Dispersion and deposition mechanisms of particles suspended in a turbulent plane Couette flow

Applied Mathematical Modelling 37 (2013) 2417–2429

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Applied Mathematical Modelling

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Dispersion and deposition mechanisms of particles suspendedin a turbulent plane Couette flow

Fahimeh Hosseiniebalam, Smaeyl Hassanzadeh, Omid Ghaffarpasand ⇑Physics Department, Faculty of Science, University of Isfahan, Isfahan 81746, Iran

a r t i c l e i n f o

Article history:Received 5 September 2011Received in revised form 29 April 2012Accepted 7 May 2012Available online 6 June 2012

Keywords:Turbulent Couette flowTurbulent Poiseuille flowParticles dispersion and depositionLarge Scale Structures

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.05.031

⇑ Corresponding author. Tel.: +98 311 7934823; fE-mail address: [email protected] (O.

a b s t r a c t

Inertial particle transfer in a turbulent plane Couette flow (C flow) was studied using DirectNumerical Simulation (DNS) of the flow combined with a Lagrangian particle trackingapproach for particles with Stokes numbers (St) 5, 25 and 125. The particle concentrationwas assumed low enough, so that the simulations were done under one-way coupling con-dition.

Our numerical simulations show that the particles suspended in a C flow (C particles)have velocities respectively smaller and larger than the carrier fluid velocities in thenear-wall and core region. The C particles in comparison to the same sized particles sus-pended in a Poiseuille flow (P particles) with a same Reynolds number (Re), have largerfluctuating activities in the core, while the opposite was observed in the near-wall region.This fact is due to the Large Scale Structures (LSSs) existing in the core region of the C flows.The LSSs also strengthen the near-wall accumulation of C particles, so that these particlesrespect to the same sized P particles have more accumulation in the near-wall region.Moreover, the LSSs affect the deposition velocity of particles whereby among the P andC particles with smallest Stokes number (St ¼ 5), the C particles have larger depositionvelocity while among the those ones with larger Stokes numbers (St ¼ 25;125) the oppo-site was the case.

Finally, it was found that although the diffusional deposition mechanism for the C parti-cles is less efficient than for the P ones, the number of deposited C particles during the timeis greater than the number of deposited P particles.

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1. Introduction

The transport mechanisms of the particles suspended in the turbulent wall bounded flows are of major concerns for someindustrial applications such as: xerography, micro-contamination, chemical industries, combustion, and air pollution con-trol. The dynamics of the motion of those particles have been clarified in several issues within the last years [1–4]). Never-theless, despite decades of efforts to describe the several features of the particle behavior in the turbulent wall boundedflows, there are some open issues (e.g. the transport mechanisms of the C particles) which were not enough investigated.Up to now, to our knowledge, only [5] employed DNS and Lagrangian particle tracking to investigate the behavior of inertialparticles in the C flows. The authors showed that the distribution of the particle velocity is very different from a Gaussiandistribution. They never considered the mechanisms of the C particles dispersion and deposition.

The C flow is generated when one of the channel plates is put into motion relative to the other one. It was first studied by[6,7] who performed extensive measurements of mean and fluctuating fluid velocities. Thurlow and Klewicki [8] found that

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2418 F. Hosseiniebalam et al. / Applied Mathematical Modelling 37 (2013) 2417–2429

when one of the walls begin to move, a fluid momentum exchange is produces between the two walls. They observed theoccurrence of elongated regions of positive and negative momentum fluctuations in the core, which they called LSSs. [9],[10], and [11] studied the LSSs of the core region with more details. They shown that LSSs extending over a very longstream-wise distance. [12–14] inferred that the existence of LSSs in the core region, causes a more significant modulationof the near-wall coherent structures. They showed that the velocity profile of the C flows in the near-wall region is organizedin narrow alternating low and high momentum regions, named streaks, which are clustered densely in the near of the mov-ing wall. This densely crowd of streaks in the near-wall region of the C flows, which is in contrast to the uniform distributionof the streaks in the same region of P flows, affect the particle transfer mechanisms.

