We review experiments and calculations of the compressible Richtmyer-Meshkov instability from a single-mode, nonlinear initial perturbation. These experiments were performed using the Nova laser. Measurements of the time-evolution of the mixing region were reported previously. We compared the experimental measurements with numerical simulations [1,2]. We found both experiment and simulation to be in good agreement with recent theories for the nonlinear evolution of the instability [3,4]. Experimental results beyond those previously presented provide additional support for the use of two phase flow models to describe the flow in the nonlinear regime. These experiments include measurement of the mixing region at additional times, including times earlier in the evolution of the instability than previously reported. We have also carried out experiments to examine the difference in the evolution of the instability from initial perturbations consisting of circular sawtooth grooves as well as rectilinear sawteeth. Our previous two-dimensional numerical simulations approximated the experimental linear grooves as circular grooves. We reasoned that the difference between the two cases would be small, based on scaling arguments, and limited to a very small region near the centerline. New experimental and numerical results confirm this. Finally, we discuss some additional issues in the derivation of the two-phase flow model used previously in describing the growth of the Richtmyer-Meshkov instability in the nonlinear phase relevant to other work presented at this meeting [5,6]

Budil, K. S.

Burke, E. W.

Farley, D. R.

Logory, L. M.

Murray, S. D.

Peyser, T. A.

Stry, P. E.

