V.K. Kedrinskii et al- Bubbly system radiation and Mach reflections

March 22-24, 2004 Symposium on Interdisciplinary Shock Wave Research Sendai, Japan

Bubbly system radiation and Mach reflections

V.K. Kedrinskii1, I.V. Maslov1, V.A. Vshivkov2, G.G. Lazareva2,G.I. Dudnikova2, Yu.I. Shokin2

1Lavrentyev Institute of Hydrodynamics,Russian Academy of Sciences, Sib. Br.,Novosibirsk 630090;2 Institute of Computantional Technology,Russian Academy of Sciences, Sib. Br.,Novosibirsk 630090

Abstract. The results of numerical analysis of axis-symmetrical statementsdevoted to active media and carried out within the framework of the so calledproblem of “acoustic” laser (hydro-acoustical analogue of physical laser) are pre-sented. It is well known that bubbly systems are able to absorb, to amplify andthen to re-radiate acoustic pulses and thus to play a role of active media. Suchkind system can be determined as “saser” (shock amplification by systems withenergy release).

Here the processes of shock wave generation by the passive spherical andtoroid bubbly clusters as well as by reactive bubbly systems were studied. Theclusters are located on an axis of hydrodynamical shock tube (ST) and are excitedby outward stationary plane shock waves (SW). It was found that this interactionresults in the effect of SW focusing inside spherical cluster, in one case, andgeneration of a circular SW with oscillating profile in liquid by a toroidal cluster,in the other. The reflection of the latter from an axis has an irregular characterthat results in the formation of a Mach-disk in the vicinity of the axis.

The other approach to the “saser” problem solution consists in a study ofwave field structure excited by reactive bubble liquid in a ST. The latter con-tains the sections of different profiles with the ”discontinuities” boundary incross sections. In this configuration, the amplification effect results from two-dimensional cumulation of the shock wave and Mach reflection in the vicinityof ST axis. The statements on the “saser” problem considered involves also theproblem of transmission of the acoustic pulse generated by the system into asurrounding liquid through the cluster–fluid interface with the least losses.

1. Introduction

The problem of active media (Zavtrak 1995, Zavtrak & Volkov 1997, Kedrinskii et al 1996)capable of absorbing and amplifying an external disturbance and then re-emitting it in theform of an acoustic pulse is one of the important problems of the so-called acoustic laser(“saser”). Numerical analysis of one-dimensional cases has shown that bubble systems, bothpassive and containing explosive gaseous mixtures, can be treated as active media (Kedrinskii

241

et al 1996). In such media, an excitation caused by bubble system interactions with shockwaves can lead to significantly amplifying the wave field and generating an intense shockpulse. There is a basis to believe that cavitating liquids including combustible componentsas well as passive and reactive bubbly systems can play a role of such active media. Forexample, “passive” cavitational cluster arose in an inhomogenious liquid (on a solid wall)under an action of tensile stresses at interaction even with weak shock wave radiates a seriesof compression pulses with a frequency of bubbly cluster pulsation and amplitudes whichare essentially higher than an incident wave ones (Kedrinskii 2000). The another exampleis pressure-liquidfied gases and combustible liquids (stored under pressure in containers)which contain the micro- or macroinhomoheneities. Their heterogenious structure can bea main reason of arising such known phenomenon as large-scale explosions of conteiners(Kedrinskii 2000). In this case one can say about the bubbly detonation effect (Sychev &Pinaev 1986). The reason is the structure-wave mechanism according which a great numberof small bubbles filled with the mixture of air and fuel vapors are formed when filling acontainer and during transportation, so a combustible fluid in a container can be consideredas bubbly reactive media. If a container crowded by such fluid is suddenly depressurized orcollided with an obstacle shock waves and rarefaction waves can form, interact and excite abubbly detonation (Kedrinskii 2000). The adiabatic explosion of a gas mixture in collapsingbubbles of reactive bubbly system as a result of its interaction with a shock wave can beconsidered too as some physical analogy of pumping process of “saser” systems. An energyrelease at bubble explosions results in a sequential amplification of a wave field during pulsepropagation in such reactive system and in a formation of so called bubbly detonation wave.Recent study (Kedrinskii et al 1998) has shown that passive bubbly systems can also amplifyshock waves if the latter are colliding. Relative maximum pressure pref/psh in a collisionplane turned out to be approximated by a simple ratio

pref

pr,l� 2 + 24.5 · k1/4

o .

