Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall

Accepted Manuscript

Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall C.D. Dritselis PII: DOI: Reference:

S0301-9322(16)30302-0 10.1016/j.ijmultiphaseflow.2017.01.004 IJMF 2524

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

20 May 2016 17 November 2016 15 January 2017

Please cite this article as: C.D. Dritselis , Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall, International Journal of Multiphase Flow (2017), doi: 10.1016/j.ijmultiphaseflow.2017.01.004

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Highlights  Particle deposition on vertical walls roughened by transverse square bars is studied.  Various values of particle response time and square bar spacing are examined.  LES coupled with immersed boundary method and Lagrangian particle tracking are used.  Particle deposition is increased due to flow changes by the bars and interception.

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ACCEPTED MANUSCRIPT Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall

C.D. Dritselis* Department of Mechanical Engineering, University of Thessaly,

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Pedion Areos, 38334 Volos, Greece

Abstract

A numerical study is presented for the effect of wall roughness on the deposition of solid spherical

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particles in a fully developed turbulent channel flow based on large eddy simulation combined with a Lagrangian particle-tracking scheme. The interest is focused on particles with response times in wall units in the range of 2.5 ≤ τp+ ≤ 600 depositing onto a vertical rough surface consisting of two-

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dimensional transverse square bars separated by a rectangular cavity. Predictions of particle deposition rates are obtained for several values of the cavity width to roughness element height ratio

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and particle response time. It is shown that the accumulation of particles in the near wall region and their preferential concentration in flow areas of low streamwise fluid velocity that occur in turbulent

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flows at flat channels are significantly affected by the roughness elements. Particle deposition onto

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the rough wall is considerably increased, exhibiting a subtle dependence on the particle inertia and the spacing between the bars. The observed augmentation of deposition coefficient can be attributed

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to the flow modifications induced by the roughness elements and to the inertial impaction of particles onto the frontal deposition area of the protruding square bars.

(Keywords: Large-eddy simulation; turbulent channel flow; particle deposition; roughness elements; immersed boundary method)

*

Tel.: 0030 24210 74337. Fax: 0030 24210 74085. E-mail address: [email protected] (Chris D. Dritselis).

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1. Introduction Turbulent gas flows containing small size particles have been investigated widely using direct numerical simulation (DNS) or large eddy simulation (LES) combined with discrete particle simulation (DPS). Based on such approaches, a significant advancement in the understanding of the

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turbulent transport and deposition of particles onto smooth surfaces has been achieved (McLaughlin, 1989; Ounis et al., 1993; Brooke et al., 1992, 1994; Pedinotti et al., 1992; van Haarlem et al., 1998; Zhang and Ahmadi, 2000; Narayanan et al., 2003; Soldati and Marchioli, 2009; Nasr et al., 2009; Marchioli et al., 2007). However, the influence of wall roughness on the

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mechanisms of particle transfer and deposition has not been investigated sufficiently.

The experiments of Sommerfeld and Huber (1999) and Sommerfeld and Kussin (2004) indicate that the statistics of both gas and particulate phases are significantly affected due to the interaction of particles with rough walls in a turbulent channel flow. The effect of wall roughness

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has been accounted for in numerical simulations based on virtual wall models leading to a very

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different overall dynamic behavior of the particles (Sommerfeld, 1992; Squires and Simonin, 2006; Konan et al., 2009). More specifically, this can be considered to be similar to that produced by

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inter-particle collisions, namely a substantial enhancement of the wall-normal particle velocity. By using sandpapers, textile materials or large-scale obstructions with regular geometry, the effect of

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surface roughness on particle deposition has been quantified in several experimental studies (Wells

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and Chamberlain, 1967; Lai et al., 1999, 2000, 2001, 2002; Hussein et al., 2009). The common outcome of these studies is that the deposition rate of particles at rough walls is higher than that at smooth walls. A lot of effort has been conducted to develop simple models that are capable of quantifying the increase of particle deposition onto rough walls observed in the experiments. Such models, in their vast majority, consider that the virtual origins of the fluid velocity and particle concentration are shifted by a distance away from the walls, and it is assumed that a particle is

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ACCEPTED MANUSCRIPT captured when it reaches an effective roughness height (Browne, 1974; El-Shoboksky and Ismail, 1980; Wood, 1981; Guha, 1997; Lai, 2005; Zhao and Wu, 2006a, 2006b). In the present study, the effect of wall roughness on the particle deposition is investigated by using LES of turbulent flows in a vertical channel with the one wall consisting of square bars separated by a rectangular cavity, as shown in Fig. 1. Lo Iacono et al. (2005), (2008) have examined

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the deposition of spherical and cylindrical particles in a similar flow configuration using LES. The present study differs in several respects from the aforementioned ones. First, the numerical treatment of the roughness elements is different, since an efficient immersed boundary method is adopted here (Fadlun et al., 2000; Orlandi and Leonardi, 2006), in contrast to a body-fitted method

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used in Lo Iacono et al. (2005), (2008). A second difference is that the present parametric study includes the influence of particle inertia, and several values of the cavity width to the roughness height ratio, thus, covering a significantly larger parameter space relative to the studies of Lo Iacono et al. (2005), (2008), who used only one set of particles in one flow configuration at a

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different Reynolds number. Finally, the present study mainly focuses on the effect of two-

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dimensional (2d) roughness elements on the particle deposition rate with respect to the associated changes in the fluid flow, which has not been investigated previously. Several numerical studies

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have used such flow configurations in order to investigate the mechanisms of momentum and heat transfer (see, for example, Vijiapurapu and Cui, 2007; Cui et al., 2008). Here, these can be

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considered simple models to study the effect of wall roughness on the turbulent transport and

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deposition mechanisms of small size heavy particles.

2. Methodology

2.1 LES of turbulent channel flow The three-dimensional (3d), unsteady, incompressible filtered continuity and Navier–Stokes equations are

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ACCEPTED MANUSCRIPT  ui  0, xi

(1)

τ ijr u i u i u j u i 1 p   ν   Πδi1 , t x j ρ f xi x j x j x j

(2)

where ui is the component of the large-scale (grid-scale) fluid velocity in the i-direction, p is the fluid pressure, ρf and ν are the fluid density and kinematic viscosity, respectively, Π is the extra

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pressure gradient in order to keep the fluid mass flow rate constant, and δij is the Kronecker delta. All the filtered quantities are denoted by an overbar. The flow configuration is shown in Fig. 1. The fluid velocity satisfies periodic boundary conditions in the x- and z-directions, and no-slip

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conditions at the walls in the y-direction. Note that u x , u y , and u z denote the resolved fluid velocity components larger than the filter scale in the x-, y-, and z-directions, respectively. The term τijr represents the effect of the subgrid-scale motions on the resolved grid-scale velocities of the carrier fluid. It is the traceless part of the subgrid-scale stress tensor, defined as

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1 τ ijr  τ ij  τ mm δij with τ ij  ui u j  ui u j . 3

(3)

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The subgrid stress τijr is modeled by using the dynamic Smagorinsky subgrid-scale turbulence

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model as

τ ijr  2CΔ 2 S Sij ,

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(4)

where S ij is the resolved strain-rate tensor and | S | = (2 S ij S ij )1/2 its magnitude. The characteristic

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length is Δ  (Γx Γy Γz)1/3, where Γx, Γy, and Γz are the grid spacings in the x-, y-, and z-directions, respectively. The model parameter C is determined by the dynamic procedure proposed by Germano et al. (1991) based on Lilly’s (1991) modification

C

Lij M ij M kl M kl

,

(5)

with Lij and Mij given as

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ACCEPTED MANUSCRIPT 





Lij  ui u j  ui u j , 2



(6) 





M ij  2Δ S Sij  2 Δ 2 S Sij ,

(7)

where (^) denotes variables calculated on the test filter twice as large as the grid spacing. The present large eddy simulations are performed by using a box filter in the physical space based on

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the trapezoidal rule, while no filtering is applied in the y-direction. The values of C are averaged over the spanwise z-direction and a cutoff is used to ensure nonnegative values of the total viscosity (i.e., the kinematic plus the turbulent viscosity) and to avoid any numerical instabilities.

The roughness elements are treated by an immersed boundary technique (Fadlun et al.,

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2000; Orlandi and Leonardi, 2006), which consists of imposing zero values to all fluid velocity components on the stationary boundary surface that does not necessarily coincide with the computational grid. This approach allows the solution of flows over complex geometries without the need of computationally intensive body-fitted grids. In accordance with the aforementioned

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studies, zero velocities are imposed in the grid points within the roughness elements. At the first

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grid point outside each square bar, all the viscous derivatives and the terms of the subgrid-scale stess tensor in the filtered flow equations are discretized by using the distance between the fluid

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velocity components and the boundary of the roughness element and not the actual mesh size (for

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more details, see Dritselis, 2014).

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2.2 Discrete particle simulation The trajectories of particles are calculated in a Lagrangian reference frame considering the

drag force. The particle equations of motion are

dx np,i dt

 vin ,

(8)

,

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(9)

ACCEPTED MANUSCRIPT n where vin and u~@p, i are the velocity of the n particle and the undisturbed fluid velocity at the particle

position xp,in, respectively. In Eq. (9), fD,in is the drag force, mp (=ρpπdp3/6) is the spherical particle mass, ρp, dp are the particle density and diameter, respectively, and bold symbols indicate vectors. Eq. (9) is appropriate for particles with diameters of the same order or smaller than the smallest length scales of the fluid motion. The coefficient cD corrects the Stokes drag force for inertial

cD =

(

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effects at non-negligible particle Reynolds numbers (Clift et al., 1978)

)

24 , 1+ 0.15Re0.687 p Re p

where

(10)

.

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The results presented here are for the case without gravity (g = 0). A safe estimation of the typical difference between the vertical particle and fluid velocities is gτp, where τp is the particle response time. For all the numerical experiments conducted in this work, the half distance between

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the channel walls is h = 0.0054 m, the bulk velocity is ub = 7.778 m/s, and the magnitude of the nondimensional gravity force is gb = g h/ub2 = 1 / Frb = 8.76×10-4 that is much smaller than the drag

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force, where Frb is the Froude number. The maximum nondimensional velocity difference gτp / ub (= τp,b / Frb ) is of the order of O(10-2) and, thus, gravity is unlikely to have much affect on particles

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in the size range to be considered with nondimensional response times of 0.239 ≤ τp,b = τp ub/ h ≤

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19.152. The lift force is not considered in the equations of particle motion in order to study the effects of wall roughness on the particle transport and deposition within a manageable parameter

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range. Other forces, such as Basset, added mass, and pressure gradient forces are discarded from the particle equation of motion. This is justified by the relatively high mass density ratio between the phases considered in the present study (Armenio and Fiorotto, 2001). Dilute flow conditions are assumed and, thus, the particle feedback in the fluid flow and interparticle collisions are not taken into account (i.e., one way coupling). Perfectly absorbing walls are considered, in order to facilitate the comparison against previous numerical studies of particle deposition onto smooth walls (see, for example, McLaughlin, 1989; Marchioli et al., 2007). 7

ACCEPTED MANUSCRIPT Therefore, the deposition of particles is identified by geometric criteria, e.g., when their centre is less than dp/2 distance from the channel walls or the roughness elements. The particles exiting the channel domain through the planes normal to the streamwise (x) and spanwise (z) directions were reintroduced in it from the corresponding opposite boundary plane with their exiting velocities. The actual mechanism of particle deposition is complicated depending on various factors, as

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for example, the surface potential energy and the possible electric charge of particle. In contrast, droplets may deform and eventually wet the surface. Such phenomena are not included in the present LESs, and it is assumed that the particles remain attached to the rough wall once they deposit. For each particle, it is examined whether intersection of the particle trajectory with any of

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the planes of the rough wall (lower horizontal plane CD, upper horizontal plane AB, outer vertical plane DE, and inner vertical plane BC) takes place based on the initial xp,i(t) and final xp,i(t+dt) particle position during a time step dt. For each possible interception, a contact time tc (0 < tc ≤ dt) and the particle position xpc,i at contact with the rough wall are calculated. If the particle position

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xpc,i belongs to any of the surfaces of the rough wall (see Fig. 1) then particle deposition occurs. In

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case multiple deposition events in more than one surface are detected during a time step, then the

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event with the smallest contact time is chosen.

