PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 10 ppt

Particle sedimentation in wall-bounded turbulent flows 381

was used for the normal direction, with ∆z+ ∼ 0.9 at the wall, and ∆z+ ∼ 7

at the center of the channel.

The particles were released homogeneously distributed in a plane at a

distance z = 0.9 H from the bottom of the channel, which corresponds to

z+ = 450, with an initial vertical velocity equal to Vt = 0.1. For each particle,

we computed the time it took to travel: (i) from z+ = 450 to z+ = 250 (center

of the channel), (ii) from z+ = 250 to z+ = 50 (buffer region), and (iii) from

z+ = 50 to z+ = 3.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.001 0.01 0.1 1 10 100

Average Settling Velocity

Particle Froude number

450 - 250

250 - 50 50 - 3

Stagnant

Fig. 8. Average settling velocity for an open-channel as a function of the particle

Froude number.

The results for different particle Froude numbers, are presented in figure 8.

When the particle Froude number was smaller than 1, and when the particles

were falling down between z+ = 450 and z+ = 250, and between z+ = 250

and z+ = 50, the average settling velocity Vs was higher than Vt. In this case,

the relation between Vs and Fp is somehow similar to the case of a vortex

array where the vortex distance is ”large” (8Rv), with an almost monotonic

decrease in the average settling velocity as Fp increases. On the other hand,

in the near-wall region, there is a maximum in the average settling velocity at

Fp ∼ 1. In the vortex array case we saw that for ”intermediate values” of Fp,

the average settling velocity had a strong dependence on the vortex spacing,

with a more complex behavior when the vortex spacing was smaller. Near the

wall the streamwise vortices play an important role and their spacing is smaller

than further away from the wall [6]. This could be a possible explanation for

the behavior near the wall. However, the behavior is quite different from the

”compact vortex array” (D = 4 Rv), and contrary to the vortex array Vs is

always higher than Vt. Clearly, the turbulence structure appears to play an

important role in determining the settling velocity.

In order to quantify the importance of the turbulence structure on the

particle motion, we analyzed the particle-fluid two-point velocity correlations.

382 M. Cargnelutti and L.M. Portela

In figures 9 and 10 are plotted, respectively, the spanwise and normal-wise

particle-fluid velocity correlation.

-0.2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Rwpwf

∆ y

Spanwise correlation at z+

=50

Fp 0.001

Fp 1

Fp 10

Fluid

Fig. 9. Particle-fluid vertical velocity two-point spanwise correlation.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500

Rwpwf

z

+

Normalwise correlation at z+

=50 and z+

=250

z

+

=50 z

+

=250

Fp 0.001

Fp 1

Fp 10

Fluid

Fig. 10. Particle-fluid vertical velocity two-point normal-wise correlation.

In the spanwise correlation plots, for the fluid auto-correlation at z+ = 50,

there is a minimum around ∆y+ = 60, which can be seen as a measure of the

vortices diameter. Even though the particle-fluid correlation is in general smal￾ler than the fluid auto-correlation, for the smallest values of Fp we notice than

the particle-fluid correlation is higher at ∆y+ ∼ 60. This seems to indicate

than the effect of the fluid structures on the spanwise direction persist in time.

On the other hand, when Fp >> 1, the velocity correlation is almost zero for

all values of ∆y+, which means that the particles ignored the presence of the

turbulence and fell down with a velocity equal to Vt.

Particle sedimentation in wall-bounded turbulent flows 383

In the normal-wise velocity correlations (figure 10) it can be seen that the

loss of correlation is not the same in the central part of the channel as in the

near-wall region. For example, for Fp = 1 the correlation is larger at z+ = 250

than at z+ = 50. This seems to indicate that the particles tend to follow in a

stronger way the larger fluid structures at the center of the channel than the

smaller structures closer to the channel wall.

In figure 10 we can also note that in both regions (center of the channel

and near wall region), there is an asymmetry in the correlations. The particles

seem to correlate more with the structures close to the top of the channel than

with those structures close to the bottom. This effect is more pronounced for

Fp < 1, where the particle-fluid correlation at z+ = 250 can be even higher

in the top part of the channel than the fluid auto-correlation. This seems to

indicate that the particles feel more the presence of the fluid structures from

the top of the channel than from below, and that they keep a ”memory” of

the fluid structure above them.

7 Conclusions

Clearly, the turbulence structure appears to play an important role in determ￾ining the settling velocity in wall-bounded turbulence. Far from the wall the

behavior is somehow similar to a vortex array with a ”large” vortex spacing.

Near the wall, the behavior is more complex and a maximum in the settling

velocity is found for Fp ∼ 1.

The precise mechanisms through which the turbulence structure influences

the settling velocity are still not clear. However, a preliminary analysis of the

two-point fluid-particle correlation shows that the particles ”feel” the normal￾wise and spanwise velocity correlation and appear to keep a ”memory” of the

fluid structure above them.

Acknowledgments

We gratefully acknowledge the financial support provided by STW,

WL—Delft Hydraulics and KIWA Water Research. The numerical simula￾tions were performed at SARA, Amsterdam, and computer-time was financed

by NWO.

References

[1] W.A. Breugem and W.S.J. Uijttewaal. Sediment transport by coherent

structures in a horizontal open channel flow experiment. Proceedings of

the Euromech-Colloquium 477, to appear

[2] W.H. de Ronde. Sedimenting particles in a symmetric array of vortices.

BSc Thesis, Delft University of Technology, 2005

[3] J. Davila and J.C. Hunt. Settling of small particles near vortices and in

turbulence. Journal of Fluid Mechanics, 440:117-145, 2001

384 M. Cargnelutti and L.M. Portela

[4] I. Eames and M.A. Gilbertson. The settling and dispersion of small dense

particles by spherical vortices. Journal of Fluid Mechanics, 498:183-203,

2004

[5] M.R. Maxey and J.J. Riley. Equation of motion for a small rigid sphere

in a nonuniform motion. Physics of Fluids, 26(4):883-889, 1983

[6] L.M. Portela and R.V.A. Oliemans. Eulerian-lagrangian dns/les of

particle-turbulence interactions in wall-bounded flows. International

Journal of Numerical Methods in Fluids, 9:1045-1065, 2003.