Characteristics of Fiber Suspension Flow in a Turbulent … · Jianzhong Lin, Suhua Shen, Xiaoke Ku Department of Mechanics Hangzhou, Zhejiang CHINA Correspondence to: Jianzhong Lin

Journal of Engineered Fibers and Fabrics 17 http://www.jeffjournal.org Volume 8, Issue 1 – 2013

Characteristics of Fiber Suspension Flow in a Turbulent Boundary Layer

Jianzhong Lin, Suhua Shen, Xiaoke Ku

Department of Mechanics Hangzhou, Zhejiang CHINA

Correspondence to:

Jianzhong Lin email: [email protected]

ABSTRACT The equations of averaged momentum, turbulence kinetic energy, turbulence dissipation rate with the additional term of the fibers, and the equation of probability distribution function for mean fiber orientation are derived and solved numerically for fiber suspension flowing in a turbulent boundary layer. The mathematical model and numerical code are verified by comparing the numerical results with the experimental ones in a turbulent channel flow. The effects of Reynolds number, fiber concentration and fiber aspect-ratio on the mean velocity profile, turbulent kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient are analyzed. The results show that the velocity profiles become full, and the turbulent kinetic energy, Reynolds stress and eddy viscosity coefficient increase, while turbulent dissipation rate decreases, as the Reynolds number, fiber concentration and fiber aspect-ratio increase. The effect of the fiber aspect-ratio on the turbulent properties is larger than that of the Reynolds number, but smaller than that of the fiber concentration in the range of parameters considered in this paper. INTRODUCTION The flows of turbulent fiber suspension in boundary layer flow are involved in many practical applications. The characteristics of the flow process and final product usually depend on the velocity profile and turbulent properties of fiber suspension. The turbulence is used to make high quality products and good energy savings in some industrial processes. However, there are still many unknowns about the flows of turbulent fiber suspension. In order to understand the flow characteristics, it is important to know the effect of fibers on the velocity profile, turbulent kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient.

There has existed some literature dealing with the fiber suspension flowing in the turbulent boundary layers. Tucker [1] used the rheological theories for fiber suspensions to examine the flow in narrow gaps. Order-of- magnitude estimates were developed for the velocity, stress and fiber-orientation components. The estimates reveal four distinct flow regimes. Limitations due to entrance effects and start-up transients were discussed, along with applications to the modeling of injection and compression molding of fiber-reinforced polymers. Andersson and Rasmuson [2] studied the flow and transition of fiber suspensions to turbulence in a rotary shear tester by Laser Doppler Anemometer (LDA) measurements. The results show that the fluctuation velocities approached those of single-phase flow with an increasing rotational speed, until they were nearly equal. The velocity profile is linear and close to the wall even with fibers present. The presence of the fibers also flattened the profiles, indicating an increased momentum transfer. Pettersson and Rasmuson [3] performed detailed measurements of mean and RMS velocities by LDA of a turbulent gas/fiber/liquid suspension in a rotary shear tester. Plotting RMS and mean velocities versus impeller speed and power input indicates that both decrease with increasing gas and fiber contents. Sun et al. [4] used the LVEL model of Phoenics software to simulate the turbulent boundary layer of medium-consistency pulp fiber suspensions. The results show that, from the inner surface of the pipeline to 0.0009m, the distributing of the velocity is linear across the water ring. The surface of the plug is quite smooth. The concentration is very high away from the inner surface, while it is low on the surface of the pipeline. Lin et al. [5] derived the equation of probability distribution function for


