STRUCTURE OF PRESSURE PULSES IN LIQUIDS WITH GAS BUBBLES

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STRUCTURE OF PRESSURE PULSES IN LIQUIDS WITH GAS BUBBLES

N. Malykh, I. Ogorodnikov

To cite this version:

N. Malykh, I. Ogorodnikov. STRUCTURE OF PRESSURE PULSES IN LIQUIDS WITH GAS BUB- BLES. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-300-C8-305. �10.1051/jphyscol:1979853�.

�jpa-00219559�

STRUCTURE OF PRESSURE PULSES IN LIQUIDS WITH GAS BUBBLES N.V. Malykh, I.A. 0G0R0DNIK0V

Institute of Thermophyeies, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 630090, USSR

Abstract.- The structure of short pressure pulses in bubbles layers in water and the dynamics of these layers under the effect of these pulses have been studied. The structures of pulses charac- teristic of dispersive media with a high-freauency "forerunner" have been experimentally obtained Experimental results are qualitatively compared with the solution of the system of an inhomogeneous wave equation and an equation of bubble motion, as well as with the high-frequency linear approxi- mation of the system, this is the Clain-Gordon equation.

I.-INTRODUCTION - The main specific feature of a liquide with gas bubbles is the great difference of its components in compressibility and density and hence the significant influence of nonlinear, dissipation and dispersion effects. Depending on the relation between the parameters of the media and the external perturbation, either a l K t h e effects manifest themselves or one fo them may dominate. With the increases of the external perturbation amplitude the influence of nonlinear effects grows. If, however, the perturbations are considered as pulses with a sharp front of the pressure increase, then as a result of the adjoi- ning liquid mass inertia, for short periods of the pulse effect or of the pulse front increase even at sufficiently large amplitudes the bybjjle does not manage to change its size significantly and to obtain velocity but it gains accelaration. In this case the dissipation effects due to viscosity and thermal conductivity do not manage to manifest themselves and the inertial effects due to accele- ration are predominant. This leads to the sound speed dispersion and determines either the short pulse structure ori.the structure in the vicinities of the pulse front.

II.- EXPERIMENTAL PROCEDURE - In a hydroacoustic vessel and a shock tube gas bubble layers in water were formed by bubbling and electrolysis. A

pulse produced by a small discharge explosion fell

to the boundary of the bubble layer from a pure liquid. The pulse structure was registered by hydrophones embedded into the bubble layer of immediately behind it. The amplitude of the pulse incident to the layer was varied by changing the distance from the explosion to the bubble layer boundary. To register high-frequency pulses minia- ture hydrophones with a linear characteristics up to high frequencies and a wide-band cathode follower were made. The dynamics of gas bubbles was stutied using high-speed recording in a shock tube with a transporent test section.

III.- DYNAMICS OF BUBBLES - The behaviour of gas bubbles under the influence of pressure pulses with steep growth front is essentially nonequilibrium due to the inertia of the joined liquid mass. This effect leads to the fact that the media parameters during and after the pulse action may greatly change with respect to equilibrium ones. Fig 1 represents the complete dynamics of notrogen bubbles from compression to collapse. Bubble compression (frames 1-2) occurs for the time period less than 500 us and cannot be resolved at the given recor- ding rate. After nonequilibrium compression the bubbles explosively expand (frames 3-8), the gas void fraction in the medium 3 ms later after the pulse action increases by a factor of 50. After overexpansion the bubbles collapse (frames 11-12) and break into 20-30 minutes bubbles (fig. 2 ) . JOURNAL DE PHYSIQUE Colloque C8, supplément au N° 11, tome 40, novembre 1979, page C8-3O0

Résumé.- On a étudié la structure de brèves impulsions de pressions dans des couches de bulles dans V e a u , ainsi que la dynamique de ces couches sous l'effet des puises. Expérimentalement, une structure caractéristique du milieu dispersif avec une haute fréquence "forerunner" a été obtenue. Qualitativement, elle a été comparée avec la solution du système d'équations d'une onde inhomogène et de l'équation du mouvement d'une bulle, ainsi qu'avec l'approximation linéaire à haute fréquence du système (équation de Clain-Gordon).

