Direct Numerical Simulation of Curly Fibers in

Aerosol Science and Technology 33:392– 418 (2000) ° c 2000 American Association for Aerosol Research Published by Taylor and Francis 0278-6826/ 00 / $12.00 + .00

Direct Numerical Simulation of Curly Fibers in Turbulent Channel Flow Mehdi Soltani and Goodarz Ahmadi DEPARTMENT OF MECHANICAL AND AERONAUTICAL ENGINEERING, CLARKSON UNIVERSITY, POTSDAM, NY 13699

ABSTRACT. Wall deposition of rigid-link brous aerosols in a turbulent channel ow is studied. The instantaneous turbulent velocity vector eld is generated by the direct numerical simulation of the Navier– Stokes equation with the aid of a pseudospectral code. It is assumed that the ber is composed of ve rigidly attached ellipsoidal links. The dynamic behavior of these elongated and irregular shaped particles is markedly different from the spherical ones. The hydrodynamic forces and torques acting on the ber are evaluated and the equations governing the translational and rotational motions of the ber are analyzed. Euler’s four parameters are used, and motions of brous particles in the turbulent channel ow eld are studied. Ensembles of 8000 ber trajectories are generated and are used for evaluating various statistics. Root mean-square ber velocities and ber concentrations at different distances from the wall are evaluated and discussed. Empirical models for the deposition rate of curly bers are also developed. The model predictions are compared with the simulation data and good agreement is observed.

INTRODUCTION Transport and deposition of aerosol particles in turbulent ows has attracted considerable attention in the past three decades due to its importance in numerous industrial elds. Air pollution control, coal transport and cleaning, microcontamination control, and xerographic processes are among the areas in which the knowledge of particle transport and deposition plays a critical role. Extensive reviews on the aerosol dispersion and depositionwere provided by Fuchs (1964), Hinze (1975), Wood (1981), Hinds (1982), Hidy (1984), and Papavergos and Hedley (1984). Accordingly, particles deposit

on a surface by diffusion, impaction, or interception mechanisms. Use of digital simulation for studying dispersion and deposition of small particles in turbulent ows was the subject of a number of studies. Turbulent dispersion of small spherical particles was studied by Ahmadi and Goldschmidt (1971). Riley and Patterson (1974) simulated the particle diffusion in a numerically integrated decaying three-dimensional isotropic turbulent ow eld. Rizk and Elghobashi (1985) analyzed motions of particles suspended in a turbulent ow near a plane wall. Recently, Maxey (1987) studied the effect of gravitational sedimentation

Aerosol Science and Technology 33:5 November 2000

on aerosol particles in a random ow eld. Direct numerical simulation (DNS) of particle deposition in wall bounded turbulent ows were performed by McLaughlin (1989) and Ounis et al. (1991, 1993). Li and Ahmadi (1992, 1993) simulated the deposition rate of aerosols in turbulent channel ows near smooth and rough walls. A computational procedure for analyzing particle transport and depositionin complex passages was described by Li et al. (1994). These studies were concerned with clarifying the particle deposition mechanisms. Squires and Eaton (1991) used a DNS of isotropic turbulence to investigate the effect of turbulence on the concentration elds of heavy particles. Brooke et al. (1992) performed detailed DNS studies of vortical structures in the viscous sublayer. Elghobashi and Truesdell (1993) studied particle motion in a numerically simulated decaying (in time) homogeneous turbulent ow. An extensive review on the use of numerical simulations to study the motion of particles in turbulent ows was provided by McLaughlin (1994). Recently, Soltani et al. (1998) studied wall deposition of charged particles in a turbulent duct ow in the presence of electrostatic charges. Studies of the motions of nonspherical particles in turbulent ows are rather scarce. Due to rotational motions of nonspherical particles in a shear eld, their dynamics are much more complicated than those of their spherical counterpart. Transport and deposition of nonspherical aerosol particles in laminar ow have been studied in the past decade. Theoretical and experimental studies on the orderly, as well as stochastic motions of ellipsoidal particles, was conducted by Gallily and his coworkers (Gallily and Eisner 1979; Gallily and Cohen 1979; Schiby and Gallily 1980; Eisner and Gallily 1982; Krushkal and Gallily 1984). The Brownian coagulation of ultra ne brous particulate aerosols was investigated by Lee and Shaw (1984). The deposition of bers in the inhalation systems of humans and animals was considered by Asgharian et al. (1988), Asgharian and

Simulation of Fibers and Turbulent Flow

393

Yu (1989), Chen and Yu (1990), and Johnson and Martonen (1993). Foss et al. (1989) and Schamberger et al. (1990) analyzes the collection process of prolate spheroidal aerosols particle by the spherical collectors. Gardon et al. (1989) studied the deposition of brous particles on a lter element. Recently, Krushkal and Gallily (1988) described the orientation distribution function of ellipsoidal particles in turbulent shear eld. Fan and Ahmadi (1995a) studied the dispersion of ellipsoidal particles in an isotropic pseudoturbulent ow eld. To describe the particle orientation, they used Euler’s four parameters instead of using Euler’s angles, which has the inherent singularity. Fan and Ahmadi (1995b) developed a sublayer model for the deposition of ellipsoidal particles in turbulent ows. Most of the earlier studies on turbulent transport and deposition were almost exclusively concerned with spherical particles, with little work reported on turbulentdepositionof straight rigid bers. Many particles in nature and in industry are, however, not spherical or straight rigid ber; cloth bers and animal hairs are just a few examples. The problem of transport and deposition of curly and twisted bers in turbulent streams is virtually an untouched area of research. In this work, a direct numerical simulation procedure for studying the dispersion and deposition processes of curly bers in a turbulent channel ow is described. A rigid-link ber model consisting of ve ellipsoidal arms is considered. For curly bers, the equation of motion including inertia, Stokes drag, and lift force is used. Using Euler’s four parameters, motions of curly bers in a turbulent ow eld are studied and their deposition velocities are evaluated. Ensembles of 8000 ber trajectories, which are released from a point source, are computed and statistically analyzed. Empirical expressions for the deposition rate of curly and straight bers are also developed. The proposed empirical equations for brous particles are compared with the simulationdata and good agreement is observed.

394

M. Soltani and G. Ahmadi


FLUID VELOCITIES FIELD In this study, the instantaneous uid velocity eld in the channel is obtained by the DNS of the Navier– Stokes equation. The channel ow code used in this study is the one developed by McLaughlin (1989). The corresponding governing equations are @vf @t

r ¢ vf = 0, f

1

f

+v¢ r v = ¡ q

(1 ) r p+m r

FIGURE 1a. Schematics of the channel ow and the computational periodic cell used.

