NOISE REDUCTION AND PATTERN FORMATION IN RAPID ...

International Journal of Modern Physics C, cfWorld Scienti c Publishing Company

NOISE REDUCTION AND PATTERN FORMATION IN RAPID GRANULAR FLOWS yDepartamento de Fsica zInstitute for Theoretical

R. BRITOy and M.H. ERNSTz Aplicada I, Universidad Complutense, 28040 Madrid, Spain Physics, P.O.B. 80.006, 3508 TA Utrecht, The Netherlands Received (received date) Revised (revised date)

Spatial uctuations in dissipative systems, such as rapid granular ows, behave very dierently from those in elastic uids. Fluctuations in the ow eld drive the linear and nonlinear instability in the density eld (clustering), while vortex structures appear and grow through the mechanism of noise reduction. The dynamics of the ow eld on the largest space and time scales is described by diusion equations with dierent diusivities for the transverse and longitudinal ow elds. The results are obtained from analytic and simulation methods.

1. Introduction

Granular matter1;2, when driven by strong external agents (gravity, shear, vibrating plates) can be viewed as a complex uid of hard particles that move ballistically and suer instantaneous and inelastic collisions. So, rapid granular ows are a uid dynamic phenomenon3;4, modeled by inelastic hard spheres or disks of unit mass, and diameter . The macroscopic equations, as well as the kinetic theory for such systems have been derived by Savage, Jenkins, Richman and Lun (see reviews in Ref. 2 ), at least for nearly elastic particles. The 7-th International Conference on Discrete Simulations of Fluids has been strongly focusing on `Dissipative Particle Dynamics' (DPD), as one of its central themes, as is clearly re ected in the present proceedings5 . It is important to position the present contribution on rapid granular ows in inelastic hard sphere uids, relative to this central theme. For historical reasons|Hoogerbrugge and Koelman introduced their model under the name `DPD'6|the generic name has become the `brand' name for a speci c many particle model with soft interactions, used in molecular dynamics (MD) to simulate complex uids on mesoscopic time and length scales. This is somewhat unfortunate and confusing, as there exist many particlebased simulation methods for complex uids, in which energy is not conserved but lost in dissipative pair interactions. Examples of such methods are event driven MD7 for inelastic hard spheres (IHS) to model rapid granular ows8, as well as the time step driven methods of (i) smooth e-mail:

brito@seneca. s.ucm.es, [email protected] 1

2 Noise Reduction and Pattern Formation in Granular Flows

particle dynamics9, based on a discretization of the equations of uid dynamics, (ii) the original DPD method, and (iii) the soft particle simulation method, based on Hertz's contact law and inelastic deformations10;4. The last one is speci cally used for simulating rapid granular ows. Steady states in such systems are only possible if the energy loss through collisional dissipation is compensated by extra energy input. This may be achieved by adding a Langevin noise source6;11 or by driving the system through external elds, gravity or shear12 . Without energy input spatially homogeneous steady states are in general unstable against spatial uctuations, and lead to clustering and microstructures8 b;c; 13, and to con gurations of densely packed vortices14;15. These phenomena occur in DPD, as discussed by Warren6c , in freely evolving IHS

uids8c;14;15;16;17;18;19, as well as in rapid granular ows modeled through Hertz's contact law20;21, and in laboratory experiments on air tables22 . The present paper is a study of two important properties of uctuations in (quasi)- steady states of rapid granular ows and their consequences: (i) uctuations drive the instabilities and lead to the formation of patterns in the density eld (clusters) and in the ow eld (vortices). (ii) uctuations are suppressed in comparison with uids of elastic particles. Consequently, a single instantaneous con guration is suciently `typical', so that ensemble and/or time averaging is not required to calculate many (but not all) macroscopic properties. In the present paper we restrict ourselves to the freely evolving IHS uid without energy input, and focus on the mechanism of noise reduction in section 2, and on pattern formation in section 3. The consequences of patterns on spatial correlations are studied in section 4.