UNT Digital Library

. ...2 ‘“ i .. .,,-,.? >UCRL-JC-1Z5747PREPRINTReview of Experimentsand Calculations of theCompressible Richtmyer-Meshkov Instability from aSingle-mode, Nonlinear Initial PerturbationT. A. Peyser, S. D. Mumay, P. L. Miller, D. R. Farley,L. M. Logory, P. E. Sky, K. S. Budil, E. W. BurkeThis paperwas preparedfor submittalto the6thInternationalConferenceon thePhysicsof CompressibleTurbulentMixingMaxseille,FranceJune1S-21,1997DISCLAIMERThis document was prepared as an account of work sponsored by an agency ofthe United States Government.  Neither the United States Government nor theUniversity of California nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the University of California.  The views and opinions of authorsexpressed herein do not necessarily state or reflect those of the United StatesGovernment or the University of California, and shall not be used for advertisingor product endorsement purposes.‘Review of experiments... ‘ 1Review of experiments and calculations of thecompressible Richtmyer-Meshkov instability horn asingle-mode, nonlinear initial pert urbat ionT.A. Peyser~ S.D. Murray, P.L. Miller, D.R. Ihrley, L.M. Logory, P.E. Stry, K.S.Budil, E.W. BurkeLawrence Livermore National LaboratoryP. O. Box 808, L-22, Livermore, California, 94551, USAAbstract: We review experiments and calculations of the compressible Rlchtmyer-Meshkovinstabilityy from a singh+mode, nonlinear initial perturbation. These experiments were per-formed using the Nova laser. Measurements of the time-evolution of the mixing region werereported previously. We compared the experimental measurements with numerical simulations[1,2]. We found both experiment and simulation to be in good agreement with recent theoriesfor the nonlinear evolution of the instability [3,4].Experimental results beyond those previously presented provide additional support for theuse of two phase flow models to describe the flow in the nonlinear regime. These experimentsinclude measurement of the mixing region at additional times, including times earlier in theevolution of the instability than previously reported. We have also carried out experiments toexamine the difference in the evolution of the instability from initial perturbations consistingof circular sawtooth grooves as well as rectilinear sawteeth. Our previous twodimensionalnumerical simulations approximated the experimental linear grooves as circular grooves. Wereasoned that the difference between the two cases would be small, based on scaling arguments,and limited to a very small region near the centerline. New experimental and numerical resultsconfirm this. Finally, we discuss some additional issues in the derivation of the tw~phase flowmodel used previously in describing the growth of the Richtmyer-Meshkov instability in thenonlinear phase relevant to other work presented at this meeting [5,6].1. Review oft he experimentThe experiments used a miniature beryllium shock tube mounted over a 700pm diameterhole made at the center of the side of a 3 mm-long, 1.5 mm-diameter cylindrical gold NovaHohlraum [7]. The shock tube was 2200 ~m long, 700pm in diameter with a 100pm wallthickness. The working material of the shock tube consisted of a 500pm -diameter, 300 pm -longsection of a high-density (1.22 g/cm3) brominated polystyrene ablator and a 500pm diameter,1900 pm long low-density (O.1g/cm3) carbon resorcinol foam payload. A schematic of theHohlraum and the attached shock tube can be obtained elsewhere in these proceedings and inthe literature [1,2,5,6]. Thermal x-ray radiation from the interior Hohlraum walls incident ontothe exposed brominated polystyrene results in a rapid ablation of material and the generation ofa strong shock (85 Mbar) which travels down the shock tube towards the perturbed plastic-foaminterface. A rectilinear sawtooth pattern was machined into the high density plastic with a highinitial amplitude (a. = 10pm) relative to the dominant wavelength (A = 23pm). The largeamplitud~t~wavelength (so/A = 0.43) initial perturbation was chosen so that the instabilitytFormorefiforrnation- Email:peyserl@llnLgov‘2 Peyseret al.would make anearly transition into the nonlinear stage. Themixing region width was measuredwith high-speed gated x-ray framing camera diagnostics using radiography side-on to the shocktube cylinder axis [1,2,5,6].2. Effect of Mach number on the spike and bubble amplitudeHigh-power laser-driven experiments make it possible to achieve extremely high Mach numbershocks. At the time the shock is incident on the interface, the Mach number is greater than orequal to 20. This is important since experiments at high Mach number exhibit the effects ofcompressibility more strongly than low Mach number experiments.The experiments were simulated using CALE, a two-dimensional arbitrary Lagrangian-Eulerian (ALE) hydrodynamics code [8]. In ALEtype codes, the mesh moves with the flow,giving added resolution in regions of high compression. Unlike purely Lagrangian codes, how-ever, advection is allowed so as to avoid mesh tangling. For the simulations presented here, weuse an initially rectilinear grid (unless otherwise noted) with 1 micron square resolution nearthe material boundary. The resolution decreases away from the boundary. Because the gridmoves with the flow, however, high resolution of the mix region is maintained. The numericalsimulations include the cylindrical region containing the plastic, the foam, the beryllium sleeveand a portion of the gold support ring. The effects of the laser drive are simulated by applyinga temperature source