Direct numerical simulation of particle interaction with ejections in turbulent channel flows

International Journal of Multiphase Flow 37 (2011) 187–197

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

Direct numerical simulation of particle interaction with ejections in turbulent channel flows I. Vinkovic a,⇑, D. Doppler a, J. Lelouvetel b, M. Buffat a a Université Lyon, Université Lyon 1, INSA de Lyon, Ecole Centrale de Lyon, CNRS UMR 5509, Laboratoire de Mécanique des Fluides et d’Acoustique. 36, avenue Guy de J H Collongue, F-69134 Ecully, France b Department of System Design Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

a r t i c l e

i n f o

Article history: Received 23 April 2010 Received in revised form 10 September 2010 Accepted 14 September 2010 Available online 25 October 2010 Keywords: Direct numerical simulation Lagrangian particle tracking Turbulent channel flow Quadrant analysis Threshold

a b s t r a c t Direct numerical simulations (DNS) of incompressible turbulent channel flows coupled with Lagrangian particle tracking are performed to study the characteristics of ejections that surround solid particles. The behavior of particles in dilute turbulent channel flows, without particle collisions and without feedback of particles on the carrier fluid, is studied using high Reynolds number DNS (Re = 12,500). The results show that particles moving away from the wall are surrounded by ejections, confirming previous studies on this issue. A threshold value separating ejections with only upward moving particles is established. When normalized by the square root of the Stokes number and the square of the friction velocity, the threshold profiles follow the same qualitative trends, for all the parameters tested in this study, in the range of the experiments. When compared to suspension thresholds proposed by other studies in the Shields diagram, our simulations predict a much larger value because of the measure used to characterize the fluid and the criterion chosen to decide whether particles are influenced by the surrounding fluid. However, for intermediate particle Reynolds numbers, the threshold proposed here is in fair agreement with the theoretical criterion proposed by Bagnold (1966) [Bagnold, R., 1966. Geological Survey Professional Paper, vol. 422-1]. Nevertheless, further studies will be conducted to understand the normalization of the threshold. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Interactions between solid particles and turbulent structures in a channel flow are an important topic in a great number of environmental systems, from sediment transport within rivers to atmospheric dispersion of pollutants or solid deposition in marine flows. Particle transport mechanisms in the vicinity of a wall are characterized by complex interactions between turbulent structures and the dispersed phase (Kaftori et al., 1995a,b). Although much work has been done on this topic Kulick et al. (1994), Rouson and Eaton (2001), there are still many aspects that are only known qualitatively, or under a limited set of parameters. It has been speculated for more than 30 years that wall generated turbulent bursts are primarily responsible for the suspension and transport of solid particles within the flow (Jackson, 1976; Sumer and Oguz, 1978; Sumer and Deigaard, 1981). However, the detailed mechanisms have not been elucidated yet. Since the development of new experimental techniques and high resolution ⇑ Corresponding author. E-mail addresses: [email protected] (I. Vinkovic), delphine.doppler@ univ-lyon1.fr (D. Doppler), [email protected] (J. Lelouvetel), marc.buffat@ univ-lyon1.fr (M. Buffat). 0301-9322/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2010.09.008

numerical simulations, a better understanding of turbulent burst structures within wall bounded flows has been achieved (Adrian et al., 2000). By experimental observations, Liu et al. (1991) found that regions containing high Reynolds stress are associated with the near-wall shear layers, terminated in regions of rolled-up spanwise vorticity, which are interpreted to be the heads of hairpin vortices. There is now a consensus among researchers that particles in the near wall region preferentially concentrate in low speed streaks (Kaftori et al., 1995b; Kiger and Pan, 2002; Marchioli and Soldati, 2002; Soldati and Marchioli, 2009). These low-speed fluid regions move through the inclined loop of the hairpin by vortex induction from the legs and the head (Adrian et al., 2000). Despite this general agreement on several features of particle behavior near the wall, numerous mechanisms for particle–wall turbulence interactions have been proposed. For example, from their experiments Kaftori et al. (1995a) suggested that funnelshaped vortices sweep along the bottom and push particles out of the way, throwing the particles out of their core and therefore creating avenues of particles to the sides of the vortex path. By means of numerical simulations in a turbulent channel at a shear Reynolds number Res = 150, Marchioli and Soldati (2002) found that particles are trapped in the near wall region by the rear-end of a quasi-streamwise vortex, which prevents particle suspension

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2. Numerical simulation 2.1. Flow The flow considered is an incompressible turbulent channel flow. The governing equations for the fluid in dimensionless form are given by:

~ :~ r u ¼~ 0

   2 @~ u 1~ ~ p þ u  1 D~ ~ ~ u ~ uþr p þ r u¼ r @t Re q 2 q 0 

Fig. 1. Channel flow. x, y, z represent the streamwise, the vertical and the transverse directions respectively.

by larger, farther from the wall, structures. Finally, Niño and García (1996) used particle and dye visualization in open channel flow experiments. The authors claim that particles are picked up from the bed by flow ejection events, trapped in the core of the shear layer and raised toward the outer regions of the wall as the flow structure is stretched out into such regions. Assertion of one of the proposed mechanisms can be fully achieved only by more quantitative experimental and numerical studies of these processes. In particular, further insight may be obtained by simultaneous Lagrangian tracking of the solid phase and the flow structure surrounding the particles. Experimentally, this has been achieved by simultaneous particle image velocimetry (PIV) for the carrier fluid, and particle tracking velocimetry (PTV) for the solid phase, by Lelouvetel et al. (2009). In that study, Lelouvetel et al. (2009) showed that particle movement away from the wall is correlated with strong ejection events. The challenge for numerical simulations is then to compute cases with high precision and sufficiently high Reynolds numbers close to experimental conditions. In light of the above discussion, the current study focuses on determining which ejection events are responsible for particle transport away from the wall. High Reynolds number direct numerical simulations (DNS) coupled with Lagrangian tracking of solid particles are used. The DNS are conducted at high shear Reynolds number Res = 590, in the range of the experimental values of Lelouvetel et al. (2009). Carrier fluid and particle characteristics are obtained simultaneously and conditioned statistical analyses are performed. The simulations allow for the identification of the ejections responsible for particle transport away from the wall. A normalized threshold for characterizing the ejections is established. Its value is compared to other studies and experiments. In future work this threshold will be used for choosing the right ejections for the simultaneous Lagrangian tracking of the solid and fluid phase. This paper presents the numerical simulations that are performed. Results confirming the preferential concentration of solid particles in low-speed streaks are briefly shown. The conditioned statistical analysis of the flow surrounding the solid phase is then described and the segregation threshold is introduced. Finally, the segregation threshold is tested on the experimental data of Lelouvetel et al. (2009) and compared to suspension thresholds proposed in the literature.