2 f

v,

where vf is the uid velocity, p is the pressure, m is the viscosity, and q is the mass density. Here, the ow is assumed to be incompressible, a constant mean pressure gradient in the x direction is imposed, and the no-slip boundary conditions on the wall are used, i.e., vf = 0

at y = §h w / 2,

(2 )

where h w denotes the channel width. All quantities are nondimensionalizedwith the aid of wall units. (Using the friction velocity u ¤ and the kinematic viscosity m , the wall units of length and time are, respectively, given by m / u ¤ and m / u ¤ 2 .) The needed computational resolution of the direct simulation increases rapidly with an increasing of the Reynolds number. For this study, a Reynolds number of 8000 based on the hydraulic diameter of the channel and the centerline velocity was used. The ow domain then had a height h of 250 wall units between the two parallel walls. The ow was assumed to be periodic with a periodic cell of 630 wall units in both spanwise (z direction) and streamwise (x direction). Figure 1a shows a schematic of the ow region and the periodic cell used in the computation. Respectively,16, 65, and 64 computational grid points in the x, y, and z directions were used. The collection points in the y direction were computed using Chebyshev series, while the grid points in the x direction were equally spaced. The choice of Chebyshev expansion in the direction normal to the channel walls increases the spatial resolution of the computation in the high shear region close to the walls where steep gradients are expected. The computational region size and grid spacing are adequate to pro-

duce the characteristic turbulent structures that are known to exist in the viscous wall region of turbulent channel ow at moderate Reynolds numbers. A time step of D t + = 0.2 was used in the ow eld simulation. Additional details of the numerical techniques were described by McLaughlin (1989). In his work, McLaughlin showed that the predicted root mean square (RMS) uctuation velocities are in good agreement with the high resolution code of Kim et al. (1987) and the experimental data. Figure 1b shows a sample velocity eld across the channel in the yz-plane at t + = 200. While the eld is random, the coherent vortical structures can clearly be seen in this gure. The motions generated by these streamwise vortices has important implications on ber transport near the wall. Relatively strong streams away and toward the wall are observed in this gure. These motions are expected to affect the ber deposition process. FIBER MODEL Here, a rigid-link ber model which is composed of ve elongated ellipsoidal links that are rigidly attached at their ends is used. The tips of the ellipsoidal links are modeled as spherical joints with a diameter equal to the ber link thickness. This model was used earlier by Fan et al. (1997) for analyzing ber removal. In the following subsection, a summary of the coordinate systems associated with the ber is presented. Coordinate Systems Figure 2 displays the associated intrinsic coordinate system, v = [n , g , f ], used to de ne



FIGURE 1b. Instantaneous velocity vector plot in the yz plane.

FIGURE 2. The ber model and the associated intrinsic coordinate system.

395

396



the relative positions of the ber nodes, and x = [ x, y, z ] is an inertial (laboratory) coordinate system. The node coordinates of the ber model in the intrinsic frame are given as

= [0, 0, 0], v = [1, 0, 1], v v 5 = [2, 0, 0], v v

1

2

v

3

4 6

= [0.5, 1, 0.5], = [1.5, 1, 0.5], = [2.5, 1, 0.5],

(3 )

where the superscripts denote the nodes shown in Figure 2. The transformation between the intrinsic and the laboratory frames is given by the linear relation x = hv .

(4 )

Here, h is the scaling factor corresponding to the physical height of the ber which is given as h = 1.633ab ,

(5 )

where b = b / a is the ellipsoidaspect ratio (ratio of the semimajor axis, b, to the semiminor axis, a). In this study, in addition to the laboratory frame x = [ x, y, z ], three additional coordinate systems are used to describe the dynamics of the brous particles. Schematic diagrams of these coordinate systems are shown in Figure 3. Here, xˆ = [ xˆ , yˆ , zˆ ] is the ber coordinate system with its origin being at the centroid of the ber and its axes being along the principal axes of the ber. The ber coordinate system is used to de ne the orientation of the ber. Similarly, the link coordinate system, xˆˆˆ = [ xˆˆˆ , yˆˆˆ , zˆˆˆ ], with its origin being at the mass center of the link and its zˆˆˆ -axes being along the major axis of each linkage is used to de ne the orientation of each link. Finally, xˆˆ = [ xˆˆ , yˆˆ , zˆˆ ], the auxiliary coordinate system (here referred to as comoving frame), is introduced. The axes of the comoving frame are parallel to the corresponding axes of the laboratory frame and its origin coinciding with that of the ber frame. The position of the ber centroid, v c , with respect to the intrinsic coordinate is given by

FIGURE 3. Schematic diagrams of inertial, comoving, link, and ber coordinate systems.

c i

v

=

P

1 2P

P j =1

mj

mj v

j i

+v

i

j +1

,

(6 )

j =1

where P = 5 is the total number of ellipsoidal links and the subscript i (i = 1, 2, or 3) denotes speci c components (along n , g , or f ) when the links have the same mass v c = [1.5, 0.6, 0.5]. Kinematics The translational displacement of the ber is given by dx = v, dt

(7 )

where x and v are the position and the velocity vectors of the ber mass center. The transformation between the comoving frame coordinates and the ber frame coordinates is given as xˆ = Axˆˆ .

(8 )

According to Goldstein (1980), the transformation matrix A = [ai j ] may be expressed as


A=

cos w sin u + cos h cos u sin w ¡ sin w sin u + cos h cos u cos w ¡ sin h cos u

cos w cos u ¡ cos h sin u sin w ¡ sin w cos u ¡ cos h sin u cos w sin h sin u

where u , h , and w are the Euler angles (the x convention of Goldstein (1980)). The time rates of change of the Euler angles are related to the angular velocities of the ber, i.e., sin h sin w sin h cos w cos h

cos w ¡ sin w 0

0 0 1

uÇ hÇ

wÇ

x

xˆ

= x x

yˆ

,

zˆ

(10) where the dot on the top of a letter stands for a time derivative and [x xˆ , x yˆ , x zˆ ] are the ber angular velocities with respect to the ber frame. The inverse transformation for evaluating time rate of change of the Euler angles is, however, singular at h = 0 or p . To cover the full rotational motion of the ber, Fan and Ahmadi (1995a,b) showed that Euler’s four parameters or quaternions, (²1 , ²2 , ²3 , g ), may be used to overcome the singularity dif culty. Therefore, here Euler’s four parameters (quaternions) as de ned by Hughes (1986) are used. The Euler angles and Euler’s four parameters are schematically shown in Figure 4. Accordingly,

[²1 ²2 ²3 ]T = e sin(X / 2), g = cos(X / 2),

(11) (12)

where e is the unit vector along the axis of rotation and X is the angle of rotation. Since the most general rotation of a rigid body is a motion with only three degrees of freedom, Euler’s four parameters are subject to a constraint given by ²12

+

²22

+

²32

+g

2

(13)

= 1.

The transformation matrix A may be restated in terms of quaternions as

A=

1 ¡ 2 ²22 + ²32 2(²2 ²1 ¡ ²3g ) 2(²3 ²1 + ²2g )

397


2(²1 ²2 + ²3 g ) 1 ¡ 2 ²32 + ²12 2(²3 ²2 ¡ ²1 g )

sin w sin h cos w sin h cos h

,

(9 )

The time rates of change of quaternions are related to ber angular velocities with respect to the ber frame, i.e., d²1 dt d²2 dt d²3 dt dg dt

1 = 2

g x xˆ ¡ ²3 x yˆ + ²2 x zˆ ²3 x xˆ + g x yˆ ¡ ²1 x zˆ , ¡ ²2 x xˆ + ²1 x yˆ + g x zˆ ¡ ²1 x xˆ ¡ ²2 x yˆ ¡ ²3 x zˆ

(15)

where t is the time. The initial conditions for the quaternions are speci ed by the initial orientation of the particle (direction cosines, ai j ). If g 6 = 0, 1 g = § (1 + 2 ²1 1 ²2 = 4g ²3

If g = 0, then ²1 = §

a11 + a22 + a33 )1 / 2 ,

(16)

a23 ¡ a32 a31 ¡ a13 . a12 ¡ a21

(17)

1 + a11 , 2

(18)

²2 =

a12 , 2²1

(19)

²3 =

a23 . 2²2

(20)

Since all solutions correspond to the same transformation matrix, the nonuniqueness of the values given by Equations (16)– (20) does not introduce any dif culty, and they lead to the same particle trajectories.

2(²1 ²3 ¡ ²2 g ) 2(²2 ²3 + ²1 g ) 1 ¡ 2 ²12 + ²22

.