3. Numerical methods

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The time evolution of the present particle-laden turbulent flows is obtained by solving

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numerically the filtered Eqs. (1) and (2) using a semi-implicit, fractional step method (see Orlandi, 2000). A second-order central finite differencing scheme on a staggered grid is used for the spatial discretization. The time integration of the discretized equations is done by a combination of an explicit low storage third-order Runge-Kutta method for the non-linear convection terms and the extra subgrid stress, and an implicit second-order Crank–Nicolson method for the linear diffusion terms. The resulting system of algebraic equations is inverted by an approximate factorization technique. The fluid momentum equations are advanced in time by using the pressure gradient at 8

ACCEPTED MANUSCRIPT the previous time step, yielding an intermediate nonsolenoidal fluid velocity field. The latter velocity field is projected by a scalar quantity onto a solenoidal one. The positions and velocities of the point particles are computed simultaneously with the numerical solution of the fluid momentum equations by performing time integration of Eqs. (8) and (9) with a second-order explicit Adams–Bashforth method. The local undisturbed fluid velocity

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n u~@p, i in Eq. (9) is approximated by the fluid velocity that results from the numerical solution of Eqs.

(1) and (2), calculated at the particle positions using sixth-order Lagrange polynomials. The effect of subgrid-scales on the particle motion is not taken into account. Thus, the particle trajectory is determined by using only the resolved fluid velocity in the LES computations. This is a reasonable

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assumption given the filtering due to the particle inertia and the moderate Reynolds number of the flow, for which there is a relatively weak effect of the unresolved on the resolved scales Armenio et al., 1999; Yamamoto et al., 2001; Vance et al., 2006; Khan et al., 2010). For more details, see Section 4 and Appendixes A and B. However, at different regimes such as lower particle inertia,

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or/and higher Reynolds number, the effect of the subgrid-scales may be significant and it should be

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properly considered (Fede and Simonin, 2006; Shotorban and Mashayek, 2006; Berrouk et al.,

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2007; Pozorski and Apte, 2009; Arcen and Taniere, 2009; Jin et al., 2010).

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4. Numerical details

All simulations are performed at the same Reynolds number of Reb (= ub2h/ν) = 5600 based

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on the bulk velocity ub (= 2/3uc) and the distance between the right smooth wall and the crest of the left rough wall 2h, where uc is the centerline velocity corresponding to the laminar plane Poiseuille flow. The values of the ratio of the cavity width w to the dimension of the square bar k examined here are w/k = 0, 1, 3, 4, 7, 9, and 19 with k = 0.2h. The extent of the 2d roughness elements in the direction perpendicular to the walls is 10 % of the wall-normal distance of the outer flow and, thus, it is more appropriate to speak of a turbulent flow over obstacles. A fixed fluid mass flow rate is considered and, thus, for all cases the velocities ub and uc are constants. For this reason, a mean 9

ACCEPTED MANUSCRIPT pressure gradient Π is imposed in the streamwise fluid momentum equation. This quantity was properly adjusted during the simulations by integrating the right hand side of the discretized u x momentum equation over the whole channel domain. The wall friction velocity uτ and the turbulent centerline velocity utc vary in the simulations. More specifically, the values of the nondimensional wall friction velocity are Uτ = uτ/ub = 6.11×10-2, 6.21×10-2, 7.53×10-2, 8.8×10-2, 10.4×10-2,

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10.41×10-2, and 8.85×10-2 for w/k = 0, 1, 3, 4, 7, 9, and 19, respectively. Consequently, the Reynolds number Reτ (=uτh/ν) is increased in the cases with roughness elements and it varies in the range of 171 ≤ Reτ ≤ 292. The wall friction velocity at the left rough wall are ranged 6.11×10-2 ≤ UτR ≤ 14.39×10-2, while at the right smooth wall it is found that 6.11×10-2 ≤ UτS ≤ 7.28×10-2.

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The dimension of the channel is Lx = 8h in the streamwise direction and Lz = πh in the spanwise direction, and the computational mesh is 201×94×65 in the x, y, and z-directions, respectively. Each roughness element is discretized by using 30 almost equidistant grid points over the range of –1.2 < y/h < –1, while 5 grid points are used in the x-direction. The rest 64 of the grid

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points are unevenly distributed in –1 < y/h < 1. For the case with both smooth walls, the grid

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resolution in wall units (denoted by the symbol +) is Γymin+ ≈ 0.48, Γymax+ ≈ 10.6, and Γz+ ≈ 7.7, which is similar to that adopted by the majority of the LES studies at the same Reynolds number

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and consistent with the requirements for obtaining accurate results (see Piomelli and Balaras, 2002).

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The grid resolution in the flows with square roughness elements on one wall is 0.49 ≤ Γymin+ ≤ 0.82, 10.7 ≤ Γymax+ ≤ 18, and 7.8 ≤ Γz+ ≤ 13.1. For all cases, the CFL number is fixed at a constant value

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of 0.5, ensuring a small enough time step and, consequently, a good accuracy for both the fluid and particle velocities. At first, numerical simulations are performed for various w/k values without considering the

particulate phase, using as initial conditions the fully developed turbulent channel flow with two smooth walls at Reb = 5600. For the latter case, the left smooth wall at y/h = -1 is also described by the immersed boundary method. Next, the trajectories of 5×105 particles are calculated in the fully developed turbulent channel flows with one rough wall. Initially, the particles are uniformly 10

ACCEPTED MANUSCRIPT distributed in the region of –1 < y/h < –0.7 near the rough wall, with velocities equal to those of the fluid at their positions. Results are obtained for six samples of particles with nondimensional diameters of dp/h×103 = 1.240, 1.754, 2.481, 3.923, 7.846, and 11.096, and response times of τp,b = (Sdp2/18ν) / (h/ub) = 0.239, 0.479, 0.958, 2.394, 9.576, and 19.152, which correspond to τp,0+ = (Sdp2/18ν) / (ν/uτ,02) = 2.5, 5, 10, 25, 100, and 200, respectively, where uτ,0 is the wall friction

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velocity for w/k = 0. Note that ub and τp,b are independent of w/k, thus, allowing comparisons to be made with varying w/k for each particle set. The values of particle diameter are ranged from 6.7 to 60 μm, and the particle-fluid density ratio is S = ρp/ρf = 1000.

The test cases examined here consist of a sufficient number of characteristic particle-laden

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flows appropriate to assess the effect of roughness elements on the deposition of point-particles by using adequate numerical methods, such as the presently adopted LES/immersed boundary method coupled with DPS. The fluid flow (LES) and the Lagrangian particulate (DPS) solvers have been extensively and successfully verified previously. For example, the mean streamwise velocity and

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the root-mean-square (rms) velocity fluctuations of the carrier phase predicted using the dynamic

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Smagorinsky turbulence model agree reasonably well with the corresponding DNS and LES results for the case of a fully developed turbulent channel flow with smooth walls at Reb = 5600 (see

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Dritselis and Vlachos, 2008, 2011a, 2011b). For the same case at a lower Reynolds number of Reb = 3000, the predictions of particle deposition based on the Lagrangian particle solver used here are in

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good agreement with other published numerical results (Dritselis et al., 2011). Finally, the accuracy

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of the numerical treatment for the roughness elements has been verified through extensive comparisons with the results of the Furuya et al. (1976) experiment for a boundary layer over 2d transverse circular rods and the DNS results of Leonardi et al. (2003) and Orlandi et al. (2006) for turbulent channel flows with square bars placed on one wall (for more details, see Dritselis, 2014). Results produced by well-resolved LES and relatively high inertial particles, as in the present study, are expected to reproduce the DNS trends not only qualitatively, but also from a quantitative viewpoint. In contrast, an inaccurate representation of particle statistics may result from 11

ACCEPTED MANUSCRIPT low resolution LES at a high Reynolds number and low response time particles. It should be noted that, when the bulk velocity in the simulations is fixed, a different Reynolds number based on the wall friction velocity is usually predicted by LES relative to the DNS one. Since turbulence does not exhibit scale similarity, it is not possible to match differently normalized Stokes numbers across a range of Reynolds numbers (Pedinotti et al., 1992). Therefore, the comparisons of LES/DPS

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against DNS/DPS should be done with caution. Nevertheless, it is generally accepted that LES can provide a reliable representation of some features of the flow field, such as velocity statistics and energy. In addition, several statistics of the dispersed phase have been reproduced adequately based on LES/DPS, as for example, Eulerian particle velocity statistics, particle concentration, preferential

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concentration, particle deposition, one- and two-particle dispersion, and Lagrangian velocity autocorrelations (see, for example, Wang et al., 1997; Armenio et al., 1999; Yamamoto et al., 2001; Lo Iacono et al., 2005, 2008; Vance et al., 2006; Dritselis et al., 2011). The interest here is mainly focused on addressing the effect of roughness elements on the concentration and deposition of

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particles. For this reason, proper statistical quantities will be used to characterize the changes

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5. Results and discussion

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induced by the rough wall on the dynamic particle behavior.