mean fiber orientation in a turbulent boundary layer, in which the correlation terms of the fluctuating velocity, fluctuating angular velocity with the fluctuating probability distribution function were related to the gradient of mean probability distribution function and the dispersion coefficients. The finite-difference method was used to solve the equation numerically. The results show that the fibers tend to align with the streamline, and the fiber aspect-ratio has a significant effect on the orientation distribution of fibers, while the effect of the distance from the wall is negligible. Xu and Aidun [6] measured the velocity profile of fiber suspension flow in a rectangular channel by pulsed ultrasonic Doppler velocimetry, and the effect of fiber concentration and Reynolds number on the shape of the velocity profile was investigated. Five types of flow behavior were observed when the fiber concentration increases or the flow rate decreases progressively. The turbulent velocity profiles of fiber suspension can be described by a correlation with fiber concentration and Reynolds number as the main parameters. It was found that the presence of fiber in the suspension will reduce the turbulence intensity and thus reduce the turbulent momentum transfer. Schmidt and Zhu [7] used digital imaging and found that the mixing layer in the channel becomes less uniform with an increase in fiber length and suspension concentration and with a decrease in the pressure drop across the flow restriction. Paschkewitz et al.

[8] calculated the orientation moments and stresses along the Lagrangian pathlines in the flow, and showed that the more commonly observed small stress fluctuations appeared to make the largest contribution to the fiber dissipation of turbulent kinetic energy and thus are responsible for the majority of the drag reduction effect. The largest contribution to fiber dissipation of turbulent kinetic energy is made by small fluctuations in the span wise shear stress component. Pettersson et al. [9] performed a detailed study of the flow behavior in the near wall region of pulp suspensions up to 4.7% using Laser Doppler Anemometry. Axial mean velocity profiles show a distinct plug flow and an increase of the plug region as the flow rate decreases and fiber concentration increases. They indicated a dilution region at 1-2 mm from the wall that is larger than the annulus region. The dilution region increases with increasing flow rate, decreasing concentration when using longer fibers. Gillissen et al. [10] studied turbulent fiber suspension flow using a direct numerical simulation

and concluded that the instantaneous stress cannot be simulated directly with the particle method due to computer restrictions. The effect of fibers on the turbulent flow is equivalent to an additional Reynolds averaged viscosity. Zhang et al. [11] studied the transport and deposition of angular fibrous particles in turbulent channel flows. For a dilute suspension of fibers, a one-way coupling assumption was used in that the flow carries the fibers, but the coupling effect of the fiber on the flow was neglected. Ensembles of fiber trajectories and orientations were generated and statistically analyzed. The results were compared with those for spherical particles and straight fibers and their differences were discussed. Effects of fiber size, aspect ratio, fiber angle, turbulence near wall eddies, and various forces were studied. Lin et al. [12] proposed a model relating the translational and rotational transport of orientation distribution function (ODF) of fibers to the gradient of mean ODF and the dispersion coefficients to derive the mean equation for the ODF, and predicted the ODF of fibers by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. The results show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation distribution of fibers, respectively. The shear stress of fiber suspension is larger than that of Newtonian solvent and the first normal stress difference is much less than the shear stress. The existence of fibers inevitably affects the turbulent flow. The investigations mentioned above did not involve the effect of fibers on the turbulent properties. In order to understand the characteristics of turbulent fiber suspensions, it is necessary to modify the equations of the averaged momentum, turbulence kinetic energy and turbulence dissipation rate. Lin et al. [13] derived the modified equations of the averaged momentum and the probability distribution function for the mean fiber orientation, and applied the equations to the channel flow. The results show that the flow rate of the fiber suspension is large under the same pressure drop in comparison with the rate of Newtonian fluid. The relative turbulent intensity and the Reynolds stress in the fiber suspension are smaller than those in the Newtonian flow. Gillissen et al. [14] studied fiber-induced drag reduction using Navier-Stokes equations supplemented by the fiber stress tensor in a turbulent channel flow. The results were used to


validate an approximate method for calculating fiber stress, in which the second moment of the orientation distribution is solved. Lin et al. [15]

solved the averaged momentum equation with the term of additional stress resulting from fibers in the flow of turbulent fiber suspensions flowing through a contraction with a rectangular cross-section to get distributions of the mean velocity, mean pressure, turbulent kinetic energy and turbulent dissipation rate. It is found that fibers reduce turbulent intensity and turbulent dissipation at a rate at the central line, but fibers enhance them over the cross section at the exit. Fibers have no effect of restraint on the turbulence in the contraction flow. However, the modified equations of turbulence kinetic energy and turbulence dissipation rate with additional terms of the fibers have not been derived and solved to determine the effect of fibers on the turbulent properties. Therefore, in the present study, we derive the modified equations of averaged momentum, turbulence kinetic energy and turbulence dissipation rate with additional terms of the fibers, and apply the equations to the turbulent boundary layer flow. We chose the boundary layer flow because it is a typical bounded flow and thus provides a building block for many practical inhomogeneous flows. EQUATIONS OF FIBER SUSPENSIONS For an incompressible fully developed turbulent fiber suspension flow, the continuity equation and momentum equation with additional fiber stress tensor are [16]:

0i

i

u

x

＝ (1)

2

2

1

1[ ( ) ]

3

i i ij

j i j

fijkl kl ij kl kl

j

u u upu

t x x x

a I ax

(2)

where iu is the velocity, p is the pressure, is

the kinetic viscosity of the suspending fluid,

/ 2ij i j j iu x u x is the tensor of rate

of strain, 3 6ln 2f n l r is the apparent

viscosity of the suspension for the dilute regime (nL3<<1) [16] ( μ is the viscosity of the suspending fluid, n is the number of fibers per unit volume, l is the fiber half-length and r is the fiber aspect-ratio),

kla and ijkla are the second- and fourth-orientation

tensor of fiber [17, 18]:

(3)

dijkl i j k la p p p p p p (4)

where ip is a unit vector parallel to the fiber’s axis, p is the probability distribution function for fiber orientation, and d p p is the probability that the fiber orientations are located between p and dp p . p satisfies the equation of conservation:

j

jj j

pu

t x p

(5)

where jp is the fiber angular velocity [19]:

i ij j ij j kl k l ip p p p p p (6)

where ij is the vorticity tensor,

( ) / 2ij j i i ju x u x , 2 21 1r r is

related to the aspect-ratio r .

AVERAGED MOMENTUM EQUATION The instantaneous velocity, pressure, tensor of rate of strain and orientation tensor can be expressed as a mean part plus a fluctuation part:

(7)

We substitute Eq. (7) into Eq. (2) and take the

average. The terms of ' 'ijkl kla and ' 'kl kla are

related to the small scale vortices which are

isotropic. Since 'ijkla and 'ija depend on the

rotation angle of the fiber, while 'ij depends on

the spatial position of the fiber, when the coordinate system turns 180o, we have

' 'ijkl kla = ' 'ijkl kla and


' ' ' 'kl kl kl kla a because of the isotropic

behavior, which means ' ' 0ijkl kla

and ' ' 0kl kla . Then we can derive the averaged

momentum equation with an additional term of the fibers as follow:

2

2

1

1)

3

i i ij

j i j

i j fijkl kl ij kl kl

j j

U U uPU

t x x x

u ua I a

x x

（

. (8)

EQUATION OF THE PROBABILITY DISTRIBUTION FUNCTION FOR MEAN FIBER ORIENTATION

We can obtain kla and ijkla in Eq. (8) by taking

average of Eq. (3) and Eq. (4):

dij i ja p p p p

(9)

dijkl i j k la p p p p p p (10)

Then we should build the equation for p .

Expressing the instantaneous velocity, probability distribution function for fiber, fiber angular velocity as a mean part plus a fluctuation part:

i i iu U u ，， i i ip p p . (11)

Substituting Eq. (11) into Eq. (5) and taking the average, we have:

( ) ( )j j jj

j j j j

p u pU

t x p x p

. (12)

Similar to the relationship between Reynolds stress and mean velocity gradient, the correlations between fluctuating probability distribution function and fluctuating velocity, fluctuating fiber angular velocity can be written as:

= ; j x j pj j

u px p

(13)

where 1/221.3 5 3x k and 1/20.7 4 15p with k

being the turbulence kinetic energy, the turbulence dissipation rate and the kinematic viscosity [20]. x and p are the dispersion

coefficients of linear and angular displacement, respectively. Combining Eq. (13) and Eq. (12) we have:

2 2

2 2

jj x p

j j j j

pU

t x p x p

＋ .(14)