(14)

398



governed by dv = fF , dt dx xˆ I c xˆ ¡ x yˆ x zˆ (Ic yˆ ¡ I cˆz ) = cˆ xFˆ , dt dx yˆ I c yˆ ¡ x zˆ x xˆ (Icˆz ¡ I cxˆ ) = cˆ yFˆ , dt dx zˆ I c zˆ ¡ x xˆ x yˆ (Icxˆ ¡ I c yˆ ) = cˆ zˆF . dt M

(21) (22) (23) (24)

Here, M is the mass of the ber, v = [v x , v y , v z ] is the translational velocity vector of the ber mass center, f F = [ f xF , f yF , f zF ] is the total hydrodynamic force (drag and lift) acting on the ber, [ Ic xˆ , I c yˆ , I c zˆ ] is the ber moments of inertia about the ber principal axes passing through the ber centroid, and [cˆ xFˆ , cˆ yFˆ , cˆ zˆF ] is the total torque acting on the ber. It should be emphasized that the translational and rotational motion in Equations (21) and (22)– (24) are expressed, respectively, in inertia and ber coordinate systems. The mass of the ber is given as P

M =

m j,

(25)

j =1

FIGURE 4. Coordinate systems de ning the Euler angles and Euler’s four parameters.

Dynamics The translational and rotational motions of a rigid-link ber moving in a general ow eld and in the absence of the gravitational force are

where m j = 43 p a 3 b q p is the mass of each ellipsoidal linkage and q p denotes the density of the ber. The moments of inertia for the ber with respect to the coordinates passing through the centroid of ber and parallel to the intrinsic coordinates may be evaluated by using the theory for the change of direction of axes of inertia (Timoshenko and Young 1956). For simplicity, considering the special case that the intrinsic frame axes are parallel to those of the comoving coordinate system, the moments of inertia tensor is given by Iˆˆ =

P

ˆj A j 0 T Iˆˆ A j 0 + II j .

(26)

j =1 j

Here, A j 0 = [amn ] is the transformation matrix between the link coordinate and comoving



Iˆˆ must be numerically evaluated. The eigenvecˆˆ and the corresponding eigenvalues, k , tors, n, must satisfy the following matrix equation:

frame at the center of each link, i.e., j j xˆˆˆ = A j 0 xˆˆ .

(27) j0

The transformation matrix A for the jth ber link is given by j0 2

1 ¡ 2 ²2 A j0 =

+ ²3j 0 2

j0 j0 j0 2 ²1 ²2 + ²3 g

j

j

j

j0

1 ¡ 2 ²3

j

j

j

j0

2 ²3 ²2 ¡ ²1 g

0 0 0 2 ²2 ²1 ¡ ²3 g 0 0 0 2 ²3 ²1 + ²2 g

j0 2

j0

j0

I¢ˆˆ nˆˆ = k nˆˆ . j0

xˆ xˆ

yˆ yˆ

j I ˆˆ ˆˆ zˆ zˆ

(1 + b 2 )a 2 j m , 5

II j = m j

2

j + zˆˆ c

j j ¡ xˆˆ c yˆˆ c

¡

j j xˆˆ c zˆˆ c

2 j xˆˆ c

j j ¡ xˆˆ c yˆˆ c

¡

2

j + zˆˆ c

j j yˆˆ c zˆˆ c

j

xˆ j = hN T (v

j

¡ v

c

(32)

),

where the transformation matrix N = [n i j ] is given as N = [ n1 , n2 , n3 ] =

The moment of inertia tensor, II j , is evaluated based on the parallel-axistheorem. Accordingly, its diagonalelements are the product of the mass of the jth link and the square of the distance of the link centroid from the corresponding axes, and its off diagonal elements are the product of the mass of the jth link and the corresponding coordinates of the centroid of j th ber link with respect to the ber centroid. That is, j yˆˆ c

(28)

ˆˆ are the unit vectors It should be noted that the ns in the directions of principal axes in the intrinsic (comoving) frame and k are the corresponding principal moments of inertia. The ber nodal coordinates in the ber frame then is given by

(29)

2a 2 j m . = 5

j0

j

ˆˆˆ nˆˆˆ m

j

j0

02 02 1 ¡ 2 ²1 + ²2

j0

principal moments of inertia of the j th ellipsoidal links as its diagonal elements. These are given by j

j

j0 j 0 j0 2 ²2 ²3 + ²1 g

where ²1 j 0 , ²2 j 0 , ²3 j 0 , and g j 0 are the Euler four parameters for the j th ber link. The inertia maˆj j trix, Iˆˆ = [ I ], is a diagonal matrix with the

I ˆˆ ˆˆ = I ˆˆ ˆˆ =

j

(31)

0 0 2 ²1 ²3 ¡ ²2 g

j0

+ ²1j0 2 j0

399

n 11 n 21 n 31

n 12 n 22 n 32

n 13 n 23 . n 33

(33)

Here, n i j is the ith component of the jth eigenˆˆ The vectors of the moment of inertia tensor I. moment of inertia tensor in the ber frame can then be evaluated as Iˆ c = N T ¢ I¢ˆˆ N,

(34)

where the diagonal matrix Iˆ c is given as j c ¡ xˆˆ c zˆˆ c

2

j j ¡ yˆˆ c zˆˆ c

j 2 xˆˆ c

j j j In Equation(30), [ xˆˆ c , yˆˆ c , zˆˆ c ] is the positionvector of the j th link centroidin the comovingframe in this special case. To obtain the ber moments of inertia about the ber principal axes, [ Ic xˆ , Ic yˆ , I c zˆ ], the eigenvalues and eigenvectors of the symmetric matrix

+

.

(30)

j 2 yˆˆ c

I c xˆ 0 0 (35) I c yˆ 0 0 . Ic zˆ 0 0 Here, I c xˆ , Ic yˆ , and Ic zˆ are the principal moments of the inertia of the ber about the ber principal axes passing through the ber centroid. Iˆ c =

400



Example The schematic diagram of the relative positions of the ber nodes with the semiminor axis of a = 5 l m and the aspect ratio of b = 25 are shown in Figure 5. According to Equation (26), the moments of inertia tensor in the comoving frame (at the instance that it is parallel to intrinsic frame) is given as 4.1 £ 10 ¡ ˆIˆ = 8.3 £ 10 ¡ 1.5 £ 10 ¡

19 20 19

8.3 £ 10 ¡ 1.9 £ 10 ¡ 1.6 £ 10 ¡

20 18 20

1.5 £ 10 ¡ 1.6 £ 10 ¡ 1.8 £ 10 ¡

19 20 18

m 2.

(36)

Using the method of diagonalization, which is to seek values of k and nˆˆ which satisfy Equation (31), the transformation matrix N is obtained as N=

0.993 ¡ 0.051 ¡ 0.1

0.082 0.083 ¡ 0.333 0.941 . 0.939 0.327

(37)

The elements of the matrix N are the direction cosines. Therefore, the angles between the ber coordinates and corresponding intrinsic coordinates are 6.7±, 109.4±, and 71±, respectively. Finally, the principal moments of the inertia of the

FIGURE 5. Schematic diagram of the relative positions of the ber nodes with the semiminor axis of a = 5 µ m and the aspect ratio of ¯ = 25.



ber is evaluated by Equation (34), i.e., Iˆc =

3.9 £ 10 0 0

¡ 19

0 1.9 £ 10 ¡ 0

18

0 0 2 £ 10¡

18

m2 .