The study of the existing differences between the C and P particle motion will be further clarified the role of the LSSs inthe C particle transport mechanisms. Moreover, it can indicate the dominant deposition mechanisms of the C particles whichhave not studied until now.

In light of the above discussion, the current study focuses on the study and investigation of the effects of LSSs, which pop-ulate the core region of C flows and affect on the near-wall coherent structures, on the C particle transfer mechanisms. TheDNS was used for simulating the flow field because it gives access to all interesting quantities. DNS in particular allows adetailed analysis of the near-wall region, where most particle transfer mechanisms take place. A Lagrangian particle trackingapproach was used to track the particles. The particle concentration was also considered low enough such that particle–particle collisions can be neglected and the turbulent flow field was not modified by the motion of the particles (one-waycoupling condition). The paper is organized as following: The transport mechanisms of P particles are reviewed in the nextsection. This review is used in the next sections when the motion of the C and P particles are compared. Section 3 presentsthe used numerical methodology containing the numerical methods for simulating the fluid flows, Lagrangian particle track-ing approach, and the developed method for evaluating the particle statistics. In Section 3.2, particle velocity profiles are con-sidered. The deposition of particles is studied in Sections 5 and 6 where we analyze respectively the instantaneous particleconcentration patterns and the mechanisms of deposition. Conclusion remarks are given in Section 7.

2. The transport mechanisms of P particles

It has been argued that the near-wall coherent structures are primarily responsible for the suspension and transport ofsolid particles. [1] demonstrated that the particles tend to accumulate preferentially in the near-wall low-speed local re-gions. The preferential concentration of particles in these regions called streaks, has been also shown by many otherresearchers [15–17] show that the streaks have a characteristic length of about 100 wall units in the span-wise directionand are associated with the occurrence of counter-rotating vortex pairs aligned in the stream-wise direction. [18] employedDNS to study the particle transfer mechanisms. They found that the streaks assist in particle fluxes by conveying particlesfrom the core to the near-wall region and vise versa through sweep and ejection events, respectively. [16] showed that strea-ky structures have an important role in particle trapping and deposition mechanisms.

Brooke et al. [19,20] also employed DNS to investigate the dispersion mechanisms of suspended particles in a turbulentchannel flow. They provided an interesting subdivision of particle flux into three components: The free-flight, the turboph-oretic, and the turbulent diffusion flux. Free-flight flux take into accounts the fraction of particles which have high enoughvelocity to execute a free-flight to the wall. Turbophoresis accounts the particle flux due to gradients in turbulence intensity,while the turbulent diffusive flux considers the particle flux due to the concentration gradients. They indicated that, depos-ited particles are often starting their free flights to the wall at large distance from the wall.

Narayanan et al. [21] defined a new type of flux for particles strongly concentrated near the wall, diffusional depositionflux. They observed that the particles segregated from the core and accumulated in the near-wall region can move towardthe wall and deposited, in some cases, by their residual turbulence fluctuations. They also showed that the deposition ofsmall particles is dominantly due to this kind of flux.

3. Numerical methodology

3.1. Direct Numerical Simulation of carrier fluids

The motion of a fluid in a simple channel is governed by the Navier–Stokes equations. Under the assumptions that thefluid is incompressible, isothermal, and Newtonian, the non-dimensional Navier–Stokes equations are:

@uj

@xj¼ 0; ð1Þ

@ui

@tþ @ui@uj

@xj¼ � 1

q@P@xiþ 1

Re@2ui

@xixj� Hdi;1; ð2Þ

where uiði ¼ 1;2;3Þ is the ith component of the velocity vector, q is the fluid density, P is the pressure of the fluid, H is themean pressure gradient required to maintain a constant flow rate and di;j is the Kronecker delta (di;j ¼ 1 if i ¼ j; di;j ¼ 0 if i – j).All the variables have been made dimensionless by the friction velocity us and the channel width h. Air in ambient conditionswas considered as the carrier fluid with density q ¼ 1:3 kg m�3 and kinematic viscosity m ¼ 15:7� 10�6 m2 s�1. The Navier–

Fig. 1. The computational domain of (a) P flow, and (b) C flow. The arrow indicates the direction of imposed boundary condition for generating the C flow(See text for details).

