Complex fluid-dynamical phenomena modeled by large-scalemolecular-dynamics simulationsWitold Alda and Witold DzwinelInstitute of Computer Science, University of Mining and Metallurgy AGH, 30-059 Kra´kow, Poland

Jacek Kitowski, Jacek Mos´cinski, and Marek PogodaInstitute of Computer Science, University of Mining and Metallurgy AGH, 30-059 Kra´kow, Polandand Academic Computer Centre, CYFRONET-Kra´kow, 30-950 Kra´kow, Poland

David A. Yuena!

Department of Geology and Geophysics, Minnesota Supercomputer Institute,University of Minnesota, Minneapolis, Minnesota 55415-1227

(Received 22 May 1997; accepted 27 July 1998)

We carried out large-scale molecular-dynamics simulations of the classical Rayleigh–Taylor~RT!phenomenon in a Lennard-Jones molecular liquid. We have observed from these simulations,involving 106– 107 particles, the development of hydrodynamic instabilities from two differentkinds of interacting particles. A free surface is introduced by deploying an overlying void. For abox with a dimension up to about 1mm and two layers having different particle sizes, the fingeringtype of instability is observed as a result of oscillations caused by the gravitational field. In thisgridless scheme, surface waves can be captured self-consistently. For equally sized particles, aspontaneous ‘‘fluctuation driven’’ mixing with a long start-up time is observed. These molecular-dynamics results suggest the possibilities of upscaling the RT phenomenon. For conducting thesenumerical experiments, which require at least;105 time steps, a single simulation would require100–200 Tflops of massively parallel computer power. ©1998 American Institute of Physics.@S0894-1866~98!01606-X#

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INTRODUCTION

Let us consider a model consisting of interacting particfor describing the dynamics of physical systems. The mamum number of particles that can be simulated by the mpowerful computers is 10215 of the Avogadro number,NA .One can hardly assume that it will ever reachNA . How-ever, in the last 30 years the number of particles in molelar dynamics has increased from 103 to 109, a realisticnumber for some types of situations. One way to enlathe simulation scale is by increasing the number of particwhile still being able to complete calculations within a resonable period of time. This involves a simplification of trules for both the interactions and motion. This approahas been applied in lattice-gas~for example, in Ref. 1! andlattice-Boltzmann-gas2 models. Another way to extend tthe macroscopic level upward is to treat particles notatoms but as mesoscopic objects, ‘‘droplets’’ of fluid, iteracting with somewhat artificial interactions. There atwo principal methods using this idea:~1! dissipative par-ticle dynamics~DPD! introduced recently for simulation ocomplex fluids3 and ~2! smoothed particle hydrodynamic~SPH!, first applied for astrophysical simulations4 and

a!Electronic mail: davey@krissy.msi.umn.edu

© 1998 AMERICAN INSTITUTE OF PHYSICS 0894-1866/98/12~6!/595/6/$15.00

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adopted subsequently to hydrodynamic problems.5 It wasrecently shown that DPD and SPH are conceptually almidentical.6

The purpose of this article is to demonstrate that weknown hydrodynamic instabilities can emerge spontaously in microscopic particle systems. These phenomare relatively difficult to obtain, since they require largparticle ensembles and/or long simulation runs. We hused a microscopic molecular-dynamics~MD! system withrare-gas-like atoms interacting via a Lennard-Jones potial. Because of the large number of particles, the sysitself may be regarded as mesoscale, covering the spatime domain of micrometers and nanoseconds. We empsize here that we do not treat the atomic parameters riously but, rather, adjust them to observe their influencethe dynamics of the system.

Earlier MD results by Rapaport,7–9 Holian andRavello,10 Cui and Evans,11 and the authors of Refs. 12–1employed much smaller numbers of particles—typicaaround 104. Those results show phenomena such asformation of eddies behind an obstacle, hydrodynamicstabilities, fracture formation, and penetration mechanwhich are usually observed in macroscale, can also betectable in the mesoscale. Due to this self-similarity, omay expect that some correlation and distribution functiocan be scale-invariant, as they are in power-law fracdistributions.16 This is the reason why in Ref. 12 the au

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thors have proposed MD as the method to be used forsoscopic simulations of phenomena, which cannot bescribed by solving partial differential equations~PDEs! dueto the discontinuous changes in the density, material prerties, momenta, and energy.