5.1 Results of the fluid phase

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In this section, representative results are shown to address the effect of roughness elements

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on the statistical properties of the fluid flow. Fig. 2 shows the mean streamlines averaged in the spanwise direction and time for various values of w/k. A single vortex is observed for w/k = 1, which occupies the whole cavity. The size of the recirculation region behind the square bar is increased for w/k = 3 and 4, and a smaller secondary vortex with opposite circulation appears adjacent to the upper horizontal wall of the cavity. In the cases of w/k = 1, 3, and 4, the flow separates at the rear edge of each element, while it reattaches on the lower horizontal wall of the cavity. At the higher values of w/k = 7, 9 and 19, a large primary recirculation zone is located 12

ACCEPTED MANUSCRIPT downstream of each square bar, which is accompanied by two smaller vortices in the corners at the front and back walls of the bar. A reattachment of the flow occurs on the vertical wall of the cavity and a flow separation takes place as the front surface of the next square bar is approached. For w/k = 19, a small recirculation area is also visible on the crest of each roughness element. It can be seen that the square bars have a strong influence on the mean flow in the vicinity of the rough wall, while

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away from it, the streamlines exhibit a weak undulation. The present LES results are in close agreement with those of Leonardi et al. (2003) who used DNS coupled with a similar immersed boundary technique, and with the LES results of Cui et al. (2008) who used a body-fitted numerical code to study the turbulent flow in similar geometries. For w/k = 7, a reattachment length Xrl of

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about 5k is predicted in this work, which is close to the value of 4.8k reported in Leonardi et al. (2003). For w/k = 19, Xrl = 5.7k is found here based on the dynamic Smagorinsky subgrid-scale model, which is also in good agreement with the DNS prediction of Xrl = 5.8k.

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5.2 Particle concentration and transport mechanisms

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The instantaneous particle number density PND is shown in Fig. 3 for various sets of particles, in order to quantify the effect of square bars on the macroscopic evolution of particle

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accumulation and deposition at the rough wall. The interest is not to account for variations within one roughness wavelength, but to address the overall effect the roughness has on the concentration

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of particles that populate the overlying flow. The distributions of PND are calculated at time tb = 33

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(t0+ = 348.11) after their release into fully developed turbulent flows, by counting the number of particles inside wall-parallel bins unevenly distributed in the y-direction. The thickness of each bin was δybin = 0.003 (δybin,0+ ≈ 0.5). The starting point of the PND profiles in Fig. 3 is different, since particles with response time of τp,b (τp,0+) = 0.0239 (2.5), 0.479 (5), 0.958 (10), 2.394 (25), 9.576 (100), and 19.152 (200) hit the y/h = -1 plane corresponding to the smooth wall when w/k = 0 at y'/h = (y+1)/h = dp/(2h) × 103 = 0.620, 0.877, 1.240, 1.961, 3.923, and 5.548, respectively.

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ACCEPTED MANUSCRIPT The present results for w/k = 0 are consistent with those found in previous studies of particle-laden turbulent channel flows with smooth walls (McLaughling, 1989; Zhang and Ahmadi, 2000; Marchioli et al., 2007). In particular, the distribution of PND shows a peak very close to the smooth wall at y'/h = 0. Its maximum value is augmented with increasing response time up to τp,b = 2.394 (τp,0+ = 25), while PNDmax is decreased as τp,b increases further. These trends are qualitatively

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maintained in the particulate flows with w/k = 1, in which non-uniform distributions of the instantaneous PND are apparent. However, the particle number density is not reduced as y'/h = 0 is approached and it is larger than in the w/k = 0 case adjacent to the rough wall for particles with τp,b (τp,0+) = 0.239 (2.5), 0.479 (5), and 0.958 (10). Away from the rough wall, PND is smaller than that

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found at w/k = 0 for all particle sets. Fig. 3 reveals a significant impact of the rough wall on the particle number density. More specifically, roughly uniform profiles of PND are seen at larger spacings between the square bars for all the τp,b values examined here.

In the absence of external forcing, the dynamic particle behavior can be established by

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complex interactions of particles with the underlying fluid turbulence in combination with the

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crossing-trajectory effect. The above proposition is true for both flows with either smooth or rough walls. For each particle ensemble, distinct PND distributions are observed with varying the spacing

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of the square bars. This is because of the flow changes induced by the roughness elements, which differ depending on w/k. For particles with τp,b ≤ 0.958 (τp,0+ ≤ 10), PND is decreased at w/k ≤ 7,

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while it is increased at w/k ≥ 9. For τp,b ≥ 2.394 (τp,0+ ≥ 25), PND is decreased monotonously with

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increasing w/k, but relatively small differences are found at w/k ≥ 7. With the exception of particles having τp,b = 0.239 (τp,0+ = 2.5), PND exhibits a maximum value at the central flow region of the channel, which becomes more evident with increasing w/k. The instantaneous profiles of PND shown in Fig. 3 may be considered to be an outcome of the effects of turbophoretic convective drift, turbulent diffusion, and particle deposition. In flat channels, the particles migrate toward the smooth walls due to the inhomogeneity of the moments of the fluid turbulence, the so-called turbophoretic convective drift (Caporaloni et al., 1975; Reeks, 14

ACCEPTED MANUSCRIPT 1983). Due to their inertia, the particles do not acquire sufficient momentum to escape the near wall region, leading to large concentrations. The increased PND can be reduced by the action of turbulent diffusion, which smooths out any spatial gradient of the particle concentration. In addition, the particles deposit at the channel walls and, thus, the flow is gradually depleted from particles reducing their concentration both locally and globally. The PND distributions in the cases

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with roughness elements indicate indirectly that the particles are transferred more quickly by the fluid motions from the core region to the rough wall. A built-up in the particle concentration near the rough wall does not occur, pointing out that a significant enhancement of particle deposition also takes place.

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Fig. 3 demonstrates the impact of rough wall on the distributions of instantaneous PND for each particle set by varying w/k, while keeping constant the values of dp/h and τp,b. Since uτ depends on w/k, the non-dimensional values of particle diameter dp+ and response time τp+ in wall units are different from dp,0+ and τp,0+. It is found that dp+ and τp+ are increased and they are given by dp+/ dp,0+

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= 1.016, 1.252, 1.464, 1.730, 1.732, 1.472, and τp+/τp,0+ = 1.033, 1.569, 2.143, 2.992, 2.999, 2.167 at

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w/k = 1, 3, 4, 7, 9, and 19, respectively. Since the bulk velocity is kept constant in the simulations, this increase in particle size and response time in wall units is unavoidable due to the flow changes

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induced by the square bars and may potentially have an impact on the PND profiles in the flows with roughness elements. However, it is evident that the substantial changes in the PND cannot be

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attributed only to the aforementioned increased unresponsiveness of the particles in the rough

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channels. For instance, Fig. 3d compares the PND distribution obtained for particles with τp,b = 2.394 at w/k = 0, 1, 3, 4, 7, 9, and 19, having τp+ = 25, 25.82, 39.22, 53.57, 74.82, 74.96, and 54.18, respectively. The obviously large differences observed in PND between the cases at w/k = 0 and 1 cannot be explained by the very small increase of the nondimensional particle response time by only 3%, but rather on the overall fluid flow modifications induced by the roughness elements with the increase of uτ, dp+ and τp+ being inseparable aspects of the flow changes. It should be noticed the PND profiles in the rough channels does not exhibit any of the well known features expected for 15

ACCEPTED MANUSCRIPT particles in flat channels having response times in the range of 25 ≤ τp+ ≤ 74.96, while these are clearly seen in the PND profiles at w/k = 0. In turbulent channel flows with smooth walls, it is generally accepted that the near wall particle concentration is non-uniform in the spanwise direction (Pedinotti et al., 1992; Rouson and Eaton, 2001; Yamamoto et al., 2001; Soldati and Marchioli, 2009). In particular, the particle

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number density is increased in the low-speed streaks, where the particles tend to line up. In order to verify whether the above behavior is preserved in the present flows with roughness elements, several realizations of instantaneous positions of particles in a region of -0.971 ≤ y/h ≤ -0.912 (5 ≤ y0+ ≤ 15) were examined. Fig. 4 shows some representative results for particles with τp,b = 2.394

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(τp,0+ = 25) at time tb = 33 after their release in the turbulent flows. Also shown in Fig. 4 are the isolines of the grid-scale fluid velocity fluctuations in the streamwise direction ux ' at the y/h = 0.969 (y0+ ≈ 5.3) plane, where the prime denotes perturbation from the averaged grid-scale quantity. The pattern of alternative low- and high-speed streaks at w/k = 0 is evident, which is closely

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related to the action of quasi-streamwise vortices over the smooth wall (see, for example, Kim et al.,

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1987; Soldati and Marchioli, 2009). The distributions of streamwise fluid velocity fluctuations at the w/k = 0 and 1 cases are similar to each other. However, the streaks start to be disrupted in the

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latter case. As w/k increases, a transformation of the elongated structures into almost circular flow

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areas occurs. It can be seen that the disturbance induced by the roughness elements gives rise to shorter structures not aligned to the flow direction, accompanied by a lengthening in the spanwise z-

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extent. The magnitude of the velocity fluctuations are increased in the rough channels; the minimum and maximum values of ux ' /uc at the y/h = -0.969 plane are: (-0.189, 0.397), (-0.283, 0.309), (-0.321, 0.44), (-0.484, 0.39), (-0.545, 0.643), (-0.484, 0.553), and (-0.687, 0.565) for w/k = 0, 1, 3, 4, 7, 9, and 19, respectively. The present LES predictions compare well with those reported in the DNS study of Leonardi et al. (2004): (-0.28, 0.37) at w/k = 1, (-0.49, 0.5) at w/k = 3, (-0.63, 0.66) at w/k = 7, and (-0.59, 0.65) at w/k = 19. Moreover, the results of the fluid phase in Fig. 4 resemble those in the flow visualizations of Leonardi et al. (2004). 16

ACCEPTED MANUSCRIPT The changes in the magnitude and characteristics of ux ' by the wall roughness are reflected on the particle behavior. In the cases with rough wall, all convective particle velocities in the streamwise direction are increased as shown in Table 1. The instantaneous particle distribution at w/k = 0 is consistent with previous findings of preferential concentration near the smooth walls of turbulent channel flows. The same tendency is observed at w/k = 1; however, the formation of

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elongated particle lines is inhibited and particle structures somehow shorter than those over a smooth wall can be noticed. As the spacing between the bars increases, particle clusters are seen instead of straight lines. The instantaneous locations of particles coincide with low-speed streaks at w/k = 0 and 1, while this trend is diminished with increasing w/k.

