numerical study of granular turbulence and the appearance of the k−53 energy spectrum without flow

El S, tNIER Physica D 80 (1995) 61-71

PHYSICA

Numerical study of granular turbulence and the appearance of the k -5/3 energy spectrum without flow

Y-h. Taguchi Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan

Received 28 April 1994; revised 12 July 1994; accepted 12 July 1994 Communicated by F.H. Busse

Abstract

A vibrated bed of powder, modeled by a vessel filled with monodisperse glass beads and shaken by a loud speaker, is investigated numerically with the distinct element method of molecular dynamics. When the bed is shaken vigourously, the displacement vectors of powder particles have a power spectrum that depends upon the wavenumber k a s k -5/3. This dependence originates in the balance of the injected and dissipative energy, analogous to Kolmogorov's proposal to explain the k -5/3 energy spectrum observed in the fluid turbulence. Furthermore, the same spectrum still appears even without flow in the powder. Thus Kolmogorov's argument appears to be more universal than believed before.

I. Introduction

Over the past few decades, non-linear physics, while previously virtually untouched due to the lack of con-

venient methods to study it, has become one of the

central topics in modern physics.

This is because the phenomenological approach sta-

tistical physics provides allows us to investigate these non-linear problems [ 1,2]. The basic philosophy un-

derlying this approach is: 'Some phenomena are independent of the details of the specific systems; hence,

the phenomenological approach will be valid.'

For example, the amplitude equation and the phase dynamics approach [ 1 ] can describe many phenomena ranging from fluid dynamics to chemical reac- tions. However, these approaches are valid only for the weak non-linear regime where only a few degrees of freedom exist.

On the other hand, in systems where many degrees

of freedom exist and couple to each other - the subjects

of most interest to non-linear physicists - can be at- tacked with phenomenological numerical approaches (e.g. the coupled map lattice [2] ). Although they ex-

hibit many interesting phenomena, they lack direct

connections to real systems. At the moment, there is

no general tool to study real non-linear systems with

many degrees of freedom. Instead of trying to find general tools, one can study

phenomena that have many degrees of freedom and appear in some real system with an easily treated model.

This is the starting point for constructing a general

theory. An example of a phenomenon that has many de-

grees of freedom is the k -5/3 energy spectrum in the fluid turbulence, first proposed by Kolmogorov [ 3 ]. In his theory, he predicted that the energy spectrum of the fluid turbulence depends upon the wave number k as k -5/3 in the high enough wave number regime,

0167-2789/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2789 (94) 00167-7

62 Y-h. Taguchi / Physica

which was confirmed later experimentally and numerically. His theory is valid for almost all kinds of fluid

turbulence: the experiments in the laboratories, the at- mospheric flows (e.g. wind), and the flows of the tide.

His success strongly supports the belief that the details are not important. However, his theory is too general to make clear the mechanism from the dynamical points of view; it does not have any direct relation-

ship to the Navier-Stokes equation which is the basic equation of fluid motion.

No one has succeeded in solidifying the foundation of his theory, since it is hard to obtain enough informa- tion about the local structures of the fluid turbulence. Experimentally, the simultaneous measurement of the velocity over the whole region is impossible. Only the

velocity at a specific point is observed as a time se- quence, which is regarded as the spatial structure of the velocity field by assuming Taylor's unjustified frozen turbulence hypothesis [ 3 ]. Numerically, although de- tailed measurements of the velocity field are easily obtained, the limited computational resources make the production of fully developed fluid turbulence difficult.

Even phenomenological approaches are hardly ever applied to the problem. The amplitude equation or the phase dynamics approach are improper to deal with the k -5/3 energy spectrum where many degrees of freedom are essential. The phenomenological numerical approach has succeeded in reproducing the k -5/3

power spectrum [ 4,5 ], but could not explain the mechanism well because they lack the direct relationship to the experiment. Hence, in order to fully understand Kolmogorov's theory, I hope to find a suitable substitution that obeys Kolmogorov's law and can be treated easily both numerically and experimentally.