F. Hosseiniebalam et al. / Applied Mathematical Modelling 37 (2013) 2417–2429 2419

Stokes equations have been discretized in an orthogonal coordinate system (x; y, and z for the stream-wise, wall-normal, andspan-wise direction, respectively) using staggered central second-order finite-difference approximations in which the ki-netic energy is globally conserved in the inviscid limit. The details of the numerical method are reported in [22].

Periodic boundary conditions were imposed on the fluid velocity field in the stream-wise (x) and span-wise (z) directions.Moreover, two different boundary conditions for two different considered fluid flows (C and P flows) were imposed in thewall-normal (y) direction. For C flow, the boundary conditions at the bottom (moving) wall were applied by forcing a con-stant value for the stream-wise velocity component while on the upper wall all velocity components (hereafter also denotedas u;v , and w) were set to zero (no-slip boundary conditions). For P flow, no-slip boundary conditions were imposed on bothof the walls.

The simulations were performed for a computational box with size Lx ¼ 4 h in the stream-wise, Ly ¼ 2 h in the wall-nor-mal and Lz ¼ 2 h in the span-wise direction. The computational domain has been discretized with 128� 128� 128 cells inthe x; y, and z directions, respectively. A sketch of the flow configuration, showing the reference system was depicted inFig. 1.

All of the simulations presented in this study have been performed with the shear Reynolds number, Res ¼ ush=m ¼ 150and with the corresponding bulk Reynolds number Reb ¼ ubh=m ¼ 3600 based on the bulk velocity ub ¼ 2:83 ms�1. The com-putational time-step in wall units was Dtþ ¼ 0:1. The fully developed C flows was generated by a DNS method described withmore details in the studies of [14] et al. The simulations in both cases were continued for a significant time, i.e. 14000 timeunits, which is sufficient to guarantee the statistical convergence of all the interesting parameters.

3.2. Lagrangian particle tracking

The particles were introduced in the simulated carrier fluids past the initial transient of 1000 time units, when the fluidswere independent of the initial conditions. The injected particles were assumed as the point-wise and rigid smooth spheres.The particle radiuses were small enough compared to the Kolmogorov length scale of the flow while those were large enoughsuch that the Brownian effects can be neglected. The density of the particles was taken to be 769 times that of the fluid den-sity. As we wish to concentrate on the effects of turbulence structures on the particle transfer mechanisms, the action ofgravity was also ignored. Under these assumptions, the drag becomes the predominant force acting on a particle [23] andparticles obey the following Lagrangian dimensionless equation of motion.

d u!

dt¼ �3

4CD

dp

qqp

!up�!� u!�� up

�!� u!� �

; ð3Þ

where up�! and u! are the particle and fluid velocity vectors respectively, dp is the particle diameter and CD is the drag coef-

ficient given by [24]:

Table 1Particle parameters.

St ¼ sþp sðsÞ dþp dpðlmÞ

5 5:66� 10�3 0.342 45.6

25 2:83� 10�2 0.765 102

125 1:42� 10�1 1.71 228


CD ¼24Rep

1þ 0:15Re0:687p

� �: ð4Þ� �

Here Rep is the particle Reynolds number based on the local relative velocity of the carrier fluid (Rep ¼dp up�!