Keeping in mind that experiments with so-called free systems could turn out to be mostcrucial, it is interesting to analyze the interaction of shock waves with bubble clusters. Waveprocesses in such free systems involve phenomena of different temporal and spatial scalesand are accompanied by the generation of shock waves with amplitudes as much as tensand hundreds of MPa. These phenomena are determined by a great number of parameters;therefore, very often it is difficult or even impossible to analyze their effects in the course ofa certain physical experiment. From this standpoint, the necessity to numerically simulatevarious states inherent in such complex acoustic active systems, including features of waveprocess occurring in them, seems to be evident.

Wave processes in passive and reactive bubbly media were studied within the frameworkof two-phase mathematical model suggested by Iordansky (1960), Kogarko (1961) and vanWijngarden (1968), and modified then in (Kedrinskii 1968, Kedrinskii 1980, Kedrinskii &Vshivkov 1996).

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2. Formulation of problem

Let a velocity jump be given at the moment t = 0 at the end of a cylindrical shock tubefilled with water and having the radius rst. The center of a bubble cluster (spherical onewith the radius Rcl or toroid one with radius Rtor) is situated on the tube central z − axis

at a distance lcl from the end.The bubble radii and their volume concentration in clusters are R0 and k0 respectively.

At a moment t > 0, the shock wave propagating along the positive direction of the z− axis

encounters the bubble cluster, rounds it, and refracts into it. It is worth noting that theshock wave velocity in the cluster significantly depends on the volume concentration k0.For example, in the case of k0 = 0.01, the velocity is equal to a few hundreds m/s, beingessentially lower than the velocity of wave propagation in the liquid. This effect turned outto play principle role for the pressure field formation in a spherical bubble cloud.

3. Focusing of Shock Wave by Spherical Bubbly Cluster

Employing a nonequilibrium two-phase mathematical model for a bubble liquid, we numer-ically simulated a plane steady-state shock wave interacting with a passive spherical cluster(Kedrinskii et al 2001). We here consider certain unexpected effects found in the courseof this analysis, which are caused by both a difference in the velocities of acoustic-wavepropagation in the cluster and in the surrounding liquid and the actual cluster shape. TheIordanski@i-Kogarko-van Wijngaarden modified set of equations is used here as a governingsystem of equations for describing wave processes in the bubble medium (Kedrinskii 1968,Kedrinskii 1980, Kedrinskii & Vshivkov 1996). Governing system (1)-(4) includs:- the conservation laws for an average pressure, density and mass velocity p, ρ, u

dρ

dt+ ρ∇�v = 0,

d�v

dt+

∇p

ρ= 0, (1)

- Raileigh equation for nondimentional bubble radius β

∂β

∂t= S, β

∂S

∂t+

32S2 = C1

T

β3− C2

β− C3

S

β− p, (2)

- the equation for temperature

∂T

∂t= δ(γ − 1)Nu

β3(1 − T )T

− 3(γ − 1)TS

β, (3)

Nu =√Pe (if Pe > 100), Nu = 10 (if Pe ≤ 100), P e = C4(γ − 1)

β|S||1 − T | ,

- the state equations for a liquid component and mixture

p = 1 +ρlc

2o

npo

[(ρ

1 − k

)n

− 1], k =

k0

1 − k0ρβ3, (4)

whereβ =

R

R0, S =

dβ

dt, p =

pl

p0, ρ =

ρmix

ρl,

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Fig. 1 Schematic diagram of a bubble cluster interacting with a shock wave and the distri-bution of pressure at the time moment t = 110 µs: (1) cluster boundary, (2) front of anincident wave, (3) isobars, (4) shock- wave front in a cluster, and (5) unperturbed domain

C1 =ρg0T0B

poM, C2 =

2σRop0

, C3 =4µ

R0√poρl

, C4 =12R0

√poρl

µ.