In this paper, the k -5/3 energy spectrum is investi-

gated with such a substitution: granular flow. Granular systems, which many physicists have recently become interested in [6-10], are systems whose number of degrees of freedom is limited enough to be integrated by the direct numerical simulations. For example, the number of degrees of freedom, even in the real experiments, is 10 6, which should be compared with Avo- gadro's number (,-~ 10 23) that represents the number of degrees freedom of typical materials like fluids, liq-

D 80 (1995) 61-71

uids, and solids. The numerical simulations of the system with one hundred degrees of freedom reproduces the non-trivial properties originating from the strong nonlinearity, as can be seen in the following. There-

fore, we can expect experimental correspondence between the numerical model and real experiments.

The organization of this paper is as follows. In Sec- tion 2, the experimental and numerical treatment of

a vibrated bed of powder is discussed. In Section 3, the results obtained numerically for a vibrated bed of powder are summarized. In Section 4, I introduce a

new model that has the experimental correspondence in order to investigate the origin of the k -5/3 law. A discussion of the results and conclusions will be presented in Section 5.

2. The experimental and numerical treatment of a vibrated bed of powder

2.1. Experimental investigation of a vibrated bed of powder

The experimental setup for a vibrated bed of powder is as follows [ 11,12]: A flat vessel with a horizontal dimension of about 10 cm is filled with granular matter: typically, monodisperse glass beads less than 1 mm in diameter. The bed is shaken vertically by an acceleration comparable to gravity. A speaker is usually used for this purpose, the frequency of the vibration is from about I0 to 100 Hz, and the harmonic dependence upon time t is assumed to be sinusoidal; that is, the vertical displacement of the bed obeys b cos toot. Although most experiments are performed with three dimensional containers, a few experiments were done in two dimensions.

When the acceleration amplitude, F = bto02, exceeds some critical value Fc slightly larger than the acceleration of gravity g, the bed exhibits several non-trivial dynamical instabilities: heaping, convection [11,12], capillarity [13], surface fluidization [26,14,15], Brazil nut (size) segregation [16-20], standing waves [21-24], and so on.

Y-h. Taguchi / Physica D 80 (1995) 61-71 63

2.2. Numerical method - distinct element method

The distinct element method (DEM), or the parti-

cle element method, was named in analogy to the fi-

nite element method. In contrast to the finite element

method, each element in DEM can move freely, but like the finite element method it represents a macro-

scopic property of the material. An individual granu-

lar particle is an element that obeys the visco-elastic

equation,

N

j=l

xiXi-- XJ l ] - d I + ' Q ( V i -- V j ) - g, (1)

where N is the total number of granular particles, xi

is the position vector of the ith particle, g is the ac-

celeration of gravity, and v is the velocity. Each par-

ticle has a diameter d. The step function O(x) causes

particles to interact only when they contact with each

other, and represents the discrete nature of the gran-

ular matter, k0 and 7/are the elastic constant and the viscosity coefficient, respectively. These visco-elastic

material constants give coefficient of restitution, e, and

collision time tcot when two particles collide head-

on; e = exp(-r /Tr/ to) and tcot = 7r/w where to = (2k0 - r / 2 ) 1/2. In this sense, the visco-elasticity is not

a real material property, but a phenomenological one.

In the limit of tcot -~ O, the DEM becomes the more

realistic hard-sphere model, which cannot be used to

simulate a vibrated bed because the hard sphere model

assumes instantaneous collisions, while the particles

are continuously in contact with each other in the vibrated bed of powder. The DEM can be also consid-

ered to be a simple molecular dynamics simulation

using the above interactions. Cundall and Struck [25] first proposed DEM in

order to investigate geological properties. DEM has

undergone several revisions. However, the simplified version used here has been shown to reproduce convection [26,27] - the most non-trivial phenomenon in

a vibrated bed of powder. The above equations are integrated by the Euler

scheme, in which the time increment At changes at

~,,,.So d.

r<r r>r r>>g Fig. 1. The process of fluidization. When/" < lc, the bed consists of the solid region. As 1" exceeds l'c, a fluid region appears, but the solid region still remains. For larger F, the bed is fully fluidized.

each step so that the displacement of each particle

does not exceeds some fixed value ot during At (see

appendix A).