� u!� �m ). Beyond

that the influence of the wall and non-stability of the flow on the drag force were neglected.The parameters of the injected particles were summarized the Table 1. We used three sets of particles with Stokes num-

bers 5, 25, and 125. The Stokes number is defined as the nondimensionalized particle response time St ¼ sþp ¼sp

sfwhere the

response time is defined as sp ¼qpd2

p

18l and sf is the characteristic time scale of the fluid flow. The C particles used in this study,because of their sufficiently large response time, were significantly affected by the LSSs of the core. Hence, the influence ofthe LSSs on the particle transport mechanisms can be studied.

A number of 100,000 particles were introduced uniformly in the whole of the channel at the initial state. The stiff equa-tions of particle motion were integrated by a 4th order Runge-Cutta method, when a same computational time step as thefluid simulations, i.e. Dtþ ¼ 0:1, was used. The initial velocities of the injected particles were equal to the fluid velocities attheir initial random positions. The fluid velocities at the grid points were interpolated to the particle position by using a tri-linear interpolation scheme. It was assumed that the particles deposited on the walls, when their distance from the wallwere less than their diameter. Besides for keeping the stationary statistically state which is necessary to study the concen-tration profile, a new particle was introduced after each particle deposition by assuming the elastic collision conditions. Theparticle tracking were continued as well as the fluid simulation (14000 time units) in order to obtain reliable all interestingstatistics.

3.3. Developed method for calculating the particle statistics

Domain division: The simulation domain, plane channel, was divided into Ns ¼ 193 wall parallel slabs, which were nonuniformly distributed along the wall-normal direction. The thickness of the sth slabs, DyþðsÞ, was determined by the follow-ing relation [4]:

Dyþ sð Þ ¼ Res

tanh cð Þ tanhcsNs

� �� tanh

s� 1Ns

� �� ; ð5Þ

where c ¼ 1:7. The largest thickness is at the channel centerline (Dyþmax = 2.84) whereas the smallest thickness is at the wall(Dyþmin = 0.361). The domain division in this way has two important advantages. First, the obtained P particle statistics withStokes numbers 5 and 25 can be compared with the studies of [4]. The authors by using a same slab division provided aquantitative data base for P particle statistics with Stokes numbers 1, 5, and 25. They used a P flow with a same Reynoldsnumber as our study. The good agreement existing between our numerical results for P particles and their data base vali-dates the numerical methodology used in this study. Second, the near-wall region where the most transport mechanismstake place was studied with more spatial resolution.

It should be noted that, the minimum thickness of the slabs allowed at the wall being limited by the radius of the particleswith St = 25 (see Table 1). Hence, the closest layer to the wall located by particles with St = 125 differs with the located layersby the other test particles.

Particles statistics: After the domain division discussed above, the number of particles and their related velocities withineach slab at each time step were determined and scored. A particle belongs to a certain slab when its center located insidethe slab. All the statistical parameters (velocities and number concentrations) were time averaged in final 1000 time unitswhen the flow system reaches to its steady state. The steady state condition can be verified by investigating the Fig. 8 wherethe time evolution of the number density of the arrived particles to the wall was studied. The profile of the arrived particlesto the wall convergences toward a mean value (horizontal line) after almost 10000 time units. The flow is in the steady stateafter this time. The reason for monitoring this quantity lies in the fact that the concentration close to the wall is the one thattakes longer to reach its steady state [4]. Finally, for obtaining the particle number density, the time averaged number ofparticles in each slab was normalized by the case of uniform particle distribution over the entire domain at the initial state.

4. Velocity statistics

The mean stream-wise velocity profiles of particles and carrier fluids were shown in Fig. 2. Evidently, when the bottomwall is put into the motion (C flow), the wall stress reduces and the location of the maximum mean stream-wise velocity of C

Fig. 2. Mean streamwise velocity.

Fig. 3. The difference of the particle and fluid mean stream-wise velocities for (a) C particles, and (b) P particles, along the distance from the wall, yþ .


flow shifts towards the wall [25]. The test particles due to their small enough response times compared to the fluid timescale, almost followed the carrier fluids. Hence, in contrast to the mean stream-wise velocity profile of P particles, whichits maximum value is in the core of the channel, the maximum mean stream-wise velocity of C particles is on the wall.