Re

Nx  Ny  Nz

150

2660

192  193  192

587

12,500

384  257  384

ð2Þ

where ~ u is the velocity vector, p the fluctuating pressure, q the fluid ~ p the mean pressure gradient that drives the flow. In density and r 0 the following, the velocity components along the x (streamwise), y (vertical) and z (transverse) directions will be denoted u, v and w, respectively. Re is the Reynolds number based on the mean velocity U at the center of the channel, the channel half height h and the viscosity m. In the above equations (Eq. (1) and (2)), all variables are dimensionless and h and U are the characteristic length and velocity scales. The computational domain consisting of two infinite parallel walls is illustrated in Fig. 1. Periodic boundary conditions are imposed on the fluid velocity field in x and z directions and no-slip boundary conditions are imposed at the walls. The calculations are performed for two Reynolds numbers, Re = 2660 and Re = 12,500. Details on the simulation characteristics are given in Table 1, where Ni and Li are the number of grid points and the domain length in direction i. Typical space and time steps are also given. At last, Res = u*h/m is the Reynolds number based on the friction velocity u*. The superscript ‘‘+” denotes quantities expressed in wall units, normalized by the friction velocity u* and the viscosity m. The incompressible Navier–Stokes equations in a turbulent channel flow are solved using a Galerkin spectral approximation (Fourier Chebyshev) and a variationnal projection method on a divergence free space as described by Pascal (1996) and Buffat et al. (2009). This code has been successfully applied to the study of turbulent channel flow by DNS using a very large number of grid points (100 million) by Laadhari (2002) and Laadhari (2007). 2.2. Particles Particles are injected into the flow at low concentrations in order to consider dilute systems. Particle–particle interactions are neglected as well as the influence of particles on the carrier fluid. Mirror conditions are applied for particle-wall bouncing. Furthermore, particles are considered to be pointwise, spherical, rigid and to obey the following Lagrangian dimensionless equation of motion:

! d~ v p ~ ~  q ~ St cy ¼ u  v p f ðRep Þ  1  qp dt

ð3Þ

d~ xp vp ¼~ dt

ð4Þ

Here, ~ v p and ~xp are the dimensionless particle velocity and position. The action of gravity is first neglected and then in a second set of

Table 1 Characteristics of the numerical simulations for the fluid flow. Res

ð1Þ

Lx  Ly  Lz

Dx+  Dy+  Dz+

dt+

4 ph 3 3 3 ph  2h  ph 2 4

9  (0.02  3)  4

0.03

7.2  (0.04  7.2)  3.6

0.033

3ph  2h 

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numerical simulations taken into account only for the solid phase. The solid particle–fluid interaction is modelled by a drag force with the correction term f ðRep Þ ¼ 1 þ 0:15Re0:687 suggested by Clift et al. p (1978). Rep is the local and instantaneous particle Reynolds number based on the local relative velocity, the particle diameter dp and the fluid viscosity. St is the Stokes number given by:

sp St ¼ sf 2 p dp

q

m

and sf ¼ 2 ; qp being the particle density. The 18qm u Stokes number characterizes the response time of a particle to fluid solicitation. The gravity parameter c is defined as: with

c¼

sp ¼

ð5Þ

sp g

ð6Þ

u

where ~ g ¼ g~ y is the gravity acceleration. This second parameter compares the particle settling velocity to the typical fluid velocity. The three parameters Re, St and c may thus play a role in particle segregation. Table 2 shows all the parameters of the particles for the numerical simulations conducted in this study. Two sets of simulations are performed, one with c = 0 and the other with c – 0. The aim of this study is to analyze particle–ejection interactions and the role ejections play in particle segregation. For this no account of gravity is needed. Nevertheless, in order to compare our results to the experiments and to confront the established conclusions to experimental conditions the action of gravity on the solid phase is introduced. It is important to notice that even when gravity is taken into account only simulations with small values of c have been performed (c 6 1.4). Once the steady state for the fluid is obtained, 200,000 particles are released at randomly chosen locations within the channel, then tracked at each time step. Steady state fluid statistics have been compared with the results of Moser et al. (1999) and Hoyas and Jiménez (2008). Details about this comparison may be found in Zamansky et al. (2010). The initial velocities of the particles are set equal to the interpolated fluid velocities at each particle location. A high order three dimensional Hermite interpolation is used for computing the fluid velocity ~ uð~ xp ; tÞ at the particle position. The time-integration of the particle motion Eq. (3) is performed using a second-order Adams–Bashforth method with the same time step as the DNS. Once the particles released, the simulations are run over several particle timescales sp. Particle statistics are sampled starting from t+  1000, counted from particle release. While velocity statistics for both phases are at stationary state, particle distribution in our computations is still developing and much longer simulation times would be required to obtain steady-state concentration profiles (Marchioli et al., 2008). The reason for this choice is that in the experiments of Lelouvetel et al. (2009) and in a number of

