Combustion, Explosion, and Shock Waves, Vol. 42, No. 2, pp. 210–216, 2006
Transformation of Shock Waves at the Interface of Bubble Media A. I. Sychev1
UDC 532.529
Translated from Fizika Goreniya i Vzryva, Vol. 42, No. 2, pp. 97–104, March–April, 2006. Original article submitted December 8, 2004; revision submitted March 11, 2005.
The transition of shock waves from a bubble medium into a liquid or into another bubble medium with different properties is considered experimentally. Data on the structure, velocity, and pressure in the shock wave incident onto the interface, transmitted wave, and reflected wave are obtained. Experimental results are compared with numerical data. Key words: bubble medium, liquid, gas, gas bubbles, shock wave, transmission, reflection, transformation.
Shock waves in bubble media have been extensively studied experimentally and theoretically. Propagation of acoustic waves in bubble media was considered in [1, 2]. The structure and properties of wave perturbations of different amplitudes and durations in bubble media were considered in [3–13]. Interaction of gas bubbles with shock waves were studied in [14, 15]. Reflection of shock waves in bubble media from a solid wall was considered in [16–22]. The transition of shock waves through the interface between a bubble medium and a liquid medium was examined in [23]. Transformation of shock waves due to their interaction with bubble screens in liquids was considered in [16, 24–26]. The activities dealing with wave processes in bubble media were summarized in [27–30]. A shock wave in an example of a nonlinear wave. During its propagation, the wave affects the ambient medium. In turn, the processes in this medium determine the structure and properties of the wave itself. A change in parameters of the medium leads to a change in characteristics of the shock wave. Shock waves in bubble media, which propagate owing to interaction of the liquid and gaseous components, are transformed as the bubble medium parameters change. The objective of the present work is to study the transition of strong shock waves from a bubble medium into a liquid or into a bubble medium with different properties. 1
Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090;
[email protected].
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The shock waves in bubble and liquid media were experimentally studied in a vertical shock tube with an inner diameter of 40 mm and a height of 4.3 m, which consisted of high-pressure and low-pressure sections with a diaphragm between them. The low-pressure section was filled by a liquid where bubbles 2.5 mm in diameter were generated as the gas passed through two systems of capillaries introduced into the liquid perpendicular to the shock-tube wall and through its end face. The volume concentration of the gas phase of the bubble media was varied within 0.5 β0 6%. The height of the column of the bubble medium was 3.55 m. The pressure on the surface of the bubble medium was equal to the atmospheric value (p0 = 0.1 MPa). The experiments were performed at a temperature T0 = 288 K. The shock waves in the bubble medium were generated by burning a stoichiometric acetylene–oxygen mixture (C2 H2 + 2.5O2) in the high-pressure section of the shock tube [31]. The shock-wave amplitude (pressure) was varied by changing the initial pressure of the gas mixture C2 H2 + 2.5O2 . The pressure of shock waves corresponded to a pressure developed above the surface of the bubble medium in the case of gas combustion in a closed volume of the high-pressure section of the shock tube [32]. The shock-wave parameters were measured by piezoelectric pressure transducers mounted along the shock tube. The signals from these transducers were registered by two digital oscillographs S9-16 (the time constant of the pressure transducers was 8.4 msec). The transducers were calibrated by shock and detonation waves in gases.
c 2006 Springer Science + Business Media, Inc. 0010-5082/06/4202-0210
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Fig. 1. Oscillograms of pressure of the incident shock wave (1), wave transmitted through the interface (21), wave reflected from the interface (11), and wave reflected from the end face of the shock tube (2): α = 0.25; gaseous N2 ; (a, b) β0 = 4%; (c, d) β1 = 4% and β2 = 1%; (e, f) β1 = 4% and β2 = 0; the distance from the interface to the transducers is −0.665 (I), −0.070 (II), and 0.060 m (III).