In order to save CPU time, I simulate a vibrated bed

of powder in which the granular particles move within

a two dimensional space with the bottom oscillating as a function of time t, bcos toot. In this representation, the acceleration amplitude F is bm 2. F is the strength

of the external driving force. When the bottom collides with a granular particle, it reflects the particle with

elastic constant k0 and no dissipation.

3. Numerical results in the vibrated bed of powder

As reported in previous publications [26,28,29], convection and surface fluidization are observed when

F exceeds the acceleration of gravity. The fluidized

region where the powder flows has a finite depth which

increases as F increases and coexists with the solid

region where the powder does not flow (see Fig. 1).

In order to understand the dynamics of the vibrated

bed of powder we must consider the granular flow in detail.

The granular flow is defined as the relative displace-

ment of respective particles. Thus, motions like trans-

lation and expansion, which do not give rise to posi-

tion exchanges among particles, are not regarded as

flow in this subsection. The configuration used to measure the flow is shown

in Fig. 2, where the two dimensional space is horizontally periodic and has an oscillating bottom. On top of the granular layer lies a lid which remains horizontal and obeys the following equation of motion:

64 Y-h. Taguchi / Physica

L h , , Lid

periodic t periodic

vibrate ÷ A h(t) Fig. 2. Configuration used in this section. The bed is horizontally periodic with a period of Lh and the lid floats above the bed remaining horizontal at all times.

d2 h .---.N M d t 2 - ~ O ( y i - h ) k o ( h - y i ) , (2)

i=l

where M is the mass of the lid, h and Yi are the vertical

coordinates (heights) o f the lid and the ith particle,

respectively. The particles collide with the lid elasti-

cally, with elastic constant k0, and are not allowed to

pass through the lid. The lid suppresses surface dif-

fusion which makes the measurement of the granular

flow in the bed difficult because the surface diffusion

causes a much larger displacement than the granular

flow does.

The height of the lid is also used to determine when

to record the positions of particles. These recorded

positions are then used to calculate the granular flow.

While undergoing vibration, the bed expands and con-

tracts repeatedly, constantly changing its volume. The

contribution of these motions to the displacements are

much larger than those of the granular flow, and mush

be removed. In order to do this, I define the measuring

times t,, to be the times when the spacing Ah( t ) (see

Fig. 2) between the lid and the bottom has a definite

value, h0. Although it becomes h0 twice during one

period, I consider only that for the contraction. This

definition enables us to measure the displacement un-

der the condition of constant volume and to exclude the contribution of volume changes to the displace-

ment o f each particle. Thus the definitions of the dis-

placement vectors are

Ax~ ") :-- x i ( t,+l ) - x i ( tn ). (3)

Typical displacement vectors in a fully fluidized bed

for F = 2.19 (too = 27r/6, b = 2.0, N = 1024, Lh =

128, e = 0.8,tcot = 0 . l , d = 2 .0 ,g = 1.0, M =

D 80 (1995) 61-71

100, h0 = 34.01 ) are shown in Fig. 3. The spatial

structure of the velocity field looks like a turbulent

fluid whose power spectrum has a k -5/3 dependence

upon the wavenumber k.

To compare the powder displacement field with the

spatial structure of the velocity field in fluid turbu-

lence, a Fourier power spectrum is calculated from the

displacement vectors. First, I divide the whole two di-

mensional space into d (horizontal) × x /3d/2 (verti-

cal) cells. Each cell moves with the bottom and is iden-

tified by the coordinates (X, Y) where X, Y are positive

integers. The cell (X, Y) covers the area d ( X - 1/2) <

x < d ( X + l / 2 ) , d ( Y - 1/2) < y < d ( Y + 1/2) ,

where x and y are horizontal and vertical coordinates

that have their origin on the bottom.

The displacement vector for each cell is given by

Ax {n) (n)

x,Y = ,/ ~ a x i , (4) iC(X,Y)

where summation runs over only the particles in cell

(X, Y). The Fourier power spectrum for Ax~x,~r ~ on each

layer is then

S ( K , Y ) - ~ x, r e x p ( - j 2 ~ ' X K / L ) , (5)

where K and L are integers, and L is Lh/d. Ax (n) is the x,g horizontal component of A ~ (~) ~'~X,r" J is a pure imaginary number. The average (.. ")n runs over 30 periods.