When an incident shock wave interacts with a bubble cluster, the excitation at differentpoints of the cluster surface occurs with a retardation due to the velocity of shock wavepropagation in liquid being finite.

The shock wave velocity in a cluster is relatively low, thereby the cluster shape affectsthe wave propagation. Therefore, the shock wave formed in the cluster (as a result of thereemission of the refracted wave absorbed by the bubbles) strongly differs from that in theone-dimensional case. The picture of a bubble cluster interfacting with a shock wave andthe distribution of the relative pressure p (in units of the hydrostatic pressure p0 = 1 atm)over the space (r, z) are shown in Fig. 1 for the moment t = 110 µs. The calculationwas performed for the incident-wave amplitude psh = 3 MPa, k0 = 0.01, R0 = 0.01 cm,Rcl = 4.5 cm, lcl = 10 cm, rst = 15 cm, and L = 40 cm. In the figures, all linear sizes areexpressed in centimeters.

As is easily seen, the shock front 4 inside the cluster is concave and the pressure gradientalong the front is high. The latter fact is associated with the unsteady nature of the shock-wave formation process in the bubble medium. Moreover, at a fixed moment, different stagesof this process turn out to be distributed over the shock front, which results in the originationof the above-mentioned gradient. At this stage, a rarefaction wave is formed outside thecluster and propagates outwards into the surrounding liquid. This wave arises due to thepressure drop beyond the refracted-wave front, which is caused by the wave absorption bythe bubbles. As follows from the calculation results [see Fig.1], the domain 5 of the bubblecluster, which is bounded by the curvilinear front of the wave generated by the bubbles,remains unperturbed in the vicinity of the far cluster boundary (z-interval is 10-14.5 cm).This occurs despite of the fact that the incident wave front 2 has already rounded the cluster

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Fig. 2 Final stage of the shock-wave focusing in the bubble cluster at t = 140 µs: (4)concave front with a pressure gradient.

Fig. 3 Intense acoustic pulse (1) generated by the cluster at t = 160 µs.

by the moment of 110 µs. The initial stage of the wave amplification becomes apparent atthis moment. This is confirmed by Fig. 2.

Indeed, the shock wave focusing region in the cluster has essentially formed by themoment of 140 µs, and the pressure gradient along curvilinear front 4 in the bubble systemis clearly seen. Due to the focusing, an intense wave with an amplitude pfoc as large as 30MPa (see Fig. 3, t = 160µs) is formed near the bubble cluster-liquid interface in the vicinityof the point z = 13.5 cm. As a result, the cluster emits shock wave 1 of the bore type, with aparabolic wave front (see Fig. 4, t = 180µs). The pressure reaches its maximum value at theparabola axis and drops fairly sharply along its branches. It is necessary to emphasize thatsuch focusing has unconventional features. Indeed, aside from the intense pressure gradientalong the wave front, it is accompanied by the absorption and then reemiting of an incidentshock wave energy by the gaseous bubbles.

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Fig. 4 Profile (1) of shock wave emitted into the liquid by the bubble cluster at 180 µ s.

Thus, at each time step, a new wave originates in the cluster, with its front having asmaller radius of curvature. The stability of the wave focusing process in the cluster wasverified by introducing a specific disturbance having the shape of a liquid sphere with a radiusrdr from 0.5 to 1 cm. The sphere was placed inside the cluster, and its center lay on thez−axis at a distance ldr < lcl−rdr from the end. The calculation showed that the pressure-field disturbance caused by this sphere is weak and decreases rapidly. Therefore, it doesnot affect the final result. As follows from the calculations, the pressure at the focal pointdepends on a number of parameters, namely, the volume concentration k0 of the gaseousphase and the radii Rcl and Rb of the cluster and bubbles, respectively. These dependencescan be approximated by the relationships (1)-(3) which were obtained for psh = 3MPa.