Table 2 Characteristics of the numerical simulations for the particles. Re

St

c

Re

St

c

2660 2660 2660 2660 2660 12,500 12,500 12,500 12,500 12,500

1 5 15 25 125 1 5 15 25 125

0 0 0 0 0 1 1 1 1 1

12,500 12,500 12,500 12,500 12,500 12,500 12,500 12,500 12,500

1 5 15 25 125 1 1 1 1

0 0 0 0 0 0.3 0.7 1.2 1.4

189

environmental applications including sediment transport particle distribution never reaches equilibrium. From an application viewpoint, statistically-developing particle concentration is thus the most probable situation to investigate as suggested by Soldati and Marchioli (2009). 2.3. Comparison with benchmark For the lower Reynolds number case (Re = 2660) particle velocity statistics have been compared with the benchmark proposed by Marchioli et al. (2008). Profiles of mean particle velocity and root mean square of particle velocity fluctuations for Re = 2660, c = 0 and St = 1, St = 5 and St = 25 are illustrated in Fig. 2a and b, Fig. 3a and b. As the Stokes number increases differences in profiles appear. However, these differences are within the variations presented in the benchmark by other groups. For the higher Reynolds number case (Re = 12,500) no available data for comparison was found. Profiles of mean particle velocity and particle Reynolds stress u0p v 0p for Re = 12,500, c = 0 and St = 1, St = 5 and St = 25 are illustrated in Fig. 4a and b, respectively. Other statistics or profiles for the other simulations can be shown upon request. 3. Results and discussion 3.1. Particle distribution and near-wall structures Fig. 5 schematically depicts the signature of the velocity field induced by a hairpin vortex on a streamwise wall-normal plane (xy plane), as suggested by Adrian et al. (2000). Here, velocity fluctua stands for the Reynolds tions are denoted by u0 , and the overbar u average. Following Adrian et al. (2000), the velocity pattern contains the following features: a transverse (i.e. spanwise) vortex core; a region of low-momentum fluid located below and upstream of the vortex head corresponding to an ejection (u0 < 0 and v0 > 0); a sweep corresponding to the region where u0 > 0 and v0 < 0; a shear layer between the sweep and ejection. As previously explained, our aim here is to quantify the interaction between the ejections and particle segregation. Therefore, the fluid velocity at the position of solid particles will be statistically analyzed in order to detect regions of the flow where u0 < 0 and v0 > 0, i.e. corresponding to an ejection. It should be noted that both for c = 0 and for c – 0 ejections are considered only from the lower wall. As reported previously in experimental studies conducted by Kaftori et al. (1995b), Niño and García (1996), Rashidi et al. (1990), Kiger and Pan (2002), or by means of numerical simulations Marchioli and Soldati (2002), Soldati and Marchioli (2009), particle positions tend to correlate with the instantaneous location of low streamwise velocity regions. This behavior is confirmed in Fig. 6, in which the instantaneous distribution of particles moving away from the wall is shown superimposed onto the mapping of the streamwise velocity in a transverse yz plane. Ascending particles tend to be sorted, such that they accumulate in ascending low-speed regions of the flow. In order to obtain additional insight on this behavior, a quadrant analysis of the conditionally sampled and unconditionally sampled fluid fluctuations is performed. This is described in the following section. 3.2. Quadrant analysis The previous section and numerous studies cited above hint to the role of ejections in the segregation of particles near the wall. A quantitative measure of this interaction between ascending particles and ejections can be obtained through the quadrant analysis. This method has been extensively applied in previous studies

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Fig. 2. Velocity profiles for Re = 2660, c = 0 and St = 1, St = 5 and St = 25, from bottom to top respectively. Line – our DNS, (M) – Marchioli et al. (2008) group UUD, (h) – Marchioli et al. (2008) group TUE, (O) – Marchioli et al. (2008) group HPU, (e) – Marchioli et al. (2008) group TUD, (+) – Marchioli et al. (2008) group ASU. (a) Mean streamwise solid particle velocity. Profiles have been shifted by 0.5 units. (b) Root mean square of solid particle streamwise velocity fluctuations. Profiles have been shifted by 0.15 units.

Fig. 3. Root mean square of solid particle velocity fluctuations for Re = 2660, c = 0 and St = 1, St = 5 and St = 25, from bottom to top respectively. Profiles have been shifted by 0.05 units. Line – our DNS, (M) – Marchioli et al. (2008) group UUD, (h) – Marchioli et al. (2008) group TUE, (O) – Marchioli et al. (2008) group HPU, (e) – Marchioli et al. (2008) group TUD, (+) – Marchioli et al. (2008) group ASU. (a) Vertical component. (b) Transverse component.

Fig. 4. Velocity profiles for Re = 12,500, c = 0 and St = 1, St = 5 and St = 25, from bottom to top respectively. Line – our DNS. (a) Mean streamwise solid particle velocity. Profiles have been shifted by 0.5 units. (b) Particle Reynolds stress u0p v 0p . Profiles have been shifted by 0.0025 units.

I. Vinkovic et al. / International Journal of Multiphase Flow 37 (2011) 187–197

191

1

0.8

-u’v’ 0.020 0.018 0.016 0.014 0.012 0.010 0.009 0.007 0.005 0.003 0.001 -0.001 -0.003 -0.005 -0.007 -0.009 -0.010 -0.012

y/h

0.6

0.4

0.2

0 2.2

2.4

2.6

2.8

3

3.2

x/h Fig. 5. Instantaneous distribution of particles () and hairpin vortex signature in a xy plane for Re = 2660, St = 5 and c = 0. Grey mapping of u0 v0 . Vectors of fluctuating velocity in the xy plane, u0 and v0 . The line and circle depict schematically the shear layer and the transverse vortex core.