Fig. 4 show the dependence of the power spectrum

upon the wavenumber K ( 1 ~ 32) and theheight from

the bottom, Y. The power spectrum near the surface

deviates from the straight line and has a flat spectrum

(white noise) in the high wavenumber region, 2 but

for small Y, the power spectrum has the correct power

dependence. The slope of the log-log plot is very close to - 5 / 3 which coincides with the slope predicted by

1 For a dense packing the particles form a triangular lattice which has a width of Lh/d = 64 and a height of N/(Lhd) = 16. Thus the height of the bed in the dense packing is ½x/3d{(N/Lhd) - l } 26.0. 2 Same behavior was recently observed in fluid turbulence vorex line near surface generates (Prof. T. Sarpkaya et al., presenta- tion in symposium: Free-surface turbulence, ASME FED summer meeting, 1994).


Fig. 3. Typical displacement vectors for a fully fluidized bed ( F = 2.19).

Kolmogorov's scaling theory. Therefore, we can con- clude that the spatial structure in the vibrated bed is

similar to the structure in fluid turbulence.

The origin of this turbulent motion in the fluidized

region should exist in the motion of solid region, be-

cause as shown in Fig. 1 the solid region becomes fluidized gradually as F increases. In order to compare

the motion in the solid region with that in the fluidized

region, the power spectrum in the solid region when

F = 1.10 (b = 1.0, h0 = 30.0, and the remaining pa-

rameters are identical to those given above), averaged

over 64 periods, is shown in Fig. 5. As seen from the

typical displacement vectors of the solid bed, (Fig. 6), almost all of the bed is solid - there are no posi-

tion exchanges between particles, but the global struc-

ture of the power spectrum is qualitatively similar; the

power spectrum in the lower layers has a K -5/3 power

law dependence, and white noise appears in the high wavenumber region for the upper granular layers.

In the following discussions, I consider this k -5/3 spectrum without flow to be the origin of the power

law behavior seen in the fluidized region and explain how the k -5/3 spectrum appears without flow.

4. The k -5/3 power spectrum without flow

In order to understand the mechanism of the k -5/3 power spectrum without flow I propose a model in

which the powder cannot flow. The elements (particles) form a triangular lattice 3 as shown in Fig. 2 and each element interacts with the six neighbors visco-

3 Hence no granular flow is allowed at all; i.e. there is no position exchange between particles.

elastically as described in Eq. ( 1 ) but without gravity

g. The bottom oscillates as before, but the lid does not obey the equation of motion given by Eq. (2). Instead,

the lid oscillates as a function of time in the same way

as the bottom does. That is, when the vertical coordi-

nate of the bottom oscillates as bcos w0t the height of

the lid is h = h0 + b0 cos ~o0t. Thus, h0 is the average distance between the bottom and the lid. The granular

particles lie on a triangular lattice between the bottom

and the lid and move obeying Eq. ( 1 ) with g = 0.

This numerical configuration corresponds to a hor-

izontally vibrated bed in real experiments. In these

experiments, one prepares a flat cell filled with glass beads and makes the two side walls oscillate harmon-

ically. Hence, the following results are expected to be

reproduced by the experiments.

First, both the lid and the bottom oscillate with the

same harmonic form, i.e. b = b0 = 1.0, g = 0.0. h0 is taken to be equivalent to the width of the triangular lattice having N/L rows, v/-3/2 x [ ( N / L ) - 1 ], and

thus the granular particles are packed without space

in between them. The remaining parameters are the

same as used in the previous section. Figs. 7 show the time development of the energy

spectrum 4

L

E(K,t) -- ( Z A v x ( t ; i , j ) i=1

2 \

exp(-j27riK/L) ~ (6) ×

j, initial ' /

4 The energy spectrum means not the spectrum of the energy but that of the velocity. Since it has the dimensionality of energy, we call it the energy spectrum.

