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Powder turbulence: direct onset of turbulent flow
Y.-H. Taguchi
To cite this version:
Y.-H. Taguchi. Powder turbulence: direct onset of turbulent flow. Journal de Physique II, EDP Sciences, 1992, 2 (12), pp.2103-2114. �10.1051/jp2:1992255�. �jpa-00247793�
J. Phys. II France 2 (1992) 2103-2114 DECEMBER 1992, PAGE 2103
Classification Physics Abs tracts
46.10 47.25A 02.60
Powder turbulence: direct onset of turbulent flow
Y-h. Taguchi
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
(Received 27 July 1992, accepted in final form 9 September 1992)
Abstract. A vibrated bed of powder with a large aspect ratio (periodic boundary) is
ex-
amined. In contrast to the bed with a small aspect ratio (with side wall), no clear convection roll is observed. The flow in the fluid phase seems to be always turbulent, because the local
vibration of powder in static state has already become spatially random before the flow starts.
This powder turbulence provides us with a new tool to study general hard turbulence.
1. Introduction.
Granular materials are among the interesting objects studied in physics [I]. For example, it is
interesting to construct the basic equations, like the Navier-Stokes equation in fluid mechan- ics. Granular materials also have a wide range of applications in powder technology [2] and
structural and earthquake engineering [3]. It is also important to study several characteristic
phenomena of granular materials, "Brazil nut" segregation [4], avalanches [5], fluidization in
a gas-fluidized bed [6], angles of repose [7], flow on an inclined slope [8] and hopper flow [9].
Besides these phenomena, the heaping, convection [10] and surface fluidization [II] in vibrated beds are interesting to study.
The vibrated bed is a set of small particles (0.01 mm < diameter < I mm) in a vessel which is shaken vertically. Typically glass spheres are used in the experiment. A shallow dish is filled with these spheres a few cm in depth. Usually the vessel is vibrated with a loud speaker as a
function of time, b coswot. Here b is an amplitude and wo is an angular frequency of vibration, respectively. When the acceleration amplitude, r = bw(, exceeds the critical value (> g), the
heaping, the convection, and the fluidization start simultaneously.
Recently numerical calculations succeeded in reproducing convective motion and surface fluidization [12, 13]. This success suggests that the model used in these calculations, "Discrete Element Method", (DEM) [15] can describe the vibrated beds.
When the simulations
are done with a small aspect ratio bounded by a side wall, the structure of convection looks like that of the normal fluid. However it is not clear whether the fluidized
bed is equal to the normal fluid under other conditions. For example, one may think that an array of convection rolls ~vill appear when the aspect ratio is large and period doublings in
convection will take place when the "external force" increases. These phenomena are widely
observed in the Rayleigh-Bdnard convection [16] and convection in EHD [17].
In this paper, we examine the vibrated bed when the aspect ratio is large and the boundary
condition is horizontally periodic. Under these conditions, no clear steady convection is ob- served. Next, we also investigate local motion in the solid phase below the fluid phase [13]. In
this solid region, a particle does not flow but vibrates. This vibration turns out to be turbulent
for(~) large r: no characteristic length in space and period doubling in time.
Since the fluid region should be more unstable than the solid region, it is natural to think that the flow in the fluid region is always turbulent. That is, the turbulent flow seems to
appear directly from the solid state which corresponds to the static state in the normal fluid.
This direct onset of turbulent flow is not common in the convection of normal fluids.
The construction of this paper is as follows. In section 2, we define the model to describe the powder and the results are presented in section 3. Section 4 includes the discussion and the conclusion.
2. Model.
We use the same model as in the previous paper [13]:
it # ~j I(d- Xi Xi () k(X; Xi d ~~ ~~
+ n(Vi Vi g, (1)
~~
Xi Xi
J"
where I(z) is a step function, N is a total number of spheres, x; is the position vector of the I-th sphere. k and
1~ are the elastic constant and the viscosity coefficient respectively. d is the diameter of a sphere, v is the velocity.