The effects of the turbulence structures on the particle dynamics were further investigated in the Fig. 3 where the differ-ences of the mean stream-wise velocity of the particles and fluid, Duþmean ¼ uþf ;mean � uþp;mean, were presented. It was observed


that the mean stream-wise velocity of P particles leads the fluid in the near-wall and lags it in the core region, while theopposite is valid for C particles. [1] argued that the slip velocities of P particles ( up

�!� u!�� ) increased dramatically in the

near-wall region where the flow mean velocity gradient is steep. The authors showed that this is due to the transverse mix-ing of stream-wise particle momentum.

The transverse migration of P particles with diameters substantially less than the Kolmogorov length scale of turbulentflow was described with more details in the studies of Lee and Wiesler [26]. They observed that the response of those par-ticles to the carrier fluid velocity fluctuations in the stream-wise direction is more significant than in the other directions likewall-normal or span-wise direction. This observation was observed in this study for C particles in the Figs. 4 and 5 (b) and (c)where the velocity fluctuations of the C particles and C flow were compared in each direction. Lee and Wiesler [26] alsoobserved that the fluctuating intensity of those P particles shifted toward the higher intensities in which direction normalto the maximum drift. In other words, those particles migrate transversely to the high stream-wise fluctuating areas. Inaccording to the Fig. 5(a), which shows that the C flow has a high intensity stream-wise fluctuating area in the near-wallregion, the C particles migrated toward the near-wall region by transverse migration phenomena.

However, the C particles in the core region have larger stream-wise velocities than in the near-wall region (see Fig. 2).Moreover, in transverse migration, a large amount of the stream-wise velocity of a particle depend on its size were retained(larger particles due to their larger inertia retain larger velocities). Therefore, the transverse migration of C particles from thecore toward the near-wall region increase the local stream-wise velocity of C particles in respect to the carrier fluid. Theopposite trend was also true in the near-wall region.

There is also an additional scenario for C particles with smallest Stokes number (St = 5). [15] proved that particles respondto the carrier turbulent fluid according to their either response time or Stokes number. Hence, C particles with small Stokesnumber (St = 5) in the core region, where existing of the LSSs were inferred, were less motivated than particles with largerStokes number (St = 125). Thus, it can be claimed that small C particles flow in the core region very close to the C fluid flowsand/or have same stream-wise velocity as the carrier fluid (see Fig. 3(b)). The C particles floating in LSSs of the core regioncause an almost large core residence time (in respect to the same sized P particles) which can affect on the particle depo-

Fig. 4. Root Mean Square velocity fluctuations for (a) normal and (b) span-wise direction.


sition mechanisms. The large core residence time of the C particles can be expected from the Fig. 5(a) where the stream-wisecomponent of turbulent intensities of the C and P flow were compared.

As it was mentioned before, the LSSs of the core region of the C flows modulate the near-wall structures. These modulatedstructures of the near-wall and the LSSs of the core region of the C flows cause that the C particles in repect to the P ones havelarger turbulent intensity in the core, while the opposite is valid in the near-wall region. Hence, the difference between theturbulence stream-wise intensity of the core and near-wall region of the P flow is greater than the C one. This larger differ-

Fig. 5. Root Mean Square of stream-wise velocity fluctuations for (a) fluids, (b) C particles with St ¼ 5, and (c) St ¼ 125.


ence may cause a more significant transverse migration of the P particles toward the wall, while the C particles spend moretime in the core. This behavior will be observed and studied again in Section 5 where the number concentration of arrivingparticles to the wall was studied.