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:1979853

Fig 1

-

Dynamics of nitrogen bubbles a f t e r pulse e f f e c t Ro = 0.35.10J m , 5 2

A, = 20.10 N/m

,

= 200.10-~s, = l o 4 , r a t e of recording=2000 f / s , d = I O O . I O - ~ m.

F i g . 2

-

Result of bubble division 17 ms l a t e r the pulse e f f e c t .

(28-302 N.V. MALYKH and I.A. OGORODNIKOV

F i g . 3 represents t h e r a d i u s vs the time dependen- ces superposed w i t h pressure

-

time ones f o r various r e l a t i o n s o f the parameters. I t may be noted t h a t as e a r l y as d u r i n g t h e wave a c t i o n small bubbles s i g n i f i c a n t l y change t h e i r parameters ( f i g . 3 a, b ) . There i s , however, a time i n t e r v a l

F i g . 3

-

Dynamics o f bubbles.

a

-

(, = d=Im, ~ ~ = 0 . 2 5 . 1 0 ' ~ m . c-Ro = 0.5.10 -2 m,0-air, A-helium, x-argon.

i n the v i c i n i t y o f a p u l s e f r o n t when bubbles have n o t y e t s i g n i f i c a n t l y d e v i a t e d from the e q u i l i b r i u m p o s i t i o n . For t h e waves w i t h s u f f i c i e n t amplitudes t h i s i n t e r v a l i s very small. Large bubbles f o r the time o f pulse a c t i o n do n o t 'manage t o d e v i a t e from t h e e q u i l i b r i u m demensions and f u r t h e r they p u l s a t e on the frequency c l o s e t o t h e i r i n t i n s i c value ( f i g . Fig.3 c). The maximum overexpankion i n t h e cases represented i n F i g . 3 a, b i s s l i g h t l y h i g h e r than

i t would be i n a p o s i t i v e p u l s e due t o the e f f e c t o f a r a r e f a c t i o n wave f o l l o w i n g a compression one.

The f a c t o f i n e r t i a l overexpansion, however, ramains and i t s v a l i d i t y i s supported by t h e nume- r i c a l c a l c u l a t i o n given i n what f o l l o w s .

I V .

-

STRUCTURE OF PULSES

-

A t the c ~ n s t a n t p o i n t o f observation ( f i g . 4) t h e s i g n a l s t r u c t u r e was observed i n a pure l i q u i d /1/ and w i t h a l a y e r o f

F i g . 4

-

S t r u c t u r e o f pulses Qo =

lo-',

d=0,3 m,

I - i n pure l i q u i d , 2-Ro=0.5.10 -4 m, 3-Ro=I.10 -4 m,

4-Ro=O. 5.10'~m

bubbles i n f r o n t o f t h e hydrophone /2,3,4/. I t i s seen t h a t t h e c h a r a c t e r i s t i c f e a t u r e o f t h e pulses i s t h e presence o f a s i g n a l e n t e r i n g t h e medium a t a sound speed i n l i q u i d . This s i g n a l , a "sonic forerunner", always e x i s t s b u t depending on t h e medium parameters may be o f various shapes. The main p a r t o f t h e p u l s e f o l l o w i n g t h e " f o r e r u n n e r "

a l s o changes i t s shape f o r v a r i o u s - s i z e bubbles ( f i g . 4 ) . With t h e increase o f t h e l a y e r thickness t h e " f o r e r u n n e r " amp1 i tude decreases, t h e "main p a r t " o f t h e p u l s e a l s o diminishes i n t h e amplitude and spreads (becomes wider) ( f i g . 5 ) . With the

JOURNAL DE PHYSIQUE C8-303

Fiu. 5 .,

-

Pulse evolution in a laver.

