EXPERIMENTAL STUDY ON THE MECHANISM OF THE RICHTMYER-MESHKOV INSTABILITY AT A GAS-LIQUID INTERFACE

423Available online at www.sciencedirect.comScienceDirectJoumal of Hydrodynamicswww. sciencedirect .com/ELSEVIER2009,21(3):423-428DOI: 10.1016/S1001-6058(08)60166-3science/jounal/10016058EXPERIMENTAL STUDY ON THE MECHANISM OF THE RICHTMYER-MESHKOV INSTABILITY AT A GAS-LIQUID INTERFACE*SHI Hong-hui, ZHANG Ga, DU Kai, JIA Hui-xiaCollege of Mechanical Engineering and Automation, Zhejiang University of Science and Technology, Hangzhou310018, China, E-mail: hhshi@ zstu.edu.cn(Received November 3, 2008, Revised December 31, 2008)Abstract: The mechanism of the Richtmyer Meshkov instability was experimentally studied in a vertical rectangular shock tube. Thevelocity of the interface driven by the shock wave was measured by a self-designed measurement system, which consists ofsemiconductor lasers, signal amplification circuits, digital oscilloscope and computer. Tests were carried out at several shock waveMach numbers. In addition, the movement of the interface and the variation of the mixed zone width with time were recorded byhigh-speed photography. The experimental results show that the interface velocity increases with the increase of the Mach number,and the distance of the interface's movement and the width of the mixed zone vary with time in a linear relationship.Key words: Richtmyer Meshkov insability, shock wave, shock tube, laser, interface1. Introductioncarried out by Meshkov2! Meyer and Blewet'3The shock wave induced instability, so-calledcarried out a numerical simulation on the R-MRichtmyer-Meshkov (R-M) instabilityh.2， occursinstability using a two-dimensional Lagrangianwhen a shock wave passes through an interfaceinviscid hydrocode and a good qualitative agreementbetween two fluids with different densities. Thiswas found with the measured experimental data byinstability has important applications in a wide rangeMeshkov4. Numerical simulation about instabilityof technology fields. For example, it can be seen inwas also carried out by Li et al.*" and Li and Chu'S.InInertial Confined Fusion (ICF)， Scramjet engine,2008, Balakumaro conducted the R-M experimentalunderwater supersonic jet, supersonic combustion, andstudy in a gas curtain.others. In addition, the R-M instability can also beIt is shown that the interface accelerated by theused to explain some natural phenomena, for instance,shock wave in the early stage is compressible andSupernova explosion.behaves almost in a linear manner, while in the laterThe R-M instability was first discovered in thestage it is nearly incompressible and behaves in anumerical research by Richtmyer in 1960, in whichnonlinear way. The main factor that makes thethe Richtmyer pulse model and the compresibleperturbation finally stabilize is the non-linear factors,theory of the R-M instability were proposed. In 1969,which include fluid compressibility, fluid property,the first experimental study on the R-M instability wasetc.. Therefore, a comprehensive research into theR-M instability phenomenon is necessary.The existing studies on the R-M instability* Project supported by the National Natural Sciencemostly focus on a gas/gas interface. For the R-MFoundation of China (Grant No. 10672144), the Naturalinstabil中国煤化工Wang et al."TScience Foundation of Zhejiang Province (Grant Noinventeof applying aY 107073).verticalYHC N M H Gtical rectangularBiography: SHI Hong-hui (1962-), Male, Ph. D.. Professorshock tube, Shi and Zhul!' investigated the R-M424instability using a CCD camera with framing rate ofThe high pressure and the low pressure sections100 fps. Based on their results, in this study, theare separated by a 0.04 mm or 0.1 mm thickinterface velocity driven by the shock wave wasaluminum diaphragm. The low pressure and themeasured by a measurement system which consists ofdischarging sections are separated by a tinfoil of 0.01semiconductor laser, signal amplification circuits,mm in thickness. The low pressure section is fileddigital oscilloscope and computer. Furthermore, thewith liquid, with two side windows for observation.movement of the interface was recorded by aAt the bottom of the