66 Y-h. Taguchi / Physica D 80 (1995) 61-71

(a) log S(K,Y) 100000 ooooooooo

,o

log ~ (C) log S(K,Y) 100000 ~ ~ , \ 10000

log K ' ' Y

(b) log S(K,Y) 00000

10000 1000 100

log K Y

(d) log S(K,Y) 100000 1 0 0 0 0 ~ 1000 IO0 10

log K ~ " " Y

Fig. 4. The dependence of the power spectrum S(K, Y) of the fully fluidized bed upon the wave number K and the height from the bottom Y ( F = 2.19). (a) Y = 1 ~ 5, (b ) Y = 6 ,--, 10, (c) Y = 11 ,-., 15, (d) Y = 16 ,-~ 20. The straight lines indicate a K -5/3 dependence. For large Y, i.e. near the surface, the spectrum becomes fiat in the high wavenumber region.

(a) log S(K,Y) (b) ~ log S(K,Y) 100 100

10 10 1 1

01 0.1 0.01 0.01

log K Y log K Y

(C) log S(K,Y) e,,,. ' ~ . oo , \ \

o

log K Y

(d) log S(K,Y) 1000 <~.. lOO %..

0.1 g N @ ~ ~

Io Fig. 5. The dependence of the power spectrum S(K, Y) upon the wave number K and the height from the bottom Y for the almost solid bed ( F = 1.10). (a) Y = 1 ~ 5, (b) Y = 6 ~, 10, (c) Y --- 11 ~ 15, (d) Y = 16,--, 18. For details, see Fig. 4.


Fig. 6. Typical displacement vectors of an almost solid bed. ( F = 1.10).

where vx ( t; i, j ) is the x component ( the direction per-

pendicular to the direction of the vibration) of the ve-

locity at the site ( i , j ) at time t. The average is taken

over the layers (y direction) and the ten initial real-

izations of the velocity with the white noise energy

spectrum. The reader should note that this is the true

energy spectrum, not the power spectrum of the dis-

placement vector as in the previous section, and so we can compare it with Kolmogorov's argument. At the

initial time t = 1.5, each particle has a small random

velocity and thus the system has a flat energy spectrum

(white noise). As time proceeds, the energy spectrum

starts to incline and comes to have power dependence

upon the wavenumber k. Furthermore, its slope is very

close to - 5 / 3 . However, in the intermediate stage, a

flat spectrum appears in the higher wavenumber region and gradually grows into the lower wavenumber

region. Finally in the late stage, the energy spectrum has become entirely flat.

This process can be understood as follows: First, the energy starts to dissipate in the high wavenumber

region where the energy dissipation rate takes on a

large value. It enables the system to reach the station-

ary state described by the Kolmogorov theory. How-

ever, the further injection of energy dominates the dis-

sipation and the system reaches thermal equilibrium:

each mode has the same amount of the kinetic energy. This is also seen in the numerical simulation of the

Navier-Stokes turbulence when the dissipation is not large enough. Therefore, the dynamics in a vibrated

bed of powder is essentially equivalent to that of fluid turbulence.

However, the k -5/3 spectrum appears only tem-

porarily, in contrast to the permanent appearance in the previous section. The dissipation in the vertically

vibrated bed may be larger because the bed is com-

pressed between the lid and the bottom. To take this

effect into account, the value of b is taken to be 0

while bo remains unchanged so that the volume of the lattice changes. Starting from the initial white noise, the power spectrum aquires a k -5/3 dependence and

then returns to a fiat one. However, this time the power spectrum recovers k -5/3 dependence from the white

noise and this process is repeated once in each period of the oscillation (Fig. 8). Here the energy spectrum

is averaged over ten periods to show the dependence upon the phase of cycle. Thus k -5/3 power spectrum

appears repeatedly if the effect of the compression is

considered.

5. Discussion and summary

In the previous section, the powder bed exhibits the k -5/3 energy spectrum. The results also suggest that

the dissipation plays the important role. The behav-

ior of the energy spectrum is equivalent to that in the Navier-Stokes numerical simulation when the dissi-

pation is not large enough. However, there is no flow at all and one cannot expect that the powder obeys the Navier-Stokes equation. So, how is this relation to the

Kolmogorov theory proposed for fluid flow? We need to reconsider Kolmogorov's argument. The

essential assumption in Kolmogorov's theory is that the balance between energy input and energy dissipation governs the dynamics of system. This means that the total amount of energy in the system is as large as the product of the energy dissipation rate, or the energy input rate, and the characteristic time scale. Fig. 9 shows the total energy, the energy input rate, and the


1 0 ! . . • . . . . . . . , . . . . . . .