(38)

It should be emphasized that throughout this study, A = [amn ] given by Equation (14) is the transformation matrix between the ber coordinate and the comoving frame, whereas A j 0 = j0 [amn ] given by Equation (28) is the transformation matrix between the link coordinate and the comoving frame. HYDRODYNAMIC FORCE AND TORQUE ON AN ELLIPSOIDAL LINK To study the motion of rigid-link bers in an arbitrary ow eld, the hydrodynamic forces and torques acting on each ellipsoidal link must rst be evaluated. It should be noted that the zˆˆˆ -axis is assumed to be along the major axis of each ellipsoidal link. With this assumption, the orientation is independent of the third Euler angle w . Therefore the required computation reduces due to this symmetry property. The hydrodynamic drag force acting on an ellipsoidal particle in an arbitrary ow eld under Stokes ow conditions was obtained by Brenner (1964). Accordingly, these involve an in nite series of uid velocity and its spatial derivatives. Retaining only the rst term of the series for small ellipsoids and including the nonlinear variation of drag, it follows that (Fan and Ahmadi 1995a) ˆˆ (u j ¡ v j ), (39) f hj = l p a 1 + 0.15Rea0.687 K¢ where the superscript j identi es the jth ellipsoidal link, f hj is the hydrodynamic drag in the inertial coordinate, l is the dynamic viscosˆˆ is the resistance tensor, and ity of the uid, K j j j j u = [u x , u y , u z ] is the uid velocity vector at the ellipsoid centroid in the absence of ellipj j soid. Here, Rea = aj u m ¡ v j is the link Reynolds number based on the semiminor axis of each link, the ow/link velocity difference, and the

401

uid kinematic viscosity m . Note that the correction factor for nonlinear drag used in Equation (39) involving Rea is identical to that used for a spherical particle. In the absence of any other method for correction, this approximation could be suitable for low ow/particle velocity differences. As was noted before, in this study a hat, a double hat, and a triple hat denote a quantity expressed in ber, comoving, and link frames, respectively. It should be emphasized that the origin of the link coordinate system and the comovingframe for the linkage is at the mass center of each ellipsoidal link. The transformation of resistance tensor for an ellipsoid of revolution is given by ˆˆˆ j0 ˆˆ = A j 0 T KA K .

(40)

Here, a superscript T denotes a matrix transpose. ˆˆˆ The link-frame resistance tensor K = [ K mˆˆˆ nˆˆˆ ] for an ellipsoid of revolution is a diagonal matrix with the diagonal elements being given as k xˆˆˆ xˆˆˆ = k yˆˆˆ yˆˆˆ

=

2

16(b

[(2b

¡ 3) ln(b +

2

2

b

¡ 1)

¡ 1)

2

b

, ¡ 1] + b (41)

k zˆˆˆ zˆˆˆ

=

2

8(b

[(2b

¡ 1) ln(b +

2

¡ 1)

2

b

¡ 1) b

2

. ¡ 1] ¡ b (42)

The hydrodynamic lift force on an arbitrary symmetric body was obtained by Harper and Chang (1968). Here, only the lift force due to streamwise ow shear which is dominant is included in the analysis. In a simple shear ow, the lift force for an ellipsoid of revolution may be expressed as f

Lj

=

2l

p m

a2

1/ 2

j

@u x @y j

@u x @y

1/ 2

ˆˆ L¢ K)¢ ˆˆ (u L j ¡ v j ), £ (K¢ Lj

(43)

where f is the lift force in the inertial coordinate, m is the kinematic viscosity of the uid,

402



ˆˆˆ ˆˆ j 0 T , G = A j 0 GA

j

u L j = [u x , 0, 0] is the reference ow for the lift, j j stands for an absolute value, and the lift tensor, L, is given as 0.0501 0.0182 0.00

L=

0.0329 0.0173 0.00

0.00 0.00 . 0.0373

hj

c ˆˆ = xˆ

16p l a 3 b (1 ¡ b 3(b o + b 2 c o ) j xˆˆˆ

+ (1 + b 2 ) w zˆˆˆ yˆˆˆ ¡ x

hj c ˆˆ yˆ

a3b

16p l = 3(b 2 c o + a

+ (b hj

c ˆˆ = zˆ

2

o)

(b

2

+ 1) w xˆˆˆ zˆˆˆ ¡ x

hj

)dzˆˆˆ yˆˆˆ ,

j yˆˆˆ j zˆˆˆ

(45)

,

(46)

,

(47)

hj

dzˆˆˆ yˆˆˆ = dxˆˆˆ zˆˆˆ

1 @u zˆˆˆ ¡ 2 @ yˆˆˆ 1 @u xˆˆˆ = ¡ 2 @zˆˆˆ

w zˆˆˆ yˆˆˆ = w xˆˆˆ zˆˆˆ

w yˆˆˆ xˆˆˆ =

1 @u yˆˆˆ ¡ 2 @xˆˆˆ

(48)

@u yˆˆ ˆ

, ˆ @zˆˆ @u zˆˆ ˆ , ˆ @xˆˆ @u xˆˆˆ

ˆ

@ yˆˆ

o

=b

o

=

(49)

.

The velocity gadient in the link frame needed in Equations (48) and (49) to compute the components of hydrodynamic forces may be obtained by using the matrix transformation,

b b

2

2

o

=¡ b

2

+

¡ 1

£ ln

£ ln

where c ˆˆ , c ˆˆ , and c ˆˆ are the components of hyxˆ yˆ zˆ drodynamic torques in the link coordinate. In Equations (45)– (47), the deformation rate and the spin tensors in link coordinate are de ned as @u ˆˆ 1 @u zˆˆˆ + yˆ , ˆ ˆ 2 @ yˆˆ @zˆˆ @u ˆ 1 @u xˆˆˆ + zˆˆ , = ˆ ˆ 2 @zˆˆ @xˆˆ

a

c

¡ 1)d xˆˆˆ zˆˆˆ

32p l a 3 b wˆˆ ¡ x 3(a o + b o ) yˆˆ xˆˆ hj

2

ˆˆˆ ˆˆ stand for dyadics expressed in where G and G the link and comotion frame, respectively. The dimensionless parameters a o , b o , and c o in Equations (45)– (47) are given by Gallily and Cohen (1979) as

(44)

The components of hydrodynamictorque acting on an ellipsoid of revolution in a linear shear ow was obtained by Jeffery (1922). For ellipsoidal link with the major axis along the zˆˆˆ -axis, the expressions are given as

(50)

2(b

b ¡

b +

2 ¡ ¡ 1

(b

2

b

2

b + b

2

b ¡

b 2

b

2

b

2

b

¡ 1)3 / 2 ¡ 1 ¡ 1

,

(51)

¡ 1)3 / 2

¡ 1 ¡ 1

.

(52)

It shouldbe emphasized that the effects of disturbance of the ow eld due to the presence of the neighboring ellipsoidal links and the particlewall interactions are ignored in this study. This approximation is reasonable for bers whose links are large aspect ratio ellipsoids and when the angles between the links are not too small. When the aspect ratio is < 5 and/or the angle beween the links is < 10±, the amount of error for calculating the drag force may increase to about 20%. In this study, the link aspect ratios larger than 10 were used in the analysis and the angle between the links are larger than 90±. Therefore the amount of error is expected to be negligibly small. OVERALL FORCES AND TORQUES ON FIBER To study the dynamic of the rigid-link ber, the total forces and torques acting on the ber centroid are needed. The total hydrodynamic forces (drag and lift), f F , at the mass center of ber is given by fF =

P

(f h j + f L j ).

(53)

j =1

The torque on the ber consists of the sum of all the torques acting on the links and the


d²1 dt + d²2 dt + d²3 dt + dg dt +

resultant torque due to moment of the forces. The later one, which depends on the position vector from the centroid of the ber to the mass center of each link, may be expressed as cR =

P j =1

[(x ¡ x j ) £ (f h j + fL j )].