In order to get jp in Eq. (14) we express the

instantaneous quantities in Eq. (6) as a mean part plus a fluctuation part, and take the average:

i ij j ij j kl k l ip p p p p p . (15)

Substituting Eq. (15) into Eq. (14) yields:

(16)

For the incompressible flow we have 0ii ＝

and 0ii , then Eq. (16) can be written as:

(17)

which is the equation of probability distribution function for the mean fiber orientation. EQUATION OF THE TURBULENCE KINETIC ENERGY Up to now there is no turbulent model for turbulent fiber suspension flows; therefore, we extend the turbulent model in the pure fluid to fiber suspensions. Here we use k-ε model because it is valid in the turbulent boundary layer for pure fluid.

The Reynolds stress tensor i ju ur ¢ ¢- in Eq. (8)

can be expressed as:

22 ( )

3ji

i j T ijj i

uuu u k

x x

(18)


where the eddy viscosity 2 /T C k with k being the turbulent kinetic

energy, the turbulent dissipation rate and 0.09C . For solving Eq. (18) and Eq. (8),

k -equation and -equation with additional terms of the fibers are given:

[( ) ]i Tj i j k

j j j k j

Uk kU u u S

x x x x

(19)

2

1 2

[( ) ]

ij i j

j j

T

j j

uU C u u C

x k x k

Sx x

(20)

where is the viscosity of the suspending fluid,

1 1.44C , 2 1.92C , 1.0k and

1.3 [21]. kS and S are the source terms

caused by the existence of the fibers. As shown in Eq. (2), the source term in the momentum equation is:

1[ ( ) ]

3f

ijkl kl ij kl klj

a I ax

(21)

Multiplying n component of Eq. (21) with mu , m

component of Eq. (21) with nu , respectively, and

then combining both, we have

1{ [ ( ) ]

3

1[ ( ) ]}

3

fm njkl kl nj kl kl

j

n mjkl kl mj kl klj

u a I ax

u a I ax

.

(22)

Substituting Eq. (7) into Eq. (22) and taking the average yields:

(23)

As shown in Eq. (8) the average form of source term is:

1)

3f

ijkl kl ij kl klj

a I ax（

(24)

Multiplying n component of Eq. (24) with mu , m

component of Eq. (24) with nu , respectively, and

then combining both, we have

{

1 1}

3 3

fm njkl kl n mjkl kl

j j

nj m kl kl mj n kl klj j

U a U ax x

I U a I U ax x

(25)

Subtracting Eq. (25) from Eq. (23) yields:

1{

3

1}

3

fm njkl kl nj m kl kl

j j

n mjkl kl mj n kl klj j

u a I u ax x

u a I u ax x

. (26)

Let m=n and multiplying Eq. (26) with 1/2 we have

(27)

Eq. (27) gives the effect of fibers on the turbulent kinetic energy. From Eq. (27) we can see that such effect is related to the apparent viscosity of the suspension, fluid density, fluctuating velocity, mean second- and fourth-orientation tensor of fiber, fluctuating tensor of rate of strain.


EQUATION OF THE TURBULENCE DISSIPATION RATE Derivating Eq. (21) with respect to mx and then

multiplying with 2 i mu x , we have

2

2

2

{

3 3

12 [ ( ) ]

3

2 ijkl ijklkl i ikl

m m j m j m

ijkl i kl kl iijkl

m m j m j m

ij ijkl i kl kl ikl

m m j m j

f iijijkl kl kl kl

m m j

f a au u

x x x x x x

a u ua

x x x x x x

I Iu a a u

x x x x x x

ua I a

x x x

2

}3 3

m

ij ijkl i kl kl ikl

m m j m j m

I Ia u ua

x x x x x x

(28)

Substituting Eq. (7) into Eq. (28) and taking the average yields:

(29)

Derivating Eq. (24) with respect to mx and then

multiplying 2 i mu x , we have

(30)

Subtracting Eq. (30) from Eq. (29) yields:

2

2

2

2 {

3 3

f ijkl ijklkl i ikl

j m m m m j

ijkl i kl kl iijkl

m m j m j m

ij ijkl kl i kl ikl

j m m m j

a au uS

x x x x x x

a u ua

x x x x x x

I Ia u a u

x x x x x

2

}3 3

m

ij ijkl i kl kl ikl

m m j m j m

x

I Ia u ua

x x x x x x

(31)

Effect of fibers on the turbulent dissipation rate is given by Eq. (31) which is complex and related to the fluid viscosity, apparent viscosity of the suspension, fluid density, fluctuating velocity, mean second- and fourth-orientation tensor of fiber, fluctuating tensor of rate of strain. Eq. (8), (17), (18), (19), (20), (27) and (31) are the basic equations of turbulent fiber suspension flow. TURBULENT BOUNDARY LAYER FLOW AND CORRESPONDING EQUATIONS A fully developed turbulent boundary layer flow is presented in Figure 1 where the thick of boundary layer is δ and the angle between the fiber’s axis and horizontal is .


FIGURE 1. Fully developed turbulent boundary layer flow and fiber.

Applying Eq. (27) and (31) to the fully developed turbulent boundary layer flow we have:

2

2

2 2

2 2

2

2

2 2

2 2

{

1

3

1

3

f xxyy xyyyk xxyy

xyyyxyyy xyyy

xy yyyyxy

yyyyyy yy

a au u vS u a u u

y y y yy

av u ua u v a v

y yy y

a au u vv a v v

y y y yy

av v va v v a v

y yy y

}

(32)

22

2 2

3 2 3

3 2 3

22 3

2 2 3

2

2

2

2

2 2

2

{2 xxyy xxyy

xyyyxxyy xyyy

xyyy xyyyxyyy

yyyy

f a au u u u

y y y yy y

au u u v v ua a

y y y yy y y

a av u v u u va

y y y y yy y y

a v v v

y y yy

S

2 3

2 3

22 3

2 2 3

22 3

2 2 3

12

3

12 }

3

yyyyyyyy

xy xyxy

yy yyyy

av v va

y yy y

a au v u v u va

y y y y yy y y

a av v v v v va

y y y y yy y y

(33)

NUMERICAL METHOD AND PARAMETERS A QUICK scheme (Quadratic Upwind Interpolation for Convection Kinetics) and a finite volume method are used to discretize and solve the equations, respectively. The QUICK scheme fits a parabola between three points to approximate the velocities at the cell faces, and can be shown to be third order accurate. The QUICK scheme represents the steep

gradients quite well, and improves rapidly as the grid is refined. However, it can produce undershoots and overshoots in regions of steep gradients. The SIMPLEC algorithm enforces mass conservation and achieves pressure-velocity coupling. The Enhanced Wall Functions given by Kader [22] is used in the near-wall region. The numerical simulation is carried out in a domain of 68 13.5 along the x and y directions, respectively. The computation grid is comprised of x y=400 400=160000 grid points. The computational grid independence has been tested. The fiber concentration is defined as f f sC V V , where fV and sV are the volume of the fibers and of the suspension, respectively. The fiber aspect-ratio is written as r . The simulations cover a range of Re=45000-75000, fC =0.0-1.0% and r =5-60. For this range of fiber (semi-dilute regime) concentration the apparent viscosity of the suspension is taken as [23]:

3 0.6634 ln ln(1/ )1

3ln(1/ ) ln(1/ )f

pf f

CnL

C C

(34)

RESULTS AND DISCUSSIONS Validation of Mathematical Model and Numerical Code In order to validate the mathematical model and numerical code, we solve the above derived equations in the turbulent channel flow of fiber suspension and compare the numerical results with the experimental ones [6]. The comparison of mean stream-wise velocity, U, is shown in Figure 2 where b is the channel width and maxU is the velocity at

the centerline. We can see that the numerical results are generally consistent with the experimental ones.

FIGURE 2. Comparison of the numerical and experimental results in the turbulent channel flow of fiber suspension ( fC =0.50% and r=60).


