particles in fluid

8/9/2019 Particles in fluid

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EPJ manuscript No.(will be inserted by the editor)

Particles in FluidsHans J. Herrmann 1 , Jose S. Andrade Jr. 2 , Ascanio D. Araujo2 , and Murilo P. Almeida 2

1 IfB, ETH Zurich, Honggerberg, 8093 Zurich, Switzerland2 Departamento de Fsica, Universidade Federal do Ceara, 60451-970 Fortaleza, Cear a, Brazil

July 25, 2006

Abstract. For nite Reynolds numbers the interaction of moving uids with particles is still only under-stood phenomenologically. We will present three different numerical studies all using the solver Fluentwhich elucidate this issue from different points of view. On one hand we will consider the case of xed parti-

cles, i.e. a porous medium and present the distribution of channel openings and uxes. These distributionsshow a scaling law in the density of particles and for the uxes follow an unexpected stretched exponentialbehaviour. The next issue will be ltering, i.e. the release of massive tracer particles within this uid.Interestingly a critical Stokes number below which no particles are captured and which is characterizedby a critical exponent of 1/2. Finally we will also show data on saltation, i.e. the motion of particles ona surface which dragged by the uid performs jumps. This is the classical aeolian transport mechanismresponsible for dune formation. The empirical relations between ux and wind velocity are reproduced.

1 Introduction

Particles in uids (liquids or gases) appear in many ap-plications in chemical engineering, uid mechanics, geol-ogy and biology [13]. Also uid ow through a porous

medium is of importance in many practical situations rang-ing from oil recovery to chemical reactors and has beenstudied experimentally and theoretically for a long time[46]. Due to disorder, porous media display many inter-esting properties that are however difficult to handle evennumerically. One important feature is the presence of het-erogeneities in the ux intensities due the varying channelwidths. They are crucial to understand stagnation, lter-ing, dispersion and tracer diffusion.

In the porous space the uid mechanics is based onthe assumption that a Newtonian and incompressible uidows under steady-state conditions. We consider the Na-vier-Stokes and continuity equations for the local veloc-

ity u

and pressure elds p, being is the density of theuid. No-slip boundary conditions are applied along theentire solid-uid interface, whereas a uniform velocity pro-le, ux (0, y) = V and uy (0, y) = 0, is imposed at the in-let of the channel. For simplicity, we restrict our studyto the case where the Reynolds number, dened here asRe V Ly / , is sufficiently low (Re < 1) to ensure alaminar viscous regime for uid ow. We use FLUENT[7], a computational uid dynamic solver, to obtain thenumerical solution on a triangulated grid of up to hundredthousand points adapted to the geometry of the porousmedium.

The investigation of single-phase uid ow at low Rey-nolds number in disordered porous media is typically per-formed using Darcys law [4,6], which assumes that a ma-croscopic index, the permeability K , relates the average

uid velocity V through the pores with the pressure dropP measured across the system,

V = K

P L

, (1)

where L is the length of the sample in the ow directionand is the viscosity of the uid. In previous studies [915], computational simulations based on detailed modelsof pore geometry and uid ow have been used to predictpermeability coefficients.

Here we present numerical calculations for a uid ow-ing through a two-dimensional channel of width Ly andlength Lx lled with randomly positioned circular obsta-cles [16]. For instance, this type of model has been fre-quently used to study ow through brous lters [27].Here the uid ows in the x-direction at low but non-zeroReynolds number and in the y-direction we impose peri-

odic boundary conditions. We consider a particular type of random sequential adsorption (RSA) model [18] in two di-mensions to describe the geometry of the porous medium.As shown in Fig. 1, disks of diameter D are placed ran-domly by rst choosing from a homogeneous distributionbetween D/ 2 and Lx D/ 2 (Ly D/ 2) the random x-(y-)coordinates of their center. If the disk allocated at thisposition is separated by a distance smaller than D/ 10 oroverlaps with an already existing disk, this attempt of placing a disk is rejected and a new attempt is made. Eachsuccessful placing constitutes a decrease in the porosity(void fraction) by D 2 / 4Lx Ly . One can associate thislling procedure to a temporal evolution and identify asuccessful placing of a disk as one time step. By stoppingthis procedure when a certain value of is achieved, wecan produce in this way systems of well controlled poros-


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2 Hans J. Herrmann et al.: Particles in Fluids

Fig. 1. Velocity magnitude for a typical realization of a porespace with porosity = 0 .7 subjected to a low Reynoldsnumber and periodic boundary conditions applied in the y-direction. The uid is pushed from left to right. The colorsranging from blue (dark) to red (light) correspond to low andhigh velocity magnitudes, respectively. The close-up shows atypical pore opening of length l across which the uid owswith a line average velocity v . The local ux at the pore open-ing is given by q = vl cos , where is the angle between v andthe vector normal to the line connecting the two disks.

ity. We study in particular congurations with = 0 .6,0.7, 0.8 and 0.9.