(54)

Here, c R is the resultant torque and x and x j are the position vectors of the mass center of ber and each ellipsoidal link in the inertial frame, respectively. It should be noted that the hydrodynamic torque acting on the jth link given by Equations(45)– (47) are expressed in the link coordinate system. Therefore the total torque acting on the ber (in the inertial frame) is given as cF =

P j =1

h

A j 0 T¢ cˆˆˆ j + c R .

vj = v + x

£ (x j ¡ x).

When the ow eld is speci ed, the ber position and orientation may be found by integrating the equations of motion given by Equations (7) and (21)– (24). NONDIMENSIONAL EQUATIONS OF MOTIONS For the case of turbulentchannel ows, using the wall units, equations of motions may be restated in nondimensional form as dx + = v+ , dt +

(58)

+

²3 x

1 2

¡ ²3 x

+

+ ²2 x

+

¡ ²1 x

yˆ

+g x

xˆ

yˆ

+

+ ²1 x

+

¡ ²2 x

¡ ²2 x

xˆ

¡ ²1 x

xˆ

+

zˆ

+

zˆ

+

+g x

+

¡ ²3 x

yˆ

yˆ

,

+

zˆ

+

zˆ

(59) +F

dx x+ˆ = dt +

I c+yˆ ¡

+

I c+zˆ ¡

I c+xˆ

I c+xˆ ¡

I c+yˆ

dx

yˆ

dt +

=

dx zˆ+ = dt +

I c+zˆ

(60) x

+

I cxˆ

I c+yˆ I cˆ+z

+

x

+

x

yˆ

x

zˆ

x

+

+

+

+

+

+

zˆ

xˆ

+

xˆ x

yˆ

cˆ x+ˆ F , I c+xˆ cˆ y+ˆ F

(61)

,

(62)

cˆ zˆ+ F , I cˆ+z

(63)

I c+yˆ

where

(56)

(57)

=

+

xˆ

dv 3f = , dt + 20p b Sa + 3

(55)

Substituting Equations (35), (53), and (56) into Equations (21)– (24), the equations of motion for the ber in a general ow eld are obtained. The velocity at the centroid of each link is then given by

g x

+

To obtain the rotational motion of the ber, the total torque in ber coordinate system is needed. This is given as cˆ F = A¢ c F .

403


au ¤ t u¤ 2 v , t+ = , v+ = ¤ , m m u m m = x xˆ ¤ 2 , x y+ˆ = x yˆ ¤ 2 , x zˆ+ = x u u

a+ = +

x

xˆ

Ic+xˆ = I c4xˆ

u¤ l m

5

, 4

cˆ x+ˆ F =

cˆ xFˆ u ¤ , l m 2

f +F =

fF . l m

Ic+yˆ = Icyˆ

cˆ y+ˆ F =

u¤ l m

cˆ yFˆ u ¤ l m

2

5

, 4

,

(64) zˆ

Ic+ˆz = I czˆ

cˆ zˆ+ F =

m , u¤ 2 (65) u¤

5

l m 4 (66)

cˆ zˆF u ¤ , l m 2 (67) (68)

Here v + is the dimensionless translational ber velocity, x x+ˆ , x y+ˆ , and x zˆ+ are the nondimensional components of rotational ber velocity, p u ¤ is the shear velocity, and S = qq f is the density ratio of the particle to the uid.

,

404



s

RELAXATION TIME The relaxation time is an important time scale of particles and is commonly used to correlate different experimental data and analytical models. The relaxation time for a sphere is de ned as the ratio of the particle mass to the coef cient of drag of the particle. The nondimensional relax+2 ation time for a sphere is s + = Sd18 , where d + is the nondimensional particle diameter. For elongated nonspherical particles there is no unique relaxation time due to the anisotropy of the drag force and the signi cance of hydrodynamic force. Nevertheless, it is advantageous to introduce a suitable scalar equivalentrelaxation time, s eq+ . Shapiro and Goldenberg (1993) suggested that the relaxation time should be de ned based on the assumption of isotropic particle orientation and the averaged mobility dyadic (i.e., inverse of translation dyadic). Fan and Ahmadi (1995a,b) used the orientation averaged translation dyadic instead of mobility dyadic. For simplicity, the we use special case where the ber coordinates coincide with the comoving frame coordinates (or u = h = w = 0). Therefore the resistance tensor for each link (in inertial frame), ˆˆ is identical to that of one in the ber frame. K, For curly bers, the equivalent relaxation time may then be de ned as eq

=

M l Kf

(69)

,

where M is the total mass of the ber given by Equation (25) and K

f

= 3(Rxˆ xˆ + R yˆ yˆ + Rzˆ zˆ ) ¡ 1 .

(70)

Here Rxˆ xˆ , R yˆ yˆ , and Rzˆ zˆ are the components of the inverse of the translation dyadic for the ber. That is, ˆ = R

5

A

j0 T

ˆˆˆ j 0 KA

¡ 1

.

(71)

j =1

s

The nondimensional equivalent particle relaxation time, s eq+ , then becomes +

eq

=

20b Sa + 2 9K

f

.

(72)

EMPIRICAL MODELS Developing an analytical expression for the deposition rate of curly bers is useful for practical applications. Recently, several empirical models for the deposition rate of straight bers were reported in the literature. Shapiro and Goldenberg (1993) proposed several empirical equations which included the effect of ber length. Kvasnak and Ahmadi (1995) modi ed Wood’s equations along the line of Shapiro and Goldenberg to obtain an improved empirical model for the deposition velocity of straight ber. Using the characteristic length of the ber, Kvasnak and Ahmadi’s model may be extended for application to curly bers. Since the curly ber is more compact than the straight ber, using the longest dimension of the ber (distance between the rst and the last nodes) may not be the appropriate length scale. Here, the nondimensional height of the ber, h + , is used as a characteristic length of the curly ber. In the absence of gravitational effects, the deposition velocity for curly bers then is given as u d+ = 4.5 £ 10 ¡ 4 s

+

eq

+ 2.5 £ 10 ¡ 3 h + 2 .

(73)

The rst term in Equation (73) is the ber deposition induced by eddy diffusion-impaction.The second term is to account for the excess ber deposition due to the interception mechanism. In an earlier work, Fan and Ahmadi (1993) developed a semiempirical equation for deposition of particles on smooth and rough surfaces based on the structure of turbulent ows. Accordingly, the limiting capture trajectory of a particle in a near wall vortex is de ned as the trajectory for which the particle touches the wall before being swept away into the core. Thus, the limiting trajectories are identi ed by the terminal condition that the center of spherical particle reaches to a distance of d + / 2 from a smooth surface. For curly bers, d + / 2 is replaced by the characteristic length, h + / 2, for the terminal condition. In addition, the coef cient of lift is evaluated based on the nondimensionalequivalentparticle relaxation time, s eq+ . Thus, an empirical equation for evaluating the deposition rate for curly bers



may be proposed, i.e., + 2 1 + 8e ¡ (s eq ¡ 10) / 32 u d+ = 0.0185 + 3/2 1 ¡ 0.725s eq / S 1 / 2 £

h+2 13.68

1 / [1 + 0.725s

+3/2

eq

/ S 1/ 2 ]

.