I. PARTICLE MODEL: IN SEARCH OF THE SOURCE OF MICROSCALEHYDRODYNAMIC INSTABILITIES

These simulations have been carried out in a twdimensional~2D! box configuration. The particles interawith each other via a Lennard-Jones potential and movaccordance with Newtonian laws. In this model two diffeent types of particles,A andB, are employed, i.e., particlewith different masses and different interaction potentiaUAA , UAB , andUBB . Each two-body interaction employeis the two-range Lennard-Jones potential in the standform with three parameters:

Uab~r i j !54e i j F S s i j

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wherea andb denote particle types,e is the well depth ofthe potential,s can be roughly treated as a nearest distabetween particles, andr i j is the actual distance betweeparticles i and j. We assume that there are two kindsparticles: the heavier have mass equal to 40 amu~the massof argon atoms! or greater, while the lighter have masshalf this size. The well depthe is set to 119.8kB joules forall interactions, which also is a valid value for argon. Thesparameter has been set in a somewhat arbitrary wLighter particles always haves equal to 3.4 Å, whereasheavier particles can be of the same size or be shruns52.8 Å. By making the parameters of heavy particlessmaller, the development of instability is considerably acelerated.

The particles all move according to Newton’s equtions of motion, which are second-order ordinary differetial equations~ODEs! solved numerically in subsequentime steps using aleapfrogalgorithm:

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wherev i and r i are, respectively, the velocity and coordnate of particlei, ml is the mass of thel-kind of particle,f i jis a force between particlesi and j, andG is the externalgravitational force acting downward. Time stepDt variesbetween 1310214 and 2310214 s, which for up to 53105 time steps in Rayleigh–Taylor simulations yieldstotal simulation time of up to 10 ns.

The particles are initially confined to the lower halfthe domain, leaving a free surface together with an emcomputational void~i.e., no particles initially!. The totalnumber of simulated particles varies from 0.723106 up to1.13106, which enables us to simulate the particle flusamples covering, initially, a rectangle of about 0.57 mmlength by a height of 0.14–0.21 mm. This number seembe somewhat optimal~for the applied short-range potentia!because simulations for greater numbers of particles~up to10 million on a 3.4530.36 mm rectangle! have not yielded

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qualitative changes in the patterns. On the other hand,smaller numbers of particles (;104) these complex mixingpatterns cannot be captured with such coarse resolutFor fewer particles (23104), as was shown in Refs. 17 an18, we have obtained only one pair of eddies becauseheavier fluid replaced the lighter as a single splashbubble.

With the classical Navier–Stokes fluid-dynamicequations,19 one can compute both the critical wavelenglcrit ~perturbations of wavelength less thanlcrit are dampedby surface tension and viscous forces! and the most un-stable wavelengthlm of disturbances, which will developsubsequently. Let us assume that the system with no inperturbation allowed be kept at constant temperature.discussed above, the macroscopic model fails to predictevolution of such a system due to the lack of driving forceOn the contrary, in MD simulation, thermal fluctuations caspontaneously initiate these instabilities.

In order to observe more complex structures, we harestricted ourselves to 2D systems only. Otherwise,three-dimensional~3D! instabilities we would need at leas108– 109 particles. In spite of this, the simulations in 2Dspace of 1 million particle ensembles in 53105 time stepsfor a broad set of parameters would, today, need peakpercomputer power. The largest simulation that has bmade involved more than 100 Tflop. Therefore the comptations, using a highly parallelized MD code, were runup to 128 processors~actually the optimal number for a10-million-particle simulation! on a Cray T3E with a 500-MHz clock at the Minnesota Supercomputing Institute. Tresults were postprocessed on 16 processors of Convexemplar SPP1600 in the Academic Computer Center, Kkow, Poland.