1

0.8

u

0.6

y/h

1.00 0.93 0.85 0.78 0.71 0.63 0.56 0.48 0.41 0.34 0.26 0.19 0.12 0.04 -0.03

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

z/h Fig. 6. Instantaneous distribution of ascending particles () in the yz plane, for Re = 2660, St = 5 and c = 0. Grey mapping of u. Vectors of fluctuating velocity in yz plane, v0 and w0 .

Kiger and Pan (2002), Marchioli and Soldati (2002), Lelouvetel et al. (2009) and is well described in the following references Lu and Willmarth (1973), Nakagawa and Nezu (1977), Nezu and Nakagawa (1993). By means of the quadrant analysis, the fluctuating fluid velocity at a given point is sorted according to four quadrants defined in the plane (u0 ,v0 ). Turbulent events associated with each quadrant are called outward interactions (Q1, u0 > 0 and v0 > 0), ejections (Q2, u0 < 0 and v0 > 0), downward interactions (Q3, u0 < 0 and v0 < 0) and sweeps (Q4, u0 > 0 and v0 < 0). 3.2.1. Quadrant analysis at a given height The quadrant analysis is applied here to the fluid velocity at a given height y+ sampled by three different conditions: the presence of solid particles, the presence of solid particles moving away from the wall and the presence of solid particles moving toward the wall. The results for Re = 12,500, St = 5 and c = 0 at y+ = 38 are

Fig. 7. Probability density functions of fluid velocity conditioned on the presence of solid particles (a), conditioned on the presence of upward moving solid particles vp > 0 (b), conditioned on the presence of downward moving solid particles vp < 0 (c). Black iso-contours represent the unconditioned PDF of fluid velocity. Re = 12,500, St = 5, c = 0 and y+ = 38.

illustrated in Fig. 7. Grey mapping represents the conditioned probability density functions (PDF) of fluid velocity, while black contours represent the unconditioned PDF of fluid velocity. From Fig. 7a, it can be seen that the PDF of fluid velocity conditioned on solid particle presence differs from the unconditioned PDF, especially in the second (Q2, u0 < 0 and v0 > 0) and fourth (Q4, u0 > 0 and v0 < 0) quadrants. This suggests that particles concentrate preferentially in regions of low-speed upward moving fluid (Q2) and high-speed downward moving fluid (Q4). If we now analyze the influence of the conditioning on solid particle velocity (Fig. 7b and c), it can be noted that most of the upward moving particles (fluid conditioned on vp > 0, Fig. 7b) are contained within the second (Q2) quadrant, which corresponds to an ejection of slow moving fluid away from the wall. This is in agreement with the observations of the previous section and with the mechanisms postulated by previous workers (Kiger and Pan, 2002; Marchioli and Soldati, 2002; Soldati and Marchioli, 2009;

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Lelouvetel et al., 2009). The observation is as well consistent with the fact that the fluid must typically be moving up to entrain the particles, but also indicates that it is preferentially done in regions of slow moving fluid. For comparison, sampling by particles moving toward the wall (fluid conditioned on vp < 0, Fig. 7c) is presented, showing a preference for high-speed sweeps of fluid moving toward the wall (Q4). 3.2.2. Influence of wall distance In the previous paragraph, preferential concentration of upward moving particles in ejections is noted for Re = 12,500, St = 5, c = 0 and y+ = 38. In order to check this statement for any distance to the wall, the part of Q2 events within all the Qi events is estimated for the PDF of the conditioned fluid velocity (Fig. 7). Fig. 8 illustrates the evolution with wall distance of this percentage for the fluid conditioned on solid particle presence and the fluid conditioned on solid particles moving away from the wall and toward the wall. These results are obtained for all the other cases simulated in this study, however, for simplicity reasons only the curves for Re = 12,500, St = 5 and c = 0 are presented here. It can be noted that, whatever the wall distance, at the position of solid particles, ejections represent up to 30% of all quadrants. In addition to this, at the position of solid particles that move away from the wall, ejections represent up to 50% of all quadrants. We therefore conclude that, whatever the wall distance, upward moving particles segregate in regions of the flow corresponding to ejections. However, at the position of downward moving particles the percentage of the second quadrant events reaches 5% for wall distances above y+  20. Even though this percentage of ejections at the position of downward moving particles is low, it never vanishes to zero, implying that whatever the wall distance, there are particles that move toward the wall within ejections. Our aim now is to separate the ejections within which all the particles move away from the wall from the ejections within which one can find particles moving toward the wall. This is described in the following section. 3.3. Segregation threshold As shown in the previous section, particles moving away from the wall are mostly surrounded by ejections. However, within

600

ejections, particles moving toward the wall are still present. This is illustrated in Fig. 9, where the cumulative distribution function (CDF) of u0 v0 in ejections conditioned on particles moving away and toward the wall, is given for y+ = 348. Once again, the results are only shown here for the case of Re = 12,500, St = 5 and c = 0, but the following conclusions can be drawn for all the other simulations. Fig. 9 illustrates that ejections conditioned on particles moving toward the wall correspond to lower values of u0 v0 = ju0 v0 j, ju0 v0 j denoting the absolute value of u0 v0 . There is a value for ju0 v0 j above which there will be almost no particles moving toward the wall surrounded by ejections. We therefore define a threshold value u0 v 0T as the local value of ju0 v0 j for ejections within which less than 2.5% of particles move toward the wall. The value of 2.5% has been chosen in order to match the percentage used in the experimental study of Lelouvetel et al. (2009) to which our data are compared as described further in this section. If locally within an ejection ju0 v 0 j < u0 v 0T , the particle can move either toward or away from the wall, and if locally within an ejection ju0 v 0 j > u0 v 0T , the particle would presumably move away from the wall. This behavior thus corresponds to a locally defined segregation threshold. Moreover, this threshold is restrictive, in the sense that almost no falling particles can be found above this value. 3.3.1. Segregation threshold normalization The threshold value for ju0 v0 j is computed for different wall distances, and for all the simulated cases described in this study (Table 2). For each simulation, we first check the preferential concentration of upward moving particles within ejections, then we compute the CDF and obtain the threshold. This threshold decreases as particle inertia decreases indicating that for small Stokes numbers particles initially entrained in an ejection follow the fluid upward. Thus, there is an evident dependence of the threshold value on the Stokes number. The threshold is therefore normalized by the following equation: 