liquid column, a 10 mm thick andhigh-speed camera with framing rate of 500 fps. The33 mm long square polytetrafluoroethylene (PTFE)experimental results were analyzed and discussed.plate is placed to ensure that the lower liquid surfacekeeps a plane shape in motion.When thehigh-pressure gas in the high pressure section reaches2. Experimental setupthe rupture pressure for the diaphragm, a shock wave2.1 General descriptionforms and moves downwards. When the shock waveFigure 1 shows the experimental setup of theimpacts on the liquid column and makes it accelerate,vertical rectangular shock tube, which mainly consiststhe R-M instability occurs. At the gas/liquid interface,of (1) a 500 mm long high pressure section fllingthe gas and the liquid are air and water, respectively.with nitrogen gas, (2) a 750 mm long low pressure2.2 Detailed experimental stepssection at atmosphere, and (3) a 200 mm longFirst, an aluminum diaphragm is placed betweendischarging section. All sections have the same squarethe high pressure and the low pressure sections and aCross section with inner width of 35 mm. The lengthtinfoil between the low pressure and the dischargingof the high pressure section is 2.5 times of that usedsections, respectively. Second, a certain amount ofby Shi and Zhuo's, for increasing the shock loadingwater is flled into the low pressure section. Third, thetime and improving the gas quality.laser and the oscilloscope are turned on, the parallelA pressure gauge and a valve are connected withbeams andne position of the Photodiode are adjusted.the high pressure section to control the pressure in theThen, the valve between the pressure reservoir andsection. The laser measurement system is placed onhigh pressure section is opened until the rupturethe both sides of the observation part of the lowpressure for the diaphragm is reached, and the instantpressure section. For the flow visualization, apressure value is recorded. Finally, immediately afterJapanese made PHOTRON FASTCAM- Super10KCthe experiment is completed, the valve is closed, andtype digital high speed camera is used.the diagrams and data of the oscilloscope are saved.2.3 Measurement principleFigure 2 shows an example of the measurementresults of the interface velocity. The measurementsystem consists of semiconductor laser (TDA635-1-G1)， photodiode (CUR52)， signal amplficationcircuits, digital oscilloscope (TDS2014B) andcomputer. The signal amplification circuita aredesigned by our research group according to therequirements of the experiment.回一、8110t△1_1I 1341141 - Pressure reservoir, 2- Valve, 3 - Hyperbaric chamber,4- Pressure gauge, 5 - Aluminum film diaphragm,26- Low pressure chamber, 7 - Observation chamber,8- Water, 9- PTFE plate, 10 - Tinfoil, 11 - Discharging2030section, 12- - Cloth cover, 13- Water container, 14- Sponge,11ms.15 - Semiconductor lasers, 16- Signal amplification circuit,urement of the interface17 -Digital storage oscilloscope中国煤化工serFig.1 Schematic diagram of the experimental setupTHCNMH G11 anc putiti, ui ouvck wave moves425downwards. When the shock wave impacts on theand 1.7, respectively. The measured△r for the threegas/liquid interface, the interface is accelerated andMach numbers are 12.4 ms, 6.8 ms and 5.0 ms,also moves downwards. The two parallel laser lightsrespectively. According to Eq.(1)， the interfacewill be cut by the movement of the interface one aftervelocity increases from 1.6 m/s to 4.0 m/s. Therefore,the other, and the optical voltage signals [10. arethe interface velocity clearly increases with theproduced and recorded by an oscilloscope, as shownincrease of the Mach numbers. The reason is related toin Fig.2. The time difference△t between two pulsethe interface acceleration when the shock wave passes.signals can be determined by the oscillograph. TheThe aceleration of the liquid column as discussed ininterface velocity V, driven by the shock wave canRef.[13] can be expressed asbe calculated according to the following relationship:_pAg=I(4)Mv=(1)Stwhere p is the pressure difference， A is thewhere L is the vertical distance between the twocross-sectional area of the shock tube, M' is theparallel laser beams, and L= 20mm。