2. Noise reduction

The relevant macroscopic variables for describing rapid granular ows are the density n(r; t), the ow velocity u(r; t), and the granular temperature T (r; t), de ned as the mean uctuation energy per particle, i.e. T = d1 hV 2 i, where d is the dimensionality, and V = v ? u is the uctuation velocity. The macroscopic equations23;24 for nearly elastic uids are given by the standard nonlinear equations of uid dynamics, except for the appearance of a sink term, ??, in the equation for the energy balance, which accounts for the collisional dissipation. To lowest order in the gradients and to lowest order in the inelasticity = 1 ? 2 (to be de ned below), the sink term is given by ? = 2 0 !T , where 0 = =2d and !(T ) is the collision frequency as derived from the Enskog-Boltzmann equation. Moreover, it has been shown23;24 that the transport coecients of viscosity and the heat conductivity of IHS uids are to lowest order in the same as those for elastic hard sphere uids. The basic explanation of noise reduction is the 'parallelization' of particle velocities in IHS uids. When two inelastic hard spheres collide, they loose a fraction = 1 ? 2 of their relative kinetic energy, where is the coecient of normal restitution3 . As a result of repeated binary collisions, the IHS-particles tend to move more and more parallel. The dominant part of the velocity of a particle v = u + V,

Noise Reduction and Pattern Formation in Granular Flows 3

ln[E(t)/E(0)]

0 α=0.6 φ=0.11

-5

-10

0 40 80 120 160 τ (number of collisions per particle)

Figure 1: (a) Noise reduction ('parallelization') of particle velocities in an IHS uid with small inelasticity ( = 0:975) and moderate density ( = 0:975) after = 700 collisions per particle. The plot shows a blow up of a microscopic con guration, where arrows represent velocities of single particles. No space or time averaging has been performed. (b) Dissipation of energy E (t) vs in IHS simulations () with N = 50000, = 0:6; = 0:11, compared with the theoretical prediction of Ref. 25 . The dashed line is the Ha's cooling law. At = 30 crossover to the nonlinear clustering regime occurs.

is given by the mean ow u. The random uctuation velocity p V isponly a small correction with a typical size given by the r.m.s velocity, hV 2i T p(`speed of sound'). Consequently, the ows are typically `supersonic' 3 with pjVj T juj, whereas in elastic uids ows are typically 'subsonic' with jVj T juj. Noise reduction can also be seen by comparing the spatial uctuations in u(r; t) in elastic and inelastic uids in more detail. At shorter wavelengths the uctuations are rapidly damped by viscosity in both systems. However, in the inelastic uid the short wave length uctuations are not replenished with energy, whereas in the elastic uid the randomizing, energy conserving collisions keep the uctuations at thermal noise level in all ranges of wavelengths. So, the dissipative dynamics of the IHS uid selectively suppresses the shorter wavelength components of the ow eld. This is illustrated in Fig. 1a, which shows a blow up of a small part of an instantaneous microscopic con guration of the ow eld at a late time. Further illustrations can be found in Fig. 1b above, as well as in gures 1a,b P in Ref. 26, which show respectively the decay of the total energy, E (t) = (1=2N ) i vi2 , and the 'kinetic theory time', i.e. the total number of collisions, (t), suered by a particle within a time t. All properties have been measured in a single realization. A dierent illustration of noise reduction is given in Fig. 2, which shows the transverse velocity correlation function G? (r; t) at three dierent times. The spatial correlation G (r; ) between the components u(r; ) of the ow eld are de ned as Z ? 1 G (r; ) = V dR u(r + R; )u (R; ); (1) and G? (r; t) is the component with u and u perpendicular to r . The data


τG

0.10 0.05

τ=40

0.00 -0.05

0.08

0.10 τ=126

0.05 0.00

0 2 4 6 8 10 r/√τ

-0.05

τ=216

0.04 0.00

0

2

4 r/√τ

6

8

-0.04

0

2

4

6

r/√τ

Figure 2: Noise reduction in transverse velocity correlation G? (r; ), at = 40; 126; 216 collisions per particle for N = 20000 IHS at coverage = 0:245 and dissipation = 0:975. Measurements are taken on a single realization.