u0 v 0T ¼

u0 v 0T

ð7Þ

u2 St1=2

The evolution with distance to the wall of the normalized threshold u0 v 0T  is shown in Figs. 10 and 11, for Re = 2660 and c = 0 (Fig. 10a), for Re = 12,500 and c = 0 (Fig. 10b) and for Re = 12,500 and c – 0 (Fig. 11). The evolution with wall distance of the normalized threshold is similar to the profile of the average Reynolds shear stress in the fluid, u0 v 0 (Fig. 10). In Figs. 10 and 11, u0 v 0 for the fluid is normalized by u2 ðSt ¼ 1Þ. The threshold presents a maximum value around y+  30 and then vanishes as y+ increases. The normalized

500

1

400

u’v’ threshold

CDF

y+

0.8 300

200

100

0.6 0.4 0.2 0

0 0

20

40

60

80

100

%

0

0.01

0.02

-u’v’ Fig. 8. Evolution with wall distance of the percentage of second quadrant (Q2). (-) – Percentage of Q2 at the position of solid particles. (N) – Percentage of Q2 at the position of solid particles with vp > 0. (M) – Percentage of Q2 at the position of solid particles with vp < 0. Re = 12,500, St = 5 and c = 0.

Fig. 9. Cumulative distribution function of u0 v0 in the second quadrant for particles moving away from the wall (j) and toward the wall (N). Re = 12,500, St = 5, c = 0 and y+ = 348. The vertical line illustrates the value of the threshold u0 v 0T .

I. Vinkovic et al. / International Journal of Multiphase Flow 37 (2011) 187–197

193

Fig. 10. Evolution with distance to the wall of the normalized threshold u0 v 0T  for different Re, St and c = 0. (a) Re = 2660, c = 0, St = 1 (h), St = 5 (M), St = 15 (O), St = 25 (}), St = 125 (w), fluid (–). (b) Re = 12,500, c = 0, St = 5 (M), St = 15 (O), St = 25 (}), St = 125 (w), fluid (–). For the fluid u0 v 0 is normalized by u2 (St = 1).

segregation threshold profiles follow the same qualitative trends. The values of the normalized threshold obtained for different Stokes

* * * * * * * * * * 400 * * * * * * * * * * 200 * * * + + +++++++ * + +++ +++++++ * ** + + ++ ++++ + +++ + ++ +** ++ ++ ++++ ** ++++++++++ + ** + ++ ++ ++++ * ** *+ + ++++*++ + 0 ** *

y+

600

0

1

2

3

-u’v’ / (u*² St1/2) Fig. 11. Evolution with distance to the wall of the normalized threshold u0 v 0T  for Re = 12,500 and different St, c. Comparison with experimental results. Numerical simulations – St = 1 and c = 1 (h), St = 5 and c = 1 (M), St = 15 and c = 1 (O), St = 25 and c = 1 (}), St = 125 and c = 1 (w), St = 1 and c = 0.7 (j), St = 1 and c = 0.3 (), St = 1 and c = 1.2 (.), St = 1 and c = 1.4 (N), fluid (–), mean value of u0 v 0T  for Re = 12,500, St = 1–125 and c = 0 (– –). Experiments – Lelouvetel et al. (2009) Re = 1000–3000, St = 4.2–13.6 and c = 1.2–2.9 (+). For the fluid u0 v 0 is normalized by u2 (St = 1).

numbers collapse to a single value away from the wall. This single value is u0 v 0T   0:2 for both Reynolds numbers (Re = 2660 and Re = 12,500). The collapse of u0 v 0T  for different Stokes numbers close to the wall is not verified, especially for the low Reynolds number case. Other normalizations have been tested and this one showed to give the best fit in both the near wall and channel center region. Collapse of data over the entire channel height, might be obtained by introducing a local (depending on wall distance) fluid timescale for the definition of the Stokes number (Eq. (5)). However, this is not the aim of the present work. Further studies are currently conducted in order to better understand the normalization and the square root dependence. Finally, for both Reynolds numbers, the threshold obtained for St = 125 does not fully overlap with the other thresholds. For high Stokes numbers, such as St = 125, particles are less influenced by the turbulent structures of the carrier fluid and have almost parabolic trajectories given by the initial and bouncing conditions. Indeed, for St = 125 particles moving away and toward the wall sample ejections equally, so that there is no statistical separation between ejections at the position of upward and downward moving particles. Therefore, the value obtained for u0 v 0T  is not a real threshold. However, this value of u0 v 0T  for St = 125 is shown in the figures since it can give an indication of the limit of u0 v 0T  . Fig. 11 illustrates the evolution with wall distance of the normalized threshold obtained for Re = 12,500 and c – 0. Here, gravity is taken into account in the equation of solid particle motion (3). However, as explained in Section 2.2 (Table 2) only cases where gravity has a small influence (small values of c) are treated here. The profiles of the segregation threshold still follow the same qualitative trends for different St and c. The values of the normalized threshold are close to the ones obtained for c = 0. Nevertheless, for c > 1, as c increases a systematic shift of the segregation threshold profiles seems to appear. Here, we analyze the role of ejections in particle segregation. As stated above, the range of tested parameters is such that c remains small which corresponds to sediment transport by suspension. Therefore, gravity is not included in the segregation threshold criteria. If situations with higher values of c were considered, corresponding to sediment transport by

