XI Session of the Russian Acoustical Society.
Moscow, November, 19-23, 2001.
N.G. Voronina F.F. Legusha
CHANGE OF AMPLITUDE OF BUBBLE OSCILLATIONS DURING ITS
RISE TO THE SURFACE
St. Petersburg State Maritime Technical University
Russia, 190008 St. Petersburg, Lotsmanskaya st., 3
Tel.: (812) 157-10-55; Fax: (812) 157-09-11
E-mail: [email protected]
The work presents the results of numerical calculations of the dynamics and acoustic parameters of gas
bubble rising in a gas-liquid mixture (GLM). On the basis of method of calculation proposed by G.A. Druzhinin,
computations of bubble oscillations and part of the energy absorbed by it were conducted. For a number of
particular situations, the coordinates of bubble entry into resonance condition were estimated. Dependences of
bubble radius, its resonance frequency, amplitude of oscillations and radiation intensity from depth were built.
A number of problems in hydro-acoustics use a screen of bubbles that rise freely from a depth
to the sea surface. Works dedicated to the acoustic properties of gas-liquid media usually investigate
the resonance behavior of bubbles. However, this approach to gas-liquid mixture as a system of
resonance scatterers may prove to be inapplicable. Bubble diameter and thus its resonance
characteristics constantly change due to decrease of static pressure and gas diffusion. Therefore, gas
bubble in liquid has to be viewed as an oscillating system with changing natural frequency in the
external acoustic field of constant frequency.
Investigation of acoustic parameters of bubble screen as a whole, including interaction of
bubbles of different size represents a complicated problem, based on the laws of statistical physics.
This work considers change of oscillation characteristics of a single bubble in the process of it rise to
the surface. According to data [1], such approximation is justifiable for GLM with gas concentration
no greater than 0.22%, where the forces of interaction of bubbles may be disregarded.
As the mathematical model, spherical bubble that rises in a boundless compressible and
viscous diphasic liquid and experiences action of external acoustic field of constant frequency was
chosen.
This work is based on the solution of Noltigk-Neppiras equation developed by G.A. Druzhinin
[2, 3]. This equation in the field of harmonic acoustic wave has the following appearance:
−3 γ
3 & 2 1 
2σ
2σ  R  

&
&
=0
RR + R +
p0 − pvap +
+ pm sin ωt −  p0 − pvap +
 
2
ρ
R
R  R0  



(1)
where R0 – equilibrium radius of bubble, ρ - density of liquid, p0 – static pressure of the medium, pvap pressure of saturated vapor, σ- surface tension on the gas-liquid interface, pm-amplitude of the external
signal, ω - frequency of the external signal, γ - Poisson's ratio.
Solution of this equation (X = R - R0) was found in the work [3] as the sum of solutions of
linear and non-linear equations.
Xëèí + Xí åëèí = Àsin(ωt + ϕ) + Bsin[2(ωt + ϕ) + ξ] + C ,
(2)
where À = −
pm
ρR0 (ω 02 − ω 2 )2 + (ωg )2