5. Particles concentration profiles

As it was mentioned before in Section 2, the total particle flux had been divided into four types: Turbophoretic, turbulentdiffusion, diffusional deposition, and free-flight fluxes. Due to the uniform distribution of the particles in the entire of thechannel at the beginning state, the turbophoretic flux is the dominant mechanism which can introduce a net drift towardsthe wall after the initiate state. In other words, particles due to turbophoretic flux, segregated from the core and accumulatedin the near-wall region during the time.

Narayanan et al. [21] showed that, the diffusional deposition mechanism plays an important role in the transfer of small Pparticles and the overall shape of small particle concentration profile is determined by opposing actions of diffusional depo-sition mechanism and turbophoretic flux. They claimed that diffusional deposition mechanism is directly related on the nor-mal component of velocity fluctuations. So that, particles with higher activities can leave the accumulation zone bydiffusional mechanisms and reduce the near-wall build up. The importance of the diffusional deposition mechanism inthe migration of small C particles toward the wall was studied in the Fig. 6 where the relatively wall-normal activity of Cand P particles with smallest Stokes number (St = 5) was compared. It was observed that C particles in the wall region havelower wall-normal activities than the same sized P particles. This fact is due to the modulated coherent structures of thenear-wall region of C flows described in previous sections.

In according to lower activities of the C particles in the wall-region, it can be expected that the diffusional depositionmechanism for C particles is less efficient than for the P particles, and so a greater near-wall built up for the C particlesin those region (near-wall) seems to be logical. This fact was confirmed in the Fig. 7 where the variations of instantaneousparticle number density (concentration profile) of C and P particles along the wall-normal direction at the steady state werecompared. It was observed that the C particles with St ¼ 5 have greater particle built up in the near-wall region than thesame sized P particles. This is not the case for larger particles (St = 25 and 125).

The C particles with larger Stokes numbers have almost a same concentration profile as the P ones. The larger particlesarrive to the wall due to their large momentum gained from the LSSs of the core. This large amount of momentum can drivethem directly to the wall. Most of the hitted particles do not have enough energy to return to the outer flow, so that theyaccumulate near the wall and increase the local turbulent activity. This phenomena can be inferred by comparing the Figs.5(b) and (c). The RMS stream-wise particle velocity profile of large particles (Fig. 5(c)) in comparison to the same profile ofsmaller particles (Fig. 5(c)) deviates toward the wall, i. e. left side. However, although the amount of momentum gained bythe lage C particles is greater than for the P ones, which is due to the geometry of C flows in the core (LSSs), bouncing off ofthe C particles from the wall is also more important than for the P particles. This balance may cause almost similar concen-tration profiles for C and P particles with large Stokes numbers (St = 25, 125).

6. Mechanisms of particles deposition

The dominant mechanisms of P particles deposition have been considered by many previous researchers which were ad-dressed in Section 2. In this section the dominant mechanisms of C particles deposition were studied. The cumulative num-ber of C particles arrived to the wall were shown in Fig. 8 as a function of the nondimensionalized time. It was observed that

Fig. 6. The differences of RMS of particle and fluid velocity fluctuations in normal direction for particles with St ¼ 5. Logarithmic scale was used for studyingthe near-wall region in more details.

Fig. 7. Concentration profile of the C and P particles with (a) St ¼ 5, (b) St ¼ 25, and (c) St ¼ 125. Logarithmic scale was used for studying the near-wallregion in more details.


the slops of the curves tend to a constant value after a certain time. This means that after a given time, i. e. particles are redis-tributed in the domain, the number of particles arrived at the wall is almost constant for every time step. This constant valuewas called deposition rate or deposition velocity. The non-dimensional deposition velocity is defined by:


kþd ¼Jx

Cmus; ð6Þ

where Jx is the mass of particles reaching to the surface per unit area per unit time and Cm is the mean bulk concentration ofparticles. The small particles due to their small response time were drifted by the coherent wall structures and most havereturned to the outer flow without arriving to the wall. Hence, their deposition velocity should be less than the largerparticles.