The following systems were considered: “bubble medium,” “bubble medium I–bubble medium II,” and “bubble medium–liquid.” The liquids were water– glycerin solutions with a volume concentration of glycerin α = 0, 0.25, or 0.5 (the viscosities of the solutions were 1.01 · 10−3 , 2.27 · 10−3, and 6.84 · 10−3 Pa · sec; the velocities of sound in the liquids, which were determined from the velocity of propagation of weak shock waves, were 1380 ± 50, 1470 ± 50, and 1580 ± 50 m/sec, respectively). The bubbles contained argon (Ar) or nitrogen (N2 ). In the systems “bubble medium I–bubble medium II,” the volume concentrations of the gas phases are β1 and β2 , respectively; in systems “bubble medium– liquid, β2 = 0, and in bubble media, β1 = β2 = β0 . The shock waves generated in bubble media were steady (Fig. 1a and b). Strong shock waves in bubble media have a fluctuating structure (Fig. 1a and b): intense pressure fluctuations are registered in the shockwave front and behind the front; in 200–300 µsec, the pressure fluctuations decay, and the pressure reaches almost a constant level. Fluctuations of the shock-wave
pressure in bubble media are a consequence of oscillations of gas bubbles. (There are no pressure fluctuations in shock waves propagating in liquids.) A stochastic character of shock-wave pressure fluctuations is caused by a random distribution of gas bubbles in the liquid. Figure 1c–f shows the oscillograms illustrating the transition of the shock wave through the interfaces “bubble medium I–bubble medium II” and “bubble medium–liquid.” In passing through the interface between these media, the shock wave is transformed: a transmitted shock wave is formed, which propagates in the second medium (bubble medium II; Fig. 1c and d) or in the liquid (Fig. 1e and f), and a shock wave reflected from the interface arises, which propagates in the bubble medium I (Fig. 1c–f). The pressure in the transmitted shock wave and in the shock wave reflected from the interface increases with increasing difference in concentrations of the gas phase in the media. Thus, the pressure of the transmitted shock wave passing from the bubble medium (β1 = 4%) into the liquid (β2 = 0) increases approximately threefold; the pressure in the
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Fig. 2. Dependences D1 (β0 ): α = 0 (1 and 2), 0.25 (3, 4, and 9), and 0.5 (5–8 and 10); gaseous N2 (1–6) and Ar (7 and 8); L = 2.72 (1, 3, and 5) and 3.25 m (2, 4, and 6–8); p1 = 2.4 (1–7 and 9) and 3.6 MPa (8 and 10); curves 9 and 10 are numerical estimates.
shock wave reflected from the interface is also approximately three times higher than the pressure in the incident shock wave. The intensity of pressure fluctuations in the transmitted shock wave propagating in the liquid (Fig. 1f) is substantially lower than that in the transmitted shock wave propagating in the bubble medium (Fig. 1d). The transmitted shock wave reaches the end face of the shock tube and is reflected from the latter. Propagating further in the bubble medium II, the shock wave reflected from the solid boundary passes through the interface and enters the bubble medium I. The velocity of shock waves in bubble media is determined by the wave amplitude and parameters of the bubble media. Shock waves in liquids (both the transmitted shock wave and the shock wave reflected from the end face of the shock tube) propagate with the velocity of sound in these liquids. Figure 2 shows the measured velocities of shock waves propagating in various bubble media (each point in this figure is a value averaged over 3 to 5 experiments). The measurements were performed by pressure transducers in two sections of the shock tube with the distance between the transducers 0.595 and 0.470 m
Sychev (the distances from the surface of the bubble medium to the middle of these intervals L were 2.72 and 3.25 m, respectively). The shock-wave velocities at these distances from the bubble-medium surface have almost identical values. The shock-wave velocity D1 decreases with increasing concentration of the gas phase in the bubble medium β0 and increases with increasing amplitude of the shock wave p1 (Fig. 2). The effect of the properties of the gas contained in the bubbles on shock-wave parameters is very weak: shock-wave velocities in systems with bubbles of monatomic and diatomic gases have similar values. The combined effect of the properties of the liquid component in bubble media on the shock-wave velocity is insignificant (see Fig. 2). Let us consider the interaction of the shock wave with the interface between the bubble media. We introduce a laboratory coordinate system whose axis is directed from left to right. Let a steady shock wave (incident wave) propagate with a velocity D1 in a quiescent bubble medium I from right to left. We write the equations of conservation of mass and momentum, which relate the parameters of the bubble medium on both sides of the shock transition: ρ10 D1 = ρ1 (D1 − u1 ), p0 + ρ10 D12 = p1 + ρ1 (D1 − u1 )2 .
