STRUCTURE OF PERIODICAL DISTURBANCES IN REAL LIQUID

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STRUCTURE OF PERIODICAL DISTURBANCES IN REAL LIQUID

V. Kedrinskii, S. Plaksin

To cite this version:

V. Kedrinskii, S. Plaksin. STRUCTURE OF PERIODICAL DISTURBANCES IN REAL LIQUID.

Journal de Physique Colloques, 1979, 40 (C8), pp.C8-293-C8-295. �10.1051/jphyscol:1979851�. �jpa-

00219557�

STRUCTURE OF PERIODICAL DISTURBANCES IN REAL LIQUID V.K. Kedrinskii and S.I. Plaksin

Institute of Hydrodynamics, Siberian Branch of the USSR Academy of Sciences, Novosibirk 630090 USSR

Abstract. - Within the framework of the nonequilibrium two-phase model the structure of supersonic waves and the cavitation development are investigated in a real liquid containing a free gas as cavitation bubbles. The calculation were carried out at the initial volumetric gas concentration Kg = 10"8, the gas bubble radius R0 = 10"3 cm and the oscillation frequency of the piston ranged from 1 to 200 kc.

1. INTRODUCTION. - The mathematical model of the motion of liquid containing gas bubbles was sugges- ted in /1-3/ and, as a rule, was used to investi- gate the structure of shock waves in liquid with specially created gas bubbles. Meanwhile, it is known /4/ that the free gas is always present in a real liquid as cavitation bubbles. The free gas parameters are very small, but the appearance of the ratefaction wave in liquid can lead to the intensive cavitation development /5/. Firstly such a problem was solved in /6/ where the numerical and experimental investigations were carried out for the parameters of rarefaction waves and the cavitation development at underwater explosion near the free surface.

2. MATHEMATICAL MODEL. - For aone-dimensional case the motion of liquid containig gas bubbles is des- cribed by conservation laws of mass, momentum, number of bubbles and energy within an accuracy of the first order by the volumetric gas concentra- tion. These laws can be written in invariants :

Here ri = u+ln(l+ p-po), r2 = -ln(l+p-p0) + u

r3 = ln(k/(l-k) R3(l+p-p0)), P = P0 + 2s/R0, W = 2a/R0P , w0 = (P /p°(P0))1 / 2/Ro.l i = R0 W° /c 0»

fj =CO/U>Q, u >p ar<e the average velocity and pres- sure, respectively ; R is the bubble radius, k is the volumetric gas concentration, s is the veloci- ty of radial bubble pulsation,^0 (P0) is the ini- tial density of the liquid component, a is the surface tension coefficient, y is the adiabatic index of gas, C Q is the sound velocity in undis- JOURNAL DE PHYSIQUE Colloque Ce, supplément au n° 11, tome 40, novembre 1979, page C8-293

Résumé. - Dans cet article, on étudie les ondes supersoniques et la cavitation qui prennent naissance dans un liquide contenant du gaz libre sous forme de bulles. L'onde de pression est produite par un piston qui oscille périodiquement. Les calculs numériques sont conduits, dans le cadre du problème à une dimension, en s'appuyant sur le modèle diphasique hors équilibre.

Initialement, la teneur en gaz est Ko = 10"8 et le rayon des bulles R Q = 10~3 cm. Le piston oscille avec une fréquence qui peut varier de 1 à 200 kHz ; l'amplitude de son déplacement peut prendre deux valeurs, qui correspondent pour la pression (dans un liquide ne contenant pas de gaz) à des amplitudes de 9 à 49 atm. On montre que l'amplitude, la durée et la forme de T o n d e de

pression dépendent fortement de la fréquence excitatrice. C'est ainsi que la diminution de cette fréquence provoque, pour T o n d e de pression, une augmentation de la fréquence, une diminution de l'amplitude et une profonde modification de la forme par rapport au modèle monophasique. Les calculs ne sont pas conduits jusqu'au moment de l'implosion.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979851

c8-294 JOURNAL DE PHYSIQUE

turbed f l u i d , Po i s t h e h y d r o s t a t i c pressure, o i s t h e frequency o f p i s t o n o s c i l l a t i o n s ; y, t a r e t h e v a r i a b l e s connected w i t h Lagrange coordinates 5, t by t h e r e l a t i o n y = (1-kg) S / t ( f u n c t i o n s and inde- pendent v a r i a b l e s are reduced t o t h e dimensionless form w i t h t h e h e l p o f w,Co).

3. STATEMENT OF PROBLEM.

-

L e t a s e m i i n f i n i t e tube bounded a t l e f t by a p i s t o n be f i l l e d by a r e a l li- q u i d w i t h a v o l u m e t r i c gas concentration KO contain- i n g u n i f o r m l y d i s t r i b u t e d c a v i t a t i o n bubbles o f r a d i u s Ro. The p i s t o n begins t o move t o t h e l e f t , and, consequently, t h e wave f r o n t as a weak discon- t i n u i t y , propagates along t h e c h a r a c t e r i s t i c s y = l t o t h e r i g h t . The f l o w r e g i o n i n t h e v a r a i b l e s y , t i s r e s t r i c t e d by t h e a x i s t=O, l i n e s y=1,0. For t > O t h e problem i s solved a t t h e f o l l o w i n g condi- t i o n s : f o r y = 0 : r l + r 2 = - 2 a o s i n t ; f o r y = 1 : r1 = r 2 = 0, r3 = l n ( k O / ( l - k o ) ) , R = 1, S = 0 ; f o r t = 0 : rl = r2 = 0, r3 = l n ( k o / ( l - K O ) ) , R = 1, S = 0. I n t e g r a t i n g t h e equation (3), we o b t a i n : k = mOR3 exp((t-1-r7)/2) where mo= i<o/(l-Ko!

l + m O ~ 3 exp( ( r l - r 2 ) / 2 )

The system ( 1 ) - ( 2 ) , ( 4 ) - ( 5 ) i s w r i t t e n i n a d i f f e - rence form and solved by t h e well-known f a c t o r i z a - t i o n method /7/.