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Table 3 Experimental characteristics, Lelouvetel et al. (2009). Test

Re

St

c

CeraP02 CeraP03 CeraP04 CeraP05 CeraG02 CeraG03 CeraG04 CeraG05 Glass02 Glass03 Glass04 Glass05

10,000 15,000 24,000 30,000 10,000 15,000 24,000 30,000 10,000 15,000 24,000 30,000

4.1 5.3 7.3 8.5 6.8 8.5 11.7 13.6 4.1 5.1 7.1 8.2

1.8 1.6 1.4 1.3 2.9 2.6 2.2 2.1 1.7 1.6 1.3 1.2

saltation for example, gravity would play an important role in the preferential concentration of particles and should be included in the segregation threshold criteria. The validity of the normalized threshold proposed here is further tested by submitting the experimental results of Lelouvetel et al. (2009) to the same analysis as the one applied for the numerical simulations. By means of PIV for the carrier fluid, and PTV for the solid phase, Lelouvetel et al. (2009) measured simultaneously solid particle and fluid velocity near the wall in an open-channel turbulent flow. Details of this study may be found in Lelouvetel et al. (2009), Lelouvetel (2008). Table 3 gives a brief description of the experimental conditions. As well as for the numerical simulations, from the experimental results we check that upward moving particles concentrate preferentially in ejections. We then calculate the CDF and determine the normalized threshold. The evolution with wall distance for this threshold is illustrated in Fig. 11 (crosses). The experimental threshold is slightly lower than the one obtained from numerical simulations. However, for a large set of parameters (see Table 3) the experimental threshold has an average value that is close to the range of values obtained numerically. This further confirms the relevance of the normalization proposed in this study. In conclusion, a threshold value of ju0 v0 j separating ejections with mostly upward moving particles from ejections with both

upward and downward moving particles is established. As particle inertia decreases the proposed threshold decreases as well indicating that for small Stokes numbers particles initially entrained in an ejection follow the fluid upward. Thus, there is an evident dependence of the threshold value on the Stokes number. When normalized by the square root of the Stokes number and the square of the friction velocity (Eq. (7)), the profiles of the segregation threshold u0 v 0T  follow the same qualitative trends, for all the parameters tested in this study and the experimental data of Lelouvetel et al. (2009). Even though the influence of the Stokes number on the segregation threshold value is evident it is still not clear why there is a square root dependence (Eq. (7)). Further studies are presently conducted in order to better understand this normalization. A laminar flow, with controlled hairpin vortices and strong ejections is used for gaining further insight. 3.3.2. Sensitivity analysis The robustness of the segregation threshold proposed here is tested by a sensitivity analysis. The influence of three parameters is studied: the total number of particles, the percentage of particles moving toward the wall within ejections and time evolution. Fig. 12a illustrates the influence of the number of particles used in the simulation on the value of u0 v0 T*, for Re = 2660, St = 5, c = 0. It can be seen that using twice or four times less particles does not change the threshold value. The threshold value u0 v 0T  proposed in this study is chosen as the local value of ju0 v0 j for ejections within which less than 2.5% of particles move toward the wall. This percentage has been chosen in order to fit the critical value used in the experiments by Lelouvetel et al., 2009, as stated in Section 3.3. When analyzing the experimental data of Lelouvetel et al. (2009), the critical value of 2.5% has been chosen as the closest value to zero for which there was no considerable qualitative statistical variation. Changing the criterion from 2.5% to 50% moves the threshold down. This is illustrated in Fig. 12b where the threshold value for different percentages going from 1% to 50% is presented for Re = 2660, St = 5, c = 0. Nevertheless, the aim of this study is to establish a local and instantaneous segregation threshold. In the context of Lagrangian tracking, this segregation threshold will further be used for choosing which

Fig. 12. Evolution with distance to the wall of the normalized threshold u0 v 0T  for Re = 2660, St = 5, c = 0. (a) Different total number of particles Np, Np = 200,000 (M), Np = 100,000 (h), Np = 50,000 (e). (b) Different criteria for the threshold determination, from left to right respectively 50%, 25%, 10%, 5%, 2.5% and 1%.

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* * * * * * * * * * 400 600

y+

* *

1

10

0

10

* *

τ*

* * * * *

200

0

** *

0

−1

10

* * * * * * * ** * **

−2

10

0

0.5

2

10

10

3

10

*

Rep

*** * * *

*

1

10

* 1

1.5

2

-u’v’ / (u*² St1/2) Fig. 13. Evolution with distance to the wall of the normalized threshold u0 v 0T  for Re = 12,500, St = 5 and c = 0 at different t+. t+  100 (h), t+  200 (w), t+  500 (}), t+  1000 (.).

turbulent events are predominant in influencing particle movement. Therefore, the initially chosen criterion of 2.5% is applied. As stated in Section 2.2, even though particle velocity profiles are stationary, the concentration profiles have not attained that condition yet, especially for the high Reynolds case, Re = 12,500. Therefore, a sensitivity analysis based on flow development is done. Fig. 13 shows how the profiles of the normalized threshold evolve in time. Particle statistics are sampled starting from t+  1000 (see Section 2.2 for more details). Thus, for the sensitivity analysis, normalized segregation threshold profiles are presented at different instants during flow development, t+  100, 200, 500 and 1000. A high Reynolds number case is chosen, Re = 12,500, St = 5 and c = 0. Profiles at different t+ are all within the boundaries of statistical error. The profiles present rather large variations since less statistical samples have been used for computing each case. There is no significant evolution of the segregation threshold during the development of the flow. The segregation threshold presented here is not influenced by the evolution of the particle concentration profile, which is related to the vertical particle flux. The particle flux at a given height depends both on particle velocity and on the number of moving particles. None of these quantities can be evaluated by only considering a local and instantaneous segregation threshold. Again, simultaneous Lagrangian tracking of fluid and particle is needed in order to gain physical understanding of particle-turbulence interaction, as it is planned in further studies.