to the edges of the plastic which extend into the hohlraum.Previously, we found excellent agreement on the time evolution of the mixing region betweenthe 2D CALE simulations and the measured results of the experiment. Fig. 1 shows the timeevolution of the material interface during the growth of the instability . The calculations suggest,in contrast with incompressible or weakly compressible flows, that the Mach numbers of thepresent flows result in spike and bubble amplitudes of roughly comparable magnitudes [9]. Asshown in Fig. la, the instability is well within the nonlinear regime by 4 ns (approximately0.5 ns after the shock was incident on the interface). The similarity in the morphology of thespikes (heavy material) and bubbles (light material) is readily apparent at this time. The spikesand bubbles shapes and amplitudes in subsequent material plots at 5, 6 and 7 ns continue toappear highly symmetric as shown in Figs. lb, lc, and Id.A detailed comparison of the instability with a simulation of a smooth, unperturbed inter-face allows us to obtain quantitative estimates of the spike and bubble amplitudes as shownin Fig. 2a. The spike zmd bubble amplitudes in the perturbed calculation are determined byanalysis of the material plots as in Fig. 1 above and comparison with the location of the un-perturbed interface at the same time in the problem. We have followed the same procedureas in our earlier work and removed the effect of the axial target decompression to facilitatecomparison between experiment and calculation on the one hand and nonlinear theory on theother hand [1,2]. Fig. 2b shows the growth of the overall mix region represented by the sum ofthe spike and bubble amplitudes (see below). The initial interface location for the unperturbedcalculation was chosen to be 290pm, i.e., at the centerline of the 20pm peak-t~valley saw-tooth perturbation, so as to conserve the total mass of the high-density brominated polystyrenein the calculation. We assume that the presence of the instability does not significantly alterthe overall hydrodynamic trajectory of the interface. This is consistent with estimates fromthe numerical simulations suggesting that the kinetic energy in the mix region is much lessthan 0.05 of the kinetic energy in the axial flow. We find that the spike and bubble amplitudesare roughly comparable (although not identical) for the duration of the calculation and theexperiment.Review of experiments...(al200 -i(b)300 5(M Axial Position(pm) w3558.0Figure 1. Density plots showing the time evolution of the material interface, from 2D CALE simul>tions, at (a) 4 ns, (b) 5 ns, (c) 6 ns, and (d) 7 ns“L++4+4d24681012141STIM (ns).,d I ! .., 40 5’ is 20TIM~(n$)Figure 2. (a) Approximate spike and bubble amplitudes (corrected for axial target decompression),es a function of time, from comparison of perturbed and unperturbed 2D CALE simulations of theexperiment; (b) Comparison of experimental data, 2D CALE simulations and results of a simple two-phsse flow model for the nonlinear evolution of the instability using two values for the coefficient ofdrag.4 Peyser et al.3. A simple two-phase flow model for nonlinear Richt myer-Meshkovgrowth in the high Mach number regimeThere are now numerous models for growth of the Rlchtmyer-Meshkov bubbles in the nonlinearregime which suggest that the evolution of the bubble velocity goes inversely with time for asingl~mode perturbation. It follows accordingly that the bubble amplitudes in these modelshave a logarithmic time dependence [10-13]. As shown above, for conditions in the presentexperiment, the spike and bubble penetrations are roughly equal. The spikes are remarkablybubble-like in their appearance and time evolution, hence the total mix width for the singlemode problem under these conditions can be approximated by a single, simple logarithmic timedependence [1,2].We repeat here our derivation of a simple tw~phase flow model with a clearer statementof the assumptions and the parameters in the model than provided earlier [1,2]. The equationof motion for a bubble of low density pP displacing a heavier fluid of density p= can be writtenas( AAv pc~ ) dU+PP ~=F’a –ticdpcu2 (1)where V is the volume of the bubble, AA/2 is the added mass coefficient, A is the frontalarea (A = md2/4 for bubble of diameter d) and cd is the coefficient of drag. Numerousexperiments on bubble rise beginning with the work of Davies et al. in the early 1950s haveshown that the bubble front retains its spherical character during its rise [14]. This is consistentwith more recent compilations of bubble geometry at high Reynolds numbers applicable to thisproblem [15]. Since we are invoking nonlinear theory only when the amplitude and wavelengthare comparable, it is permissible from the assumption of sphericity as well as from generaldimensional analysis to treat the ratio of area to volume in Eq. 1 as a characteristic scalelengthgiven by the wavelength A of the single mode perturbation. In the high velocity limit, the addedmass coefficient AA/2 is equal to one. Finally we assume further that there are no additionalforces (gravity, pressure gradient terms etc.) acting on the system. Eq. 1 then becomesdU()cd pC ~2——-z-= ~ Pc+pp(2)Eq. 2 can be rearranged and integrated to give the following expression for the bubble’ velocityu=Uo1 + Uo?nt(3)where()cd p=rn=—~ Pc+Pp(4)Finally, a further integration of Eq. 3 givesa =*+ +Lu[l+nzuo(t – to)] (5)where ~ is the initial nonlinear amplitude, m is given by Eq. 5 above and depends on thecoefficient of drag, the densities of the material and the wavelength of the perturbation, U.is the initial relative velocity of the bubbles (compared to the nominal interface location)after the passage of the shock and is the time at which the nonlinear phase of the instability~ ‘Review of experiments... 