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Numerical Simulation of Particle Deposition in Turbulent Duct Flows J. Yao,† M. Fairweather,‡ and Y. L. Zhao*,§ †

School of Energy Research, Xiamen University, Xiamen, 361005, People’s Republic of China Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, University of Leeds, Leeds LS2 9JT, U.K. § Department of Thermal Energy Engineering, College of Mechanical and Transportation Engineering, China University of Petroleum-Beijing, Beijing 102249, People’s Republic of China ‡

ABSTRACT: Particle deposition in fully developed turbulent square duct flows is simulated using large eddy simulation combined with Lagrangian particle tracking under conditions of one-way coupling, with the particle equation of motion solved with Stokes drag, lift, buoyancy, and gravitational force terms. The flow considered has bulk Re = 83 K, with three particle sizes 50, 100, 500 μm. Results obtained for the fluid phase show good agreement with the experimental data and the predictions of direct numerical simulations. The predictions for particles demonstrate that the turbulent-driven secondary flows within the duct plays an important role in the particle deposition process. Under the secondary flow effect, most particles tend to deposit close to the corners of the duct floor. It is shown that the flow particle size, drag force, shear-induced lift force, and gravity affect the particle deposition. The particle deposition velocity is found to increase with particle size, with the tendency for deposition at the duct corners increasing with the variable. From dynamic analysis, gravity most significantly affects particle deposition in the vertical direction, while drag force dominates particle deposition in the horizontal direction. The effect of the lift force becomes more significant when a particle is large or close to the duct wall. The lift force is also a contributing factor causing particles to accumulate at the corners of the duct.

1. INTRODUCTION Particle deposition in wall-bounded flows has been widely studied over the last 50 years, beginning with the pioneering work of Friedlander and Johnston.1 Since then, numerous investigations that have enhanced our understanding of particle mixing, dispersion, and deposition have appeared in the literature.2−10 Among those that investigated inhomogeneous turbulent flows, the circular pipe and the plane channel are the most widely studied wall-bounded flow geometries.11−14 Brooke et al.11 noted the possibility of particle deposition due to diffusive processes. Subsequently, they used their computations to demonstrate that an enhanced wall deposition rate could be attributed to the inertia-dependent transport of particles from the core of the channel to the near-wall region, where they tended to concentrate. Zhang and Ahmadi14 examined aerosol particle transport and deposition in a channel flow. In this work they argued the wall coherent structure plays an important role in the particle deposition process and, the shear velocity, density ratio, the shear-induced lift force, and the flow direction affect the particle deposition rate. In addition, it was demonstrated that particles deposit within low-speed streaks which suggests that such particles are neither uniformly nor randomly dispersed in the flow field. For horizontal channels, they found that gravity increases deposition by sedimentation on the lower wall. For vertical channels, with gravity in the flow direction, deposition rates were higher since, for large particles, the lift force is directed toward the walls, that is, such particles have settling velocities at least of the order of the fluid velocity. Wang and Squires13 investigated particle deposition in a channel flow using the dynamic Smagorinsky model, assuming one-way coupling between the flow field and the particles. Closest © 2014 American Chemical Society

agreement between large eddy simulation (LES) and direct numerical simulation (DNS) results was found for particles having the largest relaxation times since such particles are less likely to be influenced by the small scales of turbulence which are modeled, rather than simulated, in LES. Uijttewaal and Oliemans12 examined particle dispersion in vertical pipes of circular cross-section, using DNS and LES at friction Reynolds numbers of 360, 1000, and 2100 to compute particle deposition for relaxation times ranging from 5 to 104. The authors found that for small particles the deposition process is governed by the properties of the near-wall layer, where the wall-normal turbulence intensity is low, while for large inertial particles, turbulent dispersion determines the probability of particles colliding with the pipe wall. Despite the research noted above, further work is necessary to elucidate all the possible mechanisms by which particles deposit in inhomogeneous turbulent flows. This dictates a need for more detailed Lagrangian particle transport data than have been previously examined. Of particular interest in this work, due to its practical relevance to waste processing, are particulate flows in horizontal square ducts. A large amount of literature is devoted to the study of turbulent single-phase flows through square ducts, including experimental investigations,15−17 direct numerical simulations,18,19 and large eddy simulations.20 All of these studies have demonstrated that turbulence-driven secondary motions that arise in the plane of the duct cross-section act to transfer Received: Revised: Accepted: Published: 3329