(a) " ' . - . . \ " ,

1 ~" . . . . . . . , . . . . ~ ",,

E" "-'.-'.,'"',

~ - . . ~ -~ . • , . ~ , { '~,~ %• . . . t ~

0.1 t=2.0 . . . . . . ,..,~,.,. t=3.0 . . . . . . "'., ~'" t=4.0 .......... ' .

-5 /3 l a w . . . . .

0.01 . . . . . . . . ' . . . . . . . 10 K

10 ~ . • . . . . . . . I ' '

(b) ",.

~ - ..... ~.,,..,v,

tu~0.1 t=6.0t=5"0 _ _ _ ~ " ~ : ~ : " : " ' "

t=7 .0 . . . . . . . , t=8 .0 ..........

-5 /3 l a w . . . . .

0.01

10 (c)

1

0.1

. . . . . , i l l i ~ i i , . ,

10 100 K

"x

"x " x .

x

t= 9.0 - - ' . , t= lO.O . . . . . ' - , t=11 .0 . . . . . ' - , t=12 .0 .......... '

-5/3 l a w . . . . .

0.01 . . . . . . . . ~ ' ' . . . . 10 100 K

Fig. 7. The time development of the energy spectrum, E(K). (a) The initial state(t = 1.5) and the early stage. (b) The intermediate stage where the high wavenumber components start to flatten out. (c) The late stage. The energy spectrum shows the system has reached thermal equilibrium.

10

energy dissipation rate in the fluidized region investigated in Section 2 as a function of time t. The total energy consists of both the elastic energy between the particles, the bottom, and the lid, and the kinetic and gravitational energy of particles and the lid. The dissipation energy comes from the viscosity between

1

0.1

0.01 100

" . • . . . . . . . I ' •

4pi /3 ........... "-,,'" ,, 5p i /3 . . . . . '"-":.

2p i . . . . . -5 /3 l aw . . . . . .

, I , . . . . . .

10 100 K

Fig. 8. The dependence of the energy spectrum E(K) upon the phase of the cycle, (o0t = n~/6(n = 1,2 . . . . . 6). The k -S/3 dependence is shown for comparison.

3 5 0 0 0 , , , , ,

30000 / ................ ~.. / .......................... / .............

2500o / ........................ / ......... / ........................ '": total energy ...........

20000 . . . . . . . . . . . . . . . . . -gf~ifi ~r ~I&le- --------: . . . . . . . . 15000 energy input rate . . . . .

energy diss. rate - -

10000 ~

5 0 0 0

0 . . . .

-5000 i i i i i 150 160 170 180 190 200 210

t ime

Fig. 9. The total energy, energy input rate, and energy dissipation rate as a function of time t (See appendix B).

particles. The work done by the vibrating bottom is taken to be the energy input. The product between the energy input rate ( ~ 2 × 103) and the period of the vibration (= 6) is about 104. On the other hand, in Fig. 9, the actual total energy is about 104 , when we eliminate the gravitational potential of the grand state(~ 2 x 104). Thus, the total energy included in the bed is almost equal to the energy injected each period. This means that the balance between the energy input and energy dissipation dominates the dynamics of the system. So far, the basic assumption necessary for Kolmogorov's theory is satisfied.

The energy transport between modes with different wavenumbers is also assumed to be caused by the nonlinearity in the system, introduced by the step function O(x) in Eq. (1). In fact, Kolmogorov never used the existence of flow to derive his scaling form since


it is apparent: turbulence must flow. In the case of a

vibrated powder, the steady state which he proposed in his scaling theory can appear without flow because

the vibrated bed of powder satisfies the basic require- ments without flow.

