d i u s o f t h e c r a n k a n d w i t h t h a t t h e a m p l i t u d e o f t h e

p e n d u l u m m o t i o n a r e a d j u s t a b l e , f i n e t u n i n g o f th e c i r c u la r f r e q u e n c y o~ is

r e n d e r e d p o s s i b l e b y t h e i n f i n i t e l y v a r i a b l e s e r i e s r e s i s t a n c e . M a j o r c o r r e c t i o n s

c a n b e m a d e b y m e a n s o f t h e o u t p u t v o l ta g e of t h e p o w e r s u p p l y u n it a n d t h e

t r a n s m i s s io n o f t h e c o n t r o l m e c h a n i s m .

F ig . ~ . Pho tograp hy of t he ex per imen ta l a r r angem ent

1 5

Fig. 10. .~Photography of cyl inder, l iquid and plast ic bal ls at rest


12/14

2 8 K. Bartkowiak B. Gampert and J. Siekmann:

:Fig. 11. Flow visualization of pa th lines

Fig. 12. Photography of path lines and

theoretically determined streamlines

Fig. 13. Photography of pat h lines and

theoretically determined streamlines after

correction of position of the streamlines


13/14

O n L i q u i d M o t i o n i n a C i r c u l a r C y l i n d e r 2 1 9

F i g . 10 e x h i b i t s c y l i n d e r a n d l i q u i d a t r e s t , t h e p l a s t i c b a l l s f o r f lo w v i s u a l i -

z a t i o n a r e c l e a r l y r e c o g n i ze d . F i g . 11 s h o w s a p h o t o g r a p h y t im e o f e x p o s u r e

A t = t2 - - tl = 1 /8 s ) o f t h e t e s t a p p a r a t u s i n o p e r a t i o n . T h e w h i t e l in e s a r e th e

p a t h l i ne s . I n F ig . 12 t h e o r e t i c a l l y o b t a i n e d s t r e a m l i n e s d o u b l e t - l in e s ) a r e d r a w n

f o r c o m p a r i s o n w i t h t h e p a t h l in e s . F i g . 1 3 s h o w s a r e p r o d u c t i o n o f t h e p a t h

l in e s o f F i g . 12 , b u t w i t h a s h i f t o f t h e s t r e a m l i n e s , s o t h a t c o i n c i d e n c e w i t h t h e

p a t h l in e s is a c h i e v e d t o s o m e e x t e n t .

T h e p a t h l i n e s s , r e c o r d e d d u r i n g t h e t i m e o f e x p o s s u r e t l --


14/14

220 K. Bartkowiak, B. Gampert, and J. Siekmann: On Liquid 5fotion

[9] McCarthy, J. L., Stevens, D. G.: Inve stigat ion of the natural frequencies of fluids

in spherical and cylindrical tanks. NASA TN D-252, 1960.

[10] Petrov, A. A., Popov, Yu. P., Pnkhnachev, Yn. V.: Calculation of free oscillations

of a liquid in immovable containers by a va riational method. Zh. V. MiMF 4, 880--895

1964).

[11] Siekmann, J., Chang, Shih-Chih: Note on liquid sloshing in a container of arbitrary

shape. Z. angew. Math. Phys. 21, 830--836 1970).

[12] Chang, Shih-Chih, Wu, S. T.: On the natural frequencies of standing water waves

in a canal of arb itr ary shape. Z. angew. Math. Phys. 23, 881--888 1972).

K . B a r tl co w i a k B . G a m p e r t a n d J . S i J c m a n n

U n i v e r s i t d t E s s e n

S c h i ~ t z e n b a h n 7 0

D - 4 3 0 0 E s s e n 1

F e d er a l R e p u b l ic o / G e r m a n y

Documents

On liquid motion in a circular cylinder with horizontal axis.pdf