2 Various distributions in porous medium

The geometry of our random congurations can be ana-lyzed making a Voronoi construction of the point set givenby the centers of the disks [19,20]. We dene two disks tobe neighbors of each other if they are connected by a bondof the Voronoi tessellation. These bonds constitute there-fore the openings or pore channels through which a uidcan ow when it is pushed through our porous medium,as can be seen in the close-up of Fig. 1. We measure thechannel widths l as the length of these bonds minus thediameter D and plot in Fig. 2 the (normalized) distri-butions of the normalized channel widths l = l/D forthe four different porosities. Clearly one notices two dis-tinct regimes: ( i ) for large widths l the distribution de-cays seemingly exponentially with l, and ( ii ) for smalll it has a strong dependence on the porosity, increas-ing with decreasing porosity dramatically at the origin.Between the two regimes a crossover is visible as a peakwhich shifts between = 0 .9 and 0.8 and then stays forsmaller porosities at about l= 1, i.e., l = D.

Let us now analyse the distribution of uxes through-out the porous medium. Each local ux q crossing its cor-responding pore opening l is given by q = vl cos , where is the angle between v and the vector normal to thecross section of the channel (see Fig. 1). In Fig. 3 we showthat the distributions of normalized uxes = q/q t , whereq t = V Ly is the total ux, have a stretched exponentialform, P () exp( / 0), with 0 0.005 being acharacteristic value. This simple form is quite unexpectedconsidering the rather complex dependence of P (l) on .

0 1 2 3 4 5 6 7

l *

0

0.5

1

1.5

2

P ( l

* )

=0.6=0.7=0.8=0.9

Fig. 2. The distributions of the normalized channel widthsl = l/D for different values of porosity . From left to right, thetwo vertical dashed lines indicate the values of the minimumdistance between disks l = 0 .1 and the size of the disks l = 1.

7 6 5 4 3 2 1 0 1

log 10(

)

6

4

2

0

2

l o g

1 0

P (

)

=0.6=0.7=0.8=0.9

100

101

102

3/(1 )210

5

104

103

102

K/K 0

Fig. 3. The log-log plot of the distributions of the normalizedlocal uxes = q/q t for different porosities . The (red) dashedline is a t of the form exp( / 0 ), where 0 0.005. Inthe inset we see a double-logarithmic plot of the global uxand the straight line veries the Kozeny-Carman equation.

Moreover, all ux distributions P () collapse on top of each other when rescaled by the corresponding value of l 1 2 . This collapse for distinct porous media results

from the fact that mass conservation is imposed at themicroscopic level of the geometrical model adopted here,which is microscopically disordered, but is macroscopicallyhomogeneous at a larger scale [6].


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Hans J. Herrmann et al.: Particles in Fluids 3

3 Results on Filtration

Filtration is typically used to get clean air or water andalso plays a crucial role in the chemical industry. For thisreason it has been studied extensively in the past [21].In particular, we will focus here on deep bed ltrationwhere the particles in suspension are much smaller thanthe pores of the lter which they penetrate until beingcaptured at various depths. For non-Brownian particles,at least four capture mechanisms can be distinguished,namely, the geometrical, the chemical, the gravitationaland the hydrodynamical one [21].

Very carefully controlled laboratory experiments wereconducted in the past by Ghidaglia et al. [22] evidencinga sharp transition in particle capture as function of thedimensionless ratio of particle to pore diameter character-ized by the divergence of the penetration depth. Subse-quently, Lee and Koplik [23] found a transition from anopen to a clogged state of the porous medium that is func-tion of the mean particle size. Much less effort, however,has been dedicated to quantify the effect of inertial impacton the efficiency of a deep bed lter.

