XI Session of the Russian Acoustical Society.
Moscow, November 19-23, 2001.
D.V. Voronin, G.N.Sankin ,V.S. Teslenko
MODELING of SECONDARY COMPRESSION WAVES Under CAVITATION
CONDITIONS
Lavrentyev Institute of Hydrodynamics
Russia, 630090 Novosibirsk, academician M.À.Lavrentyev str., 15
Ph.: (3832) 332-606; Fax: (3832) 331-612;
E-mail: [email protected]
The numerical and experimental modeling of a flow behind a shock wave expanding at a free surface in
water with bubbles of gas is carried out. As is known, the shock wave in water with reflection from a free surface
generates a rarefaction wave [1]. The presence of gas bubbles essentially complicates a flow field. Behind a
falling wave the bubbles begin to shrink, their size decreases, and in environment a local non-uniformity with the
increased pressure is forming [2]. In the reflected rarefaction wave, the energy at the non-uniformity is quickly
liberated. There are compression waves, which generate a secondary shock followed the rarefaction wave. The
conditions of a secondary wave occurrence, their intensity, interaction with the initial impulse are the objects of
the present research.
1. Mathematical modeling.
The problem of cavitation influence on the flow behind a shock wave in a liquid is topical one,
since such a flow is often enough in practical cases [1]. In recent paper the possibility of secondary
shock wave generation in the vicinity of a cavitation bubble is investigated.
Let's consider a flow near to a free surface, which separates water with initial density ρ01 = 1
3
g/sm and air with initial density ρ02 = 0,001225 g/sm3. In water the complex from spherical gas
bubbles with an initial diameter d0 = 0,7 mm is weighed. Initially the medium is in equilibrium state
with zero velocity and initial pressure p0 = 1 bar. In some moment of time t = 0 a shock wave in water
begins to propagate to a free surface.
The research was carried out within the framework of hydrodynamical approach. The
equations describing a non-stationary two-dimensional motion of inviscous compressing medium
when t> 0, are based on the laws of conservation of mass, impulse and energy for continua and
describe a flow field in cases of plane and axial symmetry. To complete the system there were used
correlations specifying a shock adiabat in a liquid, and also equation of state for ideal gas (adiabat
parameter γ = 1,4). Here and in the further index «zero» concerns to a condition before an initial
shock.
We suppose, that the borders between a liquid and gas represent contact discontinuity surface,
on which the condition of pressures equality and condition of equality of velocities (normal to a
surface) taken on the different parts from a contact surface, are valid. The left border of the field (Z =
0) is considered to be closed, a condition of medium non-penetrating through it is valid (longitudinal
velocity u here is put to be equal to zero), the top and bottom (R = 0) borders are also closed
(transverse speed v = 0), the right border is open, the boundary conditions on it correspond to ones on
a free border.
The system of the equations with the given boundary conditions was solved numerically by
Harlow method of particles in cells advanced in work [3]. The bottom border of computed region is an
axis of symmetry. Number of particles in cells is a variable. The numerical algorithm provides an
opportunity of association and splitting of individual particles of a Harlow method, depending on
current meanings of the flow parameters.
Let's note, that the problem of shock wave passage through two-phase layers located at a free
surface, was solved earlier in one-dimensional and two-dimensional approaches [1], and the research
was conducted within the framework of the mechanics of inter penetrating continua, when at each
point of area there are simultaneously two phases, each of them has definite meanings of
thermodynamic parameters. Such approach imposes the certain restrictions on the flow parameters and
the character of phase interactions. For example, the size of particles of the weighed phase should be
essentially less then one of a grid cell, and the bubbles interaction can be carried out only through a
carrying phase (water). Here the task is solved in two-dimensional statement. At each point there is
only one phase. Not only in a carrying phase, but also inside each bubble the fields of velocities,
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XI Session of the Russian Acoustical Society.
Moscow, November 19-23, 2001.
density and pressure are computed. With course of time individual gas bubble can be deformed,
divided to parts, bubbles can collide or stick together. The restrictions on the bubble size are not
present.
