Nonlinear attenuation of sound waves by inertial cavitation bubbles

HAL Id: hal-01717675

https://hal.archives-ouvertes.fr/hal-01717675

Submitted on 2 Apr 2019

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Nonlinear attenuation of sound waves by inertial

cavitation bubbles

Olivier Louisnard

To cite this version:

Olivier Louisnard.

Nonlinear attenuation of sound waves by inertial cavitation bubbles.

ICU

2009 - International Congress on Ultrasonics, Jan 2009, Santiago de Chile, Chile.

pp.735-742,

Physics Procedia 00 (2008) 000–000

www.elsevier.com/locate/XXX

International Congress on Ultrasonics, Universidad de Santiago de Chile, January 2009

Nonlinear attenuation of sound waves by inertial cavitation bubbles

Olivier Louisnard

a,

*

a

Centre RAPSODEE, UMR CNRS 2392, Ecole des Mines d’Albi, 81013 Albi, FRANCE

Elsevier use only: Received date here; revised date here; accepted date here

Abstract

Acoustic cavitation fields generally involve clouds of inertial bubbles, expanding to many time their initial radius, and then collapsing. A correct estimation of the wave attenuation in such media requires a realistic estimation of the power dissipated by the oscillation of each bubble. This power loss originates mainly from thermal diěusion in the gas and viscous friction in the liquid, and is calculated numerically for a single inertial bubble, by solving the bubble dynamics in a typical parameter range at 20 kHz, using a convenient model accounting for the thermal behavior of the gas. These estimations are then injected in the nonlinear Caflish equations, which describe wave propagation in a bubbly media, conveniently recast in an energy conservation equation. A nonlinear attenuation coeĜcient is deduced and found to be several orders of magnitude higher than the linear prediction.

© 2010 Elsevier B.V.

PACS: 43.35Ei;

Keywords: acoustic cavitation, bubble dynamics, propagation in bubbly liquids, wave attenuation;

1. Introduction

The complexity and large variety of spatial and temporal scales involved in acoustic cavitation make diĜcult the derivation of a full theoretical model, accounting for the coupled eěects between the bubble field and the sound field. Recent theoretical [1] and experimental [2] progress have however allowed to restrict the range of the bubbles size involved, and the action of the acoustic field on the organization of inertial bubbles has been satisfactorily described in various configurations by particle models [3]. But conversely, the eěect of inertial bubbles on the acoustic wave propagation remains mainly unexplored.

The popular Caflish model [4], although valid in the nonlinear case, remains intractable for large multi-dimensional geometries, since it requires timedependent simulations, and presents convergence problems in the range of inertial cavitation. This is why it has often been used under the linear approximation [5–7], since in that

* Corresponding author. Tel.: +33-56-349-3062; fax: +33-56-349-3025.

E-mail address: louisnar@enstimac.fr.

Physics Procedia 3 (2010) 735–742

www.elsevier.com/locate/procedia

doi:10.1016/j.phpro.2010.01.093

Olivier Louisnard / Physics Procedia 00 (2010) 000–000

case, it simplifies into a Helmholtz equation, and allows harmonic response simulations. An attenuation coeĜcient can be obtained from the linear dispersion relation, and accounts for dissipation by a linearly oscillating bubble.

Using a linear attenuation coeĜcient for inertial bubbles is questionable, since the latter typically suěer a ten-fold expansion of their radius and may dissipate more energy than predicted by linear oscillations. To investigate this issue, we recast the nonlinear Caflish equations into a mechanical energy balance equation, in which we estimate the energy dissipation by nonlinear simulations of the bubble dynamics. A nonlinear attenuation coeĜcient is then identified and estimated.

2. Theory

2.1. Caflish equations

The Caflish model [4] is based on the equations of mass and momentum conservation:

t v t p c_l l w w ˜ ’  w w E U 2 1 _{, (1)} , l v p t U w ’ w (2)

whereȕ is the void-fraction defined by

 

3 4 . 3 N R t E S (3)

As in linear acoustics, the set of equations (1), (2) can be easily recast into an equation involving the acoustic energy density of the liquid, by multiplying (1) by p and (2) by v :

 

t V Np pv v c t _{l l} l w w ˜ ’  ¸ ¸ ¹ · ¨ ¨ © §  w w 2 2 ₂ 1 1 2 1 U U (4)

where we have assumed that the bubble number density does not vary at the oscillation time scale. Besides, the bubble radial motion equation can be described by a Rayleigh-like equation, where we leave the bubble pressure pg

unspecified for now, and each bubble suffers the local acoustic pressure fieldp(r,t), as assumed in the Caflish model:

2 3 2 4 2 l g R . RR R p R R V U §_¨  ·_¸    © ¹    _{P p} (5)

All the quantitiesR, pg and p in this equation depend on r and t, and the time derivatives represented by overdots

in this equation must be understood as partial derivatives w wtat r constant. In order to get an energetic interpretation of the bubble radial motion, equation (5) can be multiplied by the time derivate of the bubble volume

V t

w w , and noting that

2 3 2 l RR R V t U§_¨  ·_¸u w © ¹   _w

is the time-derivative of the radial kinetic energy of the liquid 3 2 2 l l K SU R R , we obtain:



2



2 4 1 l g V V 6 . K R p p R t S V t t SP w w w    w w w  R (6)

This equation is strictly equivalent to the Rayleigh equation, and corresponds physically to the theorem of kinetic energy applied to the liquid surrounding the bubble. The left-hand side represents the sum of the kinetic energy of the liquid and of the interfacial potential energy. The first two terms in the right-hand side are the work done by the gas and by the acoustic field, respectively.

2.2. Conservation of mechanical energy

The term p Vw wtcan be eliminated between equations (4) and (6), by mutiplicating the latter by N. It is readily

obtained:

 

2 2 2 2 2 16 4 2 1 2 1 R R N t V Np pv R N NK v c p t _{l l} l l g  SP V S U U w  w ˜ ’  ¸ ¸ ¹ · ¨ ¨ © §    w w ₍₇₎

Equation (7) represents the conservation of mechanical energy of the bubbly liquid: x 2



2

2



p

U

l l

c

is the elastic potential energy stored by the pure liquid per unit volume,

x 2 ₂

l

x NK v

U is the kinetic energy per unit volume of the pure liquid in its translational motion,

l is the kinetic energy per unit volume of the liquid in its radial motion around the bubbles,

x 4ʌNıR2

is the interfacial potential energy per unit volume, x pv is the acoustic intensity, or flux density of mechanical energy.

In what follows, we will assume periodic oscillations of all the fields. Averaging (7) over one cycle, the time derivative in the left side cancels and we get:



th –v



N

pv  – 

˜

’ (8)

where we have defined the two bubble dynamics dependent average quantities

0 1 , T th g V p d T t w –  w

³

t (9) 2 0 1 16 , T V RR dt T

SP

–

_³

 (10)

which physical meaning is examined hereafter.

2.3. Energy dissipation per bubble

Applying the first principle of thermodynamics to the whole bubble content, integral (9) also reads

0 0 1 1 , T T th dU dt Qdt T dt T –

_³



_³



Olivier Louisnard / Physics Procedia 00 (2010) 000–000

where U depicts the internal energy of the whole gas in the bubble. The first term in the above equation is zero for a periodic motion, so that

–  Q

th . The integral of over one cycle is necessary negative in virtue of the

second principle of thermodynamics, since the bubble is in contact with a single heat source, thus .

Q

0

th

– t

BesidesȆv defined by (10) is clearly positive, and is the period-averaged power loss by viscous friction in the

liquid as it moves radially around the bubble.

Eq. (8) can now be interpreted more clearly: the balance between the acoustic energy leaving a volume of bubbly liquid and the one reaching it is always negative, owing to thermal loss in the bubble and viscous friction in the radially moving liquid. Each bubble appears as a dissipator of acoustic energy, owing to these two phenomena. The causes of wave attenuation are thus self-contained in the Caflish model, even for nonlinear oscillations, provided that a correct model is used to describe thermal diěusion in the bubble interior. We note that Caflish and co-workers [4] proposed a conservation equation similar to (7), disregarding viscosity and assuming isothermal oscillations, in which case mechanical energy is conserved.

The integralsȆV and Ȇthcan be evaluated numerically by solving the bubble dynamics equation (5), by properly

accounting for thermal diěusion in the gas. To that aim, the bubble interior is modelled by an approximate energy conservation equation based on a thermal diěusion layer [8–10]. Figure 1 shows the results for an argon bubble of ambiant radius R0= 3 μm in water, excited by a field p = p0[1 – p* sin (2S ft)] with f = 20 kHz.

It can be seen that the power dissipated either by thermal diěusion (thick solid line) or by viscous friction (thin solid line) quickly rises in the neighbourhood of the Blake threshold, well above their value predicted by linear theory (about 6 orders of magnitude).