Another difference from the fluid is the dimensionality. In fluids, the k -5/3 energy spectrum can appear

only in three dimensions, not in two dimensions as

with the powder. However, again, Kolmogorov did

not use the dimensionality explicitly. The main differ-

ence between the two and three dimensional cases is whether or not the enstrophy5 is conserved. The en-

strophy is impossible to define in a vibrated bed of

powder because in a discrete material like a powder,

the spatial derivative (which is necessary to calculate

the enstrophy) does not exist due to the lack of con-

tinuity. Hence, for the powder, the enstrophy, which does not exist, cannot prevent the two dimensional

system from obeying Kolmogorov's argument. Thus,

in a vibrated bed of powder, the dimensionality is not

important. Furthermore, for the fluid, even in two dimensions, the k -5/3 energy spectrum can appear when

large scale motion is considered [30]. Although the

direction of the energy cascade is inverted, this result also indicates that the k -5/3 energy spectrum can ap-

pear independent of the dimensionality when the en-

strophy does not prevent the system from following

the Kolmogorov law. Assuming the k -5/3 power spectrum comes from

Kolmogorov's theory, we can explain the power spec-

trum seen in Section 2. For lower layers of the bed, the

compression is strong enough and the k -5/3 spectrum

can be clearly observed. However, the upper layers

are not compressed enough to dissipate the injected energy which causes the energy to saturate the high wave number region and causes a flat spectrum. Fig. 10

shows a schematic of the energy flow in a vibrated bed taken from the previous discussions. In contrast

to fluid turbulence, the turbulence does not develop as the velocity increases (i.e. as the higher region is considered). This difference comes from the finiteness of the degrees of freedom in a powder bed. In the fluid

5 The enstrophy is defined as (w(x , t )2 ) /2 , where w is the vor- ticity, rot v.

Energy 1 Surface

......... Energy Flow

_go Y Bottom[.~" ~ ....... ~ E n e r g y D,sslpatlon

log K Fig. 10. A schematic of the energy flow in a vibrated bed.

turbulence, the number of degrees of freedom is ac-

tually infinite and the turbulent region can grow as the energy input increases. However, the powder bed

has a limited number of degrees of freedom: about as

many as the number of particles in the bed. Hence the

turbulence does not grow, but instead the input energy

accumulates in the high wavenumber region. Similar

behavior can be seen in numerical investigations of the Navier-Stokes equation when the simulation grid

is not fine enough. Here, the finiteness of the number

of degrees of freedom is not a numerical effect but a

real effect. So far, the behavior of the turbulence dif-

fers from the fluid in this sense. Upon review of this paper, one of the referees

pointed out a new paper [31], published after this

manuscript was submitted, as well as a preprint [ 32] also pointed out by the referee, in which the power

spectrum in powder flow was found to have an expo-

nent of - 4 / 3 . This value is believed to be related to

the Kolmogorov-Obukhov scaling under dissipation

and strong fluctuations [ 33 ]. The difference from the

present results can be understood as follows: First, the

observation of this power law is not in the vibrated

bed but in the vertical powder flow. In the vertical powder flow, the dissipation occurs not only when the particles collide with other particles but also when

the particles collide with the side walls. The collision with wall is very important for the vertical flow because, without this effect, the vertical flow continues

to accelerate and there is no steady state at all. For more than two relevant dimensional parameters, the exponents can take values different from the simple Kolmogorov arguments [ 34], while both the fluid turbulence and the vibrated bed of powder have only one parameter to characterize the strength of dissipation.


Second, all observations presented in the above mentioned papers are time sequences. To relate these data to the spatial ones in the present paper, one must assume Taylor's frozen turbulence hypothesis. For the vertical flow of powder, it is unclear whether one can apply this hypothesis. For example, the flow pat- terns in Ref. [ 31 ] change rapidly compared with the

time scale necessary for the powder to flow down the vertical tube. Thus, the temporal exponents can differ from the spatial exponents.

In summary, in a vibrated bed of powder the power spectrum of the displacement vector exhibits a k -5/3

power-law behavior. This behavior comes from the local motions in the solid region which can be explained using Kolmogorov's theory. I found that the steady state proposed by Kolmogorov's theory can appear even without flow; thus his theory is more universal than believed before. Experiments should confirm

these results. This result will make the effort to understand the

k -5/3 mechanism easier, since the powder system does not flow. We can use a lattice representation, and the dimensionality is two, thus requiring smaller computational resources than the three dimensional system does. These efforts also may give rise to understanding of the general features of the strongly nonlinear systems where many degrees of freedom exist and may give us a general method to study such systems.