4. RESULTS OF CALCULATIONS.

-

The c a l c u l a t i o n s were c a r r i e d o u t a t kO=lO-8, ~ ~ = 1 0 - 3 c m , po=latm,

y = 1 . 4 , ~ = 1 . 5 ~ 1 0 ~ c m / s e c , o = 75dyn/cm f o r t h e d i f f e r e n t values o f maximum amplitudes and frequen- c i e s and r e s t r i c t e d by t h e moment of t h e bubble collapse. The change o f pressure a c t i n g t o t h e p i s - t o n w i t h time i s presented i n F i g . 1 for i t s o s c i l l a - t i o n frequencies o f 100, 50, 20, 10, 6 and 3 kc, r e s p e c t i v e l y .

With decreasing frequency t h e maximum negative pressure p e r m i t t e d by a r e a l l i q u i d s u f f i c i e n t l y decreases. This e f f e c t , according t o /6/, i s ex- p l a i n e d by t h e c a v i t a t i o n development and t h e i n - crease o f time d u r i n g which t h e t e n s i o n i n t h e medium approaches i t s maximum value. I n t h i s case the p e r i o d i c a l waves are formed ' i n t h e medi um i n s p i t e o f t h e f a c t t h a t the p i s t o n moves t o t h e l e f t y e t . However, a t frequencies o f 3 kc and

less

t h e average pressure doesn't pass t o t h e p o s i t i v e r e g i o n t o t h e moment o f t h e bubble collapse.

The c a l c u l a t i o n s show t h a t t h e i n t e n s i v e c a v i t a - t i o n development i s observed i n t h e r e g i o n b e t -

ween the p i s t o n and t h e wave f r o n t .

1, P -p.

a t m I I

(2 - ^I

Fig. 1 V a r i a t i o n w i t h t i m e o f the p i s t o n pressure depending on frequency.

The pressure d i s t r i b u t i o n i n t h e research r e g i o n , are presented i n Fig. 2 ( x

-

E u l e r coordinate) f o r t = 1.8 and t h e same frequencies. Curves 1 are t h e most close t o t h e case o f a one-phase l i q u i d .

Fig. 2 Pressure d i s t r i b u t i o n between t h e p ' s t o n and t h e wave f r o n t f o r t = 1.8, a" = 4x10-

4 ,

1

-

f = 100 kc, 2

-

f = 50 kc, 3

-

^{f = 20 kc,}

4 - f = l 0 k c , 5 - f = 6 k c , 6 - f = 3 k c

V . ^K.

KENDRINSKII and S. I. PLAKSIN

References

/1/ I o r d a n s k i i

S.V.

On t h e equations o f motion f o r l i q u i d s c o n t a i n i g gas bubbles. PMTF, 1960 n03.

/2/ Kogarko B.S, On t h e model o f c a v i t a t i n g l i q u i d . Dokl. Akad. Nauk. SSSR, 1961, v. 137 n06 /3/ Van Winjngaarden L. On t h e equations o f motion

f o r mixtures o f l i q u i d and gas bubbles.

"J. F l u i d Mech.", 1968, v. 33, p t . 3

/4/ F l i n n G. The physics o f a c o u s t i c c a v i t a t i o n i n l i q u i d s . I n book : Methods and atmaratus f o r supersonic i n v e s t i g a t i o n s , v. 1,' p a r t 6 , 1967 " M i r " .

/5/ A k u l i t s h e v V.A. Cavity p u s l a t i o n . I n book :

"Powerful u l t r a s o n i c f i e l d s " , 1968, "Izd.

"Nauka".

/6/ K e d r i n s k i i V. K. Negative pressure p r o f i l e i n c a v i t a t i o n zone a t underwater explosion near f r e e surface.

-

'Acta Astronautics", 1976, v. 3, no 7-8.

/7/ Godunov S . K., Ryabenjkii V.S., I n t r o d u c t i o n t o t h e d i f f e r e n c e scheme theory. 1962, Izd.

"Nac;kal'.

Notations

5

x,t E u l e r coordinates

5

Lagrange coordinates y ,t s u b s i d i a r y coordinates

PO i n i t i a l pressure

p O ( P o ) i n i t i a l d e n s i t y o f l i q u i d component a surface t e n s i o n c o e f f i c i e n t

Ro i n i t i a l r a d i u s o f bubble y a d i a b a t i c index gas

KO i n i t i a l v o l u m e t r i c concentration o f f r e e gas

Co sound v e l o c i t y i n i n d i s t u r b e d l i q u i d

P pressure

U mass v e l o c i t y R bubble radius

S

v e l o c i t y o f r a d i a l p u l s a t i o n o f bubble K v o l u m e t r i c concentration o f f r e e gas w,f frequency of p i s t o n o s c i l l a t i o n s r r , r i n v a r i a n t s

~ ~ , S 1 , - p , W, p, aO,mo constants