Another interesting feature is that, for the parameters used in Fig. 1, viscous dissipation becomes much larger than the thermal one, for driving pressures well above the Blake threshold, while linear theory predicts the opposite in this parameter range. To assess more clearly this issue, we calculatedȆV and Ȇthat constant p*=1.5, but varying

R0.The result is displayed in Fig. 2. Thermal dissipationȆthbecomes larger than the viscous frictionȆV for bubbles

larger than 8 μm. The two dissipated power are therefore of the same order of magnitude in the parameter range for inertial cavitation at 20 kHz.

The real power dissipated by an inertial bubble is therefore much higher than the one predicted by linear theory, by several orders of magnitude. We therefore expect the real wave attenuation in a liquid containing inertial bubbles (above the Blake threshold) to be much higher than the value calculated by linear theory. We know quantify this point.

Fig.1 Dissipated power by thermal diěusion, Ȇth [thick solid line, from Eq. (9)], by viscosity Ȇv [thick dashed line, from Eq (10)]. The thin lines are the corresponding values from linear theory. The case considered is an argon bubble of ambiant radius R0 =3 μm in water, at 20 kHz. The vertical dash-dotted line represents the Blake threshold.

Fig.2 Dissipated power by thermal diěusion, Ȇth [thick solid line, from Eq. (9)], by viscosity, Ȇv [thick dashed line, from Eq (10)], as a function of R0, for pӒ =1.5, all other conditions being the same as in Fig. 1. The vertical dashdotted line represents the Blake threshold.

2.4. Wave attenuation

2.4.1. Linear dispersion relation

In the linear case, the Caflish model can be linearized, and yields the dispersion relation [4, 7, 11, 12]:

2 2 2 0 2 2 2 0 4 , 2 l R N k c i S Z Z b Z Z Z    (11)

where the resonance frequencyȦ0 and the damping factor b are given respectively by



  

2 0 0 2 0 1 S p R Z D U ª¬  ƒ ) DSº¼, (12)





 

0 2 0 0 1 4 2 p S b R R D 2 P UZ U  ‚ )  (13)

whereĮS is the dimensionless surface tension and theĭ function depends on the thermal Peclet number, and can be

found elsewhere [13, 14]. The complex wavenumber k can be written as krí iĮ, and both kr and Į can be identified

from Eq. (11).

2.4.2. Nonlinear wave attenuation

We first examine how equation (8) casts within a linear propagation equation. We start by writing each field as a monoharmonic oscillation

 

, 1

 

. 2 i t p r t P r eZ c c., (14)

 

, 1

 

. 2 i t v r t V r eZ c c..

Olivier Louisnard / Physics Procedia 00 (2010) 000–000

0,

By linearization, it can be shown [7] that the Caflish equations reduce into a single Helmholtz equation:

2 2

P k P

’  (15)

where we have introduced a complex wavenumber k = kr – iĮ. We start the problem backwards in order to link the

attenuation coeĜcient Į with Ȇv +Ȇth.Equation (8) reads:

 

>

’˜ P'U

@

N–th– ƒ 2 1  v (16)

where we do not explicit the averages (9) and (10), and ƒ denotes the real part of a complex number. Using (2), we have iUZU =  ’P , so that (16) becomes







th v l N P ' P i ¸¸_¹  – – · ¨¨ © § ’ ˜ ’  ƒ Z U 1 2 1



(17)

Expanding, and using Helmholtz equation, we get



2 2 2







1 1 . 2 l th V k P P N i

U Z

ª º ƒ_«   ’ _» – – ¬ ¼ (18)

Taking the real part,

 



2 2 2 _l th V k P N

U Z

‚  – –



, (19)

and setting k = kr iĮ, we finally obtain:



2 r th l k P N D U Z – –V



. (20)

LinearizingȆth and ȆV, this relation is the same as the one we would obtain by taking the imaginary part of the

dispersion relation. SinceȆth and ȆV scale as |P|2 for linear oscillations (see left part of the curves in Fig. 1), this

yields a value forĮ independent of the driving amplitude |P|.

The idea of the present paper is to use Eq. (19) beyond the linear case, by using the nonlinear values ofȆth and

ȆV obtained in sec. 2.3 to calculate‚(k2), still estimatingƒ(k2) by the linear dispersion relation. More precisely, we

use:

 

2 2 0 2 2 2 0 4 l R k c S Z Z Z Z ƒ   2 , N (21)

 

2 2 2 th l N k P U Z –  ‚  –V, ₍₂₂₎

and deduce the real and imaginary parts of k = kr – iĮ from the obtained expression of k

2

.

