The Acoustic Bubble

632 Pages·1994·14.91 MB·English

by  T. Leighton (Auth.)

The Acoustic Bubble
T. G. Leighton
Institute of Sound and Vibration Research
The University, Southampton, UK
formerly at: Cavendish Laboratory and
Magdalene College, University of Cambridge, UK
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To my parents
Preface
The interaction of acoustic fields with bubbles in liquids is of interest to a wide range of people,
from those involved with the fragmentation of kidney stones and the sensitisation of explosives,
to researchers who record the underwater sound of rain-drop impact on lakes. The indications
are that the range is widening, and though there are several excellent publications which
expound aspects of the field (for example, the biomedical, chemical or erosive implications)
and others which extensively review the literature, there is a place for a somewhat gentler
approach to supplement these. The object of this book is neither to provide an exacting
grounding of any particular aspect of underwater acoustics, nor to summarise and review
historic and current literature on acoustic cavitation. It is instead written with the intention of
giving the reader a 'physical feel' for the acoustic interactions of acoustic fields with bubbles.
With the multidisciplinary nature of the field there will be some who require analogy, rather
than formulation, to appreciate a given phenomenon. Though such mental pictures are often
not rigorous, if the alternative is simply to trust the end result o fa mathematical analysis and
hope that in applying it one does so in a manner appropriate to the prevailing environment, then
the degree of adaptability that follows from having some appreciation of the physical processes
involved can only be good for the science, and the spirit. A physical understanding is also
valuable to the mathematically adept who are perhaps new to a field: it takes time to find out
where one stands in a new discipline, time in which one assimilates the origins of the
formulations so readily employed by others, and to appreciate how the assumptions which form
the basis of the mathematical models relate to the physical world. This book therefore attempts
to engender a basic level of understanding through imagery. However, whilst some readers will
find such descriptions sufficient, the field itself requires that a thorough mathematical descrip-
tion also be available, and this is provided. Because of the size of the field and the rate at which
new formulations are developed, selection has been made to include those formulations which
may best illustrate the physical mechanisms involved, and which the reader can in many
instances follow back to source. As a consequence of this approach, individual readers will
encounter sections that are too rudimentary or too involved, depending on their background,
but it is hoped that regardless ofthat background there will be much that is comprehensible and
interesting. The range of phenomena discussed is extensive, but the basic interaction of sound
with bubbles is the same. If the object is to develop a physical appreciation of this interaction,
there is much to be said for examining manifestations of it that are perhaps outside one's
immediate experience: for example, the bubble-mediated phenomena of acoustic shielding, bulk
property and parametric effects encountered in the use of clinical ultrasound are in fact current
issues on a larger scale in ocean acoustics. Even so, space limitations preclude the inclusion of
little more than an introduction to several topics, such as hydrodynamic, electrohydraulic,
optical and biological cavitation.
XI
Xll PREFACE
The book is divided into five chapters. The first incorporates a basic introduction to acoustics,
along with a description of some of the more esoteric phenomena that can be seen when high-
frequency high-intensity underwater sound is employed. The second chapter discusses the
nucleation of cavitation, and basic fluid dynamics. The third chapter draws together the
acoustics and the bubble dynamics to discuss the free oscillation o fa bubble, and the acoustic
emissions from such activity. Examples are drawn from bubble entrainment through injection,
rainfall, and wave action, and the chapter ends with a discussion of oceanic bubble populations.
In addition to the natural emissions from these bubbles, acoustic probes are often applied to
study such populations, and the behaviour of a bubble when an externally-applied acoustic field
drives it into oscillation is the topic of Chapter 4. When such a bubble is forced into a stable
oscillation, a variety of phenomena (such as radiation force interaction, rectified diffusion,
surface wave activity, microstreaming, subharmonic emission and chaotic oscillation) may
occur, and these are discussed. Energetic transient collapse may also occur. Chapter 4 closes
with a summary of the theory of a single bubble, and examples of the population phenomena
that can arise in practical sound fields. Following the discussion of these behaviours in Chapter
4, Chapter 5 outlines a variety of effects associated with acoustically-induced bubble activity.