(1) (2)
Here p0 , ρ10 , and p1 , ρ1 are the pressures and densities of the bubble medium ahead of and behind the shock-wave front, respectively; u1 is the velocity of the medium behind the shock wave. Thus, the shock wave transforms the bubble medium I from a state with the parameters p0 , ρ10 , and u0 into a state with the parameters p1 , ρ1 , and u1 (u0 = 0 is the velocity of the medium ahead of the shock-wave front). The bubble medium with the parameters p1 and ρ1 is incoming with a velocity u1 onto the interface between the bubble media I and II. The interaction of the medium flow with the interface results in formation of two waves: the shock wave that passed through the interface and propagates in medium II and the wave reflected from the interface and propagating in medium I (the reflected wave is a shock wave if the acoustic impedance of medium II is higher than the acoustic impedance of medium I; otherwise, the reflected wave is an expansion wave). Thus, the incident shock wave interacting with the interface between the bubble media decomposes into the transmitted and reflected waves. We have to solve the problem of the decay of an arbitrary discontinuity at the interface between the bubble media. We write the equations of conservation of mass and momentum, which relate the parameters of the bubble medium on both sides of the front of the
Transformation of Shock Waves at the Interface of Bubble Media transmitted shock wave propagating in a motionless bubble medium II with a velocity D21 from right to left: ρ20 D21 = ρ21 (D21 − u21 ),
(3)
2 = p21 + ρ21 (D21 − u21 )2 . p0 + ρ20 D21
(4)
Here p0 , ρ20 and p21 , ρ21 are the pressures and densities of the bubble medium ahead of and behind the front of the transmitted shock wave, respectively; u21 is the medium velocity behind the transmitted shock wave. Thus, the transmitted shock wave transforms the bubble medium II from a state with the parameters p0 , ρ20 , and u20 to a state with the parameters p21 , ρ21 , and u21 (u20 = 0 is the medium velocity ahead of the transmitted shock-wave front). We also write the equations of conservation of mass and momentum relating the parameters of the bubble medium on both sides of the front of the reflected wave propagating with a velocity D11 from left to right in the bubble medium I compressed by the incident shock wave: ρ1 (D11 + u1 ) = ρ11 (D11 + u11 ), p1 + ρ1 (D11 + u1 )2 = p11 + ρ11 (D11 + u11 )2 .
(5) (6)
Here p1 , ρ1 and p11 , ρ11 are the pressures and densities of the bubble medium ahead of and behind the front of the reflected wave, respectively; u11 is the medium velocity behind the reflected wave. Thus, the reflected wave transforms the bubble medium I from a state with the parameters p1 , ρ1 , and u1 to a state with the parameters p11 , ρ11 , and u11 . The following relations are valid at the contact boundary (interface between the bubble media) in the decay of an arbitrary discontinuity: u11 = u21 = uc ≡ u,
(7)
p11 = p21 ≡ p
(8)
(uc is the velocity of motion of the contact boundary). To obtain formulas from system (1)–(6) under conditions (7) and (8), we have to involve the equation of state of bubble media. Neglecting the change in temperature of the liquid during the shock transition, we take into account the compressibility of the liquid component of the bubble media in the acoustic approximation. We have liq 2 liq p1 − p0 = (cliq 1 ) (ρ1 − ρ10 ),
where cliq 1 is the velocity of sound in the liquid contained liq in medium I, ρliq 10 and ρ1 are the densities of the liquid ahead of and behind the front of the incident shock wave, respectively,
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liq 2 liq p − p0 = (cliq 2 ) (ρ21 − ρ20 ),
where cliq 2 is the velocity of sound in the liquid contained liq in medium II, ρliq 20 and ρ21 are the densities of the liquid ahead of and behind the front of the transmitted wave, respectively, and liq 2 liq p − p1 = (cliq 1 ) (ρ11 − ρ1 ), liq where ρliq 1 and ρ11 are the densities of the liquid ahead of and behind the reflected wave front, respectively. The compressibility of the gas component of the bubble media is written in the general case: the gas is assumed to be involved into a polytropic process. We obtain n p1 ρgas 1 = , p0 ρgas 0
where ρgas and ρgas are the densities of the gas in the 0 1 bubbles ahead of and behind the front of the incident shock wave, respectively, n is the polytropic exponent (1 n γ, where γ is the adiabatic exponent), ρgas n p 21 = , p0 ρgas 0 where ρgas and ρgas 0 21 are the densities of the gas in the bubbles ahead of and behind the front of the transmitted wave, respectively, and ρgas n p 11 = , p1 ρgas 1 where ρgas and ρgas 1 11 are the densities of the gas in the bubbles ahead of and behind the front of the reflected wave, respectively. We use the relations between the density of the bubble medium and the parameters of the liquid and gaseous components 1 − α10 α10 1 α10 1 = + gas ≈ liq + gas liq ρ10 ρ ρ ρ10 ρ10 0 0 or gas liq ρ10 = (1 − β1 )ρliq 10 + β1 ρ0 ≈ (1 − β1 )ρ10 ,
where α10 and β1 are the mass and volume concentrations of the gas phase of the bubble medium I, respectively; α10 = β1 ρgas 0 /ρ10 ; 1 1 − α20 α20 1 α20 = + gas ≈ liq + gas liq ρ20 ρ ρ ρ20 ρ20 0 0 or gas liq ρ20 = (1 − β2 )ρliq 20 + β2 ρ0 ≈ (1 − β2 )ρ20 ,
where α20 and β2 are the mass and volume concentrations of the gas phase of the bubble medium II, respectively; α20 = β2 ρgas 0 /ρ20 . Similar relations are obtained for bubble media behind the incident, transmitted, and reflected waves.