II. MODELING OF MICROSCALE HYDRODYNAMIC INSTABILITIES

Rayleigh–Taylor~RT! instability, observed in the laboratory, is a fingering instability involving an interface between two different fluids, which occurs when the lighfluid, initially in the lower layer, protrudes through thheavy top fluid~Fig. 1!. We can distinguish two kinds ofinstability: one when fluids are accelerated towards eaother ~Rayleigh–Taylor instability! and one when shock

Figure 1. Rayleigh–Taylor hydrodynamic instability.

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waves pass through an interface of fluids with vastly diffent physical properties~Richtmyer–Meshkov instability!.Despite these differences, both types of instabilities hsimilar characteristics and often occur simultaneously.instabilities are difficult to model because they easily bcome chaotic. Thus they have an intrinsic appeal andcan hardly overestimate their importance.20 The RT insta-bility patterns produced during the mixing process varytime: initially, rolls and eddies are observed, and then, thevolve into more and more chaotic structures.21 As shownin Ref. 22, also in microscale, they comprise the RayleigBenard and Kelvin–Helmholtz instability patterns. The later accompanies all the processes connected with the cterflow motion of fluid portions, while the former occursthe beginning of mixing. Due to the energy dissipatedRT instability, the vigor decreases and the system undgoes transformation to the chaotic system. Eventually thmal equilibrium is reached.

The continuum approach does not allow for the simlation of the transition from the equilibrium to nonequilibrium state with a single model. As was shown first by Rpaport for the Rayleigh–Be´rnard convection case,7,8 boththe equilibrium state~with heat advection and diffusion!and the transition to rolls can be obtained by the MD tenique with a large number of interacting particl(;104).

As mentioned above, the RT system is initially ustable. Vibrations of the particles cause an instability witmicroscopic origin. From the point of view of the macrscopic model, such a process could not occur and canmodeled only by introducing an artificial perturbation othe fluid boundary. The Kelvin–Helmholtz instabilitywhich occurs on the border of spikes~jets!, producessmaller and smaller irregularities. This leads to the qualtively different results for the various particle resolutionExisting macroscopic models of RT suffer from sevelimitations.

~1! Initial perturbations must be imposed artificially.23

~2! There are intrinsic difficulties in determining the rolestatistical heterogeneities.

~3! With the classical methods, the density discontinuitythe interface between light and heavy fluids poses sous theoretical and computational problems20 due tothe built-in assumptions as to the continuity of mattmomenta, energy, and the surface capillary forces.

The particle paradigm can be very useful for severeasons that follow.

~1! The ‘‘crossing-length-scale’’ computer experimeni.e., from micro to meso and from meso to macro, cjustify the usage of the combined SPH/DPD ‘‘quasipticle’’ approach and MD.

~2! Due to the self-similarity of the phenomena observedthe micro- and macroscale, some of the results arepected to be scale-invariant.24

~3! The effects occurring in small scales, such as dropformation, plume bursting, the influence of statisticadistributed inhomogeneities, and eddy formation, cbe investigated.

~4! We expect that the MD model via the scaling principcan produce informative quantitative results on the

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processes, which cannot be modeled classically, eby mixing of inhomogeneous fluids, convection driveby sedimentation,17,18 systems in complicated geometries, discontinuous processes, etc.

~5! MD can produce spontaneous formation of hydrodnamic instabilities, which can provide input into thmesoscopic and macroscopic models in the senseupscaling.

III. RESULTS

Compared to infinite fluid layers, which is often assumed25

a finite thickness of particle layers results in a several-fdecrease of the most unstable wavelengthlm . This is auseful property by which to select the wavelength ‘‘sentivity’’ of the mixing process. On the other hand, thgrowth rate of an optional perturbation wavelength droconsiderably.25 This may be the reason why we cannot oserve RT instability patterns for only a few particles. Tobtain mixing patterns in the microscale, faster growrates are needed.