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In order to quantify further the above observations, the probability density function (pdf) of particle number density in the region -0.971 ≤ y/h ≤ -0.912 was calculated and shown in Fig. 5 as a function of the streamwise fluid velocity fluctuations ux ' . The values of pdf(PND) were computed

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as follows: The instantaneous ux and mean U x grid-scale fluid velocities in the streamwise

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direction were interpolated at the particle positions and the corresponding fluctuation ux ' = u x –

U x was determined. The number of particles associated with each value of ux ' was calculated and

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normalized by the population of particles within the volume of observation. Fig. 5 reveals that a

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bias exists between negative and positive values of ux ' for particles with τp,b = 2.394 when w/k ≤ 1,

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as indicated by the higher values of pdf(PND) observed in the region of ux ' < 0. As w/k increases, the magnitude of positive and negative streamwise fluid velocity fluctuations, as well as the fraction of particles associated with these values are increased. The peak of pdf(PND) is decreased and it is shifted toward zero values of ux ' , and a normal (Gaussian)-like distribution is established when w/k = 4. This indicates that particles with τp,b = 2.394 are found equally in regions with low- and highstreamwise velocity fluctuations of the fluid phase. The same is also seen at w/k = 7, but two small peak values appear now in pdf(PND) at intermediate positive and negative values of ux ' . With 17

ACCEPTED MANUSCRIPT further increasing w/k, the fraction of particles related to ux ' < 0 is reduced. Eventually, at w/k = 19, the maximum value of the pdf of particle concentration is found at positive values of ux ' . The aforementioned change in the trend of particles to preferentially concentrate at certain flow regions is also confirmed by the probability representing the correlation of PND and negative streamwise fluid velocity fluctuations sampled at the particles positions in the region -0.971 ≤ y/h ≤

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< < -0.912, p(PND, ux@p ' < 0) ( pPND hereinafter). It is found that pPND is 0.701, 0.63, 0.545, 0.482,

0.462, 0.468, and 0.352 for τp,b = 2.394 and w/k = 0, 1, 3, 4, 7, 9, and 19, respectively. The instantaneous distribution of particles with τp,b = 2.394 is correlated well with flow areas having

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ux@p ' < 0 at w/k ≤ 3, while the trend is reversed with further increasing w/k. The particle clusters observed in Fig. 4 at the larger w/k values are located mainly in regions with ux@p ' > 0. The discussion so far was focused on particles with response time τp,b = 2.394 (τp,0+ = 25).

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< Table 2 summarizes the values of the probability pPND for all the particle ensembles examined here.

It can be seen that the analysis presented in this Section is valid from a qualitative point of view

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< independent of particle response time. For example, pPND > 0.5 is found in the cases with w/k ≤ 1

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< < for all particle sets, pPND > 0.5 holds at w/k = 3 for τp,b ≥ 2.394, while pPND ≈ 0.5 generally occurs at

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< intermediate w/k values. pPND < 0.5 is found at large w/k values independent of particle response

time. Consequently, the tendency of the particles examined here to preferentially concentrate in

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regions with negative streamwise velocity fluctuations is diminished in the flows with intermediate values of w/k, at which the particles are equally found in areas populated by high and low streamwise velocities of the carrier phase. Moreover, the number of particles located within regions of high streamwise fluid velocity is significantly increased as the spacing between the roughness elements increases further. Fewer particles are generally found in the region of observation with increasing w/k. The total number of particles inside the rough channel is decreased due to the particle-wall collisions 18

ACCEPTED MANUSCRIPT and, thus, the statistics shown here are in transient state and do not correspond to long time steadystate values. Nevertheless, a careful examination of several other instantaneous realizations did not reveal significant changes with respect to the results discussed above. Fig. 6 shows the instantaneous isolines of the fluctuations of the grid-scale fluid velocity in the wall-normal direction u y ' at the z/h = 0 plane for representative cases. Moreover, it shows the

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particles with τp,b = 2.394 (τp,0+ = 25) located within a region of 0 ≤ z/h ≤ 0.5. It is seen that u y ' is very weak at w/k = 1, which is also the case closer to the results of the turbulent flow in a flat channel. At w/k = 3, fluid ejections ( u y ' > 0) are observed as the flow approaches a roughness

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element. Such motions are extended into the outer flow region, accompanied by regions of fluid sweeps ( u y ' < 0). As w/k increases further, the horizontal fluid motion becomes increasingly important. At w/k = 7, very strong outward ejections of fluid are apparent, while the rest of the

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cavity is mostly populated by very intense sweeps. The minimum and maximum fluid velocities ( u y ' )min, ( u y ' )max at the z/h = 0 plane normalized by uc are: (-0.107, 0.103), (-0.103, 0.121), (0.266,

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0.15), (-0.199, 0.257), (-0.3, 0.428), (-0.295, 0.445), and (-0.35, 0.41) for w/k = 0, 1, 3, 4, 7, 9, and

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19, respectively.

The wall-normal motion induced by the roughness elements interacts with the overlying

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fluid flow, affecting also particle transfer. For non-colliding particles, the particle fluctuating motion in the y- and z-directions is at local equilibrium with the fluid flow (see Vance et al., 2006;

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Dritselis, 2016, and references therein). Thus, the cross-flow components of the particle fluctuating velocity are controlled by the drag force. Particle inertia filters out the small flow scales, reducing the magnitude of particle velocity fluctuations in the wall-normal direction. The minimum and maximum values of vy at the region shown in Fig. 6 are summarized in Table 1. It can be seen that (vy)min and (vy)max are consistent with the u y ' results in each w/k case, taking into account the filtering effect of particle inertia. 19

ACCEPTED MANUSCRIPT In a flat channel (Fig. 6a), the particles having a trajectory toward the outer flow region concentrate locally in regions with u y ' > 0, while the opposite is true for particles moving toward the smooth wall. In addition, several flow areas void of particles can be seen. These visual observations are in qualitative agreement with previous results about the efficiency of the organized fluid motions on the particle transfer (Soldati and Marchioli, 2009). The instantaneous particle

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positions adjacent to the crests of the roughness element at w/k = 1 (Fig. 6b) resemble those in the flow with smooth walls. As w/k increases, the y-component of particle velocity is significantly increased. Consequently, the particles can potentially acquire sufficient momentum to reach the vicinity of the square bars or even enter the cavities of the rough wall. At w/k ≥ 3, the particles are

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transferred close to the rough wall faster than in the w/k = 0 or 1 cases. This change in the particle behavior implies a significant impact of the roughness elements on the particle deposition. At w/k = 3 (Fig. 6c), it can be seen that the cavities are empty of particles with τp,b = 2.394. However, this does not mean that particles do not enter the cavities, but rather that they deposit rapidly once they

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are found inside them. It was verified that a significant fraction of particles with smaller τp,b values

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remain inside the cavities for sufficient time periods. Consequently, such particles may either deposit at the rough wall or escape to the outer flow region. At w/k = 7 (Fig. 6d), very strong fluid

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ejections are visible adjacent to the frontal surface of the square bars. These fluid ejections are

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mainly populated by particles having vy > 0, thus, driving the particles away from the rough wall. Since such events extend up to the outer flow region, they also oppose particles from reentering the

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cavity, resulting in regions void of particles at 2.5 < x/h < 3 and 3.5 < x/h < 4. The region near a smooth wall of a turbulent channel flow is populated by coherent

structures that create strong outward fluid motions of low speed (Q2 events, ux' < 0 and uy' > 0) and inward fluid motions of high momentum (Q4 events, ux' > 0 and uy' < 0) (Kim et al., 1987). The role of organized fluid motions in providing certain paths through which the particles can approach the smooth wall or re-entrain the central flow region has been investigated previously (see, for example, McLaughlin, 1989; Brooke et al., 1992; Pedinotti et al., 1992; Marchioli and Soldati, 2009). There 20

ACCEPTED MANUSCRIPT is a general consensus among the studies regarding the importance of the sweep and ejection events on controlling particle transport. The former events are mainly responsible for driving particles close to the smooth wall, while the latter ones control the particle flux away from it. In order to gain insight into the effect of the roughness elements on the transfer mechanisms, the particles located within Q1, Q2, Q3, and Q4 events in a region of -0.971 ≤ y/h ≤ -0.912 were counted. Fig. 7 shows the probability of the correlation between the wall-normal particle velocity vy

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and Qi events p(Qi, vy) as a function of w/k for particles with τp,b = 2.394 (τp,0+ = 25). For particles approaching the wall, p(Q4, vy<0) is smaller relative to the w/k = 0 case at w/k ≤ 3, while it is larger at w/k ≥ 4. p(Q3, vy<0) is generally attenuated in the cases with wall roughness, while p(Q1, vy<0) is

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enhanced. p(Q2, vy<0) is increased when w/k ≤ 3, and it is decreased with further increasing w/k. On the other hand, for particles moving away the wall, p(Q2, vy>0) is decreased in the cases with square bars. p(Q4, vy>0) is slightly enhanced with increasing w/k, while the opposite is true for p(Q3, vy>0). Finally, p(Q1, vy>0) is augmented in the flows with wall roughness, except in the w/k =

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1 case. It should be noted that Q3 and Q4 events correspond to fluid motions toward the wall ( u y '

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< 0). Such events assist the motion of particles having vy < 0 to reach the wall, since drag force is also negative and acts toward the wall. In contrast, Q1 and Q2 events are characterized by outward

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fluid motions ( u y ' > 0) and, thus, favor the motion of a particle moving away the wall. Fig. 7

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clearly indicates that most of the particles with vy < 0 are found in flow regions characterized by sweep (Q4) events. On the other hand, the majority of particles with vy > 0 are located within

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regions of ejection (Q2) events. The influence of particle response time on the correlations between Q2, Q4 events and the

particle fluxes away and toward the wall, respectively, is shown in Table 2. It is found that p(Q2, vy>0) is generally decreased for all particles ensembles in the rough channels, except for the w/k = 1 case with particles having τp,b = 0.958 (τp,0+ = 10). In contrast, p(Q4, vy<0) is increased in the presence of roughness elements, except in the cases with τp,b (τp,0+) = 0.479 (5), 0.958 (10) at w/k = 21

ACCEPTED MANUSCRIPT 1, and τp,b (τp,0+) = 2.394 (25) at w/k ≤ 3. Both the largest reduction in p(Q2, vy>0) and increase in p(Q4, vy<0) are found at w/k = 19. The results in Table 2 suggest that the sweep (Q4) and ejection (Q2) events are responsible for transferring particles toward and away the rough wall, irrespective of the spacing between the square bars. The outward/inward fluid motions with increasing w/k differ than those in the flat

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channel. As seen in Figs. 4 and 6, the roughness elements produce a very strong disturbance, resulting in a modification of the fluid structures adjacent to the rough wall. These local flow changes have a considerable effect on the efficiency of the organized fluid motions to transfer particles to the wall and the outer flow region, as demonstrated by the increase of p(Q4, vy<0) and

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the reduction of p(Q2, vy>0), respectively. The flow modifications due to the rough wall are better reflected on the dynamic behavior of small inertial particles. Such particles respond more efficiently to a wider spectrum of flow scales, and exhibit greater changes in the probabilities p(Q4, vy<0) and p(Q2, vy>0).

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Proper statistical descriptors were presented to demonstrate the significant impact of the

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rough wall on the preferential concentration and the particle transfer by the organized fluid motions. For this reason, the differences in the particle statistics were identified and discussed by varying

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w/k, while keeping constant the particle response time τp,b. The effect of increase of dp+ and τp+ cannot explain the changes in the particle statistics with varying w/k described above. For example,

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the formation of particles in straight lines near a smooth wall is enhanced with increasing particle

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inertia and prevails for particles with response times in the range of 3.8 ≤ τp+ ≤ 116.3 (Marchioli and Soldati, 2002, 2009). In contrast, the present study indicates that the clustering pattern changes completely from straight lines for τp+ = τp,0+ = 25 (τp,b = 2.394) at w/k = 0 to almost circular regions for τp+ = 74.9 (τp,b = 2.394) at w/k = 19, as shown in Fig. 4. The same also holds for Fig. 5, which quantifies the observations of Fig. 4. The peak of particle number density conditionally sampled near the smooth wall always occurs in negative values of streamwise fluid velocity fluctuations for particles with response times of 3.8 ≤ τp+ ≤ 116.3 (Marchioli and Soldati, 2002, 2009). This is the 22

ACCEPTED MANUSCRIPT case here for particles with τp+ = τp,0+ = 25 (τp,b = 2.394) at w/k = 0, while this trend is reversed at τp+ = 74.9 (τp,b = 2.394) at w/k = 19.