Here we will concentrate on the inertial effects in cap-ture which constitute an important mechanism in mostpractical cases and, despite much effort, are quantitativelynot yet understood, as reviewed in Ref. [24]. The effectof inertia on the suspended particles is usually quantiedby the dimensionless Stokes number , St V d2 p p / 18 ,where d p and p are the diameter and density of the par-ticle, respectively, is a characteristic length of the pores, is the viscosity and V is the velocity of the uid. In-ertial capture by xed bodies has already been describedsince 1940 by Taylor and proven to happen for invisciduids above a critical Stokes number [25]. It is our aim topresent a detailed hydrodynamic calculation of the inertialcapture of particles in a porous medium. We will disclosenovel scaling relations.

Let us rst consider the case of an innite orderedporous medium composed of a periodic arrangement of xed circular obstacles (e.g., cylinders) [26]. This systemcan then be completely represented in terms of a sin-gle square cell of unitary size and porosity given by (1 D 2 / 4), where D is the diameter of the obstacle, asshown in Fig. 1. Assuming Stokesian ow through the voidspace an analytical solution has been provided by Marshallet al. [27]. Here we use this solution to obtain the velocityow eld u and study the transport of particles numer-ically. For simplicity, we assume that the inux of sus-pended particles is so small that ( i ) the uid phase is notaffected by changes in the particle volume fraction, and(ii ) particle-particle interactions are negligible. Moreover,we also consider that the movement of the particles doesnot impart momentum to the ow eld. Finally, if we as-sume that the drag force and gravity are the only relevantforces acting on the particles, their trajectories can be cal-culated by integration of the following equation of motiond u

p

dt = (u

u

p )St + F g g| g | , where F g ( p ) |g | / (V 2 p) isa dimensionless parameter, g is gravity, t is a dimension-

D

x

y

u

Fig. 4. The trajectories of particles released from differentpositions at the inlet of the periodic porous medium cell.St = 0 .25 and the ow eld u is calculated from the analyticalsolution of Marshall et al. [27]. The thick solid lines correspondto the limiting trajectories that determine .

less time, and u p and u are the dimensionless velocitiesof the particle and the uid, respectively.

We show in Fig. 4 some trajectories calculated for par-ticles released in the ow for St = 0 .25. Once a particletouches the boundary of the obstacle, it gets trapped. Ourobjective here is to search for the position y0 of release atthe inlet of the unit cell ( x0 = 0) and above the horizontalaxis (the dashed line in Fig. 4), below which the particle isalways captured and above which the particle can alwaysescape from the system. As depicted in Fig. 4, the parti-cle capture efficiency can be straightforwardly dened as 2y0 . In the limiting case where St , since the par-ticles move ballistically towards the obstacle, the particleefficiency reaches its maximum, = D . For St 0, on the

other hand, the efficiency is smallest, = 0. In this lastsituation, the particles can be considered as tracers thatfollow exactly the streamlines of the ow and are thereforenot trapped.

We show in Fig. 5 the log-log plot of the variationof /D with the rescaled Stokes number in the presenceof gravity for three different porosities. In all cases, thevariable increases linearly with St to subsequently reacha crossover at St , and nally approach its upper limit( = D). The results of our simulations also show thatSt ( min ), where min corresponds to the mini-mum porosity below which the distance between inlet andobstacle is too small for a massive particle to deviate from

the obstacle. The collapse of all data shown in Fig. 5 con-rms the validity of the scaling law.We see in the inset of Fig. 5 that the behavior of the

system in terms of particle capture becomes signicantlydifferent in the absence of gravity. The efficiency remainsequal to zero up to a certain critical Stokes number, S t c ,that corresponds to the maximum value of St below whichparticles cannot be captured, regardless of the position y0at which they have been released.Right above St c , thevariation of can be described in terms of a power-law, (St St c ) , with an exponent 0.5. Our resultsshow that, while the exponent is practically independentof the porosity for > 0.8, the critical Stokes number de-creases with , and therefore with the distance from theobstacle where the particle is released (see Fig. 5). To ourknowledge, this behavior, that is typical of a second order


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0 1 2 3 4 5

St0.0

0.2

0.4

0.6

0.8

1.0

/ D

104

103

102

101

100

StSt c

0.01

0.1

1.0

/ D

0.5

Fig. 5. The dependence of the capture efficiency on the

rescaled Stokes number S t/ (

min ) for periodic porous me-dia in the presence of gravity. Here we use F g = 16, a valuethat is compatible with the experimental setup described inRef. [22]. The inset shows that the behavior of the systemwithout gravity can be characterized as a second order transi-tion, (St St c ) , with 0.5 and S t c = 0 .2679 0.0001,0.2096 0.0001 and 0 .1641 0.0001, for = 0 .85, 0.9 and 0 .95,respectively.

transition, has never been reported before for inertial cap-ture of particles. In order to have a more realistic model forthe porous structure we did also include disorder [26]. Herewe adopted a random pore space geometry [18] shown in

Fig. 1 and obtained the same results as for the regularcase.