Various computer simulations of Rayleigh–Taylor istability by means of MD were presented in Ref. 22. It hbeen observed that the difference in the size of superpofluid particles~smaller one for the heavier upper fluid! isthe strongest factor accelerating the mixing process in cparison with the varying temperature, Atwood number,parameters of the Lennard-Jones potential, and the shacomputational box. The second reason for enhancingmixing process and one tightly coupled to the previous ois due to an initial ‘‘shaking’’ of the particle system, causeby the acceleration fieldG being switched on. The thinlayer resulting from the collision of the two layers causthe shock wave. Particles are reflected back by the hwalls at the bottom. This results in oscillations of the entsystem, which are damped out eventually.

In these simulations we have introduced a free surfwith an overlying void, which enhances the vigor of instbility by introducing buoyant bubbles on the surface.Fig. 2 we observe in the subsequent images from two simlations that thickness of the upper layer exerts a very strinfluence on the mixing. One can detect two concurrprocesses: a periodic compression and decompressiothe entire system and instabilities appearing on the interfbetween the two layers. Due to the smaller size of the pticles from the upper layer, their numerical densityhigher. The decompressional phase can be seen in theage at 0.75 ns, where the numerical density is lower inwhole system. In the simulation with the thin heavy lay~0.036mm! light particles can move up easily forming tinsharp spikes and small bubbles~0.45 ns!. The ‘‘shaking’’causes the lighter particles to move upwards and the inbility patterns ~mushrooms! to develop. Bubbles and‘‘mushrooms’’ expand quickly diminishing the densit~0.75 ns! locally. Big bubbles form cavities, while heavparticles start to penetrate downwards~0.9 ns!. Eventuallyafter 1 ns, the bubbles tend to blow up on the surfagenerating surface waves. With the thicker heavy la~0.072mm! the process is much smoother. Due to the prence of more particles in the upper layer, light particcannot move upward so fast. The whole process starts

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The patterns formed are not as regular as they aremacroscopic simulations, but their qualitative behaviorsimilar. The mushrooms collide with each other and thundergo fragmentation. The irregularities are mainly duethe statistical inhomogeneities introduced by the fluctutions. The initial oscillations of the particle system, whicset as a driving force for instabilities, are not introducartificially and represent physically justified processes. Tsmall fluctuations on the fluid interface are agitated up adown by the oscillations. Once the light fluid pushes theavy one, it is followed by the reverse situation. Duethe thermal fluctuations, larger~lighter! particles can pen-etrate the upper layer. Smaller particles~of the same massor heavier! begin to penetrate the nearby bottom layeThey accumulate between the two neighboring mushrooformed by the larger particles. The smaller particles replathe larger ones and push the light fluid upward. The oslations and chaotic thermal motion amplify this strontime-dependent process.

For a free-surface system, we have observed thstages of transition from the open thermodynamic system

Figure 2. Particle density distribution of mixing for thicker (left columnand thinner (right column) upper layers. Density changes from red (higto blue (no particles at all). For details see the text.

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equilibrium. First, the gravitational energy is accumulatby yielding a critical state, then it is dissipated via a rollinstructure formation~Rayleigh–Be´nard type of instability—maximum dissipation!, followed by a counterflow with twodifferent fluids ~Kelvin–Helmholtz instability, the saddlepoint of the energy-dissipation curve! and finally by surfacewaves, which are close to equilibrium with a minimuamount of dissipation.

On the other hand, for particles with the same size,oscillations do not initiate the mixing. This occurs after toscillations stop. In Fig. 3, in contrast to the situation psented in Fig. 2, the numerical particle density is higherthe bottom layer~due to the pressure exerted by the upplayer!. Therefore, heavy particles cannot penetrate as eaas for the previous case. Mixing starts only after 4 ns ahas a completely different character than before. Preously, the process was initiated by the acceleration fibeing turned on. At this time the dynamics have a sponneous and critical character. For quite a long time the s

Figure 3. Particle density for fluid with equally sized particles. Due to tpressure, the density increases toward the bottom. Mixing starts afterin the right bottom corner of the box, lowering the density. Rapid dissition of energy gives rise to surface waves. Eventually the system slrelaxes to equilibrium. The number of simulated particles is 7.23105

during the 10-ns period.