5.3 Particle deposition

It is well established that the particles are transferred near the smooth wall of a turbulent

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channel flow, where a significant buildup of their concentration takes place. The particles may either re-entrain into the outer flow or deposit at the smooth wall. Two main mechanisms of particle deposition have been identified (McLaughlin, 1989; van Haarlem et al., 1998; Zhang and Ahmadi, 2000; Narayanan et al., 2003; Marchioli et al., 2007). More specifically, particles with high

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response times may acquire sufficient momentum to coast the viscous sublayer and deposit by impaction at the smooth wall. Particles with low response times usually reside in the viscous sublayer for long time periods and may deposit at the smooth wall due to diffusion effects, i.e. under the action of small turbulent fluctuations in the wall vicinity. Particles with intermediate

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response times may deposit by either mechanism.

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In this section, the effect of roughness elements on the particle deposition rate is analyzed and quantified. A proper measure for the particle deposition is the deposition coefficient or deposition

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velocity kd. This quantity is the constant of proportionality between Jp,w and Cp, where Jp,w is the rate of particle mass transfer on the deposition area, and Cp is the mean or bulk concentration of

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particles. In the present study, kd is estimated as (see, for example, Wang et al., 1997; Zhang and

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Ahmadi, 2000; Dritselis et al., 2011)

kd =

(

1 N /t Ad p,d d

) (N

p,dV

)

/ dV ,

(11)

where Np,d is the number of deposited particles in a time interval of td, Np,δV is the total number of particles located initially within a region of y* distance from the vertical crests of the roughness elements with volume δV, and Ad is the deposition area. The deposition area is given by Ad = Lx Lz +

23

ACCEPTED MANUSCRIPT 2 k Lz nsb, where nsb is the number of square bars. Similarly, the volume is given by δV = y* Lx Lz + k w Lz nsb. Fig. 8a shows the normalized deposition coefficient kd,0+ (= kd/uτ,0) as a function of particle response times τp,0+ and τp,b for various cases with w/k = 0, 1, 3, 4, 7, 9, and 19. For comparison purposes, Fig. 8a also shows the numerical results of Wang et al. (1997) and Marchioli et al. (2007), as well as the empirical correlations proposed by Liu and Agarwal (1974), kd,0+ = 6 × 10-4 (τp,0+)2,

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and by McCoy and Hanratty (1977), kd,0+ = 3.25 × 10-4 (τp,0+)2. Both empirical correlations are valid for particles in the diffusion/impaction regime with 1 ≤ τp,0+ ≤ 25. The results corresponding to the particle-laden flows in a flat channel are consistent with previous findings. For instance, kd,0+ is

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monotonously increased for particles with intermediate response times 2.5 ≤ τp,0+ ≤ 25 (0.239 ≤ τp,b ≤ 2.394). As τp,0+ increases further, kd,0+ saturates and approaches an almost constant value. The present numerical predictions of kd,0+ are generally smaller than those indicated by the empirical correlations. However, it should be noted that there is clearly an uncertainty in the estimation of

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deposition coefficient, as indicated by the large spread in the experimental data that were used (see,

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for example, van Haarlem et al., 1998; Marchioli et al., 2007). In contrast, they are in fair agreement with the DNS results of Marchioli et al. (2007), when the different Reynolds number is taken into

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account. A good agreement is achieved against the LES results of Wang et al. (1997), under the same numerical conditions and similar parameters.

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Fig. 8a clearly indicates that the deposition of particles is enhanced in turbulent flows at

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channels roughened by transversally placed square bars. Moreover, a substantial increase of the deposition coefficient takes place with increasing w/k. The variation of kd,0+ with τp,0+ at w/k = 1 is similar to that observed in the w/k = 0 case. On the contrary, a weaker dependence of kd,0+ on the particle response time is revealed at w/k ≥ 3. At w/k = 1, the largest increase in kd,0+ with respect to the flat channel is seen for particles with τp,0+ = 200 (τp,b = 19.152). At 3 ≤ w/k ≤ 7, this occurs for τp,0+ = 10 (τp,b = 0.958), while at w/k > 7 this appears for τp,0+ = 25 (τp,b = 2.394). Small particles with τp,0+ (τp,b) = 2.5 (0.239) and 5 (0.479) exhibit a maximum augmentation of kd,0+ at w/k = 4 and 24

ACCEPTED MANUSCRIPT 7, respectively, while for larger particles the maximum enhancement is consolidated at w/k = 19. The overall increase of deposition coefficient in the flows with wall roughness is consistent with previous findings (Wells and Chamberlain, 1967; Vincent and Humpries, 1978; Lai et al., 1999, 2000, 2001, 2002; Hussein et al., 2009). Fig. 8b shows the deposition coefficient normalized by the actual wall friction velocity kd+ =

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kd/uτ as a function of τp+ for various w/k cases. As expected, higher values of τp+ are seen in the flows in rough channels, while the substantial augmentation of the particle deposition is evident, but the level of increase is smaller due to the increase of uτ. The enhanced deposition coefficient can be attributed partly to the greater unresponsiveness of the particles in the rough channels. The present

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results for w/k = 0 clearly indicate that the values of kd+ are of the order of O(10-1) as τp+ is increased, in accordance with previous findings in turbulent channel flows with smooth walls. However, it is obvious that the substantially higher values of deposition coefficient of the order of

in wall units with varying w/k.

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O(1) found in the rough channels cannot be attributed only to the increase of particle response time

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Fig. 9 shows the spatial distribution of the number of particles deposited at the rough wall Np,d(x/k). A significant fraction of the particles that deposit at the rough wall is found at its outer

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vertical surface. In particular, Np,d exhibits a peak value close to x/k = 0 (point D) in the cases with small particles of τp,b = 0.239 at w/k ≤ 4. Np,d is decreased at a large part of the outer vertical surface

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of the square bar (DA), while it is increased near x/k = 1, acquiring a second local maximum value,

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which becomes greater than that at x/k = 0 when w/k ≥ 7. Moreover, Np,d is roughly constant in most of the region 0 ≤ x/k ≤ 1 (DA), and it is decreased as w/k increases. Similar observations can be made for particle sets with τp,b = 0.479 and 0.958. For τp,b ≥ 2.394, the second peak value of Np,d appearing at the trailing edge of the square bar (point A) is gradually diminished, resulting in a monotonous reduction of Np,d at the wall DA. For particles with τp,b = 9.576 and 19.152, Np,d is almost constant at the outer vertical surface of the rough wall, and it decreases with increasing w/k.

25

ACCEPTED MANUSCRIPT As discussed in Section 5.2.1, the particles can reach the rough wall rather easily due to the flow modifications produced by the roughness elements. Consequently, the observed decrease of Np,d in the region 0 ≤ x/k ≤ 1 with varying w/k can be attributed mainly to the reduction in the total area of the outer vertical surface of the rough wall. The uniform distribution of Np,d found for the larger particles is an outcome of the fact that such particles are less responsive to the surrounding

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flow scales and, thus, they can potentially hit anywhere at the wall with practically the same probability. On the other hand, small response time particles are influenced by the local fluid flow and eventually by the flow modifications induced by the roughness elements, leading to a highly non-uniform distribution of Np,d.

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Fig. 9 also shows that the number of particles deposited at the upper horizontal surface of the cavity (ΑΒ) is relatively small. This surface can be approached only by few particles with sufficiently low response times that can be captured by the main recirculation. With increasing w/k, a second smaller vortex of opposite direction to the main vortex appears, which actually opposes the

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movement of particles toward the upper horizontal surface of the cavity (AB). The vast majority of

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particle deposition occurs at the lower horizontal surface of the cavity (CD). This is true for all particle sets and w/k cases examined in the present study. The distributions of Np,d provide an

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explicit evidence of the relevance of interception on the deposition of particles at rough walls. The pronounced enhancement of particle deposition can be attributed to the direct impaction of particles

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at the frontal surface of the roughness elements. This deposition mechanism is well understood and

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it has been described previously (see, for example, Vincent and Humphries, 1978). It is closely related to the fact that a particle cannot follow exactly the trajectory of the local fluid due to its inertia. For example, when the air-particle flow approaches a square bar, the fluid will surpass the roughness element. The inertial particles will delay to respond to the changes in the fluid motion and they will probably strike onto the frontal area of the roughness element. Inside the cavity at 1 < x/k < 1 + w/k (BC), the deposition of particles is not noticeable in the part of the inner vertical surface populated by the two re-circulations. Away from the re26

ACCEPTED MANUSCRIPT circulations, the flow is almost parallel with its pattern being similar to that of the w/k = 0 case, and particle deposition is enhanced. This local increase is more obvious for particles having τp,0+ (τp,b) ≤ 25 (2.394) in the cases with w/k = 7 and 9. As the next element is approached, the particle deposition is attenuated. This is because of the small vortex at the lower horizontal surface, and the strong ejection events that occur there. Particles within intense fluid ejections may exit the cavity

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and re-entrain the outer flow or deposit at the lower horizontal surface (CD) due to the interception effect. The aforementioned trends are observed for all particles and only some quantitative differences can be noticed as a consequence of the different impact by the particle inertia.

Fig. 10 shows the total number of particles Np,dSi deposited at the outer (Si=SV,ou-DA) and the

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inner (Si=SV,in-BC) vertical surfaces, as well as the upper (Si=SH,u-AB) and lower (Si=SH,l-CD) horizontal surfaces of the cavity. At w/k = 1, particle deposition is larger in the outer vertical surface than in the lower horizontal one. This means that deposition by inertial impaction and diffusion are important mechanisms as in the case with smooth walls. However, it can be seen that the deposition

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by interception, as indicated mainly by Np,dSH,l, is also considerable. Np,dSV,ou is uniformly decreased

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in the rest of the w/k cases, as a consequence of the reduction of the overall area of the SV,ou surface. In contrast, Np,dSH,l is significantly augmented, revealing the importance of interception mechanism.

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It is seen that Np,dSH,l is increased with increasing bar spacing up to w/k = 4, while it is decreased with further increasing w/k. Particle deposition at the inner vertical surface is not activated when

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w/k ≤ 4. Np,dSV,in is increased when w/k ≥ 7, obtaining comparable values with Np,dSH,l at w/k = 19.