The non-dimensional deposition velocities, kþd , of the C and P particles used in this study reported in Table 2 and com-pared in Fig. 9. The deposition velocity value of small C particles (St = 5) was smaller than the deposition velocity value ofsame sized P particles. This value is also smaller than all the values found in the literature [21,27].

It was shown that, the dominant mechanism of small particles migration from the accumulation zone toward the wall,diffusional deposition mechanism, for the C particles is less efficient than for the same sized P particles (see previous Sec-tion). Hence, the number of C particles migrated from the accumulation zone toward the wall are less than the numberof migrated P particles and so, the small C particles (St = 5) have smaller deposition velocity than the same sized P particles.In other words, modulation of the near-wall coherent structures done by the LSSs of the core region of the C flow reduced thedeposition velocity value of small C particles respect to the same sized P particles.

Fig. 8. Cumulated number of C particles arriving to the bottom wall according to the non-dimensional time.

Table 2Deposition velocities.

St ¼ 5 St ¼ 25 St ¼ 125

C particles 5:3� 10�2 0.14 0.24

P particles 5:6� 10�2 0.1 0.19

Fig. 9. Deposition velocity versus Stokes number.


In contrast to the small C particles, larger C particles with larger Stokes numbers (St = 25, 125) have deposition velocityvalues larger than the same sized P particles. The larger C particles due to their large amount of momentum gained from theLSSs can reach to the wall easier than the same sized P particles. Hence, the deposition velocity value of C particles is greaterthan the deposition velocity value of P particles. In other words, the LSSs of the core region of the C flow increased the depo-sition velocity of larger particles.

Fig. 8 also shows that, the C particles migrate to the wall in two steps which act in tandem. In the first step, the C particlesfloating in the LSSs of the core region spend a certain time for gaining enough momentum to migrate toward the wall. Thiswas also observed in the Section 3 when the transverse migration of the suspended particles was studied. The small particlesmore followed the LSSs of the core because of their smaller response time and so, spend far more time to gain enough energywhile larger particles can gain enough momentum and left the core region sooner. The larger C particles can be arrived di-rectly to the wall by the large amount of gained momentum from the LSSs of the core.

Narayanan et al. [21] indicate that the deposited particles by free-flight mechanisms can be defined as particles havingboth a high deposition velocity and a short residence time, while particles which are deposited by diffusional depositionmechanisms are those with very small deposition velocities and very large residence times. In according to the importantrole of LSSs to drive large C particles directly toward the wall it can be claimed that the dominant mechanism of large C par-ticle (St = 25 and 125) deposition is free-flight mechanism which started from the core region. In order to clarify the dom-inant deposition mechanisms of small test particles (St ¼ 5), the Probability Density Function (PDF) of the normal componentof deposited velocities and residence time of deposited small C and P particles (St = 5) are compared in Fig. 10.

Residence time, Tþres, is defined as the time spent by a particle before deposition in a slab 3 wall unit from the wall. Forcomputing the residence time an algorithm was implemented such that, after entering a particle in the considered slab (3wall unit from the wall) a time counter for this particle start but if the particle escapes from the slab before depositionits time counter was reinitialized to zero.

Fig. 10 according to the [21] statement shows that the diffusional deposition plays a significant role for deposition ofsmall C particles. Residence times of C and P particles show a similar variation, while most of the small C particles weredeposited to the wall by lower activities than the P ones.

Fig. 10. PDF of (a) the normal component of the deposited velocities and (b) Residence time, Tþres of C particles with St ¼ 5.

Fig. 11. Cumulated number of particles with St ¼ 5 which arrived to the deposition layer of bottom wall according to the non-dimensional time.