If two particles collide frontaley with each other, they have effective an coefficient of resti- tution e
= exp(-q7r/w) and a collision time tcoj = 7r/w and w = 2k q2. g is the gravity
acceleration. tcoj is the collision time. We select k and q such that the particle has given e and tcoj.
Equation (I) treats each particle as a visco-elastic particle during collisions and as a free
particle otherwise. I(z) decides which case the particle belongs to.
One may think that a solid friction term is lacking (Coulomb friction [20]), min (tangential friction, ~x normal stress), where ~ is a positive constant. However, our model corresponds
to ~ - o~o. This means that we simulate the strong friction limit [18].
Equation (I) is not unique for giving definite tcoj and e which are independent of the collision velocity. For example, one can employ the elastic interaction calculated by Hertz (normal direction) and Mindlin [19] (tangential direction). Although their arguments are more reliable
than that used in equation (I), using them causes another difficulty. When we follow Hertz formula and require the coefficient of restitution to be independent of the collision velocity, we
must introduce an unphysical viscous interaction between the particles (See Appendix). Since Mindlin's argument assumes the Hertz interaction, it is also impossible to use Mindlin's formula without introducing an unphysical viscous term. Then we use equation (I) for simplicity. This model has already been used [12, 20] and given reasonable results.
In order to simulate the vibrated bed of powder, we put particles whose dynamics are defined in equation (I) in two-dimensional space. This space has the horizontal size of120 and is horizontally periodic. There is a "bottom" plate in the two-dimensional space which vibrates
(~ In the following, we use the term turbulence in its general meaning. Therefore we do not want to insist on the fact that the powder "turbulence" either belongs to the same universality class
as fluid turbulence or has positive Lyapunov exponent.
N°12 POWDER TURBULENCE: DIRECT ONSET OF TURBULENT FLOW 2105
as a function of time, b cos won. The bottom plate consists of completely elastic materials with the same elastic constant k as the particles. In figure 2, we show a snapshot of the simulation.
To integrate equation (I), we discretize the equation with the time interval At,
xi(I + At) = xi (t) + vi (t)At + j(ai(t) g)At~
vi(I + At) = vi (t) + (ai(t) g)At, (2)
where a;(i) is the acceleration which is caused by the interaction both among particles and between particles and bottom plate. In order to avoid to take a too large hi, we select the time step hi at each step i such that max; Azi (= 0.01 where Ax; =( xi(i + hi) xi(i) is the displacement of the I-th particle during hi.
3. ltesults.
3.I NO CLEAR ARRAY OF CONVECTION ROLLS WITH LARGE ASPECT RATIO WITHOUT WALL.
In previous papers [12, 13], the model can explain the appearance of convection observed in experiments [10]. Clear two-roll structures are observed in the model but no dynamical
property of rolls has been investigated yet.
For example, if we increase the external force (in this case, the acceleration amplitude r/g = bw(/g), are instabilities of roll structure observed? And can further increase of the external
force cause turbulent patterns? And what happens with the large aspect ratio? Such questions
must be answered. To answer them, it is most suitable to use the periodic boundary condition with a large aspect ratio. Under this condition, the motion of powder is the least influenced by the boundary condition.
The conditions of the simulations are as follows. d
= 2.0, g = 1.0, the number of particles
is 472. The values of b,wo are not fixed. r is always taken above the threshold value for the
starting of convection, rc c~ g. icoi and e are always set to be 0.5. Tbese values of e and tcoj are
taken to make the computational time short. When we use larger e or smaller tori, At becomes
small, which requires a large number of steps. On the other hand, rc is independent of e and tcoj. Therefore we can expect the mechanism of occurrence of convection to be universal for all values of e and tcoj. Then we use these values of e and icoj. In the initial condition, random
positions and velocities are given to each particle without overlapping between particles.