pf

psh� 1 + 1.56 · 103 · ko − 1.6 · 104 · k2

0 ,

for psh = 3 MPa, Rcl = 3 cm, R0 = 0.1 cm

pf

psh� 1 + 2.6 · Rcl + 0.11 · R2

cl,

for psh = 3 MPa, k0 = 0.01, R0 = 0.2 cm

pf

psh� 17.9 − 8.5 · R0 − 135.6 ·R2

0,

for psh = 3 MPa, k0 = 0.01, Rcl = 5 cm.Influence of incident wave amplitude on a shock wave amplification by spherical bubbly

cluster was analyzed for case when R0=0.2 cm, k0 = 0.01, Rcl = 5 cm. The dependenceshows that the absolute values of pressure in the focus increase monotonically and can bedescribed (beginning with psh � 1 MPa) by the following approximation

pfoc � −30 + 62 · p0.21sh

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In this case, we assumed in formulas (1), (2), and (3) that Rcl = 3cm and R0 = 0.1 cm,k0 = 0.01 and R0 = 0.2 cm, and k0 = 0.01 and Rcl = 5cm, respectively. The relationship (1)-(4) are valid for the ranges k0 = 0−0.05, Rcl = 1−5 cm, R0 = 0.05−0.3 cm, Psh = 0.4−12MPa, respectively. Analyzing the results obtained from the stand point of principles ofdesigning powerful pulsed acoustic sources, we can conclude that a passive spherical bubblecluster excited by a shock wave represents an active medium that can absorb and amplifyan external disturbance and then reemit it in the form of an intense acoustic signal. Theposition of the focal region with respect to the cluster-liquid interface can be regulated byvarying the volume concentration k0 of the gaseous phase. Therefore, the emitted-waveabsorption by the cluster in itself can be virtually excluded.

With a point of view of “acoustic laser” one can conclude that a spherical bubbly clusterexcited by shock wave is a real active medium which is able to absorb and then to re-radiatethe powerful acoustic pulse. The location of focus relative to boundary “cluster/liquid” (andhence the possibility practically to exclude an absorption of wave by a cluster after focusing)can be drived by an election of k0 value.

4. Focusing circular shock wave re-radiated by toroid bubbly cluster

The experimental and theoretical investigations on spatial shock waves excited by circularsources devote as a rule to problems of unrestricted cumulation , irregular reflection of circu-lar shock wave in a gas and liquids (Sokolov 1986, 1988; Barhudarov et al 1988, 1990, 1994;Jiang & Takayama 1998; Kedrinskii 1980). The results of such kind studies are interestingfor the problems of physical acoustics and hydroacoustics, in particular (Kedrinskii 1980).We can mention, for instance, spatial spiral charges of explosive which can be used as thesources of power hydroacoustic radiation both in water and air (Kedrinskii 1980, Pinaev etal 1999, 2001).

Here we will consider the result of numerical study of focusing of stationary shock wavewith oscillating profile. Such kind shock wave can be generated by a toroidal bubbly clustersexcited by external initiating shock wave. The pressure field structure is studied in a liquidthe state of which can be described by Tait equation. Gas phase in a toroidal cluster is anair, the adiabatic index is equal to 1,4.

Formation of sequence of Mach reflections. The calculation were carried out for thefollowing values of main parameters: k0 = 0.001 − 0.1, R0 = 0.01 − 0.4cm - bubble radiusand volumetric gas concentration in cluster, Psh = 3÷10MPa - amplitude of initiating shockwave, rst= 20 , zmax=40., lcl=10., Rtor=6, Rcirc=1 .

Characteristic isobar distribution in near field are presented on fig.5,b-d, for three instantsof time and two spatial scales for each. Values of k0 are from the left to the right. Thelevel of pressure can be estimated on the corresponding scale presented. For this calculationthe values k0 =0.01 ER0=0.1cm were used. Fig.5b presents the wave field before focusing.Fig.5c shows that already initial stage of reflection of the first maximum of wave is irregularone. Here also one can see the second maximum of shock wave and its sequent irregularreflection (fig.5d, two Mach-disks).