points, obtained from a single run, are very noisy at short times and become almost noiseless at the latest times. The qualitative considerations above have recently been turned into a quantitative analysis25 for the decay of the total microscopic energy. When the IHS uid is initialized in a homogeneous equilibrium state of elastic hard spheres, the microscopic energy is gradually (i.e. after a few mean free times) converted into R `mesoscopic', and nally into `macroscopic' kinetic energy of the ow, N ?1 dr 21 nu2, through the mechanism of noise reduction. The time dependence of the macroscopic elds, and therefore also of E (t), can be calculated from the uctuating mesoscopic hydrodynamic equations for the IHS uid14;15. For large times the decay of the energy in a d{dimensional system behaves as E (t) A ?d=2 , where is the kinetic theory time. This asymptotic prediction is in excellent agreement with the 2{D simulations of Ref. 14;15 over a wide range of densities with an area fraction in the range 0:05 ? 0:4, and a wide range of restitution coecients between 0.6 and 0.95, as shown in Ref. 25. It has been veri ed uptil the largest times realized in the simulations, where large spatial inhomogeneities are present in the density eld. It is in fact remarkable that the predictions for E (t) are still quite good for large inelasticities ( ' 0:6), although deviations between theory and simulations start to become noticeable with increasing densities, as illustrated in Fig. 1b. For strongly inelastic systems ( = 0:4) the simulated value of E (t) at large t is an order of magnitude larger than the theoretical prediction. The prediction, E (t) ' E0=(1 + !0 t)2 ' E0 exp(?2 ), obtained by Ha3 for the short time decay in a homogeneous cooling state (indicated in Fig. 1b as a straight dashed line) breaks down as soon as inhomogeneities in the density25;20;21 start to appear. Here !0 is the Enskog collision frequency in the initial state. It would be very interesting to make a quantitative analysis of the results found for E (t) in the time step driven 3?D simulations of Chen et al.21, as well as those obtained by Herrmann and Muller using event driven MD simulations for inelastic hard disks20 .


The dependence of E (t) and (t) on the actual time t is only known for short times (homogeneous cooling state). Computer simulations in the nonlinear clustering regime have shown25 that (t) becomes again linear in t, with a slope that is smaller than in the initial (elastic) equilibrium state, and seems to depend only on density and inelasticity .

3. Diusive ows and clustering densities.

The goal of this section is to explain the observed patterns in density and ow eld, in terms of the macroscopic equations for the IHS uid on the largest spatial scales. In this so-called `dissipative' regime27, where the wavenumber k , there are no sound modes; all components of the ow eld decay diusively,

@ u? (k; ) ' ?D? k2 u? (k; ) @ ul (k; ) ' ?Dk k2ul (k; ):

(2)

Here u? and ul are respectively Fourier transforms of the transverse components of the ow eld (vorticity) and of the longitudinal one (divergence). The dimensionless p wave number k is measured in units of l0 , where l0 = T=!(T ) is the mean free path. These results can be derived straightforwardly from the linearized hydrodynamic p equation for the free IHS uid27;19;26;28. The transverse diusivity, D? = =nl0 T , is independent of the granular temperature. The longitudinal diusivity Dk is explicitly calculated in Refs. 15;28. The latter depends mainly on the inelasticity, = 1 ? 2 , and on the pressure, and reduces at low densities to Dk = 1= +2(1 ? 1=d)D? . Equations (2) shows that all Fourier components of the ow eld u (k; ) are stable, and remain microscopic, bounded for all times by the initial thermal noise. Next we compare the vortex patterns, found in MD simulations of IHS|which resemble a dense uid of `hard' vortex structures14 |with the patterns found when solving the diusion equations (2) in real space, with random initial conditions. The equations are discretized on a square lattice with periodic boundary conditions. The initial velocity at each site is an independent random vector u(r; 0), drawn from a Gaussian distribution with initial temperature T0 , and diusivities Dk = 20D? , all chosen as in the corresponding MD simulations. If the inelasticity is small (so that Dk D? ), the dynamics selectively suppresses the longitudinal components ul (k; ) as compared to the transverse components u?(k; ). Consequently, the ow eld shows predominantly growing vortices. When the number of IHS particles is small, typically a few thousand16;17;18;26, the ow eld shows a small number of vortices, growing with a diameter ? ( ) p 8D? . As soon as the sizey of the dominant vortex satis es, 2? L, there is interference with its periodic images and the system makes a transition to a `sheared state' at a time of order ? = L2 =32D? . This state, illustrated in gure 7 of Ref. 16 y In

p

Ref. 26 the size of a vortex Lv a 8D? is identi ed as the location of the minimum in G? (r; t) (see also Fig. 2). Calculation yields that a ' 2 within a 10% margin for Dk =D? in the range 2 to 100.