First let's consider task about passage of a bipolar impulse through individual gas bubble far
from a free surface (i.e. the volumetric concentration of bubbles is rather small, and their interaction
on the certain interval of time can be neglected). The impulse is made from (i) a wave of compression
with amplitude about 110 bar and length of a wave - about 5 mm and (ii) a rarefaction wave of the
same length and with amplitude -110 bar. The distance between waves is 2 mm. The size of
computational area: Z x R = 35 mm x 2 mm. At the moment t = 0 the impulse is at the left border of
the region. The initial coordinates of the bubble center: Z = 13,3 mm, R = 0 mm.
On Fig.1. the field of pressure is represented at the moment of time t = 5,6 µs. Fig.1.a
represents a background picture of a pressure field. Fig.1.b and Fig.1.c correspond to pressure profiles
in vertical and horizontal cross sections, depicted on the background picture. To this moment the wave
of compression has passed the bubble, its amplitude thus has changed insignificantly. As it is visible
from the figure process of the bubble collapsing began. Its sizes decrease, and the pressure of gas
inside grows (wave A on Fig.1.c), considerably exceeding amplitude of an initial impulse in a wave of
compression. The rarefaction wave through the bubble has not passed yet.
Fig.1. The pressure profiles at t = 5,6 µs.
Fig.2. The pressure profiles at t = 8,8 µs.
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XI Session of the Russian Acoustical Society.
Moscow, November 19-23, 2001.
On Fig.2. the pressure field is represented at the moment of time t = 8,8 µs. To this moment
the rarefaction wave has passed the bubble. In the second wave of the bipolar impulse, liberation of
energy occurs, accumulated in spatial non-uniformity (bubble) due to the first wave. As it is visible
from the figure, the bubble quickly extends, and around it the wave of compression is formed, which
amplitude has one order of size in comparison with an initial wave of compression.
On subsequent period of time the qualitative picture of the flow corresponds to Fig.2. The
complex of two waves of compression was generated, the rarefaction wave is located between them.
In the course of time secondary wave of compression catches up with the rarefaction wave and
extinguishes it.The variation of an initial bubble diameter and parameters of a bipolar impulse in the
certain ranges does not change a qualitative picture of the flow. The strength of a secondary
compression wave can be insignificant, and in some period of time can exceed capacity of the
initiating wave. If the initial bubble size exceeds critical one, the secondary wave of compression
practically will not be formed. The critical size defines by intensity of an initial impulse and time of
delay between waves of compression and rarefaction.
Let's consider a problem, when a bubble is located at a free surface. The initial impulse
represents just a wave of compression moving perpendicularly to a free surface. A wave of
compression accumulates energy in the bubble. With reflection from a free surface it will be
transformed to a rarefaction wave, which passing through the bubble liberates energy. In this case
situation is similar to a bipolar impulse considered above. The variation of distance from a free surface
up to the bubble is similar to change of distance between initial waves of compression and rarefaction.
The case is investigated also, when two bubbles are located successively one behind the other. For
large distances between them the situation essentially does not vary in comparison with considered
above. When they are located very close one to the other, their interaction becomes essential. The
secondary wave of compression, formed by a complex of two bubbles, turns out more powerful, than
from individual bubble, having the same volume as a total one for the complex. If the bubbles are
located on intermediate distances one from other, the second bubble is disposed on periphery of a
cavitation trace from the first. Formed from the second bubble, the wave of compression is weaker,
and the interaction of secondary waves complicates a picture of the flow. A total impulse not so
significant than in case of smaller distances.
The similar picture arises with a parallel bubble arrangement (case of plane symmetry). One
more flow property reveals here: when the bubble volumetric concentration exceeds critical meaning,
the initial impulse cannot overcome a barrier from water-gas medium, and the reflection of waves
from the barrier becomes determining for the whole process (analogue to [1]).
2. EXPERIMENT
Cavitation is induced by means of flat electromagnetic generator of acoustic pulses with
aperture of 70 mm in water [4]. The pressure in the pulse is controlled by voltage of the generator Ug
from 5 to 10 kV. The slanting fall of the acoustic waves on the free surface is under investigation. The
high-speed rotating mirror camera ÑÔÐ-1 ì is used for shadow recording of the wave picture and
bubble dynamics.