August 21, 2013 December 9, 2013 January 30, 2014 January 30, 2014 dx.doi.org/10.1021/ie4027499 | Ind. Eng. Chem. Res. 2014, 53, 3329−3341

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Figure 1. Schematic of the computational domain.

fluid momentum from the center of the duct to its corners, thereby causing a bulging of the streamwise velocity contours toward the corners. They have also established that the Reynolds normal and shear stresses contribute equally to the production of mean streamwise vorticity. When compared with single-phase flows, there are only a few detailed numerical studies dealing with particle-laden turbulent flows in ducts. To date, three groups have, however, undertaken simulations in such geometries. Winkler et al.21 applied large eddy simulation, coupled with Lagrangian particle tracking, to study the preferential concentration of heavy particles in a duct flow, with their work focusing on particles with a low response time (St = 0.25−8). Sharma and Phares22 also used direct numerical simulation and Langrangian particle tracking to study secondary flow effects on particle transport and deposition in a square duct flow. However, in the latter work, the effects of gravity were neglected despite consideration of the behavior of large heavy particles in such flows. Both groups also focused on low Reynolds number turbulent flows (Reτ = 360 and 300, respectively, based on the mean friction velocity and the duct width). More recently, particle dispersion23 and resuspension24 mechanisms in a duct flow at a high Reynolds number (Reτ = 10 500) have been investigated by the applied large eddy simulation of the present authors. As a consequence, given this limited amount of research, the particle deposition mechanism in square duct flows, particularly at high Reynolds numbers and across the wide range of particle Stokes numbers of practical relevance, has not been fully elucidated. So far, by simulation, neither single phase nor particle-laden flow at bulk Re = 83 k has been found. In this work, LES coupled with a Langrangian particle tracking technique is used to study particle deposition in a fully developed turbulent square duct flow (Reynolds number, 83 k; particle sizes, 50− 500 μm; and associated Stokes numbers, 3.23−287). The turbulent flow field is predicted using a large eddy simulation, with a Lagrangian particle tracking approach used to solve the particle equation of motion which simulates particle trajectories as a result of the Stokes drag, lift, buoyancy, and gravitational forces acting on the particles. Predicted flow fields are

compared with available experimental data, with good agreement found. Results for particle-laden flows demonstrate that the secondary flow in the duct plays a significant role in the particle deposition process, with the particle size, drag force, shear-induced lift force, and gravity all influential. For deposition in both the vertical and horizontal directions, the results show that for all particle populations, particle deposition can be explained by a preferential deposition mechanism.

2. MATHEMATICAL MODEL 2.1. Flow Configuration. A schematic diagram of the duct geometry and coordinate system used is given in Figure 1. The flow considered was three-dimensional and described using a Cartesian coordinate system (x, y, z) in which the z axis was aligned with the streamwise direction, the x axis was in the direction normal to the floor of the duct, and the y axis was in the spanwise direction. The corresponding velocity components in the (x, y, z) directions are (u, v, w), respectively. In modeling this flow, the boundary conditions for the momentum equations were no-slip at the duct walls. The specification of inflow and outflow conditions at the open boundaries of the duct was avoided by assuming that the instantaneous flow field was periodic along the streamwise direction, with the pressure gradient that drives the flow adjusted dynamically to maintain a constant mass flux through the duct. The Navier−Stokes equations were solved numerically in a square domain of size h × h × 4πh in the x, y, and z directions, respectively, which in terms of wall units gives Lx+ = Ly+ = 3860, and Lz+ = 48506. The length of the duct was sufficiently long to accommodate the streamwise-elongated, near-wall structures present in wall-bounded shear flows, with such structures rarely expected to be longer than approximately 1000 wall units.25 The physical domain was discretized using between 4.5 × 105 grid points for the low Reynolds number simulation, and 5.9 × 105 grid points for the high Reynolds number case. All discretizations were uniform in the streamwise (z) direction, whereas in the vertical and spanwise directions (x and y, respectively) grid points were clustered toward the walls; in 3330