(74)

In the subsequent sections, it is shown that Equations (73) and (74) are in good agreement with the simulated deposition velocities for curly bers. SIMULATION PROCEDURE In this section, a brief summary of the simulation procedure for evaluating the translation and rotation of a ber in a three-dimensional shear eld is provided. In the following analysis, it is assumed that the initial ber translational and rotational velocities are equal to that of the uid at the location of the ber centroid. To calculate the Stokes drag force on the ber, the instantaneous uid velocity at the center of each ber link is needed. The instantaneous velocity eld is obtained by the direct numerical simulation of the Navier– Stokes equation. In this simulation, partial Hermite interpolation is used to obtain the uid velocity component at the center of each link in the streamwise and spanwise directions, while a direct spectral sum is used for the direction normal to the walls. Balachandar and Maxey (1989) and Kontomaris et al. (1992) reported that the partial Hermite interpolation is the most accurate scheme for marginally resolved turbulent ow simulations. The computationalalgorithm consists of the following steps. 1. Identify the size of ber link thickness, 2a, and the aspect ratio, b . 2. De ne the position of the nodes in the intrinsic coordinate system. 3. Find the mass center of the ber. 4. Evaluate the principal moments of inertia of the ber with respect to the principal axes of the ber, and de ne the position of the nodes in ber coordinate (principal axes of the ber) by using Equations (26)– (35).

405

5. Find the locationand orientationof each link centroid in ber coordinate. 6. Set the initial location and orientation of the ber centroid in the inertial coordinate. 7. Use Equation (9) to obtain the initial A matrix. 8. Evaluate the initial values of Euler parameters from Equations (16)– (20). 9. Evaluate the transformation matrix A according to Equation (14). 10. Find the locationand orientationof each link centroid in the inertial coordinate. 11. Evaluate the transformation matrix A j 0 for each link according to Equation (27). 12. Evaluate the uid velocity, u j , and the uid velocity gradient, @u k+ / @xl+ , at each link centroid. Then transform the uid velocity gradient to the link coordinate system by using Equation (50). 13. Use Equation (57) to nd the velocity of the ber at each link centroid. 14. Compute the forces and torques on each link by using Equations (39), (43), and (45)– (47). 15. Evaluate the total forces and torques on the ber in inertial coordinateby Equations(53) and (55). 16. Use Equation (56) to transform the torque from inertia frame to ber coordinate system. 17. Solve the equation of motions, Equations (58)– (63), for obtaining the new particle position and Euler parameters. 18. Return to step 9, and continue the procedure until the desired time period is reached. The code uses a Lagrangian approach and solves the numerical integration of equations of motion for rigid-link bers undergoing translation and rotation.The fourth-order Runge– Kutta scheme is used for the time rate of change of the quarternions in Equation (59). An implicit Euler backward method is used for the integration of the translational and rotational equations of motion for the ber. A time step of D t + = 0.1 for integrating the equations of motion is used. The

406



size of D t + is quite small compared to the period of bers of about 2p in the viscous wall region. RESULTS AND DISCUSSION In this section, digital simulation results for dispersion and deposition of curly bers in turbulent channel ow are presented. The deposition of bers with a uniform concentration near the wall is rst analyzed. This is followed by the analysis of ber dispersion and deposition from plane and point sources. Uniform Concentration In the rst set of simulations, the bers are initially released with a uniform concentration between 1 and 5 wall units and a linearly varying concentration within 1 wall unit from the wall. For each ber thickness, ensembles of 8000 ber trajectories are generated and a time duration of 100 wall units is used for evaluating the corresponding ber deposition velocity. The air is assumed to be at 297±K with the kinematic viscosity of m = 1.5 £ 10 ¡ 5 m2 /s and a density of q = 1.12 kg/m3 . A friction velocity of 3.7 cm/s, which is a common value for clean room applications, is also assumed. Particle-to-air density ratios of S = 1000 and 2000 are used in these digital simulations. It is assumed that once a ber reaches the wall it will stick to it. This criterion is checked at each time step by evaluating the distance of the tips of each ber link from the channel walls. To observe the pattern of curly ber motions and the corresponding ber orientations,several simulations are performed. Sample trajectories of bers with a = 20 l m, b = 12 (s eq+ = 1.7) are displayed in Figures 6 – 8. Trajectories of spheres with identical nondimensional relaxation time released from the same locations as these bers are also shown in these gures for comparison. (Note that the solid lines correspond to the trajectories of ber centroid.) Figure 6 shows that the ber and the sphere have a very similar path at the beginning; however, the sphere trajectory is separated from that of the ber and drifts to the side during their transport along the channel.

FIGURE 6. Sample ber and sphere trajectories in the xz plane.

Figures 7 and 8 show the yz and xy projections of the trajectories. (Note that y + = 125 is the upper wall in these gures.) It is observed that both the sphere and the ber move away from the upper channel wall but along different paths. These results imply that a particle’s shape affects its trajectories signi cantly. Figures 9 and 10 show the orientation time histories of a curly ber with a = 8 and b = 20.

FIGURE 7. Sample ber and sphere trajectories in the yz plane.



407

FIGURE 8. Sample ber and sphere trajectories in the xy plane.

FIGURE 10. Sample ber orientation time history in the vicinity of the upper channel wall.

For every 10 time steps, the ber trajectory is recorded. These gures illustrate the way a ber translates and rotates under the action of hydrodynamic forces and torques in the simulated turbulent channel ow. Figure 9 shows the ber trajectory away from the upper channel wall in the xy plane. The relatively slow rotation of the ber about its principal axes can be seen in this gure. Figure 10 shows the xy projection of the

ber trajectory in the vicinity of the upper channel wall. A rapid rotational motion is observed in this gure. This is because as the ber approaches the wall, the mean ow shear rate increases signi cantly, which results in a high rate of ber rotation. It should be emphasized that unlike spherical particles, the trajectory of a brous particle is a strong function of its initial orientation as it enters the ow region of nearwall vortices. To study the deposition velocity of curly bers, a series of computer simulationsfor bers with different link thicknesses, densities, and aspect ratios were performed. The simulations were performed on a SPARC-5 workstation with 128-M memory and a speed of 110 MHz. For a typical run of 500 time steps (100 wall units of time) for 8000 identical bers, the total CPU time on a SPARC-5 computer was about 5000 min (approximately 4 days). Sample variations of the number of deposited bers versus time for different aspect ratios and link thicknesses are shown in Figure 11. Ensembles of 8000 ber trajectories are used for each ber size and density. This gure shows that the number of deposited bers reaches to a steady slope and then begins to level off as the ber concentration in the domain is depleted. It

FIGURE 9. Sample ber orientation time history near the upper channel wall.

408



TABLE 1. Variations of the deposition velocity vs ber thickness u d+ S = 1000

FIGURE 11. Variation of the number of deposited bers versus time for different aspect ratios.

is observed that the number of deposited bers increases as the ber link thickness increases. For a xed link thickness, Figure 11 also shows that the number of deposited bers increases as the aspect ratio increases. The dimensionless deposition velocity for particles released with a uniform concentration Co near a surface is given by u d+ = J / C o u ¤ ,

S = 2000

a (l m) b

= 12

b = 20 b

6 8 10 12 20

0.00025 0.0004 0.00047 0.001 0.0032

0.00031 0.0009 — 0.0027 0.0077

0.00027 0.00045 0.00075 0.0013 0.005

= 12

For a xed density ratio, the deposition velocity increases as the aspect ratio increases. This table also shows that for a xed ber size, the deposition velocity increases somewhat as the density ratio increases. Figure 12 shows variations of deposition velocity with the equivalent particle relaxation time for brous particles of various densities. The equivalentparticle relaxation time used here is de ned by Equation (72). The ber height, h, varies with the link tickness, a, in accordance with Equation (5). In Figure 12 the dot-dashed and solid lines are predictions of the empirical models given by Equations (73) and (74) for deposition of brous particles. The simulation data

(75)

where J is the particle mass ux to the wall per unit time. The deposition velocity, u d+ , in Equation (75) may be estimated as u d+ =

Nd td+

No yo+

,

(76)

where No is the initial number of particles uniformly distributed in a region within an initial distance yo+ from the wall and Nd is the number of deposited particles in the time duration td+ . The time duration should be selected in the quasi-equilibrium condition when Nd / td+ becomes almost constant. The simulated deposition velocitiesfor bers with different link thicknesses, aspect ratios, and densities are listed in Table 1. It is observed that the deposition velocity, u d+ , increases as the ber thickness increases.

