##### Document Text Contents

Page 1

Adiabatic Waves in

Liquid-Vapor Systems

Page 2

International Union of Theoretical

and Applied Mechanics

G. E. A. Meier' P. A. Thompson (Eds.)

Adiabatic Waves in

Liquid-Vapor Systems

IUTAM Symposium Gbttingen,

28.8. - 1. 9.1989

Springer-Verlag

Berlin Heidelberg New York

London Paris Tokyo Hong Kong

Page 215

210

the coherence function ~ remain larger than 0.67. The corresponding

data may be straightforwardly interpreted in terms of a single wave

whose velocity is C and whose spatial amplification coefficient is :

Ln H

d

d being the distance between successive probes (here 0.2 m)

2. In some part of the test section, the data exhibit some of (in most

cases all) the following characteristics :

There are significant differences between

significantly from one pair of probes

Cc and

to the

CS ' and Cs varies

next. The cross

correlation function may present two neighboring peaks.

- H varies significantly form one pair of probes to the next, and the

the product of Hmin by Hmax is of the order of 1.

- Coherence "accidents" (very low isolated values) may occur.

As it can be shown by computing the apparent velocity and gain resulting

from the superimposition of two waves (Boure, 1988), this is exactly

what is to be expected when two modes are coexisting, in which case Cs

and H lose their significances.

The existence of two modes is confirmed by plotting (fig. 1) the data for

the wave velocity and the drift flux as functions of the void fraction (the

data is tabulated in Boure, 1988). According to the theory (Boure, 1988)

the significant wave velocity is supposed to be the velocity C-W relative

to the mixture center of volume with the definitions :

WG and WL being the average velocities of the two phases. On fig. 1, the

points are strikingly distributed into two families which correspond to the

two modes. One mode, referred to hereafter as mode 3 (the subscripts 1 and

2 being reserved for pressure waves), is predominant at small void

fractions (~< 0.25), while the other (mode 4) is predominant at larger

void fractions (~> 0.30). The two modes probably coexist at least for

0.20 < ~ < 0.40.

Page 216

c - W (em/s)

30~------+-------,-------~-------+--~

~~------~--------~--------~------~--~

10~------T---~~1--------r-------+----

a

OL-------~------~------~------L-~

5 (em/s)

5~------~------~------~~~--~~

a

O~------~------~--------~------~--~

.1 .2

c- W= 5~

Fig. 1 Wave velocities (top) and drift flux (bottom). Data

(segments and crosses) and conjoint correlations (solid lines).

211

The existence of two modes appears also on the plot of the amplification

coefficient as a function of the frequency or of the wavelength. This plot

is not reproduced here because only orders of magnitude can be derived from

the available data for mode 4.

It is pointed out that, as shown by fig. 1, the domains of predominance of

the two modes correspond to different relationships between the drift flux

& and the void fraction~, i.e. to different flow regimes. It is suggested

that mode 3 may result from individual interactions between otherwise

independent bubbles, while mode 4 involves interactions between individual

bubbles and swarms of bubbles (in active interactions themselves).

Page 430

435

pressure (p )x~o penetrates about 30 times deeper into the cavitation region at an increa-

se of £ by a factor 10.

The spherical cavity cluster

Spherical cavity clusters are typically generated by focused acoustic waves, and from

experiments R = 3 mm, £ = 0.3 mm are realistic values at f = 20 kHz. Assuming that

ao = 10 11m and Pm = 20 kPa we find from (1) and (13) that when Ipi » I Peg; I the

effects of sphericity are of minor importance as the stress penetration is increased by

only 10 % compared with the planar case.

However, changing the intercavity distance to f = 1 mm causes the tensile stress and

the cavity growth to penetrate to the cluster centre as shown in figs. 6 and 7, respecti-

vely.

Discussion

The development of a planar cavity cluster from a uniform distribution of micro--cavi-

ties is connected to the imposed stress field which decays from the cluster boundary, at

first essentially exponentially [1], but as the cavities grow the decay becomes stronger

because the cavity radius influences the second term in (7). However, the change of the

cavity radius vs. position in the cluster turns out to remain almost exponential over a

long period of time. The penetration of the stress into the cluster depends primarily on

the inter--cavity distance f and increases significantly with £, but still the decay con-

stant is of the order of one to a few inter-cavity distances only. During cavity growth

the reflection coefficient for tensile stress waves is found to approach -1 very quickly,

and thus the cluster is essentially comparable to a compliant wall with an acoustic

impedance approaching zero. Consequently, the stresses at the cluster boundary be-

come small during cavity growth.

The calculations indicate that realistic values of the tensile stress occuring at the

cluster boundary are of the order of the critical stress for normal cavitation nuclei, and

it means that in (7) it may be a very crude approximation to consider I pi> > I Pegl.

In addition differences of initial cavity radius with position may influence the cluster

development.