Appendix A. Numerical method to integrate the equation of motion

The method employed to integrate the equation of the motion Eq. ( 1 ) is the Euler scheme,

x i ( t + At) = x i ( t ) q- v i ( t ) A t d- ½ai ( t ) (A t ) 2 , (A.1)

Vi(t d- At) = v i ( t ) + a i ( t ) A t , (A.2)

where ai is the acceleration of ith particle. Eq. (A.I) enables us to relate the time increment At to Axi( t ) :--

[ x i ( t + At) - x i ( t ) [ and Ayi ( t ) ==-1 y i ( t + At) - Yi (t) I, where xi and Yi are the two components of xi, respectively. Consequently

Atxi = V/v2i q- 2axiA xi -- vxi , (A.3) axi

~ / 2 .~_ 2ayiA Yi -- Uyi Uyi Atyi = , (A.4)

ayi

where Vxi and Vyi are the x and y components of vi,

and axi and ayi are the x and y components of ai. Using these equations, the representation of At with

Axi, At takes the value of minxl Atxi with the fixed A xi and A y i. In the present simulation, A xi = A y i =

a = 0.005d.

Appendix B. The calculation of the energy and dissipation

Acknowledgements

The author would like to thank Prof. H.J. Her- rmann for sending him Ref. [ 31 ] before publication. Dr. D. Lohse is acknowledged for the useful discussions. The author also acknowledges B.J. Buchalter for critical reading of the manuscript. This study is supported by Hosokawa-Powder Technology Founda- tion, Foundation for Promotion of Industrial Science, and Grant-in-Aid for Encouragement of Young Sci- entists (05740252) from the Ministry of Education, Science, and Culture, Japan. All of the Linux devel- opers are acknowledged here, too. Without the Linux project, this paper would never have been published.

In this appendix, the definition of several quantities appearing in Fig. 9 are explained.

The total energy consists of the kinetic energy, the elastic energy, and the gravitational potential energy.

The kinetic energy Ekineti c is defined as

N 1 2 Ekinetic = ~ MI)li d -Jr" Z 1 2 ~Vi , (B.1)

i=l

where Vtid is the velocity of the lid. The gravitational potential energy Ug can be expressed easily, too,

N

Ug = Mgh + Z g y i . (B.2) i=1


Here one should r emember h is the height ( the vertical

coord ina te ) o f the lid. The elast ic energy Uet consists

o f those among the particles, the bo t tom plate, and the

lid,

Uel = ~ k 0 O ( d - I x i - x j I) ( d - I x i - x j I) 2 ~, i,j

N

+ Z 0 ( b c o s toot - Yi) ( b c o s toot - y i ) 2 i=l

+ ~ _ ~ O ( y i - h ) ( y i - h ) 2 • (B .3)

i=l

Thus the total energy Etot can be expressed as Etot =

Ekinetic Jr- Ug + Uel.

The energy input rate Einpu t is the amount o f work

done by the bo t tom per unit t ime,

N

E(t) . _ 1 Z [ k o { O ( b c o s ~ o t _ y i ( t ) ) inpu, - A-"tt

i=l

X ( b c o s a~ot - y i ( t ) ) }

x b { c o s w o ( t + At) - c o s w t } ] . (B .4)

Typically, At is taken to be 1 x 10 -5. The energy dissi-

pat ion Edi.~ due to the viscosi ty be tween part icles i and

j is "Q f I (12i -- 13j) • d ( x i - x j ) I when two particles contact wi th each other. Us ing d ( x i - x j ) / d t = ( vi -

v j ) , we get Edis = rl f ( v i -- v j ) 2 d t . In the present dis-

cret izat ion subst i tut ing v i ( t + At) = v i ( t ) + a i ( t ) A t ,

,7 O ( d - I x , - xj I) Edi, = --~ i,j

x {AvZi jA t+ Au 0 • a i j (A t ) 2 + la~(At)3}, (B .5)

where A v i j = vi - v j and Aai j = a i -- a j .

These above vari abl es are related by, Etot ( t + A t ) =

E,ot( t ) + {Einput( t ) - Edis( t ) } A t + O ( A t 2 ) .

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numerical study of granular turbulence and the appearance of the k−53 energy spectrum without flow

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