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Fig. 5 Pressure fields for t = 110 (b), t = 130 (c) and t = 180 µ s(d) (left pictures has largerscale)

Fig.6,a-d, presents an example of typical irregular reflection for 4 values of k0. Here onecan see that the width of Mach-disks is equal about to 4-5 cm and they have a zone of highpressure restricted by a system of close isobars which can be determined as a disk core. Theisobars in front of first core belong to the front of first wave maximum. The isobars of a wavefront reflected are pressed to an axis by the wave front of the next maximum. The results ofMach-disk radius (Rmach) dynamics versus its coordinate Zmach show that its value growsmonotonically.

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a. b.

c. d.

Fig. 6 Isobar distribution in Mach-disks for: d - k0=0.1, c - k0=0.05, b - k0=0.01, a -k0=0.001

Unexpected effects were found at the study of influence of radius magnitude Rcirc oftoroid cross section on the parameters and structure of wave field generated in a liquidwhen toroid radius Rtor remains the same (fig.7,a-d, from the left to the right). The resultswere received for Rcirc=0.5, 2, 4 and 6 cm at psh=3MPa, Rtor=6cm, k0 =0.01 ERb=0.1 cm.Topology of the flow is not practically changed for the three first values of Rcirc. The analysisshows that the pressure amplitude inside the disk core increases versus Rcirc: p =81.7 forERcirc=0.5 cm, 99.4 for Rcirc=1 cm, 166 for Rcirc=2 cm, 258 for Rcirc=3 cm, 386 for Rcirc=4cm, 568 for Rcirc=5 cm and 859 for Rcirc=6 cm.

In the fourth case, Rcirc= 6 cm, internal boundary of toroid is closed in a point. Thelatter essentially changes the dynamics and structure of pressure field: the front of the waveemitted by toroid occurs a concave surface with the pressure gradient. The cumulation offlow in the vicinity of the axis tends to the formation power single shock wave with amplitudewhich is higher than the incident wave approximately by 30 times (fig.4,d, 859 atm).

The analysis of the wave field structure has shown that function p(Zmach) for pressurein the Mach-disk core has the maximum which increases versus k0. In the vicinity of thetoroid plane this pressure in 6-7 times higher then the amplitude of shock wave psh ignitingthe bubble system in toroid. The analysis of the data has shown that for diapason ofk0 = 0.01 ÷ 0.1 the pressure in a core of Mach-disk decreases monotonically versus Zmach

but on the distance of 20 cm from the toroid plane it is higher of psh in 2-2.5 times.

5. Mach reflection in ST with reactive bubbly liquid

In the present section the features of the wave structure in active bubble systems for shocktubes with sudden changes in the cross sections and with a one-phase liquid waveguide is

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Fig. 7 Influence of Rcirc value on evolution of wave structure generated by toroid :a) Rcirc = 0.5cm, b) Rcirc = 2cm, c) Rcirc = 4cm, d) Rcirc = 6cm.

Fig. 8 Geometry of the shock-tube: 1) annular channel; 2) working section; 3) waveguide.

analyzed (Kedrinskii et al 2002).

Formulation of the problem and modified two-phase model. Sudden changes in the crosssection of a shock tube of radius RST (Fig. 8) are produced by changes in shock-tube profile(transition from region 2 to region 3) or/and by an inside coaxial rigid cylinder (radius rcyl

and length Lcyl), which forms an annular channel 1 filled with a two-phase mixture similarlyto region (2). The condition LSW ≤ Lcyl is satisfied (LSW is the characteristic distanceat which the steady-wave regime of bubbly detonation initiated at the left butt-end of theshock tube is established). This inner ST geometry stimulates the effects of wave collisionand focusing (the pressure in region 2 is changed by varying the distance L between the butt-end and the wall) in the bubble medium on intensification of the acoustic pulse generatedby waveguide 3 (radius rout).