3.4. Comparison with suspension thresholds Suspension thresholds are an important quantity for applications in hydraulics. They are often presented in the Shields diagram (van Rijn, 1984). Therefore, the local segregation threshold obtained in this study is compared to the experimental suspension thresholds of Niño et al. (2003), van Rijn (1984), Lelouvetel et al. (2009), and to the theoretical relationship proposed by Bagnold (1966). The classical Shields curve for incipient particle motion is also plotted in Fig. 14 as a reference for the initiation of bedload transport.

Fig. 14. Comparison between the average normalized threshold obtained in this study and the critical suspension thresholds obtained by other authors. () – numerical simulations for Re = 12,500, different St and c = 1. (h) – numerical simulations for Re = 12,500, St = 1 and different c. Open symbols ( and h) correspond to a threshold criterion of 97.5%. Full symbols ( and j) correspond to a threshold criterion of 50%. (+) – Lelouvetel et al. (2009). (Thick line –) – Niño et al. (2003). (– –) – van Rijn (1984). (    ) – Bagnold (1966). (–.–) – Bagnold (1966) with the correction term proposed by Clift et al. (1978). (Thin line –) – Shields (1936).

3.4.1. Shields diagram is presented in Fig. 14, in the parameter space  The comparison  Rep ; s , where Rep and s* denote respectively a particle Reynolds number

Rep ¼

u dp

ð8Þ

m

and a dimensionless bed shear stress

s ¼

qu2 gðqp  qÞdp

ð9Þ

For the comparison, the segregation thresholds obtained in this study and by Lelouvetel et al. (2009) are averaged over the wall distance. The standard deviation obtained by this averaging is used for the estimation of error bars. In addition to this, the dimensionless bed shear stress s* is plotted using the relation:



s ¼



q u0 v 0T  u2 St1=2 gðqp  qÞdp

ð10Þ

For the estimation of Rep , first a threshold friction velocity u*threshold is calculated:

uthreshold ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u0 v 0T  u2 St 1=2

and then, Rep ¼

uthreshold dp

m

ð11Þ

is obtained.

3.4.2. Comparison and discussion As shown in Fig. 14, the threshold curve resulting from the present simulations first decreases as Rep increases and then increases for Rep > 3. The threshold points for Rep < 3 correspond to the cases (Re = 12,500, St = 1, c = 0.3), (Re = 12,500, St = 1, c = 0.7), (Re = 12,500, St = 1, c = 1), (Re = 12,500, St = 1, c = 1.2) and (Re = 12,500, St = 1, c = 1.4) respectively as Rep increases. For a given set of Re and St parameters, as c increases, stronger local values of ju0 v0 j define the threshold. These higher values of ju0 v0 j, correspond to higher values of u*threshold and therefore to higher values of Rep , but to lower values of s*. For Rep > 2, the threshold points correspond to increasing St parameter (St = 1, 5, 15, 25, 125) for

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c = 1 and Re = 12500. For this set of points, as the Stokes number increases, particle inertia increases. Therefore, there are more downward moving particles in regions of upward moving flow, and the threshold value for ju0 v0 j is higher. The higher the St, the less correlated the particle and fluid motions are. That is why the threshold increases with the Stokes number. The threshold point corresponding to Rep ¼ 55 is obtained for (Re = 12,500, St = 125; c = 1). As mentioned in the previous section, for St = 125 particles are no longer respond to turbulent fluctuations of the flow. They have almost parabolic trajectories given by the initial conditions. As mentioned in Section 3.3.1, the segregation threshold predicted by our simulations is in good agreement with the experimental segregation threshold obtained by Lelouvetel et al. (2009). In both cases the segregation threshold is estimated in the same way and the average threshold values collapse for the set of parameters studied here. When compared to the limit of entrainment proposed by van Rijn (1984) or Niño and García (1996), our simulations predict a much larger threshold. Still, the same order of magnitude is obtained despite different approaches and threshold definitions. We do not consider full particle trajectories and mean bed shear stress but rather the particle and fluid instantaneous and local behaviors. The differences in the obtained thresholds may therefore be due to (i) the measure used to characterize the fluid behavior and (ii) the criterion chosen to decide whether particles are influenced by the surrounding fluid. Indeed, we measure the fluid behavior by using a value of u0 v0 while the other authors measure the mean bed shear stress. As for the second reason of the difference (ii), for van Rijn (1984) or Niño and García (1996), suspension is supposed to be present when some particles have long trajectories without hitting the ground, while in our study the chosen criterion is that almost all particles (97.5%) must have vp > 0. If we chose a less restrictive criterion, for example 50% of particles that must have vp > 0 (see Section 3.3.2 for the sensitivity analisys), the threshold proposed here is shifted towards lower values of u0 v 0T  and therefore lower values of Rep and s*. This is illustrated by the full symbols in Fig. 14. This shift implies roughly better fit of the different thresholds for higher Stokes numbers but fails for St = 1. For intermediate values of Rep , the segregation threshold obtained by our simulations and the experimental segregation threshold of Lelouvetel et al. (2009) are in fair agreement with the suspension criterion proposed by Bagnold (1966). In defining its theoretical criterion, Bagnold (1966) assumes that the vertical velocity fluctuations of the flow, v0 , must balance the settling velocity, for the particle to be suspended. In addition to this, Bagnold (1966) assumes that the vertical velocity fluctuations of the flow are of the order of u*. In this sense, Bagnold’s criterion for defining the suspension threshold is closer to the segregation criterion used in this study. There are however two main differences. The first resides in the fact that we chose to define a limit in terms of u0 v0 rather than in terms of v0 . The second difference is that Bagnold (1966) proposes a global fluid quantity for estimating v 0 , (v0  u). Moreover, without any additional explanation, Bagnold, 1966 states that for all large grains exceeding about 2 mm, s* should remain constant for all large material irrespective of size. This explains the constant value of s* obtained by Bagnold (1966) for Rep > 20. No such behavior is observed in this study for the local segregation threshold defined here. In Fig. 14, we also plot the theoretical relationship proposed by Bagnold (1966) in the case when the drag force is corrected by the term proposed by Clift et al. (1978). As shown in Fig. 14, when the correction is introduced, the constant value obtained for s* seems to be attained for much higher values of Rep . In conclusion, the local segregation threshold proposed here roughly agrees with the global suspension thresholds proposed in other studies. Even though different criteria for both the fluid