In allliquid mass, g。is the acceleration of the gravity.measurements, the distance between the interface andhe pressure difference can be obtained by thethe first laser beam is 5 mm.incident shock wave Mach number at the rupture pointof the diaphragm and the reflected shock wave Machnumber from the gas/liquid interfacel4. The relation3. Experimental results and analysisbetween the interface velocity and the shock wave3.1 Interface velocity at different Mach mumbersMach number will be discussed in the next section.In the experiment, the nitrogen gas is used as the3.2 Evolution of the interface and mixed zonedriving gas. Different Mach numbers are obtained byFigure 3 shows the high-speedphotographs ofchanging the thickness of aluminum films (0.04 mm,the R-M instability on an air/water interface at the0.08 mm, 0.1 mm, respectively). The height of theshock wave Mach numbers of 1.2, with the height ofliquid column is 0.14 m. According to the shock tubethe liquid column of 0.18 m and the time intervaltheory",，the shock wave Mach number can bebetween two adjacent photos of 2 ms. The startingcalculated by the following formulasmovement of the interface is shown in Fig.3(a).M=.-1+1+1B(2)2Y2% PB= B{1-(x-1)(9)(二-1)/P: Pia4 P1(a(bc) (d(C) () (g)、27r[(x-1)+(x +)(4)}2xX0-D) (3)Pwhere M is Mach number, γ，γ4 are thadiabatic coefficients of the driven and driving gases,respectively, P2/P\ is the pressure ratio between thedriving and driven gases after the rupture ofdiaphragm, p:/p\ is the pressure ratio between the(h)0(k)0m) (间)driving and driven gases at the rupture of diaphragm,q,a are the speed of sound of the driven andFig.3 Photographs of R-M instability at a rectangularair/water interface. The upper is air and the below isdriving gases, respectively.中国煤化工goes downwards.The calculations according to the above twoexpressions and the rupture pressure ratio show thatTY HCNMHGthe corresponding Mach numbers with the threeIn" rIgs.s(a)-3(e), une Inerrace is accelerateddifferent thickness aluminum diaphragms are 1.2, 1.5downwards. In Fig.3(0), an obvious deformation of the426interface can already be seen, with spikes and bubbles.and turbulent mixing as shown in Figs.5(g)-5().Figure 3(g) clearly shows that the R-M instabilitydevelops on the whole interface, as a multimodeinstability, with multiple spikes and bubbles ofvarious heights. In Fig.3(k)-Fig.3(n), one sees themixing in the form of water fog, as a result of theKelvin-Helmholtz instability and its own tensionduring the deformation process of the interface. Thespikes show more prominently the influence of thesurface tension. The spikes and bubbles arefractured and broken.(a)b)c) (d) (e)(0. (a)、(6)_ (C) (d (e)__ 0(8) () ()Fig.5 Photographs of R-M instability at a rectangularair/water interface. The upper is air and the below iswater. Shock wave ( M=l.7 ) goes downwards.Interframe time is 2 ms0.016|。M=1.7(g)h)i)j) (k(0.012M-1.2Fig.4 Photographs of R-M instability at a rectangular昌0.008water. Shock wave ( M=1.5 ) goes downwards.0.004Figure 4 shows the high-speed photographs whenthe shock wave Mach number is increased to 1.5. Inthis case, it is obvious that a single mode instability is1/ msdeveloped. The spike is formed as shown inFig.6 Relationship between interface position Z and time 1Figs.4(d)-4(f), then it is developed through to the end.It is of significance to note that in a rectangular tube, asingle-mode R-M instability can also be developed. In0.008M=1.7the experiments of Shi and Zhuoh.s, only multimodes M=1.5instabilities were found. Here it is shown that the0.006M=1.2increase of shock loading time may produce thesingle-mode instability. The high-speed photographswhen the shock wave Mach number is increased to 1.7are shown in Fig.5. After the strong shock wave0.002impacts on the interface, two different processes comeinto play. The first is the R-M instability in which a8single spike is developed, as shown in Figs.5(a) - 5(f)and three bubbles are developed as shown in Figs.5(g)中国煤化工- Fig.5(i). This kind of instability can be defined asCNMHGwidth h and time tpseudo-single-mode instability. The second process isMYHthe liquid degradation including breakup, evaporation427Figures 