Considering a spherical cluster it appears that when I pi> > I Peg I the effect of spheri-

city is small at cluster radii and inter--cavity distances as expected from experiments,

Page 431

436

but moderate increase of e causes stress penetration and cavity growth throughout the

cluster. Thus the sphericity term in (13) is close to being significant. The cavities are

calculated to develop strongest at the cluster boundary, but from experiments it ap-

pears that those at the center (the focal point) develop strongest [7]. This indicates

that though the cavities reach critical size almost simultaneously [6] those at the center

relax first to supercritical conditions, and so, Peq decreases (the tensile stress increases)

in radial direction ·which causes growth of the central cavities at the expense of the

outer ones. In (13) Peg causes reduction of the third term, thus strengthening the

influence of the second term. It suggests that Peq is an important parameter for ex-

plaining the spherical cluster development. A major problem in this context is that

very little is known about the initial size distribution of cavitation nuclei that can be

used to make reliable calculations of its effect on the cluster development.

References

1.

2.

3.

4.

5.

6.

7.

Hansson, 1.; Kedrinskii, V.; Morch, K.A.: On the dynamics of cavity clusters. J.

Phys. D: Appl. Phys., 15 (1982) 1725-1734.

v. Wijngaarden, L.: On the collective collapse of a large number of gas bubbles

in water. Proc. VIth Int. Congr. Applied Mech. Munich 1964 (H. G6rtler ed.)

854-861, Springer Verlag.

Morch, K.A.: On the collapse of a cavity clusters in flow cavitation. Springer

Ser. in Electrophys., 1: (1980) 91-100.

Hansson, 1.; Morch, K.A.: The dynamics of cavity clusters in ultrasonic (vibrato-

ry) cavitation erosion. J. Appl. Phys., 51 (1980) 4651-4658 and 52 (1981) 1136.

Chahine, G.: Pressure field generated by the collective collapse of cavitation

bubbles. Proc. IAHR Symp. Operating Problems of Pump Stations and Power

Plants. Amsterdam 1982. Paper 2, 1-l4.

Morch, K.A.: On cavity cluster formation in a focused acoustic field. J. Fluid

Mech. (1989) 201, 57-76.

Ellis, A.T.: Techniques for Pressure Pulse Measurements and High-Speed Pho-

tography in Ultrasonic Cavitation. Proc. N.P.L. Symposium on Cavitation in

Hydrodynamics 1955. Paper 8, 1-32, Her Majesty's Stationery Office, London.

Adiabatic Waves in

Liquid-Vapor Systems

Page 2

International Union of Theoretical

and Applied Mechanics

G. E. A. Meier' P. A. Thompson (Eds.)

Adiabatic Waves in

Liquid-Vapor Systems

IUTAM Symposium Gbttingen,

28.8. - 1. 9.1989

Springer-Verlag

Berlin Heidelberg New York

London Paris Tokyo Hong Kong

Page 215

210

the coherence function ~ remain larger than 0.67. The corresponding

data may be straightforwardly interpreted in terms of a single wave

whose velocity is C and whose spatial amplification coefficient is :

Ln H

d

d being the distance between successive probes (here 0.2 m)

2. In some part of the test section, the data exhibit some of (in most

cases all) the following characteristics :

There are significant differences between

significantly from one pair of probes

Cc and

to the

CS ' and Cs varies

next. The cross

correlation function may present two neighboring peaks.

- H varies significantly form one pair of probes to the next, and the

the product of Hmin by Hmax is of the order of 1.

- Coherence "accidents" (very low isolated values) may occur.

As it can be shown by computing the apparent velocity and gain resulting

from the superimposition of two waves (Boure, 1988), this is exactly

what is to be expected when two modes are coexisting, in which case Cs

and H lose their significances.

The existence of two modes is confirmed by plotting (fig. 1) the data for

the wave velocity and the drift flux as functions of the void fraction (the

data is tabulated in Boure, 1988). According to the theory (Boure, 1988)

the significant wave velocity is supposed to be the velocity C-W relative

to the mixture center of volume with the definitions :

WG and WL being the average velocities of the two phases. On fig. 1, the

points are strikingly distributed into two families which correspond to the

two modes. One mode, referred to hereafter as mode 3 (the subscripts 1 and

2 being reserved for pressure waves), is predominant at small void

fractions (~< 0.25), while the other (mode 4) is predominant at larger

void fractions (~> 0.30). The two modes probably coexist at least for

0.20 < ~ < 0.40.

Page 216

c - W (em/s)

30~------+-------,-------~-------+--~

~~------~--------~--------~------~--~

10~------T---~~1--------r-------+----

a

OL-------~------~------~------L-~

5 (em/s)

5~------~------~------~~~--~~

a

O~------~------~--------~------~--~

.1 .2

c- W= 5~

Fig. 1 Wave velocities (top) and drift flux (bottom). Data

(segments and crosses) and conjoint correlations (solid lines).