Figure 3: Snapshot of (a) weakly ( = 80) and (b) strongly ( = 160) clustered density elds in IHS simulations with N = 50000; = 0:4; = 0:9. Crossover from the HCS to the nonlinear clustering regime occurs at c ' 68. Every third particle is represented by a dot.

and gure 5 of Ref. 26, exhibits two antiparallel shear layers, which are a consequence of nite size eects resulting from the rather unphysical periodic boundary conditions, and not an interesting `phase' of rapid granular ows. Moreover, it is a strictly linear phenomenon, that also appears in the solution of the diusion equations (2) with Dk D? , and with random initial conditions, after suciently long times. Next, we consider the spatial density uctuations, using the linearized hydrodynamic equations for the Fourier modes. In the dissipative regime (k ) these equations yield 1 ( 0 ?Dk k 2 ) ?D k2 2 e u (3) n(k; ) = ? ikC l (k; 0) + e n n(k; 0) + O(k ): Here C1 is some constant and Dn depends on pressure and inelasticity, but not on transport coecients. It reduces in a d{dimensional system at low densities to Dn ' d= (see Ref. 28). Equation (3) shows the onset of the clustering instability, based on a linear stability analysis. It has been discovered by Goldhirsch and Zanetti8c; , and studied by many authors16;19;14;15;26 using MD and Direct Simulation Monte Carlo methods, as well as in experiments using air tables22. One sees that the density

uctuations increase at an exponential rate, driven by an initial uctuation in the longitudinal ow eld (ul (k; 0) 6= 0). However, even if ul (k; 0) = 0, the density uctuations would grow, due to a nonlinear instability, driven by the viscous dissipation term (dux =dy)2 in the energy balance equations8c;29 . Equation (3) also suggests that it may take a long time before the clustering instability appears, because the rst term in the RHS of (3) has a small coecient of O(k), which must dominate over the other terms.

Noise Reduction and Pattern Formation in Granular Flows 7 0.001

0.0010 Realization 1, 2, 3 Theory (Ref. 19)

Theoretical Simulations 1/V

0.000

G

G

0.0005 0.0000 -0.0005

0

50

100 r

150

200

-0.001

0

25 r

50

Figure 4: Transverse velocity correlation function G? (r; t) a) in an IHS-simulation with = 0:245; = 0:9; = 80, where three realizations are shown (thin lines) together with the theory of Ref. 15 (thick line). The crossover time is c ' 70. b) from the simulation of diusion equations (2), with Dk = 20D? = 0:125, averaged over 100 runs (), compared with the theoretical result of Eq. (A.5) (thick line) and three typical realizations. The dashed line is the nite size correction discussed in the Appendix. The small arrows in b) represent the values of ? and k respectively.

Figure 3 illustrates the clustering instability and the relative dierences in time scales. At = 80, where the ow eld shows a high density of well developed vortices (see gure 1a of Ref .14), the density eld is barely inhomogeneous. At = 160 the density eld shows macroscopically large inhomogeneities/clusters. The linear stability analysis suggests that the spatial uctuations in the temperature eld decay exponentially fast at all wavelengths, and have a small amplitude of O(k=) in the dissipative k-regime.