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uiu͠ j − u∼iu∼j , and hence Lij = Tij − τ̃ij = uiu͠ j − u∼iu∼j . This expression is known as Germano’s identity and involves only resolved quantities. To evaluate C, some form of relationship between the model constant values C and C2(∼) at the grid- and test-filter levels must be specified. Based on the hypothesis that the cutoff length falls inside the inertial subrange, the expression generally used is C2 = C2(∼) . Such a subrange is not, however, guaranteed to occur in wallbounded or low Reynolds number flows, with the largest deviations from universality of the subgrid scale motions occurring in the regions of weakest resolved strain. Values of the model parameter at different filter levels are therefore likely to differ, and to account for this28 the following is proposed:

particular, and for all the simulations, the point closest to the wall was placed at x+ or y+ = 0.37−6.39, with on average five nodes in the near-wall region (x+ or y+ < 10). Other simulations using an increased total number, and alternative distributions, of nonuniformly distributed nodes were also used to give better resolution near the walls of the duct, further details of which can be found in refs 23 and 24. These sensitivity studies did, however, demonstrate that the discretizations noted above resulted in turbulence statistics, in those regions of the duct of interest herein, that were independent of grid resolution. The flows investigated had bulk Reynolds numbers, Reb = wbh/ν, of 83 k, defined using the cross-stream area averaged streamwise velocity, with equivalent friction Reynolds numbers, Reτ = uτh/ν, of 3860. 2.2. Large Eddy Simulation. In LES only the large-scale parts of the velocity and scalar fields are computed and the effects of the subgrid scales are modeled. To achieve this, a spatial filter is applied to the equation of motion: the spatial filter of a function f = f(X, t) is defined as it convolution with a filter function, G, according to f ̅ (X , t ) =

∫Ω G(X − X′; λ(X ))f (X′, t ) dX′

⎛ ⎞ ε ⎟ C 2(∼ ) = C 2⎜1 + 2 a ⎝ 2 2 Δ∼ 2 || ∼s || || ∼s || ⎠

(4)

In eq 4, ε has the dimensions of dissipation and, assuming the flow to have only one length scale l and velocity scale ν, ε ≈ ν3/ l, with ν taken as the bulk velocity and l as the half-width of the square duct. Equation 4 assumes that the scale invariance of C can only be invoked if the cutoff falls inside an inertial subrange, and when this occurs the modeled dissipation should represent the entire dissipation in the flow. Conversely, in the high Reynolds number limit, the dissipation is only determined by ν and l, so that the ratio of ε to Δ∼2∥s∥̅ 3 measures how far the flow is from scale preserving conditions. This equation is a first-order expansion of other scale-dependent expressions for C, for example, Porte-Agel et al.,29 which also use a single length and velocity scale. Lij = Tij − τ̃ij = uiu͠ j − u∼iu∼j and eq 4, with contraction of both sides with the tensor ∼s , then give

(1)

where the filter function must be positive definite to maintain filtered values of scalars such as mass fraction within bound values and the integration is defined over the entire flow domain Ω. The filter function has a characteristic width of λ, which, in general, may vary with the position. Applying eq 1 to the Navier−Stokes equations for an incompressible Newtonian fluid with constant properties, under the hypotheses that filtering and differentiation in space commute, gives

a a [2 2 (C 2Δ)2 || s ̅ || || s ij̅ a ||∼ ∼sij − Lija∼sij ] * C = a 2 ε + 2 2 Δ∼ 2 || ∼s || || ∼s ||

Continuity:

2

∂uj̅ ∂xj

=0 (2)