Bubble detection, sonoluminescence, sonochemistry and pulse enhancement are included. The
chapter closes with a discussion of cavitation erosion, and cavitation bioeffects, relating
particularly to the clinical use of ultrasound. The emphasis throughout the chapter is on
understanding the mechanisms through which the effects may arise and, where alternative
mechanisms have been proposed, to assess them in the light of the available experimental data.
In order to convey these ideas I have drawn upon my own research, much of which was
undertaken at the Cavendish Laboratory, Cambridge, and my thanks goes to those people there
who have assisted me over the years in my work, but above all for their friendship. I would
especially like to acknowledge my gratitude to John Field, for his friendship and support over
the years. I count myself very lucky in that, in addition to John Field, I have the privilege to
thank the following people for their invaluable friendships and criticisms of sections of this
book: Phil Nelson, Steve Thorpe, Joe Hammond, Mike Buckingham, Charlie Church, Christy
Holland, Roy Williams, Stan Barnett, Hugh Pumphrey, Ron Roy, Andrew Hardwick, Chris
Beton, and Knud Lunde. I could not have been more fortunate in meeting such people. For their
advice and assistance during my research I would also like to thank Alan Walton, Mike
Pickworth, Phillip Dendy, Larry Crum, Martin Lesser, Karel Vokurka, Nick Safford, Colin
Seward, Jim Pickles, Kelvin Fagan, Dave Johnson, Ray Flaxman, Arthur Stripe, Bob Marrah,
Roger Beadle, Rik Balsod, Alan Peck and Dick Smith. I hope they share with me fond and
perhaps amusing memories of my requests for sometimes unorthodox assistance. I am very
grateful to Sue Hellon for the patient assistance she has given me with the references, and to
Chris Rice, Maureen Strickland, Paul White, Chris Morfey, Steve Elliott, Frank Fahy, Mike
Fisher, Peter Davies, Mike Russell, Anne Barrett, Dean Thomas and Terry Vass for their
encouragement since my arrival at the Institute of Sound and Vibration Research, Southampton.
For their help since the move I also thank Shiela and Vic Fisher, and John and Kath Morrant.
And Siän Lloyd Jones, who read the proofs.
I would very much like to thank the Master of Magdalene and Lady Calcutt, and the Fellows
of Magdalene College, who extended immeasurable friendship, and trust, and who from the
first gave such enthusiastic support for a young man's interest in acoustic bubbles.
And for their support Annie, Nick and Linda, Justine and John, Anne, Peter, Deniol, Hilda,
Patty, and John and Christine.
T.G. LEIGHTON
June 1992
Symbols and Abbreviations
Symbols
a acceleration of a particular piece of fluid
a a unit vectors along the incident and scattered directions respectively
ly s
a(t) time-dependent amplitude of a purturbation which is the spherical
nm
harmonic of order w, m, superimposed on a spherically symmetric
pulsating sphere
a„ reduced forms of a(t)
nm
au #2 coefficients associated with derivation of the mechanical index
A amplitude of the particular integral describing steady-state displacement
response of a general oscillator (may be complex)
A(t) amplitude function associated with velocity potential of raindrop crater
A plane area perpendicular to a given axis
Λ the ratio A = Rmzx/Ro used to characterise bubble oscillation
ε t
A amplitude of spherical harmonic of order n superimposed on a stationary
n
spherical shape (n = 1,2, 3 ...)