Fiber Orientation Distribution Function Additional fiber stress tensor is related to the fiber orientation distribution function as shown in Eq. (2)-(4). The relationship between the orientation distribution functions ψ and the angle φ included between the fiber principal axis and the x-axis for varying y/δ is shown in Figure 3 in which most fibers are oriented between 0o and 30 o, i.e., the fibers tend to orient, on average, to the flow direction. The reason is that the large gradient in mean velocity near the wall exerts a large torque on the fibers, making the fibers rotate to a state experiencing the smallest torque.

FIGURE 3. Orientation distribution function of fibers for varying y/ δ( fC =0.50%, Re=75000 and r=5).

Effect of the Reynolds Number on the Flow Characteristics The effect of Reynolds number on the mean velocity is shown in Figure 4 where the experimental result for pure fluid [24] is also given. The experiment was performed with outflow velocity of 15m/s and outflow relative turbulent intensity of 0.03%. In the figure 0U is the outflow

velocity and * /wu is the wall friction

velocity with w being the friction stress on the wall.

It can be seen that the numerical results are consistent with the experimental ones for the pure fluid at Reynolds number of 75000. The velocity profiles show a fully developed turbulent flow. Even though the velocity profiles become full as the Reynolds number increases in the range of 45000 <Re<75000, the Reynolds number has insignificant effect on the velocity profile.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

16

(U0-U

)/u

*

y/ No fibers ●:Re=75000; fiber suspension:▲: Re=75000;

■: Re=6500;▼: Re=55000;◆: Re=45000

FIGURE 3. Variation of the mean velocity with the Reynolds number for fC =0.10% and r=5.

The variations of turbulence kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient with the Reynolds number are shown in Figures 5, 6, 7 and 8, respectively. The experimental results for pure fluid [24] are also given. The numerical results are consistent with the experimental ones for the pure fluid at Reynolds number of 75000. The magnitude of Reynolds number is directly proportional to the flow rate when the length scale and fluid viscosity remain constant. We can see that the turbulent kinetic energy decreases gradually from the wall to the outflow as Figure 5. The turbulent kinetic energy increases with increasing the Reynolds number. As shown in Figure 6 there is a maximum of the Reynolds stress around y/δ=0.05. The Reynolds stress increases from the wall to y/δ=0.05, and then decreases to zero from y/δ=0.05 to the outflow. The Reynolds stress increases with increasing the Reynolds number. From Figure 7 it can be seen that the turbulent dissipation rate decreases from the wall region to the outflow. The Reynolds number does not have an obvious effect on the profile of turbulent dissipation rate. Figure 8 shows the variation of eddy viscosity coefficient with the Reynolds number. The maximum of the eddy viscosity coefficient appears around y/δ=0.3. The eddy viscosity coefficient increases from the wall to y/δ=0.3, and then decreases to zero from y/δ=0.3 to the outflow. The eddy viscosity coefficient increases as the Reynolds number increases. At this range of parameters, the effect of Reynolds number on the flow behavior is about the same as that of Newtonian fluid, because the presence of fibers has an insignificant effect on the turbulent intensity and the momentum transfer is dominated by turbulence.


0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

k/u*2

y/ No fibers ●:Re=75000; fiber suspension:▲: Re=75000; ■:

Re=6500; ▼: Re=55000;◆: Re=45000 FIGURE 4. Variation of the turbulence kinetic energy with the Reynolds number for fC =0.10% and r=5.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

u'v

'/u*2


Re=6500; ▼: Re=55000;◆: Re=45000

FIGURE 5. Variation of the Reynolds stress with the Reynolds number for fC =0.10% and r=5.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

/u

*3


Re=6500; ▼: Re=55000;◆: Re=45000

FIGURE 6. Variation of the turbulent dissipation rate with the Reynolds number for fC =0.10% and r=5.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

m/u

*


Re=6500; ▼: Re=55000;◆: Re=45000

FIGURE 7. Variation of the eddy viscosity coefficient with the Reynolds number for fC =0.10% and r=5.