$0=10-3, ~ ~ = 1 . 1 0 - ~ m

I

-

i n pure l i q u i d , 2-d=3.5.10-~rn,

increase of the amplitude A. of the incident pulse the "gravity center" of the "main part" of the pulse moves closer t o the "furerunner" ( f i g . 6 ) . The oscillograms in Fig. 6a d i s t i n c t l y shows the unhannonic shape of pressure pulsations with acute vertices c h a r a c t e r i s t i c of nonlinear relaxing media The ampl i tude of pulsations incrases up t o value of the signal i n the media with the pulse ampl i tude.

V .

-

CLAIN-GORDON EQUATION

-

Consider a liquid w i t h gas bubbles of the same s i z e . Neglecting the gas mass, the mixture density w i l l be in the fonn p1 = ( I - @ ) p, where p = liquid density, @ = gas void f r a c t i o n . Substituting the density expression t o the equation of mass and momentum conservation transform the system by the method reported by D . Crighton and J . Williams / l / . A1 1 terms depen- ding on @ a r e replaced t o the right-hand s i d e and then the wave operator i s separated. As a r e s u l t the system reduces t o an inhomogeneous wave equa- t i o n . Adding t h i s equation by the bubble motion equation and the relation between the gas void f r a c t i o n $I and the bubble radius R we obtain the i n i t i a l system of equations

Fig. 6

-

Structure changes with ampl itude, 1$~=7. ~ ' = 1 . 1 0 -4 m, d=35

5 2 5

1-Ao=26.10 N/m

,

2-40.105, 3-55.10

,

4-67. l o 5 , 5-89.

lo5,

6-200.10 5

.

- - a

(pu

- a

In (1-$1 + -

a

pum)

+

- 1 - - Z ( l - @ ) ~ u

a'

2

a t ax a x

2

a x

c8-304 N.V. MALYKH and I.A. OGORODNIKOV

where P=pressure i n a wave, C=sound speed i n a l i q u i d w i t h o u t bubbles, u = c o e f f i c i e n t o f l i q u i d v i s c o s i t y , y= a d i a b a t i c index, N=bubble number per u n i t volume, Index o denotes the i n i t i a l s t a t e Equation (1) i s i d e n t i c a l t o the t o t a l system o f equations up t o the pressures a t which l i q u i d i s s t i l l considered t o be l i n e a r , i .e. when P=pC 2

.

The advantage o f t h i s form o f w r i t i n g the equation i s the p o s s i b i l i t y t o c l e a r l y i n t e r p r e t t h e e f f e c t o f a g a s phase t o l i s u i d i n terms o f the sources whose e f f i c i e n c y i s determined by t h e equation o f bubble motion i n a wave f i e l d . The main r o l e i n

t h e r i g h t - h a n d s i d e o f equation (1) i s played by t h e f i r s t terme b e i n g o f a monopole character. I t should be noted, however, t h a t a t very small values o f gas v o i d f r a c t i o n ( 4 = the c o n t r i b u t i o n o f t h e monopole and t h i r d - q u a d r i p o l e terms may become o f one order o f magnitude, I f i n t h e f i e l d o f t h e i n c i d e n t wave the bubble s i z e changes s l i g h t l y , i .e. W/Ro << 1, equation ( 1 ) and (2) may be l i n e a r i z e d , then from the system o f ( I ) , (2) and (3) we o b t a i n one equation

a2

P ^{2 t}

a2 -

^{? = k2}

+ ^j

^{P ( T )} exp ( - u ( t - r ) )

72

^{a x}

a t

o

S i n w0 (t

-

^{T) d r} (4)

where

T h i s equation i s v a l i d f o r any r e l a t i o n s between t h e c h a r a c t e r i s t i c times o f bubble changes T and e x t e r n a l p e r t u r b a t i o n T . The d i s p e r s i o n r e l a t i o n o f t h i s equation i s o f the form