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Fig. 11 shows the variation of the total number of particles deposited at each surface of the rough wall Np,dSi with the particle response time. In general, a small dependence of Np,dSV,in, Np,dSV,ou, and Np,dSH,l on τp,b is observed for particles with τp,b ≥ 2.394 (τp,0+ ≥ 25). Fig. 11a shows that Np,dSV,in is reduced except in the w/k = 19 case. It also reveals that small inertial particles deposit frequently at the inner vertical wall at small bars spacings. On the other hand, Fig. 11b shows that Np,dSV,ou is increased up to approximately constant values with increasing τp,0+ in all w/k cases. Fig. 11c shows that Np,dSH,l is similarly enhanced, except in the w/k = 1 and 19 cases. The results for the particles 27

ACCEPTED MANUSCRIPT deposited at the upper horizontal surface of the cavity are not shown here, as Np,dSH,u is very small (0 ≤ Np,dSH,u / Σ Np,dSi ≤ 0.013). The results in Fig. 11 suggest that the analysis on the contribution of each surface of the cavity to the deposition of particles with response time of τp,b = 2.394 (τp,0+ = 25) presented above also holds for the other particle ensembles studied here. Some rather minor differences in the distributions shown in Fig. 10 do occur due to the particle inertia, however,

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without affecting the observations from a qualitative point of view. In order to quantify further the particle deposition onto a rough wall, the total particles entering Np,en and exiting Np,ex the cavity, and those deposited Np,d were monitored during the simulations and shown in Fig. 12. Almost all the particles with τp,b = 19.152 that are transferred into

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the cavity are eventually deposited, as indicated by Np,d/Np,en ≈ 1, irrespective of the w/k value. This is also seen for particles with τp,b = 9.576, but at smaller spacings of the square bars w/k < 9. With decreasing τp,b, Np,d/Np,en is decreased, obtaining the smallest values in the w/k cases with particles having τp,b = 2.394. It was verified that the particles could enter and exit the cavities of the rough

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wall several times. For this reason, Np,en can be larger than Np,d, leading to values of Np,d/Np,en

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smaller than unity. The latter behavior becomes more evident at low particle response time and large bars spacing. The fraction of particles that were captured by the local re-circulations and

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remained inside the cavity for relatively long time periods Np,in/Np,en was very small, of the order 1% of the particles entering the cavity. The vast majority of the particles that entered the cavity by

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did not deposit at the rough wall were eventually re-introduced into the outer main flow. The

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population of particles escaping from the cavity can be estimated approximately as Np,ex/Np,en ≈ 1 – Np,d/Np,en. Fig. 12 clearly indicates that, for the present geometric configuration consisting of square bars placed transversally at the wall, large inertia particles are effectively deposited, while the capture efficiency is attenuated with decreasing particle response time and increasing bars pitch. Particle deposition by inertial impaction and diffusion are expected to be effective also in the flows with roughness elements. However, the relative importance of these mechanisms would depend not only on the particle response time, as in the case with smooth walls, but indirectly also 28

ACCEPTED MANUSCRIPT on w/k, through the changes in the fluid flow produced by the square bars. The large wall-normal velocities of particles denote that the deposition at the rough wall by inertial impaction is strengthened. A fraction of particles with insufficient momentum are transferred in the vicinity of the rough wall and eventually deposited by diffusion. However, the presence of roughness elements does not allow long residence times of particles in the rough wall proximity, thus, inhibiting the

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diffusion mechanism. The deposition by diffusion would be significant for a fraction of particles that deposit at the outer vertical surface of the rough wall, and for small particles that are captured by the local re-circulations. However, as discussed earlier, very few particles remain inside the cavity for long time periods, corroborating the enhanced role of inertial impaction mechanism in the

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observed augmentation of particle deposition at the flows with a rough wall.

Summarizing, a particle approaching the rough wall may either strike the frontal surfaces of the square bars or hit their outer and inner vertical surfaces. The results in Figs 9-12 clearly indicate that a major mechanism responsible for the increase of deposition coefficient in the flows at rough

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channels is interception. This mechanism is more efficient for large inertia particles with τp,b (τp,0+)

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≥ 9.576 (100) and for intermediate spacings of the square bars 3 ≤ w/k ≤ 7, while its effectiveness is decreased with further increasing w/k. The latter can be attributed to the reduction of the total area

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of the frontal surface of the bars, which affects the number of particles obstructed. Moreover, some particles may avoid deposition due to interactions with strong ejections that occur in the cavity near

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the frontal surface of the bars at large w/k values. The opposite observations can be made for the

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deposition of particles at the inner vertical surface of the rough wall. Its total area is increased with increasing w/k, contributing to the augmentation of particle deposition mostly by inertial impaction. The total deposition area is increased in the flows with wall roughness. However, as shown in Figs 9-11, the deposition at the rear surface of the bars is very small when compared with that at the other surfaces of the rough wall. Consequently, the increase of the total deposition area is not accompanied by a similar augmentation in the particle deposition. For example, the maximum value

29

ACCEPTED MANUSCRIPT of the total deposition area occurs at w/k = 1 and the minimum one at w/k = 19, but kd,0+ in w/k = 1 is smaller than that in the w/k = 19 case.

6. Conclusions The effect of wall roughness on the particle deposition in a fully developed turbulent channel flow was investigated based on large eddy simulation coupled with a Lagrangian particle-

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tracking scheme. The focus was on particles with response times in the range of 2.5 ≤ τp+ ≤ 600 depositing onto a vertical surface roughened by two-dimensional transverse square bars separated by a rectangular cavity. Several values of the ratio of the cavity width w to the roughness element

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height k and particle response time were examined.

Particle accumulation near the rough wall is considerably influenced. More specifically, the instantaneous distributions of particle concentration are relatively uniform near the rough wall, which is closely related to the increased particle deposition. This is in contrast to the non-uniform

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particle concentration in the wall-normal direction found here at w/k = 0 and widely observed in the

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case of smooth channels (McLaughling, 1989; Zhang and Ahmadi, 2000; Marchioli et al., 2007). The modified organized fluid motions effectively provide certain paths through which

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particles are transferred toward the rough wall and the outer flow region. In accordance with the case of a flat channel (Marchioli and Soldati, 2002, 2009), the particle fluxes toward and away the

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rough wall are statistically well correlated with strong sweep and ejection events, respectively.

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However, the modified organized outward/inward fluid motions are different from those observed in the turbulent channel flow with smooth walls and they are more effective in affecting the particle motion.

The tendency of particles to preferentially concentrate in flow regions of low streamwise fluid velocity is diminished with increasing the spacing of the square bars. The formation of particles in straight lines seen at w/k = 0, which is consistent with previous works (Pedinotti et al., 1992; Rouson and Eaton, 2001; Yamamoto et al., 2001; Soldati and Marchioli, 2002, 2009), is 30

ACCEPTED MANUSCRIPT gradually changed into a circular pattern close to the rough wall with increasing w/k, while the particles are located mostly within regions of positive streamwise fluid velocity fluctuations. This results in a different clustering pattern of particles as compared to what is known to occur in flat channels. It was shown that particle deposition is substantially increased in the presence of roughness

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elements due to modifications in the fluid flow and to the direct impaction of particles at the frontal surface of the square bars.

The present study confirms previous findings about the significance of the interception mechanism in the augmentation of particle deposition at the rough wall (see, for example, Lai et al.,

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1999, 2000, 2001, 2002; Hussein et al., 2009). It reveals further that the efficiency of interception mechanism depends both on particle inertia and the spacing between the square bars. The interception mechanism is found to be more efficient for intermediate rib spacings 3 ≤ w/k ≤ 7. Its effectiveness is attenuated with further increasing w/k, which can be attributed to the decrease of the

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total area of the frontal surface of the roughness elements and to the strong ejection events that

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occur in the cavities of the rough wall at large values of w/k. The former can obviously influence the total number of particles obstructed, while the latter points out that some particles can avoid

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deposition by interception because of their interaction with the strong outward fluid motions observed near the frontal surface of the square bars. In this frame, the interception mechanism is

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also less effective for small inertial particles with τp,b ≤ 2.394 (τp,0+ ≤ 25) that can potentially interact

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with a wider spectrum of fluid flow scales, and may escape more easily from the cavities of the rough wall and re-enter the outer flow region. Deposition due to direct inertial impaction is known to be significant in flat channels,

especially with increasing particle response time (McLaughlin, 1989; Brooke et al., 1992; Pedinotti et al., 1992; van Haarlem et al., 1998; Zhang and Ahmadi, 2000; Narayanan et al., 2003; Marchioli et al., 2007; Dritselis et al. 2011), which is also found to be valid in the present flows. This deposition mechanism becomes more efficient in channels roughened by transversally place square 31

ACCEPTED MANUSCRIPT bars. This is because the particles can obtain sufficient momentum by their interaction with the modified flow structures, while they find a smaller resistance as they approach the rough wall owing to the disruption of the viscous layer. The presence of roughness elements does not allow long residence times of particles near the rough wall and, thus, the deposition by diffusion mechanism is inhibited with its significance being limited to a fraction of small inertial particles

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that are captured by the local re-circulations. The role of vortices inside the cavities of the rough wall on the particle deposition was also analyzed. For w/k = 1, the single vortex is rather weak and has little direct effect on the particle motion and deposition. At intermediate and larger values of 3 ≤ w/k ≤ 19, a small fraction of

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particles is captured by the main recirculation and deposit at the upper vertical surface. The secondary vortices appearing adjacent to the upper and lower vertical surfaces of the rough wall with increasing w/k oppose the movement of particles and, subsequently, their deposition onto these surfaces. In all cases, a very small deposition takes place below the main recirculation. Finally, it

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was demonstrated that the augmentation of deposition surface is not accompanied by a similar

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increase of particle deposition at transversally placed square bars.

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Acknowledgment

Partial financial support from the Association EURATOM–Hellenic Republic during 2008-2016 is

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gratefully acknowledged. The content of this paper is the sole responsibility of its author and it does

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not necessarily represent the views of the European Commission or its services.

Appendix A. Effect of grid resolution In order to conduct accurate LES, the grid resolution must be chosen carefully by considering the physics of the fluid flow and the complexity of the flow geometry. In Appendix A, the effect of grid resolution on the LES results of the carrier phase is presented for a representative case of the fully developed turbulent channel flow with square roughness elements at one wall and 32

ACCEPTED MANUSCRIPT w/k = 7. It is recalled that all the results shown here were obtained using 201×(30 + 64)×65 grid points in the x, y, and z directions, with 30 points located uniformly within -1.2 < y/h < -1 and 64 points unevenly distributed in -1 < y/h < 1. The present LES grid resolution is much coarser than those used in similar studies of a two-dimensional channel flow with asymmetric two-dimensional roughness based on DNS (Leonardi et al., 2003; Leonardi et al., 2004; Orlandi and Leonardi, 2006;

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Orlandi et al., 2006). In order to verify whether the LES results of the carrier phase are grid converged, the following computational meshes were also examined: 81×(30 + 64)×65, 201×(30 + 32)×65, 201×(15 + 64)×65, and 201×(30 + 64)×33. In all cases, the Reynolds number based on the bulk velocity was Reb = 5600 and the channel domain was [0, 8h]×[-1.2h, h]×[0, πh].