To describe wave processes in bubble hydrodynamic shock tubes, we also use Iordansky–Kogarko–van Wijngaarden model which, for the reactive gas phase, can be supplemented

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Fig. 9 Stationary shock-wave profiles in passive (a) and reactive (b) bubble media.

with a chemical-reaction kinetics equation of the type of the Todes equation [22], a moregeneral kinetic equation [23], or the simple condition of an instantaneous adiabatic explosionof gas in bubbles at their constant volumes. An analysis of various approaches to describingwave processes in active media with bubbles filled with an explosive gas mixture showedthat a simplified formulation can be used, ignoring the reaction kinetics in bubbles duringtheir compression by shock waves [24]. This

formulation assumes that when the gas inside the bubbles is heated to the ignitiontemperature, the reaction (on reaching the relevant degree of bubble compression R0/R

∗)proceeds instantaneously: there is an adiabatic explosion in a constant volume (constantbubble radius R∗) with an instantaneous jump of pressure and temperature in the detonationproducts:

p∗ = ρ∗(γ∗ − 1)Qexpl and T∗ = Qexpl/cv.

The adiabatic exponent for gas phase also changes instantaneously and becomes equalto that of the detonation products γ∗. After that, the process develops under new “initial”conditions without change of the mathematical model.

In the calculations, the liquid phase is water; up to the adiabatic explosion the gas-phase(2H2 + O2) is an ideal gas with an adiabatic exponent of 1.4. At the moment t = 0, astationary shock wave with the amplitude p(0) is generated on the left shock-tube wall.

Typical structures of steady-state shock waves in the passive and active media are givenin Fig. 9a and Fig. 9b, respectively. Here the horizontal dashed lines correspond to thespecified jumps in pressure p(0) at the shock tube butt-end that initiate the wave process.In both cases, p(0) = 10 atm, the gas-phase volume fraction was k0 = 0.01, and the bubbleswere of the same size (R0 = 0.1 cm). For the indicated parameters, the establishmentof a steady-state regime, for example, for a bubble detonation wave (maximum amplitudeapproximately 150 atm) is recorded at a distance of x ≈ 15 cm from the left wall of the shocktube. A stationary wave in the passive medium (Fig. 9a) and the “tail” of a detonation wave(Fig. 9b) running in the bubble system with detonation products (passive system) have aclassical oscillation structure. A system of precursors is generated ahead of the detonation

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Fig. 10 Spatial pressure field (a) and isobars (b) at the moment t = 225 µsec in the shocktube for expansion and focusing of the bubble detonation wave.

leader and the shock-wave front.

Further analysis of the wave field structure in the shock tube was carried out for thevalues k0 = 0.01, rcyl/RST = 0.5 and R0 = 0.1 cm. Taking into account the complexity of theexpected wave pattern in a shock tube with discontinuities in cross section, we performed apreliminary numerical analysis of the wave structure features for two formulations: reflectionfrom an annular wall in a shock-tube channel without a rod and expansion of a bubbledetonation wave behind the butt-end of a coaxial rod of length of Lcyl in a “semi-infinite”shock tube filled with an active bubble medium.

Expansion and Focusing of the Bubble Detonation Wave. We consider the wave structureof the pressure field formed in the shock tube when the detonation wave expands in a bubblemedium behind the butt-end of the coaxial rod. When the wave leaves the annular channel 1and enters section 2 of the semi-infinite shock tube behind the rod butt-end a rarefactionregion occurs (Fig. 10a). Obviously, at some distance from the rod butt-end, unloading isnot so large and an amount of energy sufficient for bubble compression to the temperatureof mixture ignition is preserved in the wave. Not only does the energy radiated by theexploding bubbles compensate for the losses by the wave but the wave can also be amplifiedby focusing, which is confirmed by calculation results (Fig. 10). In Fig. 10a, one can seea rarefaction region near the rod butt-end detonation wave does not arise at this stage ofthe process. However, since the active mixture in the bubbles is preserved in this region,subsequent pulsations of these bubbles can lead to detonation.

In Fig. 10, the focus zone is well defined: at a certain distance from the rod butt-end,a Mach disk forms, whose characteristic width exceeds 1 cm for a relatively small focusingradius. The Mach configuration moves along the axis, and in the case considered (Fig. 10b),it is at a distance of 3 cm from the rod butt-end. Obviously, there is some optimal geometryof the shock-tube cross section for which the pressure amplitude in the disk reaches somemaximum value as well as the resulting shock wave generated in the waveguide by the activebubble medium.