4. Spatial oscillations in the ow eld correlation functions 4.1. Average over broken symmetry states

As long as the system is homogeneous, the large majority of realized con gurations are close to a typical mean con guration on account of the noise reduction mechanism. However, as soon as the appearance of a vortex or a cluster breaks the translational symmetry of a MD realization, certain properties may wildly uctuate around the predicted theoretical mean over the dierent realizations. One of these properties is the transverse velocity correlation G? (r; t), as shown in Fig. 4a for three dierent realizations (thin lines) containing vortices and clusters. One expects that agreement with the theoretical prediction (thick line) of Refs. 14;15 is recovered when averages are performed over a number of realizations, suciently large to restore the translational symmetries. This is however not feasible in MD simulations, where a single run, required to produce each of the three simulations in Fig. 4a, typically takes 24 CPU hours on a Digital Alpha machine. As explained in section 2, the behavior of the ow eld u(r; t) on the largest

8 Noise Reduction and Pattern Formation in Granular Flows -2

-3

r

-4 -5

-2

Realization 1, 2, 3 Theory (Ref. 19)

0

1 log r

2

log [G||+1/V]

log G||

-2

r

-3 -4 -5

-2

Theoretical Simulations

0

1 log r

2

Figure 5: As in Fig. 4a,b for the longitudinal velocity correlation function Gk (r; t). The average in Fig. 5b is performed over 50 realizations. Dierent realizations are essentially indistinguishable from the average.

space and time scales, is well described by the much simpler diusion Eqs. (2), with random initial values, where the coupling to the unstable density eld can be neglected. We therefore measure G?(r; t) also by simulating the diusion equation (2). The results for three arbitrary initializations, shown in Fig. 4b, also exhibit strong uctuations, similar to those in Fig. 4a. However, performing an ensemble average over 100 initializations yields excellent agreement with the theoretical prediction. This prediction for G? (r; t), calculated in the appendix by solving the diusion equations (2) is qualitatively, but not quantitatively, the same as the one calculated from uctuating hydrodynamics14;15. Both correlation functions only coincide on the largest r{ and t{scales. The reason is that uctuating hydrodynamics takes the additional internal noise of the rapid short wavelength degrees of freedom in the N particle system into account as well. The longitudinal correlation function Gk (r; ) appears to be very insensitive to the presence of vortices, and barely shows any oscillations. The agreement with theory, shown in Fig. 5a,b for MD and diusion equations respectively, is very good except at the largest distances (r > k ). As the diusion length k ' 32 is close to the maximum, 21 L, of the r-range, there occur interference eects of the system with its periodic images, which presumably account for the small deviations between theory and simulations, near the right end of the r-interval.

4.2. Long lived spatial oscillations

Following a time sequence of snapshots of a ow eld con guration, we observe that large IHS systems, specially at higher densities, frequently contain rather longlived local con gurations with densely packed, more or less rotationally symmetric vortices with the short range order of a square lattice. When G? (r; t) is measured in these regular con gurations, it shows long lived spatial oscillations. For instance, the location of the rst and second maximum in


0.5 G||

0.0

G

-0.5

0

10

20

30

40

50

r

Figure 6: Left: Illustration of a purely rotational ow eld, which is a solution of the diusion equations (2). Right: Correlation functions Gk (r; t) and G? (r; t), measured in the con guration in Fig. 6b,. The transverse correlation function shows oscillations with the same periodicity as the vortices, while the parallel one is almost insensitive to the presence of vortices.

G? (r; t) of gure 2c in Ref. 14 is about the same for = 40; 60 and 80. This can be understood as follows. Let us represent a regular vortex eld on a square lattice as u(x; y; t) = A(t)(? sin(ay); sin(ax)) for several n{values. Substitution of this form into the diusion equation for the vorticity r u shows that it is an eigenfunction with a time dependent amplitude A(t) = exp(?D? a2 t). This suggests that spatial structures, locally resembling eigenstates of the vorticity equation, are rather long-lived, and do not decay to the mean values predicted by the theory within the duration of the simulation. In support of this explanation we have prepared divergence free initial ow elds u(x; y; 0) = f? sin(ny=L); sin(nx=L)g which are eigenstates of the vortex diusion equation in real space with time dependent amplitudes, as illustrated in Fig. 6a. The transverse velocity correlation G?(r; ), measured on this lattice, exhibits a regular stable spatial oscillation with a period R = 2L=n, equal to the size of a vortex{antivortex pair. The same relation was observed in the IHS simulation of Ref. 14. In summary: the IHS dynamics at the largest spatial and temporal scales are controlled by two diusion equations. They are a direct re ection of the noise reduction mechanism in the IHS dynamics, which makes the particles move more and more parallel. Many of the observed phenomena in rapid granular ow are caused by this dynamics, the most interesting of which is the time decay of the total energy E (t)25.