(5)

where C2* is a provisional value for C2, that is, its value at the previous time step.27 Equation 4 gives a simple expression for C2 whose evaluation requires only minor modifications to the approximate localization procedure. The advantage of the method is that it is well conditioned and avoids the spiky and irregular behavior exhibited by some implementations of the dynamic model and, as the resolved strain tends to zero, C2 also tends to zero, while C2(∼) remains bounded. Equation 5 also yields smooth C2 fields with no need for averaging, and the maxima of C2 are of the same order of magnitude as Lilly’s30 estimate for the Smagorinsky model constant. The approach does not, however, prevent negative values of the model parameter, with such values set to zero to prevent instability. Negative values of the subgrid scale viscosity are similarly set to zero. Test-filtering was performed in all space directions, with no averaging of the computed model parameter field. The ratio Δ∼/Δ was set to 2, and the filter width determined from Δ = (ΔxΔyΔz)1/3. Computations were performed using the computer program BOFFIN. The code implements an implicit finite-volume incompressible flow solver using a collocated variable storage arrangement. Because of this arrangement, fourth-order pressure smoothing, based on the method,25 is applied to prevent oscillations in the pressure field. Time advancement is performed via an implicit Gear method for all transport terms, and the overall procedure is second-order accurate in space and

Momentum: ⎡ ∂ui̅ uj̅ ∂uj̅ ⎞⎤ ∂τij ∂ui̅ ∂ ⎢ ⎛⎜ ∂ui̅ 1 ∂p ̅ ⎟⎥ − + =− + ν⎜ + ∂xj ∂t ∂xj ρ ∂xi ∂xj ⎢⎣ ⎝ ∂xj ∂xi ⎟⎠⎥⎦ (3)

In eq 3, ρ is the constant fluid density, ui is the velocity component in the xi direction, p is the pressure, ν is the assumed constant kinematic viscosity, and τij = uiuj − ui̅ uj̅ represents the effect of the subgrid scale motions on the resolved scale motions. This term, known as the subgrid scale stress, must be modeled in order to solve the filtered equations. The dynamic subgrid scale stress model26 was used in this work, implemented using the approximate localization procedure27 together with the modification proposed by di Mare and Jones.28 This represents the subgrid scale stresses as the product of a subgrid scale viscosity, νsgs, and the resolved part of the strain tensor, with νsgs evaluated as the product of the filter length Δ times an appropriate velocity scale, taken to be Δ∥s∥. of the subgrid scale stresses is ̅ The anisotropic part a given by τaij = −2(CΔ)2∥s∥s ̅ i̅ j, where the model parameter C must be determined. In the dynamic model this is achieved by applying a second filtering operation, denoted by ∼, to eq 3. In the test filtered equation the subgrid scale stresses are: Tij = 3331

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Table 1. Parameters Relevant to the Simulations of Particle Deposition dp/μm 50 100 500

gravity/N

dp+

buoyancy/N

−9

−10

1.60 × 10 1.28 × 10−8 1.60 × 10−6

6.41 × 10 5.13 × 10−9 6.41 × 10−7

4.83 9.65 48.26

dt

=

⎛ ⎞ ρ 3 ρ CD (V − Vp)|V − Vp| + ⎜⎜1 − ⎟⎟g 4 ρp d p ρp ⎠ ⎝ + 1.615d pμ Res0.5c ls

[(V − Vp) × ω] |ω|

(6)

where V is the fluid velocity and ρ its density, Vp is the particle velocity, and ρp is its density, dp is the particle diameter, and g is gravity. The term ω = ∇ × V is associated with fluid rotation, Res = ρ·d2p|ω|/μ is the particle Reynolds number, and c = Fls/ Fls,Saff represents the ratio of the extended lift force to the Saffman force,36,37 which is given by ⎧ 0.0524(β Re )0.5 for Re > 40 p p ⎪ ⎪ c ls = ⎨(1 − 0.3314β 0.5) e−Rep /10 + 0.3314β 0.5 ⎪ ⎪ for Rep ≤ 40 ⎩