(
) 2 
A2  2ω 2 + 3γ + 1 ω 0 
2 
B= − 

 ωg
2
2,
 ω0 − ω 
ϕ=-arctg 

 2

A2   (3γ + 1)  −  ω
2 
2   2ω 0  

C= 
, ω0 – Minaert's frequency, g = ω0δ, δ - damping constant,
2 R0
 2ωg

2
2.
ω 0 − 4ω 

, ξ=-arctg 
42
(
)
2 R02 ω 02 − 4ω 2 + (2ωg )
2
2
,
XI Session of the Russian Acoustical Society.
Moscow, November, 19-23, 2001.
Analysis of the solution shows that the presence of the non-linear terms in the equation of bubble
oscillations leads to distortion of harmonic oscillation. Non-linear distortions describe the second
harmonic of bubble oscillations with the frequency double that of the driving force, thus coinciding
with the resonance frequency (2Ω=ω0). At that two maxima of amplitude are observed at frequencies
of 2Ω and Ω.
However, as it was pointed out above, in the process of rising in a cloud bubble size
constantly changes and thus the bubble resonance frequency varies. Method developed by
V.K.Goncharov and N.Yu. Klement’eva [4] allows to compute change of resonance frequency of
bubble as it rises to the surface. The system of differential equations derived in [4] was solved
numerically using Runge-Kutta method. The resulting dependences of bubble radius, rising speed and
resonance frequency from time and path of rise of hydrogen bubble with initial diameter 50 and
100 ì m rising from the depth of 25m were graphed.
The obtained curves show that bubble rising in liquid reaches the resonance frequency at a
particular depth. The exact depth depends on the initial characteristics of bubble. At all other depths
the energy absorption by bubble is significantly lower than the absorption in the resonance conditions.
Consequently, it is possible to surmise that sound-proofing properties of bubble screen as published in
a number of works [5,6] are overestimated and these properties change with depth.
Graphic representation of the expression (2) for bubbles with initial radius 50 ìm in the field
of monochromatic wave (ω = 10 and 100 kHz) is shown on Fig. 1 and 2. Both graphs show amplitude
of oscillations of rising bubble (firm line) and its resonance value (broken line). In the first case during
the reviewed time period (100 s) bubble does not achieve the resonance conditions. According to the
conducted calculations, the energy it absorbs does not exceed 5% of the energy absorbed by the
bubble that has resonant diameter for the given conditions. Figure 2 shows the case when at some
stage of bubble rise resonance can be observed and the absorbed energy reaches its maximum.
We can justifiably suppose that in real situations there may be a combination of conditions
when the rising bubble having reached the resonance size does not have time to enter a stable-state
resonance condition of forced oscillations since transition into such state requires time t≥3τ, where τrelaxation time of the system. Since there is no time for the transition process to end, the bubble can be
either more, or less effective scatterer compared to resonance bubble depending on the phase
difference between the driving force and natural oscillations.
On the basis of computations, relaxation times of bubbles for the given cases were obtained.
Calculations show that the presence of significant energy losses in the resonance state of the bubble
leads to very small values of τ. This means that hydrogen bubbles produced by electrolysis of sea
water rising from depth of 25m will always achieve the resonance conditions.
Figure 1. Dependence of amplitude of bubble oscillations from rising time. Initial radius 50 ìm, initial
depth 25 m, frequency of external signal 10 kHz, calculations without taking into account forces of
interaction of bubbles in cloud.
43
XI Session of the Russian Acoustical Society.
Moscow, November, 19-23, 2001.
Figure 2. Dependence of amplitude of bubble oscillations from rising time. Initial radius 50 ìm, initial
depth 25 m, frequency of external signal 60 kHz, calculations without taking into account forces of
interaction of bubbles in cloud.
For comparison similar calculations were conducted foe the bubbles with the same parameters
rising in the center of bubble cloud with a greater concentration of gas phase. If the concentration of
gas is greater than 0.22%, the interaction of bubbles in the mixture becomes inevitable and associated
flow of liquid captured by each bubble leads to uncertainty of rising speed. In the center of the cloud
the rising speed of bubbles is independent of their size and equals 0.35 m/s [7]. Figures 3 and 4 present
the results of calculations for bubbles with the initial parameters corresponding to the situations
examined on figures 1 and 2 that rise in the center of a cloud with high gas concentration (rising speed
is constant and equals 0.35 m/s).
Figure 3. Dependence of amplitude of bubble oscillations from rising time. Initial radius 50ìm, initial
depth 25 m, frequency of external signal 10 kHz, bubble rising speed 0.35m/s.
Figure 4. Dependence of amplitude of bubble oscillations from rising time. Initial radius 50 ì m, initial
depth 25 m, frequency of external signal 60 kHz, bubble rising speed 0.35m/s.
44
XI Session of the Russian Acoustical Society.
Moscow, November, 19-23, 2001.
REFERENCES
1.
Feuillade C. The attenuation and dispersion of sound in water containing multiply interacting air bubbles.JASA, v.99 (1996).
2. Druzhinin G.A., Logachova E.Å., Semyonova I.S. Spectra of periodic and nonperiodic pulsation of
cavitational bubbles, Scientific Symp. «Physics and engineering of ultrasound», Saint Petersburg. 1997,
p.271-272. (In Russian).
3. Druzhinin G.A. Nonlinear acoustic, http://www. phys spbu. ru.
4. Goncharov V.K. Klement’eva N.Yu. Modeling the dynamics and conditions of sound scattering by gas
bubbles floating up from deep – water oil and gas deposits - Àcoust J. v. 42, ¹3, p.371-377.
5. Physics of sound in the sea (U.S. Government Printing Office, Washington D.C. 1946, reprinted by
Peninsula Publishing, Los Altos C. A.)
6. Fox F. E., Curley S.R., Larson G.S. Phase velocity and absorption measurements in water containing air
bubbles, JASA, 27, 1955 p.534.
7. McDougall T.J. Bubble plumes in stratified environments.- J. Fluid Mech. v. 85, part 4, 1978.
45