Appendix

We calculate the structure factors Sa and spatial correlations Ga of the ow


elds from the solutions of the macroscopic diusion equations with random initial conditions. We start with the structure factors 2 (A:1) Sa (k; ) hjhjuua ((kk;; 0))jj2ii = e?2Da k2 ; a where a denotes the longitudinal (a =k) or transverse (a =?) components of the

ow eld. To keep the calculations simple we restrict ourselves to two dimensions. The average h i refers to an average over random initial con gurations, drawn from a Gaussian distribution. The second equality is obtained by solving the diffusion equations (2). The relevant set of wave numbers is obtained by imposing periodic boundary conditions, yielding k = 2n =L with n = 1; 2; . The fully isotropic second rank tensor eld S (k; ), where ; = fx; yg denote Cartesian components, is then obtained as

S (k; ) = bkbk e?2Dk k2 + ( ? bkbk )e?2D? k2 :

(A:2)

The spatial correlation functions follow by Fourier inversion, Z G (r; ) = (2dk)2 eikr S (k; ) = rb rb Gk (r; ) + ( ? rb rb )G? (r; ); (A:3) where G (r; ) is again an isotropic tensor eld that can be split into two scalar components Gk(r; ) and G? (r; ). We consider rst the transverse one, i.e. the components of G; (r; ), perpendicular to r, Z dk ikr n 2 ?2D?k2 o (kb br) e + (1 ? (kb br)2 )e?2Dk k2 : (A:4) G? (r; ) = (2)2 e After some calculation, the resulting correlation function becomes ?r2 =8D? ?r2 =8Dk ? e?r2 =8D? G?(r; ) = e 8D ? e : (A:5) 2r2 ? The result for Gk (r; ) is obtained by interchanging D? and Dk . Similar integrals have been calculated in Refs. 14;15. The correlation function can be cast into a scaling18 form,

8D? G? (r; ) = F? (x; h) = e?x + (e?x ? e?x=h )=2x; with

(A:6)

x = r2 =8D? ; h = Dk =D? ; (A:7) and an analogous formula for Gk (r; ). In the limit where Dk goes to in nity, we recover the analog of the incompressible theory of Ref. 14 with ul (k; ) = 0. If the two diusivities are equal, the last term vanishes. However, this is not the case in typical IHS simulations, where h = Dk =D? is large in general, specially for small inelasticities. Consequently, the exponentials in (A.5) behave very dierently. For


p r in the range, ? r k , where a = 8Da with a = fk; ?g are the diusion lengths, there exists an intermediate asymptotic regime where G?(r; ) behaves essentially as (?1=2r2). Similarly, the longitudinal correlation function behaves in this range as Gk (r; ) 1=(2r2) + 1=(8D? ). Finally, we derive a nite size correction30 to (A.3), as the statistical accuracy is very high when solving the diusion equation with random initial conditions. Instead of (A.3) we need to consider, Z X (A:8) G (r; ) = V1 eikrS (k; ) ' (2dk)2 eikr S (k; ) ? V : k

This can be understood as follows. The discrete Fourier sum in (A.8) does not include the k = 0 term, as S (k = 0; ) = 0 because the total momentum, P = u(k = 0; ) = 0, is conserved and the system is initially prepared at P = 0. In the integral on the second line, the wavenumber k = 0 is included, and contributes S (k ! 0; ) = according to (A.2). Consequently, the (k = 0) contribution, 1=V , needs to be subtracted30 on the RHS of (A.5). Acknowledgements It is a pleasure to thank D. Frenkel for an invaluable comment and helpful correspondence. We also thank T.P.C. van Noije, M. Hagen and I. Pagonabarraga for stimulating discussions, M. Hagen for sending his plots of the ow eld and J.A.G. Orza for his help in performing the numerical simulations. The authors also acknowledge nancial support from the Oces of International Relations of Universidad Complutense and Universiteit Utrecht. One of us (R.B.) acknowledges support to DGICYT (Spain) number PB97-0076.

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