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Figure 4. The process of single-instability creation, appearing spontaneously as a result of statistical fluctuations with a wavelength much shorter tlcrit .The absolute value of the velocity field is shown.

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tem seemed to be stable. Meanwhile, the potential andnetic energies increased almost linearly. The ‘‘phatransition’’ appears after 4 ns, i.e., 23105 time steps. Ascan be observed in Fig. 4, it begins, in fact, slightly earliThe cause of the mixing was due to stochastic fluctuatioshown in the center of the second~small! picture. The de-velopment becomes apparent just after the accumulatiosufficient energy. The start-up time of such a process isallowed by classical fluid-dynamics theories. This can obe obtained from the stochastic character of the partimethod. When the fluids exchange their positions~about 6ns after the initial stage!, they move in opposite directionand Kelvin–Helmholtz patterns can then be observed~seeRef. 22!.

Development of the instability depends very strongon the different sizes of the particles, but also on the mratio of light to heavy particles and the thicknesses ofrespective layers. Due to the collective influence of thparameters, the process of mixing can be acceleratedcertain combinations of them. In Fig. 5 we can seegrowth of mixing layer for three different mass ratios. Tfastest growth is for the 100:20 mass ratio because of c

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structive vibrational interference of the entire system wthe interface layer.

IV. CONCLUDING REMARKS

The unstable initial conditions endow the RT mixing prcess with excess energy. This energy is distributed inform of RT mixing patterns. The combined effect of diffeent particle sizes~the s parameter in the Lennard-Jonepotential! and oscillations caused by the gravitational fiecan considerably accelerate the mixing. Because offact, RT patterns can be observed in the microscale. Wthe size of the particles is the same for both fluids, differionly in mass, and the critical wavelength is comparablethe computational box size and greater than the fluid-lathickness, the mixing phenomenon is quite different frothat observed for macroscopic fluids of infinite or thicfluid layers. The mixing is delayed, and the processserved subsequently becomes critical. This result, i.elong accumulation of energy and rapid mixing, maysomehow related in the macroscale to the longer-timesprocesses investigated in other fields by, e.g., geophysi

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~motion of lithospheric layers,26 earthquakes, etc.! and as-trophysicists ~the processes accompanied with stelevolution and supernovae explosions!.27

Our results have provided the following conclusion

~1! Initial perturbations produced at the microscale calso be propagated into the macroscale. This upscaphenomenon would also apply to the nature of statical heterogeneities, which can be directly investigaby the MD model.

~2! The MD approach, when applied to a RT phenomencan be used for investigations of the transition frounstable to stable equilibrium for open thermodynamsystems. This can yield information on the naturedissipative systems and self-organized critical strtures.

This paradigm based on particles28 can be used not only instatistical physics but also in other areas as well. Therefwe feel that the particle-based method will become mpopular with ever-increasing computer power. A lacknew ideas for solving very large-scale problems wsharply varying properties by standard partial-differenequation techniques will also spur progress along thisrection of upscaling.

Figure 5. Growth of the mixing layer for different mass ratios, 40:2100:20, and 200:20, and a constants ratio, 0.28:0.34. This is comparedwith a simulation consisting of equally sized particles.

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ACKNOWLEDGMENTS

The authors thank Jennifer Dudley of the Minnesota Supcomputing Institute for help in the visualization. Morsnapshots and mpeg animation can be found on the folling World Wide Web pages: http://banzai.msi.umn.edu ahttp://www.icsr.agh.edu.pl. The authors also thank the Mnesota Supercomputing Institute and the Academic Coputer Centre CYFRONET-Krako´w for the computer time.This research has been supported by the U.S. Departmof Energy and by the Committee for Scientific Resea~KBN! under Grant No. 5678 in Poland.

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