4 The Mechanism of Saltation

Aeolian of sand is a major factor in sand encroachment,dune motion and the formation of coastal and desert land-scapes. The dominating transport mechanism is saltationas rst described by Bagnold [28] which consists of grainsbeing ejected upwards, accelerated by the wind and -nally impacting onto the ground producing a splash of new ejected particles. Reviews are given in Refs. [29,30].

A quantitative understanding of this process is howevernot achieved.The wind loses more momentum with increasing num-

ber of airborne particles due to Newtons second law untila saturation is reached. The maximum number of grains awind of given strength can carry through a unit area perunit time denes the saturated ux of sand q s . This quan-tity has been measured by many authors in wind tunnelexperiments and on the eld, and numerous empirical ex-pressions for its dependence on the strength of the windhave been proposed [31,32]. In previous studies theoreti-cal forms have also been derived using approximations forthe drag in turbulent ow [33,34]. All these relations areexpressed as polynomials in the wind shear velocity uwhich are of third order, under the assumption that thegrain hopping length scales with u [3135]. The velocity

>

u p

> >

u x( y) y

x

Fig. 6. Setup showing the mobile wall at the top, the velocityeld at different positions in the y -direction and the trajectoryof a particle stream (dashed line).

prole in a particle laden layer has also been the object of measurement [36,37] and modellization [38]. The completeanalytical treatment of this problem remains out of reachnot only because of the turbulent character of the wind,but also due to the underlying moving boundary condi-tions in the equations of motion. More recently, a deter-ministic model for aeolian sand transport without heightdependency in the feedback has been proposed [39]. De-spite much research in the past [40] there remain manyuncertainties about the trajectories of the particles andtheir feedback with the velocity eld of the wind.

Here we present the rst numerical study of saltationwhich solves the turbulent wind eld and its feedback withthe dragged particles [41]. As compared to real data, ourvalues have no experimental uctuations neither in thewind eld nor in the particle size. As a consequence, wecan determine all quantities with higher precision than

ever before, and therefore with a better resolution closeto the critical velocity at which grains start to be trans-ported.

To get a quantitative understanding of the layer of airborne particle transport above a granular surface, wesimulate the situation inside a two-dimensional channelwith a mobile top wall as shown in Fig. 6 schematically.Here a pressure gradient is imposed between the left andthe right side. Gravity points down, i.e., in negative y -direction. The y -dependence of the pressure drop is ad- justed in such a way as to insure a logarithmic velocityprole along the entire channel in the case without parti-cles, as it is expected in fully developed turbulence [42].

More precisely, this prole follows the classical formux (y) = ( u/ )ln( y/y 0) , (2)

where ux is the component of the wind velocity in thex -direction, u is the shear velocity, = 0 .4 is the vonKarman constant and y0 is the roughness length whichis typically between 10 4 and 10 2m. The upper wall of the channel is moved with a velocity equal to the velocityof the wind at that height in order to insure a non-slipboundary condition.

Inside the channel we have air owing under steady-state and homogeneous turbulent conditions. The Rey-nolds-averaged Navier-Stokes equations with the standardk model are used to describe turbulence. The solutionfor the velocity and pressure elds is obtained through dis-


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Hans J. Herrmann et al.: Particles in Fluids 5

cretization by means of the control volume nite-differencetechnique [7].

Having produced a steady-state turbulent ow, we pro-ceed with the simulation of the particle transport along

the channel. Assuming that drag and gravity are the onlyrelevant forces acting on the particles, their trajectory canbe obtained by integrating the following equation of mo-tion:

du pdt

= F D (u u p) + g ( p )/ p , (3)

where u p is the particle velocity, g is gravity and p =2650 kg m 3 is a typical value for the density of sandparticles. The term F D (u u p ) represents the drag forceper unit particle mass where

F D = 18 p d2 p

C D Re24

, (4)

d p = 2 .5 10 4 m is a typical particle diameter, Re d p |u p u | / is the particle Reynolds number, and thedrag coefficient C D is taken from empirical relations. Eachparticle in our calculation represents in fact a stream of real grains. It is necessary to take into account the feed-back on the local uid velocity due to the momentumtransfer to and from the particles.