First we draw the displacement vector which represents only the relative motion between
particles that we are interested in. At each time i = T/4 + nT (n is integer), where T
= 27r/wo
is a period of external force, the bottom plate returns to the origin of vibration. Therefore we
can see only the relative displacement between particles by considering the spatial displacement
from i = T/4 + nT to i
= T/4 + (n + m)T,
Axj (n, m) e x;(T/4 + (n + m)T) xi(T/4 + nT). (3)
When drawing the displacement vector, we omit the global displacement which corresponds to the motion of the center of total mass of particles. This means £~ Ax;(n, m)
= 0.
Figure I shows the typical displacement vectors (m
= 2). The length of displacement vect.or is larger than d. This means that the motion is not a simple local motion. Some part.icles change places with other particles. Therefore we call it "flow" in order to distinguish it. front the local motion. This flow is essentially the same as the flow in convection observed in previous
papers [13, 14].
11-~1('tl Fl@jfj~(il@j
,/ ,,, ,,,
--.;----ll/m,, ,m,,~r. ,-:...~--~--
ilwl@llflltlfl)ill
fjjjflfl~jj@@@§j§j(jj~jj,,,;1,~',iijii==j
'II
''' '' Cfi,?~'f, (/Cf7J4ff (1$1)]3q '' ;$;.')I'@'
,'~ ',',,,
~,,... ~_, ,, ,,
' .""""' "SfT,~/.,,~-~,-. ..,,,,rz,u=~,=,=jj,,~ ,,
~- ,
j,,'j)@$il~j@l'j~~" ~~~
j,~~._~~jj/-jj,j;,jjjy,l~,/;.<'""'~'li'"~~"~~~~"~~
jj@jj)j')(jjj[j)()[@(
,,,,,,,,,,..,,,,,.,..,,,,,. ,,-,,...,,,,,,,,,,,
Fig-I- Turbulent flow, Ax,(n, m
= 2), in granular materials. (b = 1.2,T
= 6.0). Time runs from upper to lower.
Contrary to the expectation, we can never observe a clear array of convection rolls. The flow
always looks turbulent. The situation is similar even when the wall exists. Convection rolls
are local12ed only near the wall and far from the wall, the flow is turbulent.
One may think that it is very strange that the flow of particles is always turbulent. Air and water particles behave very differently although they also have random motion, "thermal
motion". However, the random motion of powder particles differs from the thermal motion in normal fluids.
As pointed out by Stavans [14], it is impossible to distinguish thermal motion from cooper- ative motion in powders. The amount of both "thermal" and "cooperative" motion is almost
the same. This means that in powder dynamics, lamellar flow is not obtained by the coarse
graining procedure. It can appear only when 'random'motion itself becomes coherent.
In the next section, we will clarify the reason why the flow of powder is always random.
3.2 TURBULENT VIBRATION IN TIIE SOLID PHASE BELOW THE FLUIDIZED PHASE. In tllis
section, we take b and wo in such a way that convection exists only in the surface region [13]
and a solid region remains below the fluidized region. In the following, we mainly investigate
wbat happens in the non-convecting region below the convecting region when the convection is concentrated near the surface. This enables us to know the
reason why the turbulent flow
N°12 POWDER TURBULENCE: DIRECT ONSET OF TURBULEN~' FLOW 2107
o om omm o oJrocrcom
Fig.2. Snap shot of vibrated beds. (b = o.3, T
= 2.o) Double circles indicate the fourth layer which is used in figures 3 and 4. The number of particles in the fourth layer is 64. They are numbered from
left to right. This numbering £ is used as the spatial axis in figureA.