The d i s p e r s i o n r e l a t i o n o f equation (5) i s

The Green f u n c t i o n i s

The f i r s t term describes a "sonic forerunner" pro- pagating a t sonic v e l o c i t y i n a l i q u i d w i t h o u t bubbles. The second terme depends on t h e pulse and media parameters. I t should be noted t h a t equation (5) i s v a l i d n o t o n l y f o r small p e r t u r b a t i o n s when AP/Po << 1 s i m i l a r t o equation (4), b u t f o r

a r b i t r a r y p e r t u r b a t i o n amp1 i tudes. To d e r i v e equation (5) i t i s necessary t h a t i n equation ( 2 ) t h e terms 3/2 R2 p, 4v

AIR,

p , ( ~ ~ / R ) ~ ~ m a y be neglected as compared t o R

k'

p. It i s p o s s i b l e when t h e c o n d i t i o n s BR/Ro<< 1, 4

v

-r/pRo << 2 1

2 2

and T /T << 1, a r e met, which enables t o r e l a t e t h e parameters o f t h e media w i t h t h e p e r t u r b a t i o n d u r a t i o n and amplitude AR

2

^T 2 P/Rop

.

T h i s evalua- t i o n enables t o estimate t h e times w i t h respect t o t h e pulse beginning a t t h e given media parameters when t h e Clain-Gordon eauation i s v a l i d . Thus f o r

-4 -2 5

Ro = 10 m and Ro = 0.5. 10 m a t P-10 n/m2,r = 31

5 2

and 1580

us,

and a t

P

= 100.10 n/m T = 3 and

V1.- CALCULATION RESULTS.

-

Fig. 7 represents t h e r e s u l t s o f t h e numerical c a l c u l a t i o n o f t h e system o f ( I ) , ( 2 ) and ( 3 ) . As seen, i n t h e i n i t i a l l a y e r t h e bubble behaviour i s n o n l i n e a r ( 7 a I I ) and t h e bubble o v e r d i s t r i b u t e s r e l a t i v e t o t h e i n i t i a l s i z e a f t e r t h e p u l s e e f f e c t . I n t h e l a y e r a t a distance o f 36 cm t h e bubble behaviour s i g n i f i c a n t l y changes, t h e bubble p u l s a t e around i t s e q u i l i b r i u m p o s i t i o n ( 7 a I I I ) . The p l o t o f t h e instantaneous pressure p r o f i l e (7c) shows t h a t a t f i r s t t h e s i g n a l w i t h a sonic speed i n l i q u i d a r r i v e s a t t h e p o i n t o f

-

observation which i s f o l l o w e d by t h e r a r e f a c t i o n

For t h e frequences w << wo equation (4) reduces t o wave, then t h e r e a r r i v e s t h e "main" p a r t o f t h e t h e Clain-Gordon one p u l s e w i t h t h e d u r a t i o n o f about 50 us. T h i s

demonstrates t h e c h a r a c t e r i s t i c f e a t u r e s o f t h e 1 a 2 p - a 2 p = - h 2 p

(5) s i g n a l observed i n t h e experiment. F i g . 7c r e p r e -

G - 7

sents t h e instantaneous p r o f i l e o f gas v o i d f r a c t i o n

JOURNAL DE PHYSIQUE

Fig. 7 - Results of numerical calculation Po=10 ~/m' 5 - pressure in environmental

media, -

3

3

~ ~ 1 1 . 1 0 ,

= =

50.10-~ s.

a1 - shape of initial perturbation, aII- solution of Clain-Gordon equation, a 111- solution of the system.

As seen, with the displacement of the wave at a distance of 12

CT

the gas void fraction of the first layer was increased by the factor of 2.5 fig. 7d gives the comparison of the numerical calculation of the system with the Clain-Gordon equation. As seen, at the times of 150 us ( 3 c ~ ) the solution of the Clain-Gordon equation coincides with that of the system. The difference in the curves is due to the fact that for the frequencies below

w =

AC the Clai-Gordon equation does not describe the waves motion.

Thus considering a linear equation and allo- wing for only inertia effects, it is possible to qualitatively describe the structure of a short compression pulse in a liquid with bubble.

REFERENCE

/1/ Crighton (D.G.) and Ffowecs Williams (J.E.)

J. Fluid Mec., 1969, 36, 585