5begins. Given the similarity between spike and bubble amplitude growth in the high Machnumbers of the present problem, Eq. 5 can be rewritten as an equation for the total mix widthevolution w where WOis equal to two times the instantaneous nonlinear amplitude and UOisthe instantaneous spike (or bubble) velocity relative to the nominal interface at the time ofonset of nonlinearity.w = WO + ~ln[l +mUO(t–tO)] (6)~Since we do not know the values of the initial nonlinear amplitudes, relative velocities or timeof onset of nonlinearity either a priori or directly from the experimental data, we make use ofthe 2D numerical simulation to obtain estimates of these quantities. An examination of thecalculations suzzests that the instability begins to exhibit nonlinearity at approximately 3.8 ns.“to 4.3 ns. Fig. 3 shows the density and velo&y plots at 3.8 ns from the 2D CALE simulations.. .At thk time, we find from the simulations that the total mix width wo = 18pm and the velocityDensity(c@).50Velocity(pmhs)110I ❑ l3.05 0X5Axial position (pm)340Figure 3. Density and velocity plots at 3.8 ns showing onset of nonlinear amplitudes from 2D CALEsimulationof the spike and bubble relative to the nominal interface UO= 14pm /ns. A least squares fit ofthe data shows further that the behavior of the measured mixing region admits a logarithmictime dependence with m = 0.096 pm’1. Assuming densities in Eq. 4 of p. = 0.4 g/cm3 andPP = 1.6 g/ems consistent with the numerical simulation, we find a value for the appaentcoefficient of drag C.j x 2.8 + 0.4 where the uncertainty is due to the scatter in the data aswell as the simplifying assumption used in the model. Predictions from the nonlinear theoryfor two vrdues of the coefficient of drag are shown above in Fig. 2b. Interestingly, the coefficientof drag thus inferred from this experiment is consistent with published vrdues for an air bubblein water at high Reynolds numbers [15].4. Additional experimental data on the nonlinear mix widthMeasurements from additional experiments carried out subsequent to the results reported previously are shown in Fig. 2b above. As with the previous results, the total mix width is inferredfrom a detailed analysis of the 5 – 95% transmission of x-rays across the nominal mix region.The experimental data (squares) are in good agreement with both the numerical simulation(circles) and Eq. 6 above using the quoted parameters. These data points were all obtainedwith initial perturbations that were rectilinear in nature as shown in a high-resolution scanningelectron microscope image in Fig. 4a. We have also plotted the measured mix width from ex-periments in which the initial perturbations were circular in nature as shown in Fig. 4b. There, ‘6 Peyser et al.Figure 4. High-resolution scanning electron microscope image of (a) the original rectilinear sawtoothperturbations and (b) the curvilinear perturbationsappears to be little difference between the two perturbation types which affirms the use Of a two-dimensional axisymmetric code to simulate the instability growth from the non-axisymmetricperturbations used in the experiment. We have also found little difference in the growth of themix region between numerical simulations having cylindrical and Cartesian geometries. Thkis not surprising from simple dimensional considerations since the radius of curvature in thecalculations is large compared with the amplitude and wavelength of the perturbation.Acknowledgement. The authorswishto thank the operationsstaffat the Nova laserfacility for theirexpert technical assistance. In addition, we wish to thank H. Louis, T. Demiris and their colleaguesfor their sseistancewith the target fabrication and characterization. This work was performed underthe auspicesof the U.S. Departmentof Energy by the LawrenceL1vermoreNational Laboratory underContract No. W-7405-ENG-48.References[1]T.A. Peyser et al., in: Proceedings of the Fifth IWCTM, 257 (1996).[2] P.L. Miller et al., in: Proceedings of the Twentieth ISSW, 581 (1996).[3] T. A. Peyser, et aL, Phys. Rev. Lett. 75,2332 (1995).[4] U. Alon et al., Phys. Rev. Lett. 72,2867 (1994); J. Hecht et al., Phys. Fluids 6,4019 (1994).[5] D.R. Farley et al., these proceedings.[6] L.M. Logory et al., these proceedings.[7] E.M. Campbell, Lasers and Particle Beams 9,209-231 (1991).[8] R.T. Barton, in Numerical Astrophysics, 482 (1985).[91 D. Shvarts. D. Orrm, personal communication (1997).~lil][11][12][13][14][15]J. Hecht, ‘U. Alon, D. Shvarts, Phys. Fhrids 6:4019-4030 (1994).S.W. Haan, Phys. Rev. A 39,5812-5825 (1989).J.W. Jacobs and J.M. Shelley, Phys. Fluids 8,405 (1996).Q. Zhang and S. Sohn, Phys. Fhrids 9,1106 (1997).R.M. Davies and G. Taylor, Proc. Roy. Soc of London A 200,375 (1950).R. Clift, J.R. Grace and M.E. Weber, Bubbles, Drops and Particles, Academic Press, NY(1978).Technical Information Department  • Lawrence Livermore National LaboratoryUniversity of California • Livermore, California  94551

Review of experiments and calculations of the compressible richtmyer-meshkov instability from a single-mode, nonlinear initial perturbation

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Review of experiments and calculations of the compressible richtmyer-meshkov instability from a single-mode, nonlinear initial perturbation

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