Effect of the Fiber Concentration on the Flow Characteristics Effect of fiber concentration on the mean velocity, turbulence kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient for Re=75000 and r =5 are shown in Figures 9, 10, 11, 12 and 14, respectively. The experimental results [24] for pure fluid are also given. In Figure 9 the velocity profiles become full as the fiber concentration increases in the range of 0.1%< fC <1%. Figure 10 shows that the turbulent kinetic energy increases over the boundary layer with an increasing fiber concentration. The existence of fibers will hinder the momentum transfer through decreasing turbulence intensity and, on the other hand, enhance the momentum transfer by providing a solid link between adjacent fluid layers. The overall momentum transfer is determined by the competition of these two effects. The curves in Figure 11 illustrate that the variation of the fiber concentration does not change the shape of the Reynolds stress profile. The maximum of Reynolds stress is located around y/δ=0.1, and the Reynolds stress increases as the fiber concentration increases because the primary effect of increasing the fiber concentration is to increase the magnitude of the fiber extra stresses. From Figure 12 we can see that the turbulent dissipation rate is increased with a decreasing fiber concentration. This means that the presence of fibers, at this range of concentration, results in the increase of viscosity, which causes the decrease of velocity gradient in the wall region. The turbulent energy equation is a balance between turbulent energy production and turbulent dissipation, which can be illustrated by comparing Figure 10 and Figure 12, i.e., the


turbulent kinetic energy increases and turbulent dissipation rate decreases with an increasing fiber concentration. Perlekar et al. [25] carried out a high-resolution direct numerical simulation of decaying, homogeneous, isotropic turbulence with polymer additives, and revealed that the drag reduction corresponds to the reduction in the turbulent dissipation rate. Therefore, the drag reduction can be induced by fibers in a turbulent boundary layer, and the effect of increasing the fiber concentration is an increase in the drag reduction effectiveness because the drag reduction performance is closely correlated to resistance to extensional motions, and for fibers the extensional viscosity is proportional to the fiber concentration. The effective viscosity increases with an increasing fiber concentration, which results in the increase of turbulent kinetic energy and decrease of turbulent dissipation rate. The variation of the eddy viscosity coefficient with the fiber concentration is shown in Figure 13 from where it can be seen that the eddy viscosity coefficient increases with increasing the fiber concentration. Generally, the effect of fiber concentration on the turbulent properties is larger than that of the Reynolds number.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

16

(U0-U

)/u*

y/ ●: No fibers; fiber suspension:▲: fC =0.1%; ■: fC =0.4%;▼: fC =0.7%;◆: fC =1%

FIGURE 8. Variation of the mean velocity with fiber concentration for Re=75000 and r =5.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

k/u*2


FIGURE 9. Variation of the turbulence kinetic energy with fiber concentration for Re=75000 and r =5.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

u'v

'/u*2


FIGURE 10. Variation of the Reynolds stress with fiber concentration for Re=75000 and r =5.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

/u

*3


FIGURE 11. Variation of the turbulent dissipation rate with fiber concentration for Re=75000 and r =5.


0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

m/u

*


FIGURE 12. Variation of the eddy viscosity coefficient with fiber concentration for Re=75000 and r =5.

Effect of the Fiber Aspect-Ratio on the Flow Characteristics The distribution of mean velocity profile, turbulence kinetic energy, Reynolds stress, turbulence dissipation rate and eddy viscosity coefficient along the lateral direction at different fiber aspect-ratios are shown in Figures 14, 15, 16, 17 and 18, respectively, for Re=75000 and

fC =0.10%. Changing the aspect ratio modifies both the dynamic behavior of the fibers and the fiber extra stress. We can see that the fiber aspect-ratio has a scant effect on the mean velocity profile in Figure 14. The turbulent kinetic energy increases with increasing the fiber aspect-ratio as shown in Figure 15. It is easy for the fibers with a large aspect-ratio to increase the momentum transfer through providing a solid link between the adjacent fluid layers [26, 27]. The Reynolds stress, as shown in Figure 16, also increases with increasing the fiber aspect-ratio and appears a maximum around y/δ=0.05. From Figure 17 it is found that the turbulent dissipation rate decreases with increasing fiber aspect-ratio. The fibers with a large aspect-ratio align more easily along the flow direction, which results in the augment of turbulent dissipation rate. The effect of increasing the fiber aspect ratio is an increase in the drag reduction effectiveness, which is consistent with the conclusion given by Paschkewitz et al. [28] using direct numerical simulation for a turbulent channel flow. This effect can be explained by examining the behavior of the fibers in the shear flow. As the aspect ratio is increased, the total extra shear stress at the wall is decreased, enhancing the drag reduction effectiveness. Additionally, as the aspect

