HAL Id: jpa-00219558
https://hal.archives-ouvertes.fr/jpa-00219558
Submitted on 1 Jan 1979
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.
ON RESPONSE CURVES OF BUBBLES THE OSCILLATIONS OF WHICH ARE DAMPED BY
SOUND RADIATION
E. Cramer, W. Lauterborn
To cite this version:
E. Cramer, W. Lauterborn. ON RESPONSE CURVES OF BUBBLES THE OSCILLATIONS OF
WHICH ARE DAMPED BY SOUND RADIATION. Journal de Physique Colloques, 1979, 40 (C8),
pp.C8-296-C8-299. �10.1051/jphyscol:1979852�. �jpa-00219558�
ON RESPONSE CURVES OF BUBBLES THE OSCILLATIONS OF WHICH ARE DAMPED BY SOUND RADIATION
Cramer, E. and Lauterborn, W.
Drittes Physikalisch.es Institut, Univevsitat Gottigen, Burgestvasse 42-44, D-Z40O Gottingen, F.R. Germany.
Abstract.- Forced oscillations of spherical bubbles in a compressible, viscous liquid (water) are calcu- lated numerically. The response of a bubble distribution to a sound field with a fixed frequency and an example of the pressure distribution around a single bubble during oscillation are given.
To understand (and perhaps utilize) the emission of noise in liquids irradiated by sound of high intensity beyond the cavitation threshold a know- ledge of the response of bubbles of different sizes to sound fields of different frequencies can be expected to be of great help. Response curves have already been given for bubbles oscillating in an incompressible liquid /l/. We now want to present numerical calculations for bubbles in a compressible liquid. The compressibility becomes important as soon as bubble-wall velocities beco- me comparable with the speed of sound in the liquid.
The bubble model : a) Bubble-wall motion
A very successful modification of the Rayleigh equation that takes into account liquid compres- sibility was obtained by Gilmore / 2 / on the basis of the Kirkwood-Bethe approximation / 3 / . This approximation consists in assuming that
2
the quantity r(h+u / 2 ) , with u the liquid velo- city and h the enthalpy, is propagated unalte- red along the outgoing characteristic
dr = (u+c)dt ; that is
In this equation c = c(p) denotes the speed of sound in the liauid, and D/Dt = 3/9t + u3/3r is the material derivative. The derivatives with respect to r in the second term of equa- tion (1) can be eliminated with the aid of the momentum equation, Du/Dt = -3h/3r, and the
continuity equation
The result then leads to the following equation for the bubble wall, the Gilmore equation
R, H and C denote here the values of the quanti- ties r, h and c at the bubble wall, with
With the equation of state
and
Résumé.- Les vibrations forcées des bulles sphériques dans un liquide compressible et visqueux (de l'eau) sont calculées numériquement. La réponse d'une distribution de bulles à un champ acoustique avec une fréquence fixe et un exemple pour la distribution de pression autour d'une bulle sont montrés.
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:1979852
JOURNAL DE PHYSIQUE
H becomes
b) Pressure d i s t r i b u t i o n (see /4/)
From t h e Kirkwood-Bethe hypothesis we g e t t h a t
holds a t any p o i n t o u t s i d e t h e bubble w i t h
-
21n (l+Bu)BUBu (1+BU)
I
(41where
1/2
The pressure p i s given by
2n
which f o l l o w s from t h e equation o f s t a t e .
The method :
Equation (2) i s solved by a Runge-Kutta method e v a l u a t i n g m a i n l y the steady-state s o l u t i o n . For t h e c a l c u l a t i o n o f t h e pressure along an outgoing c h a r a c t e r i s t i c t h e s t a r t i n g c o n d i t i o n s R, H, and U are taken from t h e steady-state s o l u t i o n o f t h e r a d i u s - t i m e curves.
Results :
F i g u r e 1 shows a s e t o f response curves o f bubble d i s t r i b u t i o n e x c i t e d by a sound f i e l d w i t h a f i x e d frequency o f 23,56 kHz and d i f f e r e n t sound pressu- r e amplitudes pm. For n o t t o o h i g h values o f p, resonances may be observed which can be associated w i t h some n a t u r a l numbers r and s. The expression r / s i s then c a l l e d t h e o r d e r o f t h e resonances.
The case s = 1, r = 2, 3,
...
denotes t h e harmo- nics, t h e resonances w i t h 1 = r and s = 2, 3,. . .
a r e c a l l e d subharmonics and t h e resonances w i t h r = 2, 3,
.. . .
and s = 2, 3,. . .
a r e c a l l e d u l t r a - harmonics ( a l s o u l trasubharmonics).O f special i n t e r e s t a r e t h e ultraharmonics o f t h e order 3/2, 5/2, 7/2
. . .
as they g i v e r i s e t o a subharmonic component a t one h a l f t h e d r i v i n g frequency i n t h e spectrum o f t h e hubble o s c i l l a - t i o n . These subharmonic components may be respon- s i b l e f o r t h e f i r s t subharmonic i n t h e measured spectra o f a c o u s t i c c a v i t a t i o n noise. F i g u r e 2 shows an example o f t h e steady-state s o l u t i o n o f t h e o r d e r 3/2 and F i g u r e 3 shows t h e r e s u l t i n g pressure d i s t r i b u t i o n around t h i s bubble and t h e amplitude spectra o f t h e pressure i n t h r e e d i f f e - r e n t distances r.References
/1/ Lauterborn, W., J. Acoust. Soc. Amer.
2
(1976) 283.
/2/ Gilmore, F.R., C a l i f o r n i a I n s t . Techn. Rep.
NO. 26-4 (1952).
/3/ Kirkwood, J.G., Bethe, H.A., Osrd Rep. No 588 (1942).
/4/ Akulichev, V.A., e t al., Sov. Phys. Acoustics 13 (1967) 321.
FIGURE
- 1 :Frequency response curves f o r a bubble d i s t r i b u t i o n i n water a t a driving sound f i e l d of 23,56 kHz f o r d i f f e r e n t sound pressure amplitudes pm. The numbers occurring above the peaks of the curves a r e the order of the resonances. Rmax i s the maximum radius of the steady- s t a t e solution.
FIGURE 2
:Example of a steady-state solution a t the resonance of the order 3/2.
Upper curves
:driving sound f i e l d with the frequency of 23,56 kHz and the sound pressure amplitude of
0,8bar, and the resulting radius-time curve. Lower curve
:amplitude and phase spectra of the (periodic) solution.
FA= 2 3 . 6 KHZ F/FO= 0.6031 0 RN= 82.0 MU€ GAMMA= 1.33
PM= 0.800 BAR PO- 1.000 BUR
JOURNAL
DE PHYSIQUE
FR-23.58 nnz TO- 0.08988 IIS nN-82.00 W E PI- 0.BDOOO ERR
FIGURE 3 :