till)lllfl~tlllllllltl'i=itlii/
~))))))(i<,())j()ijil)ji,i~)=:.:'.'lijj()j)
ljj))(,yfl))~jl>[itfl[ii;.illll
flfj)jll.I.i?Wi.>)~
~ ~~~#~~i~'~~.~~~,~i~'~'~'~~~#~~-~ '~~~"-(~~,, ,,'(S~~~/~~~~..~~~~-
'~~~~~ijf;I.I.I.I.)..~/(/ji.'.<.=;jjj$j,
~~~~~""~~°~~~i~
~l~l"l~l~l°i°;li,.S,illili?% ,jjj~j=j;=jj;==~j~=__,~,j~.=,
jjj))i§@lfl#()jiflflil)i;.:i((j@fl)il:'illlli-I(ij~
,,,,....,,,,; .,,,,ez,,,,, ;, I..,, ,e,,. .,, ,,,".. i,,-
.::,~,,.,.<::: --=rv,','.'.<.'."~,""" .--'r.""" rrrTzr?rr.""','"
Fig.3. Displacement vectors of particles, Ax,(n,m
= 1). Since magnification factor is 20, the length of Ax, is too short for the particles to change places with each other. Their relative positions,
like the order in one layer, remain unchanged (b = o.2,T
= 2.o).
appears directly from the static state.
Figure 2 shows the snapshot of a vibrated bed. In the following, the horizontal direction is denoted by ~ and the vertical one, by y. As you can see, particles are regularly packed in a
triangular lattice. Except for the region near the surface, the particles cannot slide on each other. Since the depth of the convecting region is less than the size of a particle, almost the whole region is the non,convecting. Now many interesting features can be seen even in the
non-convecting region.
Figures 3 shows displacenlent vectors of particles (m = I). However, now, the flow is local
and does not cause any exchange of particles. The size of a displacement vector is much smaller
than the diameter of a particle. When one sees only one snapshot it looks like fluid turbulence.
Some eddy-like structure is observed and spatial intermittency is seen. However if the sequence of figures is considered, these turbulent motions turn out to be frozen spatially. Actually, they
are vibrations rather than flows, because particles are confined to fixed positions. We may call this "spatially frozen turbulence". This is not
a true turbulence, because its temporal motion is periodic.
In order to distinguish these nlotions from the flow observed in the previous section, we do not call them "flow" in the following. The flow in the previous section is really a flow because
particles can change place with each other. However, motion in this section is nothing but local vibration.
In order to see how the change of external force influences the motion of each particle, we consider the fourth layer of the triangular lattice shown as double circles in figure 2 and plot
the y-components of the displacement vector Ax;(n, m = I) of each particle belonging to the layer. Figure 4a shows the contour plot of the y-component of the displacement vector shown in figure 3,
Ayi(n) = x12(T/4 + (n + I)T) x12(T/4 + nT), (4)
where x12 is the y-component of xi Here £ is not the numbering shown in equation (I), but the order in the fourth layer shown as double circles in figure 2. In figure 4a n is taken to be the vertical axis and £ is the horizontal axis. You can clearly see that the spatio-temporal
patterns are spatially frozen and have period two in time.
In order to analyze the patterns in detail, we take their spatio-temporal Fourier spectrum
(Fig. 5a),
S(IP,Q) =( £ £ Ayi(n) exp(-j(I(£ an) (~, (5)
n i
where j is a pure 1nlaginary number, Ii and Q are the wave number and the angular frequency, respectively. (Here we use It and Q instead of their lower cases to avoid confusion since lower cases, k and u~, are already used.) In this plot, spatially homogeneous components are
onlitted (that is, the conlponent of wave nunlber K
= 0). Figure 5a is also a contour plot
and nornlalized by the nlax1nlunl value. It also exhibits period two in t1nle (Q/~ = l.0),
In addition, sortie peaks can be seen in wave nunlber space. The values are close to 2'~/L,
where L is the number of particles in the fourth layer, 64, and s is integer. For exanlple, 1(/~ = 2~/L = 0.03,0.06,0.13,0.25, for s
= 1,2,3,4; This may be spatial period doubling.
S1nlilar figures are shown for the different sets of values of b and
u~o. Roughly speaking,
as b increases with fixed u~o, the following is observed. First, the spatial structure vanishes and beconles turbulent in space. This nleans that the Fourier spectrum in wave number space beconles broad and conies to have no peak. Next, temporal structures develop. Period two
(Q/~ = l.0) beconles period four (Q/~ = 0.5) and then period three (Q/~'3 0.66) appears.