ratio is increased, extensional stresses are increased and this effect may also be responsible for the higher levels of drag reduction. The variation of eddy viscosity coefficient with the fiber aspect-ratio is given in Figure 18, we can see that the eddy viscosity coefficient first increases from the wall along the lateral direction, and then decreases to zero at the outflow. The eddy viscosity coefficient enhances with the increase of the fiber aspect-ratio and has a maximum around y/δ=0.3. Comparing the contribution of the Reynolds number, fiber concentration and fiber aspect-ratio we can see that the effect of the fiber aspect-ratio on the turbulent properties is larger than that of the Reynolds number, but smaller than that of the fiber concentration in the range of parameters considered in this paper.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

16

(U0-U

)/u*

y/ ●: No fibers; ▲:r =5; ■: r =10;▼: r =30; ◆: r =60

FIGURE 13. Variation of the mean velocity with a fiber aspect-ratio for Re=75000 and fC =0.10%.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

k/u*2

y/ ●: No fibers; ▲:r =5; ■: r =10;▼: r =30; ◆: r =60

FIGURE 14. Variation of the turbulence kinetic energy with a

fiber aspect-ratio for Re=75000 and fC =0.10%.


0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

u'v

'/u*2

y/ ●: No fibers; ▲:r =5; ■: r =10;▼: r =30; ◆: r =60

FIGURE 15. Variation of the Reynolds stress with a fiber

aspect-ratio for Re=75000 and fC =0.10%.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

/u

*3

y/ ●: No fibers; ▲:r =5; ■: r =10;▼: r =30; ◆: r =60 FIGURE 16. Variation of the turbulent dissipation rate with a


0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

m/u

*

y/ ●: No fibers; ▲:r =5; ■: r =10;▼: r =30; ◆: r =60

FIGURE 17. Variation of the eddy viscosity coefficient with a


CONCLUSIONS The characteristics of fiber suspension flow in a turbulent boundary layer are different from those in the turbulent single phase boundary layer. In order to explore the effects of the Reynolds number, fiber concentration and fiber aspect-ratio on the mean velocity profile, turbulent kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient, the equations of averaged momentum, turbulence kinetic energy, turbulence dissipation rate with the additional term of the fibers, and equation of probability distribution function for mean fiber orientation are derived and solved numerically in the fiber suspension flowing in a turbulent boundary layer. In order to validate the mathematical model and numerical code, the derived equations are solved in the turbulent channel flow of fiber suspension, and some of the numerical results are consistent with the available experimental results. The results show that the velocity profiles become full, and the turbulent kinetic energy, Reynolds stress and eddy viscosity coefficient increase, while turbulent dissipation rate decreases, as the Reynolds number, fiber concentration and fiber aspect-ratio increase. The effect of the fiber aspect-ratio on the turbulent properties is larger than that of the Reynolds number, but smaller than that of the fiber concentration in the range of parameters considered in this paper. ACKNOWLEDGEMENT This work is supported financially by the research grant from the research fund for the Doctoral Program of Higher Education of China (No. 20120101110121) REFERENCES [1] C. L. Tucker, Journal of Non-Newtonian Fluid

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AUTHORS’ ADDRESSES Jianzhong Lin Suhua Shen Xiaoke Ku Department of Mechanics 38, Zheda Road Hangzhou Zhejiang 310027 CHINA

Documents

Characteristics of Fiber Suspension Flow in a Turbulent … · Jianzhong Lin, Suhua Shen, Xiaoke Ku Department of Mechanics Hangzhou, Zhejiang CHINA Correspondence to: Jianzhong Lin