On the Fig. 3 the results of the photo-recording with rate of 250000 frames per second is
shown. The pictures show reflection of acoustic pulse wave from generator for Ug=8 kV. The angle
between the first flat wave (2) and the free surface was 15 degrees. In reflected wave with inverted
amplitude produce cavitation near the free surface. The second pressure pulse (3) acts on the
cavitational bubble and accelerate and synchronizes its collapse. The spherical shock waves (4) from
bubble collapse resulting in interferention of pressure disturbance between wave fronts (5) and (6).
The further bubble growth is observed with the camera rate of 62500 frames per second.
It is important to note that the free surface is moved under the influence of the acoustic pulse,
that is in agreement with another results [5]. Hence, the second wave is reflected by curved surface
and the relative position of two reflected waves is different before and after reflection. On the Fig. 4
the pressure measurements with pvdf hydrophone are shown. The comparison for the free surface and
steel rigid boundary and for different position of the hydrophone is done. The similar results is
described in [6] for cavitation in focusing shock acoustic wave reflected by the free surface.
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XI Session of the Russian Acoustical Society.
Moscow, November 19-23, 2001.
Fig. 3. The photo-recording of cavitation for slanting fall of double acoustic pulse on the free surface in
water (angle of 15 degrees, Ug = 8 kV). 1 – free surface, 2 – the first wave, 3 – the second wave, 4 - shocks from
bubble collapse, 5 – the first reflected wave, 6 – the second reflected wave.
Fig. 4. The pressure F) reflection from the free surface and S) reflection from the steel rigid boundary,
measured in the center of the aperture for Ug = 6 kV. a) 15 mm, á) 20 mm under the surface, 2 – the first
compression pulse, 3 – the second compression pulse, SCW – secondary cavitation waves.
3. CONCLUSION
The experimental and theoretical results show the origin of secondary cavitation shock waves
in heterogeneous media like a water with gas bubbles. The amplitude of secondary wave is of the same
order of magnitude and can be even higher in comparison with the amplitude of initial pulse. The
initial bubble radius and delay time between waves of compression and rarefaction are important
parameters in the problem. The volume concentration of bubble and average distance between them
should be taken into account for multi bubble systems under the influence of the acoustic pulse.
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XI Session of the Russian Acoustical Society.
Moscow, November 19-23, 2001.
This work was supported by grants No. 01-02-06444 and No. 00-02-17992 from RFBR.
REFERENCES
1.
2.
3.
4.
5.
6.
Nigmatullin R.I. Dynamics of multiphase media. P. 1, 2. M.: Nauka. 1987. 360 pp. (In Russian).
Teslenko V.S. Shock-wave breakdown in liquid. Kinetics of stimulated acoustic scattering at focusing of
shock waves. Technical Physics Letters. 1994, V20, N5. (In Russian).
Agureikin V.À., Krukov B.P. A method of individual particles for the computation of flows of
multicomponent media with large deformations. // Numerical methods of the mechanics of continuous
media. 1986. v. 17, N 1. pp. 17- 31. (In Russian).
Teslenko V.S., Sankin G.N., Drozhin À.P. Luminescence in water and glycerine in the field of spherically
focused and plane shock-acoustic waves. //Combustion, Explosion and Shock Waves, 1999, v.35, N6,
pp. 717-720. (In Russian).
Besov A.S., Zaitsev V.V. Research of an initial stage of cavitational rapture of suspension in pulse
rarefaction waves. // Akustika neodnorodnyh sred (Dinamika sploshnyh sred), Institute of hydrodynamics,
1997, issue 112, pp. 43-54. (In Russian).
G. Sankin, R. Mettin, W. Lauterborn and V. Teslenko. Secondary acoustic waves in shock induced
cavitation bubble clouds. // Submitted to the XI Session of the Russian Acoustical Society. November 1923, 2001, Moscow, Russia. (Http://www.akin.ru/ras)
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