FIGURE 12. Variation of curly ber deposition velocity with ber relaxation time for different density ratios.


of McLaughlin (1989) for spherical particles with S = 713 and the sublayer model prediction of Fan and Ahmadi (1995b) for ellipsoidal particles are reproduced in this gure for comparison. In addition, the present direct simulation results for spherical particles with S = 1000 are also shown in Figure 12. It is observed that the deposition velocities for brous particles with S = 1000 are much higher than those for spherical particles with S = 1000 and S = 713. The reason is that the interception process for curly bers is more effective than those for spherical ones. Figure 12 shows that the sublayer model prediction of Fan and Ahmadi (1995b) for ellipsoidal particles are lower than those for curly bers with link aspect ratio of 12. Note that the simulations data of Fan and Ahmadi (1995b) are for aspect ratios of b = 2, and 5. To compare with these results, an equivalent aspect ratio for curly bers is needed. For curly bers, an equivalent aspect ratio may be evaluated by the ratio of the maximum dimension of the ber to the ber link thickness. The rigid-link ber model used in this study has a maximum dimension of L max = 2.74 h. Using Equation (5), the equivalent aspect ratio b eq = L max / 2a is about 2.24b , where b is the link aspect ratio. For example, for the link aspect ratio of b = 12, the equivalent aspect ratio of the curly ber is b eq ’ 27. This aspect ratio is several times larger than those used by Fan and Ahmadi (1995b). As was noted before, a large increase in aspect ratio leads to a substantial increase in deposition rate due to interception mechanism. For s eq+ > 0.1, it is observed that the deposition velocity increases sharply as the equivalent particle relaxation time increases. This is due to the turbulent eddy-impaction effect, which becomes important for this range of relaxation time. Figure 12 also shows that the deposition velocities decrease as the density ratio increases. This is because, for a xed s eq+ , the ber size decreases as the density ratio increases. The interception of smaller bers by the surface is less ef cient, which causes the deposition velocity to decrease. From Figure 12 it is observed that


409

FIGURE 13. Variation of curly ber deposition velocity with ber relaxation time for different aspect ratios.

the deposition velocities for bers with different density ratios as predicted by Equations(73) and (74) are in close agreement with the present digital simulation. Figure 13 shows variations of deposition velocity with particle relaxation time for brous particles with different aspect ratios. A density ratio of S = 1000 and b = 12 and 20 are used in these simulations. The empirical model predictions of Equations (73) and (74) are also presented in this gure for comparison. It is observed that, as the aspect ratio increases, the deposition velocity of bers increases signi cantly. The reason is that the ber size (length) increases with aspect ratio and the interception process becomes more effective. This is consistent with the observations of Shapiro and Goldenberg (1993) and Fan and Ahmadi (1995b), who noted the importance of interception process on elongated ber deposition. Figure 13 also shows that the proposed empirical equations for curly bers are in good agreement with the simulation data for different aspect ratios. Plane and Point Sources In the second set of simulations, dispersion and deposition from plane and point sources of curly

410


bers at a distance of 5 wall units from the channel walls are studied. Here, an ensemble of 8000 bers (4000 on each side of the channel) with a = 20 l m, b = 12, and a density ratio of S = 1000 are used. Fiber trajectories are evaluated for a duration of 200 wall units of time and statistically analyzed. The mean, the standards deviation (SD), and the absolute maximum and minimum ber positions are computed. Note that Y 0 + = 125 ¡ j Y + j , which is the distance from the wall, is used in several gures in this section. The trajectory statistics for bers with a = 20 l m, b = 12, and a point source distance of 5 wall units from the upper wall are shown in Figure 14. The trajectory statistics for spherical particles with identical nondimensional relaxation time is shown in this gure for comparison. The nature of the spreading of bers and spheres could be clearly observed from this gure. It is seen that the bers and spheres spread about §35 wall units with respect to their mean in the time duration of 200 wall units. Figure 14 also indicates that the mean trajectories move away from the wall with ber moving slightly further away. These results show that the ber’s dispersion behavior in the channel is quite similar to that of spheres.

FIGURE 14. Fiber trajectory statistics for a point source at a distance of 5 wall units from the upper wall.


FIGURE 15. Concentration of bers versus distance from the wall at time t+ = 200 for a particle source at a distance of 5 wall units from the upper wall.

To study the concentration pro le in the channel, the height of the channel is divided into equally spaced bins. At the last time step (t + = 200), the number of bers in each bin is counted. Here a bin size of 1 wall unit and bers with a = 20 l m, b = 12 are used. Note that when a ber touches the wall, it is considered removed from the ow domain. The resulting concentration pro le for a point source of bers at a distance of 5 wall units from the channel walls is shown in Figure 15. As expected, the maximum concentration occurs at the initial source distance from the wall. This gure also shows that the bers disperse, with some reaching the core of the channel. Figure 16 shows the variation of the streamwise ber and uid mean velocities for a source distance of 5 wall units from the wall. Here, bers with a = 20 l m and b = 12 (s eq+ = 1.7) are used. The mean velocity for a sphere with an identical relaxation time is shown in this gure for comparison. It is observed that the bers and the spheres lead the ow at close distances near the wall. These particles carry much of their streamwise momentum as they cross the viscous sublayer. Thus they have a velocity higher than the local uid velocity near the wall. When the


FIGURE 16. Variations of streamwise mean velocity components.

particle distance from the wall exceeds several wall units, these large particles lag the ow. This gure also shows that the ber mean velocity is somewhat larger than that of the sphere away from the wall but it is slightly smaller at shorter distances. Figure 17 displays variations of the RMS ber and uid velocity components near the wall. Here, bers with a = 20 l m and b = 12 are

FIGURE 17. Variations of RMS velocity components.


411

used. The components of RMS velocities for spherical particles with identical relaxation time are also shown in this gure for comparison.Figure 17 shows that the ber RMS vertical velocity components become larger than that of the uid ones at distances smaller than a few wall units from the wall. This is mainly due to the effect of turbulence near wall eddies, which play a major role for moving large particles toward the wall. In general, Figure 17 also shows that the RMS ber velocities are slightly higher than those of the sphere. The exception is the axial RMS components at short distances from the wall. In an earlier work, Ounis et al. (1993) has shown that the distribution of the initial locations of depositedspherical particles in the plane parallel to the wall form distinct lines. This observation further con rmed the importance of near wall coherent eddies on particle deposition process. A similar numerical experiment is performed for the curly ber deposition process. Starting with a uniform distribution of bers on a plane at a distance of 5 wall units from the upper wall of the channel, the initial locations of the deposited bers (in a duration of 200 wall units of time) are evaluated. In this simulation, bers with a = 20 l m and b = 12 are used and the results for rigid-link bers are shown in Figure 18. It is observed that the initial

FIGURE 18. Distribution of initial locations of deposited bers in the xz plane.

412



location of deposited bers are scattered around lines along the channel. These bands correspond to the location of streamwise vortices shown in Figure 1. This implies that the rigid-link bers are mainly deposited by downsweep motions generated by the coherent vortices. Figures 19– 21 show sample trajectories for two bers with initial location in the z + ’ 100 band in Figure 18. These bers deposit on the upper wall in a time duration of 200 wall units. The trajectories for spherical particles which are released at the same locations as those bers and have identical relaxation times are shown in these gures for comparison. Figure 19 shows

FIGURE 20. Sample ber and sphere trajectories in the yz plane corresponding to the band at z+ ’ 100.