211

The existence of two modes appears also on the plot of the amplification

coefficient as a function of the frequency or of the wavelength. This plot

is not reproduced here because only orders of magnitude can be derived from

the available data for mode 4.

It is pointed out that, as shown by fig. 1, the domains of predominance of

the two modes correspond to different relationships between the drift flux

& and the void fraction~, i.e. to different flow regimes. It is suggested

that mode 3 may result from individual interactions between otherwise

independent bubbles, while mode 4 involves interactions between individual

bubbles and swarms of bubbles (in active interactions themselves).

Page 430

435

pressure (p )x~o penetrates about 30 times deeper into the cavitation region at an increa-

se of £ by a factor 10.

The spherical cavity cluster

Spherical cavity clusters are typically generated by focused acoustic waves, and from

experiments R = 3 mm, £ = 0.3 mm are realistic values at f = 20 kHz. Assuming that

ao = 10 11m and Pm = 20 kPa we find from (1) and (13) that when Ipi » I Peg; I the

effects of sphericity are of minor importance as the stress penetration is increased by

only 10 % compared with the planar case.

However, changing the intercavity distance to f = 1 mm causes the tensile stress and

the cavity growth to penetrate to the cluster centre as shown in figs. 6 and 7, respecti-

vely.

Discussion

The development of a planar cavity cluster from a uniform distribution of micro--cavi-

ties is connected to the imposed stress field which decays from the cluster boundary, at

first essentially exponentially [1], but as the cavities grow the decay becomes stronger

because the cavity radius influences the second term in (7). However, the change of the

cavity radius vs. position in the cluster turns out to remain almost exponential over a

long period of time. The penetration of the stress into the cluster depends primarily on

the inter--cavity distance f and increases significantly with £, but still the decay con-

stant is of the order of one to a few inter-cavity distances only. During cavity growth

the reflection coefficient for tensile stress waves is found to approach -1 very quickly,

and thus the cluster is essentially comparable to a compliant wall with an acoustic

impedance approaching zero. Consequently, the stresses at the cluster boundary be-

come small during cavity growth.

The calculations indicate that realistic values of the tensile stress occuring at the

cluster boundary are of the order of the critical stress for normal cavitation nuclei, and

it means that in (7) it may be a very crude approximation to consider I pi> > I Pegl.

In addition differences of initial cavity radius with position may influence the cluster

development.

Considering a spherical cluster it appears that when I pi> > I Peg I the effect of spheri-

city is small at cluster radii and inter--cavity distances as expected from experiments,

Page 431

436

but moderate increase of e causes stress penetration and cavity growth throughout the

cluster. Thus the sphericity term in (13) is close to being significant. The cavities are

calculated to develop strongest at the cluster boundary, but from experiments it ap-

pears that those at the center (the focal point) develop strongest [7]. This indicates

that though the cavities reach critical size almost simultaneously [6] those at the center

relax first to supercritical conditions, and so, Peq decreases (the tensile stress increases)

in radial direction ·which causes growth of the central cavities at the expense of the

outer ones. In (13) Peg causes reduction of the third term, thus strengthening the

influence of the second term. It suggests that Peq is an important parameter for ex-

plaining the spherical cluster development. A major problem in this context is that

very little is known about the initial size distribution of cavitation nuclei that can be

used to make reliable calculations of its effect on the cluster development.

References

1.

2.

3.

4.

5.

6.

7.

Hansson, 1.; Kedrinskii, V.; Morch, K.A.: On the dynamics of cavity clusters. J.

Phys. D: Appl. Phys., 15 (1982) 1725-1734.

v. Wijngaarden, L.: On the collective collapse of a large number of gas bubbles

in water. Proc. VIth Int. Congr. Applied Mech. Munich 1964 (H. G6rtler ed.)

854-861, Springer Verlag.

Morch, K.A.: On the collapse of a cavity clusters in flow cavitation. Springer

Ser. in Electrophys., 1: (1980) 91-100.

Hansson, 1.; Morch, K.A.: The dynamics of cavity clusters in ultrasonic (vibrato-

ry) cavitation erosion. J. Appl. Phys., 51 (1980) 4651-4658 and 52 (1981) 1136.

Chahine, G.: Pressure field generated by the collective collapse of cavitation

bubbles. Proc. IAHR Symp. Operating Problems of Pump Stations and Power

Plants. Amsterdam 1982. Paper 2, 1-l4.

Morch, K.A.: On cavity cluster formation in a focused acoustic field. J. Fluid

Mech. (1989) 201, 57-76.

Ellis, A.T.: Techniques for Pressure Pulse Measurements and High-Speed Pho-

tography in Ultrasonic Cavitation. Proc. N.P.L. Symposium on Cavitation in

Hydrodynamics 1955. Paper 8, 1-32, Her Majesty's Stationery Office, London.