Finally the spectrum becomes broad in both time and space except for the peak at period
four in time and K
= 0 (Ho&vever, since the component of Ii
= 0 is removed in our plot, the peak looks as being located at Ii > 0.). The peak at period four in time does not prevent the
structure from being turbulent. When the global spectrum is broad, we can call it turbulent.
For example, although the turbulent state in the Kuramoto-Shibashinsky equation [21] has a peak in the wave number space, its spatial structure is called turbulent.
N°12 POWDER TURBULENCE: DIRECT ONSET OF TURBULENT FLOW 2109
n n
19.o 19.o
14. 5 14.5
9.5 9.5
4.8 4.8
0.0 0. 0
0. 0 15. 8 51.5 4?.5 63.0 0. 0 15.8 31. 5 4?. 5 65.0
a) i b) 1
~ n
19,o 19,o
14. 3 14. 3
9. 5 9. 5
4.8 4.8
0.0 0. 0
0.0 15.8 31. 5 4?.3 63.0 0. 0 15.5 51.0 46.5 62.0
c) £ ~) £
FigA. Space-time contour plots of y-components (ayt(n)) of displacement vectors of particles
which belong to forth layer. T
= i-o, and (a)
= o.3, (b)b
= oA, (c)b
= o-s, id) b
= o.55, (e)
= o.6.
ayt(n) is normalized such that the maximum value is unity. The higher intensity of grey patterns indicates the higher values.
n 19. o
14. 3
9 5
4 8
0. 0
0. 0 15.8 31.5 4?. 3 63.0
I e)
Fig. 4 (continued)
In summary, the scenario for the vibration to come to turbulence is the following. First, the local vibration of powder becomes spatially turbulent. Temporally, it is a periodic motion.
Then the temporal motion becomes unstable, exhibits period doubling and becomes spatio- temporal turbulence.
Now we can understand why the "flow" observed in the previous section is always turbulent, when it appears. When we pay attention to the amount of Ayi(n), it is still less than the size of the particle, although it is not shown in figures. The particles still remain at their original points. Their motions are restricted around their origin. The order of particles in the fourth
layer will never change. No flow can be observed. In spite of that, their motion has already become turbulent. Since a much stronger vibration must be applied to the powder in order to
cause the "flow", it is unrealistic that coherent lamellar motion will appear in the fluid phase.
This turbulent vibration will be the origin of the direct onset of turbulent flow.
Similar phenomena (turbulent vibration) were recently found in the sound propagation in sand [22]. Even if the input sound is harmonic, the propagating sound turns out to be turbulent and has a broad spectrum. This may have some relationship with our results.
However, I am not sure whether "turbulent" vibration and "turbulent" flow are true turbu- lence. In order to confirm this point, we must check the appearance of chaos or anomalous diffusion. The only thing we can say is that both "turbulent" vibration and motion clearly
differ from simple white noise. They seem to have some coherency.
N°12 POWDER TURBULENCE: DIRECT ONSET OF TURBULENT FLOW 2111
a/7r a/7r
1. o i. o
0. 8 0. 8
0.6 0.6
0.4 0.4
o, o.
o_ 0.
0.0 0.2 0.4 0.6 0.8 1.0 o-o °.2 °.4 °.6 °.8 1.°
~~j/~ Klir
a) b)
11/7r a/~r
i-o i~
0.8 0.8
~'
0.6 0. 4
0. 4
~'~
0.2
°'(
~ ~ ~ ~ ~ ~ ~ ~ ~ j ~ 0.0
0.0 0.2 0.4 0.6 0.8 1.0
K/~r ~/
c) d) ~
Q/7r
0 8
0 6
0 4
0 2
o. o
0.0 0.2 0.4 0.6 0.8 1.0
e) K/~r
Fig.5. contours of spatio-temporal Fourier spectrum of figure 4.