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Fig. 11 Spatial pressure fields (a, c, and d) and isobars (b) in the shock tube and waveguidefor a bubble detonation wave focused and reflected from the wall (shaded areas correspondto the rod butt-end and the wall) for t = 215 (a and b), 225 (c) and 235 (d).

Shock tube with two “discontinuities” in cross section. Calculations show that in thegeneral formulation, the wave radiated into the waveguide can be amplified using the effectof focusing (irregular Mach-reflection of bubbly detonation wave) and the wall effect, forexample, by choosing the parameter L such that the Mach configuration forms near theinterface (in the examples below, L = 4 cm).

The waveguide radius rout can be used as a second control parameter, whose value mustbe chosen in accordance with the radius of the Mach disk so that a one-dimensional waveforms in the waveguide channel. For example, if the waveguide radius is equal to the rodradius (rout = rcyl = 2.5 cm and rcyl/RST = 0.5), the pulse amplitude in the waveguide is200 atm. When the exit channel radius rout decreases to the size of the Mach stem (roughly1 cm), the wave amplitude reaches 500 atm.

The calculation results shown in Fig. 11 support the above assumption on the possibilityof considerable amplification of the wave generated in the waveguide by the active bubblemedium as the bubble detonation wave propagates in the shock tube with “discontinuities”

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in section. At t = 215 µsec, the pressure at the center of the focus (xf ≈ 16.5 cm) is in excessof 600 atm (Fig. 11a and b). In 10 µsec (Fig. 11c and d), the focus is displaced by 2 cm,and the amplitude in the focus increases to 800 atm (wall effect). Figure 11e and f showsthe pressure distribution in the waveguide at t = 235 µsec. At this time, the wave crestin the waveguide is at a distance of roughly 1 cm from the interface and had a maximumamplitude of approximately 600 atm on the channel axis followed by a slight decrease (to500 atm) at a distance of r ≈ 0.9 cm from the axis (for rout = 1 cm). An abrupt decreasein pressure is observed in the millimeter zone at the waveguide wall. It is seen that in thehydrodynamic shock tube with “discontinuities” in cross section filled with bubbles of anexplosive gas mixture, a strong pressure pulse is generated in the waveguide upon excitationof a bubble detonation wave.

6. Conclusions.

With a point of view of “acoustic laser” one can conclude that a spherical bubbly clusterexcited by shock wave is a real active medium which is able to absorb and then to re-radiate the powerful acoustic pulse. The location of the focus relative to the boundary“cluster/liquid” (and hence the possibility practically to exclude an absorption of wave bya cluster after focusing) can be derived by an election of k0 value.

The study of pressure field dynamics for the axis-symmetrical problem about an interac-tion of incident stationary shock wave with a bubbly toroidal cluster was carried out. It wasshown that a sequence of Mach-disks is formed in a liquid as a result of focusing and irregularreflection from an axis of SW with oscillating profile emitted by cluster. Mach-disks have acore with nonuniform distribution of pressure on its radius. Data on the dynamics of diskradius and maximum pressure in a core versus volumetric gas concentration were received.

The phenomenon of wave amplification in channels with “discontinuities” in cross sectionswas studied in an axisymmetric formulation. This phenomenon is observed when a wave isfocused at the butt-end of a rigid rod aligned coaxially to the channel and is reflected fromthe wall. In this configuration, the amplification is due to two-dimensional accumulation onthe axis of the shock wave after it leaves the annular channel. In the focusing spot, a Machconfiguration forms.

The geometrical characteristic of the shock tube allow one to control (in a certain range)the amplification coefficient and the position of the focusing spot. In particular, the wavecan be focused in the vicinity of the rigid annular wall (in the region of passage through theinterface to the waveguide) and amplified upon reflection. If the waveguide radius is equalto the radius of the Mach-disk, the amplitude of the radiated wave is maximal.

Acknowledgement. The work is supported by SS-2073.2003.1 and IP-SB-RAS 22 grants.

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