We see in Fig. 6 the trajectory of one particle streamand the velocity vectors along the y -direction. Each timea particle hits the ground it loses a fraction r of its energyand a new stream of particles is ejected at that positionwith an angle . The parameters r = 0 .84 and = 36oare chosen from experiments [43,44].

We see in Fig. 7 the plot of q s as function of the windvelocity u. Clearly, there exists a critical wind velocitythreshold u t below which no sand transport occurs at all.This agrees well with experimental observations [28,32].Also shown in Fig. 7 is the best t to the numerical datausing the classical expression proposed by Lettau and Let-tau [32],

q s = C Lg

u2(u u t ) , (5)

where C L is an adjustable parameter. We nd rather goodagreement using t parameters of the same order as thoseof the original work [32] and a threshold value of ut =0.35 0.02. This is in fact, to our knowledge, the rsttime a numerical calculation is able to quantitatively re-produce this empirical expression and it conrms the va-lidity of our simulation procedure. Other empirical rela-tions from the literature [3335] can also be used to tthese results. In Fig. 7 we also show that for large valuesof u asymptotically one recovers Bagnolds cubic depen-dence. Close to the critical velocity ut interestingly we ndthat a parabolic expression of the form

q s = a(u u t )2 (6)

ts the data better than Eq. (6), as can be seen in Fig. 7and in particular in the inset. In the limit u >> u t oneobtains the classical behavior of Bagnold [31], as veriedby the dash-dotted line in Fig. 7 and which is consistentwith Refs. [3235]. The limit u u t , however, yields the

0.3 0.4 0.5 0.6 0.7

u *

0.00

0.05

0.10

0.15

0.20

0.25

q s0.01 0.1 1.0u *u t

0.001

0.01

0.1

1.0

q s

2.0

Fig. 7. Plot of the saturated ux q s as function of u . Thedashed line is the t using the expression proposed by Lettauand Lettau [32], q s u2 (u u t ), with ut = 0 .35 0.02.The full line corresponds to Eq. (6) and the dashed-dotted lineto Bagnolds relation, q s u3 [31]. The results shown in theinset conrm the validity of the the power-law relation Eq. (6),q s (u u t )2 . The critical point is u t = 0 .33 0.01.

quadratic relation for the ux given in Eq. (6). Physicallythis is due to the fact that close to u t the laminar compo-nent is relevant.

5 Conclusion and Outlook

We have found [16] that although the distribution of chan-nel widths in a porous medium made by a two-dimensionalRSA process is rather complex and exhibits a crossover atl D, the distribution of uxes through these channelsshows an astonishingly simple behavior, namely a square-root stretched exponential distribution that scales in asimple way with the porosity. Future tasks consist in gen-eralizing these studies to higher Reynolds numbers, othertypes of disorder and three dimensional models of porousmedia.

We also presented results for the inertial capture of particles in two-dimensional periodic as well as randomporous media [26]. For the periodic model in the absenceof gravity, there exists a nite Stokes number below whichparticles never get trapped. Furthermore, our results in-dicate that the transition from non-trapping to trappingwith the Stokes number is of second order with a scalingexponent 0.5. In the presence of gravity, we show that(i ) this non-trapping regime is suppressed (i.e., St c = 0)and ( ii ) the scaling exponent changes to 1. We intendto investigate in the future the possibility of non-trappingat rst contact and the effect on the capture efficiency of simultaneous multiple particle release.

We nally also showed results of simulations [41] givinginsight about the layer of granular transport in a turbulentow. The lack of experimental noise allows for a precise


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study close to the critical threshold velocity u t that leadus to a parabolic dependence of the saturated ux. Thepresent model can be extended in many ways includingthe study of the dependence of the aeolian transport layer

on the grain diameter, the gas viscosity, and the solid oruid densities. This would allow to calculate, for instance,the granular transport on Mars and compare with the ex-pression presented in the literature [35].

We thank the Max Planck Prize for nancial support.

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