FIGURE 19. Sample ber and sphere trajectories in the xz plane corresponding to the band at z+ ’ 100.

the xz projection of the bers at different time steps as they translate and rotate in the random velocity eld. Figures 20 and 21 show the zy and xy projections of the ber trajectories. In contrast to the bers, these gures show that the spherical particles do not deposit on the wall. As was noted before, the curly bers have large capture ef ciency, which makes their deposition more likely when compared with the spherical ones. This implies that the particle shape plays a major role on the transport and the deposition of brous particles in turbulent channel ow. Close-upsof the ber orientationsnear the upper wall for three deposited bers are shown in



413

FIGURE 22. A close-up of the ber orientation near the upper wall.

tions given in (73) and (74) with the experimental data reported by Shapiro and Goldenberg (1993) and Kvasnak and Ahmadi (1995) are presented in this section. Shapiro and Goldenberg (1993) studied deposition of glass bers with averaged diameters of 0.93 and 1.86 l m and various lengths ranging from 2 to 50 l m in a pipe. Deposition velocities for three ow Reynolds numbers of Re = 3 £ 104 , 5.4 £ 104 ,

FIGURE 21. Sample ber trajectories in the xy plane corresponding to the band at z+ ’ 100.

Figures 22– 24. These gures show the ber orientations in the xy plane at three time steps before their deposition on the upper wall. Here, the ber trajectories are recorded for every 10 time steps. These gures show that these curly bers deposit on the wall by one of their end nodes reaching the wall. COMPARISON WITH EXPERIMENTAL DATA Comparisons of the deposition velocities for curly bers as predicted by the empirical Equa-


414



and 8.7 £ 104 (corresponding to u ¤ ’ 0.6, 1, and 1.6 m/s, respectively) were reported. Kvasnak and Ahmadi (1995) measured the oor deposition rate for 5 l m glass bers with their length varying between 20 to 100 l m in a horizontal channel. The ow Reynolds number based on the hydraulic diameter of the channel was 1.7 £ 104 , which corresponded to a friction velocity of 0.27 m/s. Variation of deposition velocity as predicted by empirical Equations given in (73) and (74) with the equivalent relaxation time are shown in Figures 25 and 26. The model predictions and the experimental data for 0.93 l m diameter bers are shown in Figure 25, while those for 1.86 l m diameter bers are shown in Figure 26. To compare the model predictions with the experimentaldata, the gravitationalsedimentation, s eq+ g + (where g + = gm / u ¤ 3 ), is added to Equations (73) and (74). Here, a particle-to- uid density ratio of S = 1820 is used in the model predictions. The ber height, h, varies with the link aspect ratio, b , in accordance with Equation (5). Four friction velocities u ¤ = 0.27, 0.6, 1, and 1.6 m/s corresponding to the ow conditions of Kvasnak and Ahmadi (1995) and Shapiro and Goldenberg (1993) are used in the analyses. It is observed that the empirical model pre-


FIGURE 25. Comparisons of the model predictions for curly bers with the experimental results of Shapiro and Goldenberg for 0.93 µ m diameter straight glass bers.

dictions are lower than the experimental data. This is because the curly bers are more compact than the straight bers used in the experiment. As expected, the deposition velocity for curly bers is less than that of straight bers.

FIGURE 26. Comparisons of the model predictions for curly bers with the experimental results of Shapiro and Goldenberg (for 1.86 µ m diameter straight glass bers) and Kvasnak and Ahmadi (for 5 µ m diameter straight glass bers).


For a xed s eq+ , the deposition velocity increases as the Reynolds number (or shear velocity) decreases. These gures show that the highest deposition velocities occur for u ¤ = 0.27 and the lowest ones are those for u ¤ = 1.6 m/s. This is because g + is proportional to u ¤ ¡ 3 . Therefore the contribution of the gravitational sedimentation changes accordingly. To extend the applicability of the empirical Equations (73) and (74) to straight bers, the proper characteristic length must be used. As was noted by Kvasnak and Ahmadi (1995), the maximum dimension of the ber L + = 2ab is the appropriate characteristic length. Thus the deposition velocity for straight ber may be estimated by replacing h + by L + in Equations(73) and (74). Figure 27 shows the variation of deposition velocity with relaxation time as predicted by Equation (73) for straight and curly bers. The sublayer model predictionof Fan and Ahmadi (1995b) for ellipsoidal particles with an aspect ratio of b = 5 is also shown in this gure for comparison. As noted before, the equivalent aspect ratio for curly bers is b eq = 2.24b . To compare the results of straight and curly bers, their maximum dimensions must be identical. This can be achieved by choosing the aspect ratio of the straight ber 2.24 times larger than that


415

for the curly ber link. Here, the aspect ratios of 5 and 2.23 are used for straight ber and curly ber link, respectively. As expected, the deposition velocities as predicted by Equation (74) for straight bers are higher than those for curly bers. Furthermore, the predicted depositionvelocities for straight ber are in good quantitative agreement with the sublayer model predictionof Fan and Ahmadi (1995b). CONCLUSIONS In this work, a numerical simulation method for analyzing the deposition rate of brous particles in a turbulent channel ow is described. The ber equation of motion includes the lift force and turbulent dispersion effects. Two sets of simulations are performed. Starting with an initially uniform concentrationnear the wall, the deposition process of brous particles is studied. The deposition rate of rigid-link bers for various densities and aspect ratios are evaluated. In the second set of simulations, a plane source of bers at a distance of 5 wall units from the upper channel wall is considered. Several simulations are performed and ber trajectory statistics are evaluated, compiled, and statistically analyzed. Empirical expressions for estimating the deposition velocityof curly bers are also developed. For different densities and aspect ratios, the model predictions are compared with the simulation results. Based on the presented results, the following conclusionsmay be drawn:

A ber trajectory in the near wall turbulent

FIGURE 27. Comparisons of the model predictions with the sublayer model of Fan and Ahmadi (1995b).

ow depends strongly on its shape and its initial orientation. For a xed equivalent relaxation time, the deposition rate of curly bers is larger than that of spherical particles and is smaller than that of straight bers. For a xed relaxation time, the depositionrate of curly bers increases as the aspect ratio of ber links ( ber size) increases. For a xed relaxation time, the depositionrate of curly bers increases as the density ratio decreases.

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Curly bers acquire considerable rotational

speed in the vicinity of the wall due to strong ow shear. The air ow turbulencehas a signi cant effect on curly ber trajectories, and the dispersion pattern of curly bers in the bulk of turbulent channel ow is quitesimilar to that of spheres. The particle leads the ow at the close distances near the wall. The streamwise mean velocity for a ber is somewhat higher than that for a sphere with an identical equivalent relaxation time away from the wall. The RMS velocitiesof particles are lower than those of uid away from the wall. The RMS ber velocities are slightly higher than those of the spheres. Near wall turbulence, streamwise eddies strongly affect the ber deposition process. The initiallocationsof deposited bers is concentrated in the bands where the coherent vortices form strong streams toward the channel wall. The proposed empirical equations for practical applications appears to properly predict the trend of variation of the deposition rate of curly and straight bers. Comparisons with the available experimental data for straight bers show that the model predictions have the correct trend.

The nancial support of the Department of Energy through the University Coal Research program at NETL and the New York State Science and Technology Foundationthrough the Center for Advanced Material Processing (CAMP) of Clarkson University throughout this work is gratefully acknowledged.

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Received March 29, 1999; accepted November 30, 1999.