Figure 3 displays the attenuation coeĜcient Į calculated this way (thick solid line), in the same conditions as for Fig. 1, for a typical [15] void fractionE0 = 10-4

.

The attenuation coeĜcient rises abruptly for acoustic pressures just above the Blake threshold, and becomes about 4 orders of magnitude larger than its linear value. The behaviour of

the kr curve above the Blake threshold, which closely followsĮ, comes from the fact that ‚(k2) >> ƒ(k2) so that

D



~

k_r .

Fig.3 Real part (dashed) and imaginary part (solid) of the wavenumber k. The thin horizontal lines are predictions from linear theory (11) and the thick lines are results calculated from Eqs. (21), (22). The vertical dash-dotted line represents the Blake.

3. Summary and discussion

Inertial bubbles dissipate much more energy than a linearly oscilating bubble, both by thermal diěusion in the gas and viscous dissipation in the liquid. The wave attenuation in an inertial cavitation field is therefore much larger than the linear value predicted by the classical dispersion relation (by typically 4 orders of magnitude).

A typical value Į = 200 m-1

can be read on Fig. 3, which means that the wave amplitude would suěer a 10-fold decrease on about 1 cm. This is the order of magnitude of the cavitation zone observed near a tip transducer. Of course more precise predictions should involve at least the spatial variations of the bubble concentrations, and 2-D or 3-D eěects, for example conical structures [2, 16, 17]. Besides, it is seen that, contrarily to the linear prediction, the attenuation coeĜcient increases with the wave peak-amplitude. Thus, increasing the source vibration amplitude would not necessarily produce a more extended bubble field since the attenuation would also increase. This saturation phenomenon is well-known in cavitation experiments [17].

To conclude, we note that our estimation of the nonlinear attenuation opens the way to more realistic simulations of the coupled evolution of the cavitation field and the acoustic field. The idea underlying such simulations would be halfway between a fully nonlinear simulation of the Caflish equations, which requires painful temporal integration, and a fully linearized model which, as shown above, underestimates the wave attenuation by several orders of magnitude.

References

[1] R. Mettin, in Oscillations, Waves and Interactions, edited by T. Kurz, U. Parlitz, and U. Kaatze (Universit¨atsverlag G¨ottingen, 2007),

pp. 171–198.

[2] R. Mettin, in Bubble and Particle Dynamics in Acoustic Fields: Modern Trends and Applications, edited by A. A. Doinikov (Research

Signpost, Kerala (India), 2005), pp. 1–36.

[3] U. Parlitz, R. Mettin, S. Luther, I. Akhatov, M. Voss, and W. Lauterborn, Phil. Trans. R. Soc. Lond. A 357, 313 (1999). [4] R. E. Caflish, M. J. Miksis, G. C. Papanicolaou, and L. Ting, J. Fluid Mech.153, 259 (1985).

[5] G. Servant, J. L. Laborde, A. Hita, J. P. Caltagirone, and A. G´erard, Ultrason. Sonochem.10, 347 (2003).

[6] R. Mettin, P. Koch, W. Lauterborn, and D. Krefting, in Sixth International Symposium on Cavitation CAV2006 (paper 75)

(Wageningen (The Netherlands), 2006), pp. 125–129.

Olivier Louisnard / Physics Procedia 00 (2010) 000–000

[8] R. Toegel, B. Gompf, R. Pecha, and D. Lohse, Phys. Rev. Lett. 85, 3165 (2000). [9] T. J. Matula, P. R. Hilmo, B. D. Storey, and A. J. Szeri, Phys. Fluids 14, 913 (2002). [10] B. D. Storey and A. Szeri, Proc. R. Soc. London, Ser. A 457, 1685 (2001). [11] L. L. Foldy, Phys. Rev. 6 , 107 (1944). 7

[12] L. van Wijngaarden, in 6

th

Symposium on Naval Hydrodynamics (OĜce of Naval Research, Washington DC, 1966), pp. 115–128.

[13] A. Prosperetti, L. A. Crum, and K. W. Commander, J. Acoust. Soc. Am. 83, 502 (1988). [14] A. Prosperetti, J. Acoust. Soc. Am. 61, 17 (1977).

[15] F. Burdin, N. A. Tsochatzidis, P. Guiraud, A. M. Wilhelm, and H. Delmas, Ultrason. Sonochem. 6, 43 (1999). [16] A. Moussatov, C. Granger, and B. Dubus, Ultrasonics Sonochemistry 10, 191 (2003).

[17] C. Campos-Pozuelo, C. Granger, C. Vanhille, A. Moussatov, and B. Dubus, usson 12, 79 (2005).