6 and 7 show the movement of thebeam measurement and high-speed photography. Itinterface position (Z ) and the variation of the widthcan be seen that the increasing tendency of the(h) of the mixed zone against time t at different shockinterface velocity with the Mach number is nearly thewave Mach numbers. It can be seen in Fig.6 that thesame, although their levels see some difference due tointerface moves with uniform speeds after 2 ms andthe different mass of the driven liquid column, whoseFig.7 shows that the mixed zone width also increasesheight is 0.14 m in the laser beams measurement andlinearly with time. The mixed zone width at the shock0.18 m in the high-speed photography. The results inwave Mach number of 1.7 is greater than that at theFig.8 confimm that greater water mass results in lessother Mach numbers of 1.2 and 1.5 at the same time,velocity. Quantitatively, an increase in water columnwith also a faster increase rate.height of 4 mm results in a decrease in velocity ofThe mixed zone, width h of the R-M instabilityabout 0.5 m/s.can be expressed asolThe measurements error by the laser beammethod can be estimated as follows. Differentatingh,=a,Agt2 = 2a;AZ(5)Eq.(1), one obtainswhere A=(A-p2)/(p +p2) is Atwood number,dv,_ dL_ dOt(6)L OtP，P2 are densities of the heavy and light fluids,respectively, g is the acceleration of the gravity, Zwhich means that the error in velocity measurement isis the movement distance of the interface, and a, isequal to the error in distance measurement minus theconstant, where subscript i stands for b or s, thaterror in time measurement. Since the distance betweenis, bubble or spike. From the experimental resultsthe two laser beams is set as L= 20mm and dL isshown in Fig.3, we have c, = 0.052，a, = 0.052 ,estimated to be l mm since the diameter of the laserbeam is1 mm, so that dL/L is 5%. Thea,1a, = 2.5, which are basically consistent with themeasurement error for time comes mainly from thetheoretical results ofa。=0.05 and a,/a,=3determination of Nt according to points when theobtained by Alon et al!"6two photo-voltages begin to drop and to rise. For theIt can be concluded from the above analysistypical example shown in Fig.2, Ot ≈6.25ms andbased on Figs.6 and 7 that the movement distance ofdSt≈0.5ms, so that d△t/Ot is 8%. Therefore thethe gas/liquid interface with time satisfies the relationvelocity measurement error dy/v, is about 3%.Zxt at different Mach numbers. And the mixedzone width at the later stage of the R-M instabilityvaries with time in a linear relationship hxt during4. Conclusionsa certain period of time.The experimental results show that the interfacevelocity increases with the increase of the Machnumber, and the movement distance of the interfaceand the width of the mixed zone have a linearrelationship with time.(1) The tendency of the variation of the interfacemovement velocity with the Mach number is the sameusing different testing systems: semiconductor laser-Laserand high-speed camera.+High-speed photography(2) This experiment shows that the movementdistance of the gas/liquid interface and the mixed zone2.width vary with time in a linear relationship, thatis, Zxt ,hxt . The linear relationship does notFig.8 Measurement results of the interface velocity using twochange with the increase of the shock wave Machdifferent experimental methodsnumber.(3) In a rectangular shock tube, although the four3.3 Comparison of results obtained with differentcorners may produce extra disturbances on thelp. .mndencendn-single-mode andexper imental methodsFigure 8 shows the measured results of themulti中国煤化工oped.interface velocity using two different methods: laserTYHCNMHG428gas/liquid interface[J]. Chinese Journal of TheoreticalReferencesand Applied Mechanics, 2007, 39(3): 417-421(inChinese).[1] RICHTMYER R. D. Taylor instability in shock [9] SHI Hong-hui, ZHOU Qi-wei. Shock wave inducedacceleration of compressible fluinstability at a rectangular gas/iquid interface[CI. Proe.Appl Math9601.le luds[小J. Commun. PureISSW26. Gottingen, Germany,CI. Proc.App Math,1960. |, 13: 297-319.8en Cemany, 2000 Paper 0940.[2MESHKOV E. E. 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