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Fig. 13 shows the effect of grid resolution on the mean streamwise fluid velocity ( u x )x,z,t and the rms fluid velocity fluctuations ux,rms in the streamwise and uy,rms wall-normal directions, where ( )x,z,t denotes averaging over the streamwise and spanwise directions and time. Similar results were obtained for the spanwise component uz,rms, but they are not shown here for clarity of

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presentation. It can be seen that the results using 81×(30 + 64)×65 grid points do not differ largely

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from those using 201×(30 + 64)×65 grid points, revealing that the refinement of the computational mesh in the x-direction is less significant. Each roughness element is discretized in the x-direction

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by using two grid points for the former mesh, while five grid points are utilized for the latter mesh.

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A good agreement is also observed for the results obtained by using 201×(30 + 32)×65 and 201×(30 + 64)×65 grid points. This indicates that 32 grid points are sufficient to discretize the outer flow

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region -1 < y/h < 1, while the first grid point is located in y+ < 1 with respect to the plane of the crests of the roughness elements due to the non-uniform grid distribution in the wall-normal direction (see Section 4). The results produced based on 201×(15 + 64)×65 points are close to those of the simulation with 201×(30 + 64)×65 points, suggesting that the discretization of the roughness element using fewer grid points in -1.2 < y/h < -1 has a small influence on the distributions of the mean and rms velocities of the carrier phase in the outer flow region -1 < y/h < 1. It is seen that the mesh refinement in the z-direction has a somehow stronger impact than that in the x-direction, as 33

ACCEPTED MANUSCRIPT indicated by the relatively larger differences between the results of the 201×(30 + 64)×33 and 201×(30 + 64)×65 meshes. A fine resolution in the z-direction is required in order to account properly for the modifications of the fluid flow structures by the roughness elements. Fig. 13 reveals that the differences in the distributions of ( u x )x,z,t, ux,rms and uy,rms are generally small with grid refinement in the x, y and z directions, and that the results based on the computational mesh

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with 201×(30 + 64)×65 grid points agree reasonably well with the DNS results based on a much finer grid resolution (Orlandi et al., 2006). Consequently, it can be concluded that the results of the carrier phase are grid converged.

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Appendix B. Effect of the subgrid scale fluid velocity on the particle motion

In point-particle LESs, only the filtered fluid velocity seen by the particles is available and, consequently, a subgrid error is introduced in the Lagrangian motion of particles. The effect of subgrid scale fluid velocity fluctuations has been neglected in several LES studies of particle-laden

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turbulent flows (see, for example, Wang et al., 1997; Yamamoto et al. 2001; Lo Iacono et al., 2005,

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2008; Vance et al., 2006). All these works indicate that the statistics of the particulate phase can be predicted satisfactorily in well-resolved LES under the assumption that the particle response time is

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larger than the smallest resolved time scale.

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More specifically, the effect of filtering velocity fields on the particle motion was extensively analyzed in Yamamoto et al. (2001). They found that a grid spacing of Γx+ = 45, and

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Γz+ = 15 could reproduce particle dispersion correctly within an error of 0.3% of the characteristic length scale. Similarly, Armenio et al. (1999) found that the results obtained when the particle motion is computed using a well-resolved LES velocity field are in good agreement with the DNS. More specifically, the dispersion statistics of tracer particles differed by less than 8%, while the errors were significantly smaller in the case of real inertial particles with τp / τk > 1, where τk is the Kolmogorov time scale. Moreover, they attributed the differences mainly to modeling than subgrid errors. In Wang and Squires (1996), the effect of subgrid scale fluid motions on the particle 34

ACCEPTED MANUSCRIPT trajectory was considered in a turbulent channel flow with smooth walls at Reτ ≈ 180. The largest difference in the deposition results in the cases without and with the effect of subgrid scale fluid velocity fluctuations was less than 10% for particles with τp+ = 2, while the discrepancies were significantly decreased with increasing particle inertia. The present grid resolution of the turbulent channel flow with smooth walls is consistent

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with the requirements for conducting well-resolved LES and for obtaining accurate results (see Piomelli and Balaras, 2002). It should be noted that the adopted grid spacing is also smaller than that indicated by Yamamoto et al. (2001) for obtaining accurate dispersion results, especially in the streamwise direction. As shown in Appendix A, the mean streamwise fluid velocity and the rms

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fluid velocity fluctuations are in good agreement with the DNS results based on a much finer grid resolution, indicating that the LESs of the turbulent channel flow with rough walls are indeed well resolved. The fact that the LESs in this work are well resolved is further supported by the values of eddy viscosity νt modeled by using the dynamic Smagorinsky model. The quantity νt serves as an

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indicator of the effect of the subgrid-scale fluid flow motions that are actually modeled during the

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simulations. It is found that νt is 7.5-80% of the fluid viscosity and, thus, νt yielded by the LES model is always smaller than ν. In addition, the dynamic model contributes about 10% of the

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molecular dissipation rate over much of the channel for w/k = 0, while the contribution is increased

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up to 30-45% in the cases with rough wall. The subgrid scale dissipation rate is calculated as τij S ij and the molecular dissipation rate as – (2/Reb) S ij S ij , where τij is given by Eqs. (3) and (4). Based

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on the above analysis, it can be concluded that that the grid size used in the present LES study is adequate to resolve a significant part of the energy of the turbulent flow in the cases with smooth or rough walls at Reb = 5600 (for more details, see Dritselis, 2014). Table 3 summarizes the values of particle response time examined in the present study normalized by the smallest resolved time scale τξ near the rough wall at the y/h = -0.85 plane. The time scale τξ is calculated in a similar manner as the Kolmogorov time scale using the grid-scale viscous dissipation provided by LES. It can be seen that the particles used here have response times 35

ACCEPTED MANUSCRIPT of the same order or larger than τξ and, thus, the particle motion is controlled by the fluid velocity that is being computed as part of LES (Balachandar, 2009). The grid-scale viscous dissipation in the rough channels can be 6-7 times larger than that in the smooth channel (see also Orlandi et al., 2006) and, thus, the dissipative length ξ and time τξ scales are decreased, leading to smaller and faster smallest resolved eddies in the flows with square roughness elements. The particles are less influenced by the fluid motions and increasingly unresponsive to the subgrid scale velocity

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fluctuations as compared with the w/k = 0 case. It can be seen that τp/τξ > 1 generally holds, except for particles with τp,b = 0.239 (τp,0+ = 2.5) at w/k = 0 and 1. Note that larger values of τp/τξ are generally obtained, for example, if τξ is calculated in the y/h = -1 plane of the crests of the

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roughness elements or if the dissipation rate averaged over the entire channel is used in the computation of τξ, or if the extra contribution to the dissipation rate provided by the dynamic subgrid scale model is taken into account in the calculation of the dissipative scales. Therefore, it can be safely assumed that the subgrid scale fluid velocity fluctuations are unlikely to have a

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measurable effect on the deposition of particles onto rough walls in the present study. A small

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impact may be anticipated for the small inertial particles with τp,b = 0.239 (τp,0+ = 2.5) at w/k = 0 and 1, however, without altering the discussion and the conclusions made in this work.

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Table 3 also shows the particle diameter made non-dimensional using the smallest resolved length scale ξ. As expected, dp/ξ is increased in the turbulent flows with rough wall due to the

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reduction of ξ. It can be seen that the particles used here are smaller than or comparable to ξ.

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Similarly, the values of the particle diameter in wall units are increased dp+/dp,0+ (= uτ/uτ,0) = 1.016, 1.252, 1.464, 1.730, 1.732, and 1.472 for w/k = 1, 3, 4, 7, 9, and 19, respectively. Since one-way coupling is assumed in the present simulations, the particles sets are well suited for performing point-particle LESs as discussed in Balachandar (2009). The minimum and maximum values as well as the averaged value over the particulate phase of the particle Reynolds number were monitored during the simulations. The time variation of Rep depends on the particle response time and the w/k value. The typical range of the maximum particle 36

ACCEPTED MANUSCRIPT Reynolds number is 0.859 ≤ Rep,max ≤ 30.971 for w/k = 0, 1.129 ≤ Rep,max ≤ 32.592 for w/k = 1, 2.618 ≤ Rep,max ≤ 35.071 for w/k = 3, 2.651 ≤ Rep,max ≤ 41.757 for w/k = 4, 2.777 ≤ Rep,max ≤ 51.081 for w/k = 7, 2.633 ≤ Rep,max ≤ 47.880 for w/k = 9, and 1.939 ≤ Rep,max ≤ 47.821 for w/k = 19. The lower and upper limits of Rep,max correspond to particles with the lowest (τp,b = 0.239) and highest (τp,b = 19.152) response times, respectively. It can be seen that the values of Rep reflect the changes

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in the fluid and particulate velocities induced by the square roughness elements and they are consistent with the requirements of the point-particle approximation and the adoption of Eq. (9).

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63. Vijiapurapu, S., Cui, J., 2007. Simulation of turbulent flow in a ribbed pipe using large eddy simulation. Numerical Heat Transfer A 51, 1137–1165. 64. Vincent, J.H., Humphries, W., 1978. The collection of airborne dusts by bluff bodies. Chem. Engng Sci. 33, 1147–1155. 65. Wang, Q., Squires, K.D., 1996. Large eddy simulation of particle deposition in a vertical turbulent channel flow. Int. J. Multiphase Flow 22, 667–683. 66. Wang, Q., Squires, K.D., Chen, M., McLaughlin, J.B., 1997. On the role of the lift force in turbulence simulation of particle deposition. Int. J. Multiphase Flow 23, 749–763. 67. Wells, A.C., Chamberlain, A.C., 1967. Transport of small particles to vertical surfaces. Br. J. Appl. Phys. 18, 1793–1799. 68. Wood, N.B., 1981. A simple method for the calculation of turbulent deposition to smooth and rough surfaces. J. of Aerosol Science 12, 275–290. 69. Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, T., Tsuji, Y., 2001. Large eddy simulation of turbulent gas-particle flow in a vertical channel: effect of considering inter- particle collisions. J. Fluid Mech. 442, 303–334. 70. Zhang, H., Ahmadi, G., 2000. Aerosol particle transport and deposition in vertical and horizontal turbulent duct flows. J. Fluid Mech. 406, 55–80. 71. Zhao, B., Wu, J., 2006a. Modeling particle deposition from fully developed turbulent flow in ventilation duct. Atmospheric Environment 40, 457–466. 72. Zhao, B., Wu, J., 2006b. Modeling particle deposition onto rough walls in ventilation duct. Atmospheric Environment 40, 6918–6927.