b resistive dissipation constant
btot total resistive dissipation constant
fevp resistive dissipation constant in the volume-pressure frame
&RF radiation dissipative constant in the radius-force frame, equal to Re{Z^}
£RP radiation dissipative constant in the radius-pressure frame
bv$ radiation dissipative constant in the volume-force frame
fevp radiation dissipative constant in the volume-pressure frame
byp thermal dissipation constant in the volume-pressure frame
byp viscous dissipation constant in the volume-pressure frame
b amplitude acoustic attenuation constant
B bulk modulus
B adiabatic bulk modulus
s
BT isothermal bulk modulus
B bulk modulus of the bubbly liquid
c
Z? bulk modulus of the bubble-free liquid
w
2?bub a component of bulk modulus incorporating the pressure-mediated volume
change which is due to the entire bubble population
B/A second-order nonlinearity ratio of a liquid
c speed of acoustic waves
xiii
XIV SYMBOLS AND ABBREVIATIONS
c speed of sound in a bubbly medium
c
Cd speed of sound in a solid wall
c speed of sound in the gas contained within the bubble
g
en speed of wavelets in Huygens construction
Co speed of sound at infinitesimal amplitudes
CL speed of sound in the liquid at bubble wall
c speed of sound in the bubble-free liquid
w
c shock speed
s
C(Ö a function of the acoustic frequency with the dimensions of speed
C concentration of gas dissolved within a liquid
Cc* initial uniform gas dissolved within a liquid at time t = 0
CR dissolved gas concentration in liquid at bubble wall
C^ dissolved gas concentration in liquid at bubble wall when bubble has
equilibrium radius (r = R )
0
C specific heat capacity of gas within bubble at constant pressure
p
C , molar heat capacity of gas within bubble at constant pressure
pm
C specific heat capacity of gas within bubble at constant volume
v
Qap electrical capacitance
d d radiation damping constant
ra
Jth thermal damping constant
dyis viscous damping constant
dtot total damping constant: </ t = d d + dth + d^
to ra
D diffusion coefficient for dissolved gas within a liquid
Dak thermal diffusivity of air
Ddip dipole strength defined specifically for bubble sources (see text)
Ddi ,i initial dipole strength defined specifically for bubble sources (see text)
P
Dg thermal diffusivity of a gas
A thermal diffusivity of a liquid
D/Df = d/dt + v.V, the so-called 'material derivative', a differential which takes
account of changes both in time and position
e exponential constant, «2.718
/ ratio of bubble gas density to liquid density at equilibrium
foop fractional Doppler shift
fufz functions that are solutions to the wave equation
/ the complex scattering function of the bubble
F force
F force applied to a 'spring'
a
F force exerted by a 'spring'
s
F amplitude of driving force (may be complex)
A
FBI Primary Bjerknes force
FD drag force on a rigid sphere in potential flow
F amplitude of general driving force of an oscillator
0
E^cxt ^e vector sum of the external forces, both volume and surface, per unit
mass on an infinitesimal element of liquid
F time-averaged radiation force
SYMBOLS AND ABBREVIATIONS XV
Ftrav time-averaged radiation force on a bubble in a travelling-wave field
FBI time-averaged radiation force on a bubble in a standing-wave field
f^art time-averaged radiation force exerted by a pulsating bubble on a particle
Frad2 time average of F^®,
Frad2 instantaneous value of radiation force on bubble 2 due to field radiated by
bubble 1
{ΐτ\ Froude number
F(Q a time-dependent function representing the difference between the
concentration of dissolved gas at the bubble wall, and the concentration far
from the bubble
g dimensionless multiplicative factor which accounts for effect of surface
tension on bubble stiffness
g gravitational acceleration
Gth a dimensionless constant associated with thermal damping
Gi,G2,G3,G4,G5 constants associated with thermal damping such that:
Gi a parameter proportional to the square of the ratio of the thickness of the
layer where conduction will cause significant temperature changes, to the
acoustic wavelength in the gas
Gi a parameter proportional to the square of the ratio of the bubble radius to
the thermal penetration depth
A distance from a plane surface (e.g. transducer faceplate, liquid surface)
Aent entrainment depth
Amax maximum depth to which dirty bubbles are carried in ocean
A depth at which equilibrium occurs for gas flux from oceanic bubble
0
Aobs depth of observation below the sea surface
AR a substitution parameter, AR = (r3 - R3)/3
A saturation depth for gas flux from oceanic bubble
s
H significant wave height
s
H liquid enthalpy
//D Henry's constant
H time-dependent locus of a plane gas/liquid interface
i i2 = -l
/ acoustic intensity