The main reason of the particle escaping from the wall region is the particle activity and Figs. 7 and 10(a) show that thesmall C particles have lower activities than the P ones. Therefore, it can be expected that the small C particles entered to thewall region more deposited on the wall respect to same sized P particles. This was confirmed in Fig. 11 where the number ofsmall C and P particles arrived to the deposition layer, ndep, during the time is compared. The deposition layer is either thenearest layer or sum of layers to the wall with diameters equal to the particle diameter. As it was mentioned before, a par-ticle was assumed deposited when enter to this layer. However, although the number of small P particles arrived to the wallis greater than the same sized arrived C particles (kþd;P > kþd;C), the deposition of those C particles is more important than the Pones.

These observations verified again the important role of the coherent structures of the near-wall region. The modulation ofthose structures in the C flow done by LSSs of the core region decreases the escaping energy of the C particles and increasestheir local deposition.

7. Conclusions

In this study, Direct Numerical Simulation of a turbulent plane Couette flow was combined with Lagrangian particletracking to study the dispersion and deposition mechanisms of C particles. A data-base for P flows and P particles were alsoobtained by a same trend for validation the used method and comparison the results. Ensembles of particles with Stokesnumbers 5, 25, and 125 were tracked under the assumption of one-way coupling and were statistically analyzed. The follow-ing are the main conclusions that can be drown from this paper:

1. For the suspended particles in the class of turbulent channel flows, when the bottom wall is put into motion (C flows), thewall stress gradually reduces and the location of the maximum mean stream-wise velocity of particles shifted from thechannel centerline (P particles) toward the moving wall (C particles).

2. The mean stream-wise velocity of C particles lags the carrier fluid in the near-wall and leads it in the core region, whilethe opposite observed for P particles. This difference between the mean stream-wise velocity of particles and the carrierfluid, which was due to the particle transverse migration, is more efficient for larger particles.

3. Due to the LSSs of the core region of C flow, C particles in the core have larger turbulence activities (RMS velocity fluc-tuations) than the P particles while the opposite is observed in the wall region. Hence, the difference of turbulent inten-sities between the core and near-wall region for C particles is smaller than for P particles, so that the C particles spentmore time in the core in respect to the same sized P particles. In other words, LSSs increase the time floating of particlesin the core region.

4. The near-wall modulated structures in C flows reduced the wall normal turbulent intensities of C particles, so that thediffusional mechanisms for small and medium C particles in this region are less efficient than for the P ones. This maycause a more C particle built up in the near-wall region. Hence, the C particle built up in the near-wall region is greaterthan the P particle built up in the same region.

5. The deposition velocity value of small C particles is smaller than the depostion velocity value of same sizd P ones whilethe opposite is true for medium and large particles. The number of small particles which can leave the accumulation zoneand migrate toward the wall may be reduced by the modulated coherent structures of the near-wall region of the C flows.On the other hand, a large amount of momentum gained by LSSs of the core, may increase the number of larger C particlesarrived to the wall.

6. The modulated coherent structures of the near-wall region reduced the intensity of C particle re-entrainment mecha-nisms from the wall region. Hence, the number of deposited C particles is larger than the number of deposited P ones.


The C particles also deposited by velocities smaller than the depositing velocities of P particles while the residence timesof both considered particles are almost the same.

As a final conclusion, this study shows that the LSSs of the core region of the C flows have significant roles in the particletransport mechanisms. The used particles size in this study were in the micrometer range and our feature study can be theinvestigation of the LSSs effects on the nanoparticles motion. However, DNS with more spatial resolution is also needed inthis case.

Acknowledgments

We would particularly like to express our appreciation of the excellent cooperation of Prof. Paolo Orlandi and Dr. MatteoBernardini from Dipartimento di Ingegneria Meccanica e Aerospaziale, Universita di Roma ’La Sapienza’, Rome, Italy, espe-cially for preparing the DNS simulations. The support from the office of Graduate Studies at the University of Isfahan, Iranis also gratefully acknowledged. Omid Ghaffarpasand undertook this work with the partially support of the ICTP Programfor Training and Research in Italian Laboratories, Trieste, Italy.

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Documents

Dispersion and deposition mechanisms of particles suspended in a turbulent plane Couette flow