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We obtain the formulas for the pressure in the transmitted and reflected waves as a function of the pressure in the incident shock wave liq p β1 p1 2 ρ10 1+ − (cliq 1 ) p0 p0 1 − β1 p0 1/n 1/n p ρliq cliq 1/2 p0 /p1 − p0 /p 1 × + − 1 10 liq p/p0 − p1 /p0 p0 ρliq c 20 2 liq 1/n 1/2 β2 2 ρ20 1 − (p0 /p) (cliq 2 ) 1 − β2 p0 p/p0 − 1 liq p β1 1 2 ρ10 = −1 1+ (cliq ) p0 1 − β1 1 p0 1/n 1/2 1 − (p0 /p1 ) × ; p1 /p0 − 1
× 1+
(9)
the velocity of the shock wave incident onto the interface β1 cliq 1 1+ D1 = (cliq )2 1 − β1 1 − β1 1 1/n −1/2 ρliq 10 1 − p0 /p1 × ; (10) p0 p1 /p0 − 1 the velocity of the shock wave that passed through the interface β2 cliq 2 1+ (cliq )2 D21 = 1 − β2 1 − β2 2 1 ρliq 1 − (p0 /p) n −1/2 × 20 ; (11) p0 p/p0 − 1 the velocity of the wave reflected from the interface liq β1 liq 2 ρ10 1 + D11 = cliq (c ) 1 1 − β1 1 p0 1/n 1/n −1/2 − (p0 /p) (p0 /p1 ) × ; (12) p/p0 − p1 /p0 the velocity of the flow behind the front of the incident shock wave β1 (p1 /p0 − 1) 1+ u1 = liq liq (cliq )2 1 − β1 1 c1 ρ10 /p0 ×
1/n 1/2 ρliq 10 1 − (p0 /p1 ) ; p0 p1 /p0 − 1
(13)
the velocity of the flow behind the transmitted and reflected waves (p/p0 − 1) β2 u = liq liq (cliq )2 1+ 1 − β2 2 c2 ρ20 /p0 ×
1/n 1/2 ρliq 20 1 − (p0 /p) . p0 p/p0 − 1
(14)
The calculated parameters of the incident shock liq liq waves were analyzed for ρliq 10 = ρ20 ≡ ρliq and c1 liq = c2 ≡ cliq for two limiting cases with n = 1 (isothermal process; model of “thermal equilibrium of the gas bubbles and the liquid”) and for n = γ (adiabatic process; model of “thermally insulated gas bubbles”). The analysis shows that the models considered are equivalent for strong shock waves (∆p/p0 1 and ∆p = p1 − p0 ): under the present test conditions (see Fig. 2), the dimensionless difference in results calculated by both models is within 4%. For shock waves of moderate amplitudes (∆p/p0 ≈ 1) and for weak shock waves (∆p/p0 1), the dimensionless different in results calculated by both models reaches 10 and 20%, respectively, and the choice of the numerical model is determined by thermal relaxation of gas bubbles in the shock wave [16, 33, 28, 29]. The calculated velocity of the shock waves incident onto the interface is plotted in Fig. 2 (the “thermal equilibrium” model was used, which offers simpler formulas). The calculated results are in agreement with experimental data. Note that Eq. (10) for the velocity of shock waves in bubble media in the limiting cases (with n = 1 and n = γ) agrees with the formulas derived in [7, 29] and [4], respectively. The calculated velocity of waves formed by transformation of the incident shock wave at the interface also agree with experimental data. Thus, e.g., for the case shown in Fig. 1c [with the incident shockwave velocity D1 = 223 m/sec (experiment) and D1 = 239 m/sec (calculation)], the velocity of the wave reflected from the interface is D11 = 1017 m/sec (experiment) and 1088 m/sec (calculation), the velocity of the transmitted wave is D21 = 616 m/sec (calculation), the velocity of the medium behind the front of the incident shock wave is u1 = 9.4 m/sec, and the mass velocity behind the fronts of the transmitted and reflected waves (the velocity of motion of the interface) is u = 7.3 m/sec (calculation). Figure 3 shows the measured pressure in the wave reflected from the interface and in the wave that passed through the interface (each point in this figure is a value averaged over 3 to 5 experiments). The pressures in the reflected and transmitted waves have similar values and increase with increasing difference in concentrations of the gas phase of the media. The calculated dependence offers a reasonable description of experimental results. Note that the transmitted and reflected shock waves in the experiments of [23] had different amplitudes. Apparently, the lack of coincidence of pressures in the waves at the interface is caused by unsteadiness of the incident shock wave. Note also that the experi-