/ f reference acoustic intensity
rc
/index mechanical index
Im frequency-weighted index, given by V/index
j mass flux of a fluid (mass crossing unit area in unit time)
J flux of gas dissolved in liquid
/„ Bessel function of order n
k wavenumber = 2π/λ
k wavevector (\k \=k)
k\ wavevector from source 1 of a pair
ki wavevector from source 2 of a pair
xvi SYMBOLS AND ABBREVIATIONS
:т*
I wave vector from an image
fcęomp complex wavenumber of sound in a medium containing a uniform
distribution of bubbles
*с wavenumber of a coherent acoustic field within a sparse distribution of
scattering bubbles
łCw wavenumber in bubble-free seawater
кв Boltzmann's constant
к spring constant
k polytropic stiffness of a bubble for radial displacement
R
k-vp stiffness of a bubble for volume-pressure frame (effects of surface tension
not included)
к$
adiabatic limit of #vp (effects of surface tension not included)
№ isothermal limit of Jfcyp (effects of surface tension not included)
к\т stiffness of a bubble in volume-pressure frame
К a function in the expansion of the radial oscillation
thermal conductivity of gas within bubble
*g
Ki thermal conductivity of the liquid
I length
b width of thermal boundary layer
length of an unloaded spring
/о
L a length
U average penetration depth of a bubble cloud
U. diameter of a liquid drop
discontinuity length
£dis
U diameter of active area of a hydrophone
h diameter of a molecule
^ms thickness of acoustic boundary layer over which microstreaming acts
dimensions of an acoustic source
^s» ^-oc> L*
distance between the point of bubble formation and the wall
Lnf/
peak-to-trough average wave height
^wave
a function of the acoustic frequency with the dimensions of length
MD
electrical inductance
^ind
m inertia of a general oscillator
тъ mass of gas contained within a bubble
m mass of a gas molecule
g
all inertial components of a pulsating bubble excluding the radiation mass
ГП{
t*»rad radiation mass in the radius-force frame
m&d radiation mass in the radius-pressure frame
m$ radiation mass in the volume-force frame
nć$ radiation mass in the volume-pressure frame
mass of a vapour molecule
/Wy
mx mass contained in one acoustic wavelength
M the acoustic Mach number
m an integer
n an integer
SYMBOLS AND ABBREVIATIONS XV11
number of bubbles per cubic metre in a population of identical bubbles
rib
nfizJioWo number of bubbles per unit volume at depth z having radii between R0 and
Ro + dRo
r* number of bubbles per cubic metre per micrometre increment in radius
number density of gas molecules within the bubble
number density of liquid molecules near a liquid surface (specifically, near
the bubble wall)
no equilibrium, constant number density of vapour molecules during
isothermal and isobaric collapse conditions for a vapour cavity
Avogadro's number « 6.023 x 1023
N total number of bubbles
b
total number of vapour molecules that must condense out to obtain
isothermal and isobaric conditions within a collapsing vapour cavity
N total number of moles (especially contained within a bubble)
m
p momentum of acoustic wave
px momentum associated with one acoustic wavelength
p pressure
p atmospheric pressure at level of sea surface
atm
PA the amplitude of the oscillation of the pressure of the permanent gas phase
within the bubble
p a constant with dimensions of pressure
c
p instantaneous gas pressure within a pulsating bubble
g
p pressure of gas phase within the bubble at equilibrium
g>e
p , initial gas pressure inside cavity before Rayleigh-like collapse under
g m
hydrostatic pressure, when R = R and R = 0
m
Pg,max maximum pressure reached by a gas during collapse
Pi instantaneous total pressure within a pulsating bubble
Pi pressure within the bubble at equilibrium
>c
Pimpact pressure radiated by initial impact of raindrop
Pin instantaneous pressure inside a bubble undergoing shape oscillations
Pout instantaneous pressure outside a bubble undergoing shape oscillations
PL liquid pressure just outside a bubble or cavity, at the wall
PL,, external pressure outside the cavity when it begins to collapse
p the liquid pressure which brings about the onset of nucleation from a
n
conical crevice
PN partial pressure of nitrogen in the water far from the bubble
2
Pin pressure at wall inside bubble
Prad,abs radiation pressure for absorption
Prad,ren radiation pressure for reflection
p vapour pressure within the bubble
v
p _h water-hammer pressure
w
ρ small-amplitude pressure fluctuation inside bubble
ε
ρ pressure due to surface tension (the Laplace pressure)
σ
Poo pressure in the liquid far from the bubble
p hydrostatic liquid pressure outside the bubble
0
p' the pressure existing at some boundary within the liquid
Pr» Ρθ» Ρφ the three principal stresses in the spherical coordinate directions