and solid phases are applied and even though different quantities are used for estimating the threshold, the orders of magnitude are roughly the same. This section gives an indication on the global behavior of the local segregation threshold proposed here. Nevertheless, this is not the main point of this study. As stated in the introduction to this section, the comparison with global suspension thresholds is only briefly explored as a first step towards hydraulics applications.

4. Conclusion Direct numerical simulations for high shear Reynolds number, coupled with Lagrangian particle tracking are performed to examine the mechanisms of solid particle transport within a turbulent channel flow, and to study the characteristics of ejections that surround solid particles moving away from the wall. The results show that there is preferential concentration of particles in regions of ejections and sweeps, confirming previous studies on this issue Kaftori et al. (1995a) and Niño and García (1996). As already shown by Kiger and Pan (2002), we find that particles moving away from the wall are surrounded by upward moving fluid, and mostly by ejections. However, ejections are found at the position of particles that move toward the wall as well. The difference between ejections at the position of upward and downward moving particles is linked to the absolute value of the Reynolds stress tensor, ju0 v0 j. Particles moving toward the wall are surrounded by low values of ju0 v0 j. A threshold value separating ejections with only upward moving particles from ejections surrounding both upward and downward moving particles is established. The profiles of this segregation threshold follow the same qualitative trends, for all the parameters tested in this study when normalized by the square root of the Stokes number and the square of the friction velocity. These trends of the normalized segregation threshold are confirmed by analyzing the experimental data of Lelouvetel et al. (2009). The segregation threshold proposed here is compared to suspension thresholds from the literature. When compared to the limit of entrainment proposed by van Rijn (1984) or Niño and García (1996), our simulations predict a much larger threshold value. This may be due to the difference in (i) the measure used to characterize the fluid and (ii) the criterion chosen to decide whether particles are influenced by the surrounding fluid. The theoretical suspension criterion proposed by Bagnold (1966) is closer to the definition of the segregation threshold used in this study. For intermediate values of Rep , the segregation threshold obtained by our simulations and the experimental segregation threshold of Lelouvetel et al. (2009) are in fair agreement with the suspension criterion proposed by Bagnold (1966). The main result of this study resides in the correlation between strong values of u0 v0 and the transport of particles away from the wall. The correlation can be normalized by the square root of the particle Stokes number. Even though the influence of the Stokes number is evident it is still not clear why there is a square root dependence. Further studies will be conducted in order to better understand this normalization. In particular, a laminar flow with controlled hairpin vortices and strong ejections will be used for testing the relevance of the segregation threshold definition and the normalization. The original aspect of this study resides in the local analysis of the interaction between upward moving particles and ejections. We learn from this study that upward moving particles are surrounded by upward moving fluid and mostly by ejections. More precisely, ejections with u0 v 0T  > 0:2 are strong enough for influencing particle trajectories and entraining them upward. This threshold value can further be used for conditional sampling and

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the detection of strong ejections at the solid particle position. Statistical properties of solid particles surrounded by ejections above the threshold can be analyzed. In future work, solid particles surrounded by ejections above the threshold will be tracked simultaneously in order to better understand particle–turbulence interactions and transport mechanisms. For example, when presenting their transport mechanism Kaftori et al. (1995a) chose the following threshold:

u0 v 0 =u0 v 0 > 1:6 In this study, we give some advantages for choosing u0 v 0T  > 0:2. No major advancement is presented here in terms of understanding the interaction between particles and large-scale turbulent structures. However, the original aspect of this study is that the direct numerical simulations are conducted in the range of the experimental parameter values (Lelouvetel et al., 2009). In addition to this, we show a sensitivity analysis and the details of the threshold behavior that will further be applied to selecting relevant turbulent events in the particle–ejection interaction study. At last, the segregation threshold proposed in this study can be used for modelling purposes in sediment transport applications. For instance, the segregation threshold can be applied to the estimation of particle residence time within ejections. Acknowledgments The authors express their gratitude to F. Laadhari who kindly provided his initial fields for the DNS. This work was granted access to the HPC resources of CINES under the allocation 2009-c200,902,560 made by GENCI (Grand Equipement National de Calcul Intensif). Numerical simulations were also performed on the P2CHPD parallel cluster. The authors are therefore grateful to C. Pera for the administration of the computational tool. References Adrian, R., Meinhart, C., Tomkins, C., 2000. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54. Bagnold, R., 1966. An approach to the sediment transport problem from general physics. Geological Survey Professional Paper 422-1. Buffat, M., LePenven, L., Cadiou, A., 2009. An efficient spectral projection method on divergence-free subspaces for transition analysis in wall bounded flow. Tech. rep., Laboratoire de Mécanique des Fluides et d’Acoustique, Université Claude Bernard Lyon 1. Clift, R., Grace, J., Weber, M., 1978. Bubble, Drops and Particles. Academic Press. Hoyas, S., Jiménez, J., 2008. Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511. Jackson, R., 1976. Sedimentological and fluid-dynamic implications of the turbulent bursting phenomenon in geophysical flows. J. Fluid Mech. 77, 531– 560.

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