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ACCEPTED MANUSCRIPT Table 1. Minimum and maximum values of the streamwise velocity fluctuations vx' of particles located within -0.971 ≤ y/h ≤ -0.912, and of the wall-normal velocity vy of particles located within 0 ≤ z/h ≤ 0.5. w/k=0

w/k=1

w/k=3

w/k=4

w/k=7

w/k=9

w/k=19

(τp+=τp,0+)

(τp+=1.033τp,0+)

(τp+=1.569τp,0+)

(τp+=2.142τp,0+)

(τp+=2.993τp,0+)

(τp+=2.999τp,0+)

(τp+=2.167τp,0+)

2.5

(-0.189, 0.365)

(-0.288, 0.350)

(-0.348, 0.376)

(-0.476, 0.355)

(-0.498, 0.543)

(-0.484, 0.442)

(-0.529, 0.476)

0.479

5

(-0.195, 0.375)

(-0.273, 0.369)

(-0.346, 0.387)

(-0.478, 0.350)

(-0.546, 0.495)

(-0.512, 0.372)

(-0.527, 0.439)

0.958

10

(-0.882, 0.388)

(-0.266, 0.384)

(-0.374, 0.378)

(-0.465, 0.333)

(-0.540, 0.423)

(-0.466, 0.360)

(-0.514, 0.369)

2.394

25

(-0.192, 0.427)

(-0.290, 0.400)

(-0.312, 0.349)

(-0.415, 0.325)

(-0.514, 0.427)

(-0.455, 0.328)

(-0.525, 0.320)

9.576

100

(-0.216, 0.446)

(-0.293, 0.399)

(-0.308, 0.366)

(-0.317, 0.306)

(-0.328, 0.338)

(-0.313, 0.289)

(-0.361, 0.281)

19.152

200

(-0.252, 0.440)

(-0.298, 0.348)

(-0.251, 0.318)

(-0.332, 0.277)

(-0,341, 0.275)

(-0.295, 0.293)

(-0.525, 0.238)

τp,b

τp,0+

0.239

(vy,min, vy,max)

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(vx',min, vx',max)

2.5

(-0.099, 0.108)

(-0.121, 0.136)

(-0.210, 0.210)

(-0.258, 0.233)

(-0.263, 0.446)

(-0.336, 0.396)

(-0.315, 0.438)

0.479

5

(-0.093, 0.104)

(-0.117, 0.134)

(-0.176, 0.213)

(-0.222, 0.221)

(-0.230, 0.335)

(-0.303, 0.348)

(-0.275, 0.371)

0.958

10

(-0.088, 0.097)

(-0.107, 0.123)

(-0.163, 0.201)

(-0.165, 0.198)

(-0.200, 0.278)

(-0.215, 0.312)

(-0.202, 0.309)

2.394

25

(-0.082, 0.083)

(-0.096, 0.105)

(-0.118, 0.164)

(-0.135, 0.162)

(-0.154, 0.218)

(-0.167, 0.214)

(-0.132, 0.261)

9.576

100

(-0.063, 0.057)

(-0.070, 0.072)

(-0.075, 0.088)

(-0.093, 0.098)

(-0.096, 0.127)

(-0.101, 0.126)

(-0.096, 0.135)

19.152

200

(-0.053, 0.042)

(-0.049, 0.048)

(-0.058, 0.069)

(-0.067, 0.076)

(-0.079, 0.098)

(-0.081, 0.100)

(-0.082, 0.101)

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0.239

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ACCEPTED MANUSCRIPT Table 2. The probability of particle number density sampled at particle locations with negative < streamwise fluid velocity fluctuations pPND , and the variation with respect to the w/k = 0 case of the

probabilities of the correlation between ejection events and the particle flux away the wall p(Q2,vy>0), and sweep events and the particle flux toward the wall p(Q4,vy<0). τp,b

τp,0+

w/k=0

w/k=1

w/k=3

w/k=4

w/k=7

w/k=9

w/k=19

(τp+=τp,0+)

(τp+=1.033τp,0+)

(τp+=1.569τp,0+)

(τp+=2.142τp,0+)

(τp+=2.993τp,0+)

(τp+=2.999τp,0+)

(τp+=2.167τp,0+)

2.5 5 10 25 100 200

0.663 0.682 0.684 0.701 0.719 0.706

0.609 0.625 0.653 0.630 0.608 0.593

0.437 0.461 0.475 0.545 0.569 0.539

0.370 0.383 0.421 0.482 0.525 0.495

0.403 0.380 0.379 0.462 0.506 0.512

0.382 0.365 0.388 0.468 0.486 0.493

0.313 0.286 0.281 0.352 0.436 0.470

-

-0.037 -0.018 0.016 -0.001 -0.058 -0.090

-0.105 -0.087 -0.056 -0.029 -0.056 -0.095

-0.112 -0.124 -0.077 -0.041 -0.046 -0.069

-0.200 -0.232 -0.189 -0.085 -0.080 -0.117

-0.159 -0.179 -0.144 -0.081 -0.129 -0.109

-0.316 -0.359 -0.290 -0.198 -0.176 -0.147

-

-0.001 -0.012 -0.040 -0.059 0.001 0.026

0.113 0.069 0.031 0.014 0.027 0.066

0.087 0.068 0.047 0.020 0.071 0.086

0.148 0.113 0.074 0.026 0.071 0.104

0.233 0.205 0.173 0.075 0.101 0.109

<

0.239 0.479 0.958 2.394 9.576 19.152

0.239 0.479 0.958 2.394 9.576 19.152

2.5 5 10 25 100 200

2.5 5 10 25 100 200

0.169 0.141 0.079 0.030 0.093 0.120

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0.239 0.479 0.958 2.394 9.576 19.152

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Γp(Q4, vy<0) - sweeps

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p PND

PT

Table 3. Particle response time and particle diameter normalized by the smallest resolved time τξ and length ξ scales computed near the rough wall at y/h = -0.85. τp,0+

τp,ξ = τp/τξ 2.5 5 10 25 100 200

w/k=1

w/k=3

w/k=4

w/k=7

w/k=9

w/k=19

0.867 1.734 3.467 8.667 34.672 69.344

0.931 1.862 3.724 9.310 37.244 74.487

1.265 2.531 5.061 12.652 50.611 101.223

1.460 2.921 5.841 14.602 58.414 116.829

1.766 3.533 7.066 17.665 70.665 141.330

1.813 3.627 7.254 18.133 72.538 145.075

1.648 3.296 6.592 16.480 65.926 131.852

0.123 0.174 0.246 0.389 0.777 1.099

0.127 0.180 0.255 0.403 0.805 1.139

0.148 0.210 0.297 0.469 0.939 1.328

0.159 0.226 0.319 0.504 1.009 1.427

0.175 0.248 0.351 0.555 1.110 1.569

0.178 0.251 0.355 0.562 1.124 1.590

0.169 0.240 0.339 0.536 1.072 1.516

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0.239 0.479 0.958 2.394 9.576 19.152

w/k=0

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τp,b

dp,ξ = dp/ξ 0.239 0.479 0.958 2.394 9.576 19.152

2.5 5 10 25 100 200

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2h

w

(flo di r ec. on)

B

z (Lz=πh) y

x (Lx=8h)

A

cavity C

square bar

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upper horizontal surface

inner ver. cal surface

lower horizontal surface

D E

outer ver. cal surface

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Fig. 1. Geometry and flow configuration.

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λ=w+k

x

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(c)

(d)

(e)

(f)

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(b)

Flow direc+on

(a)

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Fig. 2. Mean streamlines averaged in the spanwise direction and time for w/k = 1 (a), 3 (b), 4 (c), 7

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(d), 9 (e), and 19 (f).

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Fig. 3. Distribution of particle number density for various values of w/k and particle sets with τp,b =

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0.239 (a), 0.479 (b), 0.958 (c), 2.394 (d), 9.576 (e), and 19.152 (f).

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Fig. 4. Contours of the grid-scale fluid velocity fluctuations in the streamwise direction at the y/h = -0.969 plane (y0+ ≈ 5.3) and the instantaneous positions of particles at time tb = 33 in a region of -

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0.971 ≤ y/h ≤ -0.912 (5 ≤ y0+ ≤ 15) over the smooth wall (a) and over the rough wall with w/k = 1 (b), 3 (c), 4 (d), 7 (e), 9 (f), and 19 (g). Empty spaces indicate low-speed regions. Particle response time τp,b = 2.394 (τp,0+ = 25).

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Fig. 5. The probability density function of the particle number density in a region of of -0.971 ≤ y/h ≤ -0.912 (5 ≤ y0+ ≤ 15) as a function of the grid-scale fluid velocity fluctuations in the streamwise

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direction for w/k = 0 (a), 1 (b), 3 (c), 4 (d), 7 (e), 9 (f), and 19 (g). Particle response time τp,b = 2.394

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(τp,0+ = 25).

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Fig. 6. Isolines of the grid-scale fluid velocity fluctuations in the wall-normal direction u y ' at the

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z/h = 0 plane and the instantaneous positions of particles at time tb = 33 in a region of 0 ≤ z/h ≤ 0.5

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for a flat channel (a) and rough channels with w/k = 1 (b), 3 (c), and 7 (d). Solid and dashed lines denote u y ' ≥ 0 and u y ' < 0, respectively. Red color indicates particles with vy ≥ 0, while blue color

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corresponds to particles with vy < 0. Particle response time τp,b = 2.394 (τp,0+ = 25).

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Fig. 7. Probability of the correlation between quadrant events Qi and the wall-normal velocity vy of

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particles with τp,b = 2.394 (τp,0+ = 25) in a region of -0.971 ≤ y/h ≤ -0.912 (5 ≤ y0+ ≤ 15) as a function

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of w/k for particle fluxes toward (a) and away the wall (b).

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Fig. 8. Particle deposition coefficient kd as a function of particle response time for various values of the spacing between the square bars w/k, normalized by the wall friction velocity uτ,0 of the w/k = 0 case (a), and the actual wall friction velocity uτ (b).

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Fig. 9. Number of particles deposited at the rough wall in a time period of td = 66 as a function of

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the distance x/k for τp,b = 0.239 (a), 0.479 (b), 0.958 (c), 2.394 (d), 9.576 (e), and 19.152 (f).

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Fig. 10. The normalized number of particles τp,0+ = 2.394 (τp,0+ = 25) deposited at each surface Si of the rough wall in a time period of td = 66 as a function of the spacing between the square bars w/k. SV,ou: outer vertical surface, SV,in: inner vertical surface, SH,u: upper horizontal surface, SH,l: lower

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horizontal surface.

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Fig. 11. The normalized number of particles deposited at each surface Si of the rough wall in a time period of tb = 66 as a function of particle response times τp,b and τp,0+ for various values of the spacing between the square bars w/k. SV,ou: outer vertical surface (a), SV,in: inner vertical surface (b), and SH,l: lower horizontal surface (c).

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Fig. 12. Ratio of the particles deposited Np,d to those entering the cavity Np,en of the rough wall in a time period of td = 66 as a function of the spacing between the square bars w/k for various values of

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the particle response time τp,b.

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Fig. 13. Distributions of ( u x )x,z,t (a) and Ux,rms, Uy,rms (b) for square roughness elements with w/k = 7 predicted using various computational meshes. Lines: present LES results based, symbols: DNS results of Orlandi et al. (2006). All quantities are averaged over time and the streamwise and spanwise directions ( )x,z,t, and normalized by the bulk velocity ub.

55