Transformation of Shock Waves at the Interface of Bubble Media
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with a fixed amplitude is incident onto the interface between the bubble media, the mass velocity behind the fronts of the transmitted and reflected waves (velocity of motion of the interface) depends on the ratio of concentrations of the gas phase of the bubble media and increases with increasing concentration of the gas phase in both medium I and medium II. Finally, we should note that, in addition to the interaction of the shock wave with the interface between the bubble media, Eqs. (9)–(14) describe the transformation of the shock wave at the interface between various media: • for β1 = 0, transition of the shock wave from a liquid to a bubble medium; Fig. 3. Dependences pi /p1 (β2 ): α = 0.25, gaseous N2 , β1 = 4%, p1 = 2.4 MPa, and pi = p11 (1), pi = p21 (2), and pi = p (3) (calculation).
mental data of [23] and the results of the present experiments were obtained for different parameters of shock waves and bubble media, which makes it impossible to directly compare the results. The dependences of the ratio of pressures in the transmitted and incident waves on the ratio of concentrations of the gas phase in the media, which were obtained in [23] by solving the problem of the decay of an arbitrary discontinuity (numerical calculation) and the calculated dependence plotted in Fig. 3 are qualitatively similar. An analysis of the calculated results shows that the pressures in the transmitted and reflected waves in the case of the incident shock-wave transition from the bubble medium to the liquid increase with increasing amplitude of the incident shock wave and with increasing concentration of the gas phase of the bubble medium. If the shock wave passes from the liquid to the bubble medium, the pressures in the transmitted and reflected waves increase with increasing amplitude of the incident shock wave and decrease with increasing concentration of the gas phase of the bubble medium. The velocity of the wave formed owing to shockwave reflection from the interface between the bubble medium and the liquid and the velocity of the transmitted wave formed by the incidence of the shock wave onto the interface between the liquid and the bubble medium increase with increasing amplitude of the incident shock wave and decrease with increasing concentration of the gas phase of the bubble medium. The velocity of the medium behind the front of the incident shock wave increases with increasing amplitude of the wave and with increasing concentration of the gas phase of the bubble medium. If a shock wave
• for β2 = 0, transition of the shock wave from a bubble medium to a liquid; • for β1 = 0 and β2 = 0, transition of the shock wave liq from a liquid with the parameters ρliq 10 and c1 into liq liq a liquid with the parameters ρ20 and c2 ; liq • for β2 = 0, ρliq 20 ≡ ρm , and c2 ≡ cm , reflection of the shock wave propagating in a bubble medium or in a liquid (for β1 = 0) from a solid surface liq (for ρliq 10 c1 ρm cm , shock-wave reflection from an absolutely rigid surface is observed; ρm and cm are the densities of the material of the reflecting surface and the velocity of sound in the solid, respectively).
The transition of the shock wave from a bubble medium into a liquid or into another bubble medium with different properties can be used to generate shock waves with prescribed parameters in liquids and bubble media. This work was supported by the Russian Foundation for Basic Research (Grant No. 01-03-32351).
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