St(τp+) −4

3.45 × 10 1.33 × 10−3 3.07 × 10−2

3.23 12.38 287.26

effect by at least 2 orders of magnitude.42 Other forces, such as the hydrostatic force, Magnus effect, Basset history, and added mass forces, were similarly neglected due to their being orders of magnitude smaller than the effects considered.43 A fourth-order Runge−Kutta scheme was used to solve the equation of motion, given the initial particle location and velocity. Particles were initially distributed randomly throughout the duct, corresponding to an initially uniform wall-normal particle number density profile. The initial particle velocity was set equal to the fluid velocity, interpolated to the particle position. Particles were assumed to interact with turbulent eddies over a certain period of time, that being the lesser of the eddy lifetime and the transition time. For particles leaving the duct in the streamwise direction, periodic boundary conditions were used to reintroduce them into the computational domain. The total number of particles within that domain at any given time ranged from 3000 to 500 k, with the precise number employed being sufficient to ensure statistical independence of the results which was tested using the method.44 Particle and fluid densities were set to ρp = 2500 kg m−3 and ρ = 1000 kg m−3, with the kinematic viscosity of the fluid ν = 1.004 × 10−6 m2 s−1. The particle relaxation time is τp = ρpd2p/ (18 μf·f D), where f D = 1 + 0.15Rep0.687. The nondimensional particle response time is defined as the particle Stokes number, St = τpu2τ /ν, where the uτ is the shear velocity (defined as uτ = (τw/ρ)1/2, with τw being the wall shear stress). Three particle diameters were considered, namely dp = 50, 100, and 500 μm, with corresponding particle relaxation times, Stokes numbers, and other relevant parameters given in Table 1.

time. A constant time step was chosen, requiring the maximum Courant number to lie between 0.1 and 0.3, with this enforced for reasons of accuracy. Time-averaged flow field variables reported later were computed from running averages during the computations. The code has been applied previously to the LES of a wide range of flows including plane and round jets in a crossflow,31,32 flow over a swept fence,28 turbulent sprays,33 and particle flows in ducts.23,24 Further details of the code and the methods embodied within it may be found in refs 33 and 34. 2.3. Lagrangian Particle Tracking. Particle motion was modeled using a Lagrangian approach35 in which the particles are followed along their trajectories through the unsteady, nonuniform flow field. The analysis was simplified by assuming that the particle-laden flow was dilute, interactions between particles were negligible, the flow and particles were one-way coupled (i.e., the effect of particles on the fluid was neglected), all particles were rigid spheres with the same diameter and density, and particle-wall collisions were elastic. Under these assumptions, the motion of a spherical particle within a flow is governed by the force balance equation: dVp

τp (s)

3. RESULTS AND DISCUSSION 3.1. Flow Field Analysis. In Figures 2 and 4, solutions at Reb = 83k are compared with the experimental data of Brundrett and Baines.15 These results demonstrate that the present LES predictions are in good agreement with the results of other studies of duct flows. The comparison for the streamwise mean velocity shown in Figure 2, which gives velocities normalized by the bulk flow velocity along the lower wall bisector (y = h/2), therefore, shows good agreement between the simulations and the results of alternative study. In addition to the mean streamwise velocity, the turbulence-driven secondary flow and the turbulence statistics are important quantities by which the quality of the simulations can be assessed. The secondary velocity vectors and the contours of mean streamwise velocity in the cross-section (x−y) plane are shown in Figure 3. The secondary velocities (Figure 3a) convect mean-flow momentum from the central region to the corner region along the corner bisectors. This results in the bulging of the streamwise velocity contours toward the corners which can be seen in the Figure 3b. A fair degree of symmetry over the quadrants has been obtained. Figure.3a displays the secondary flow velocity vectors in the lower left quadrant, revealing the two streamwise, counter-rotating vortrices in each corner that characterize the flow. It is seen that the secondary flow moves from the core of the duct toward its corner. Close to the corner it turns rapidly with a small radius of curvature

(7)

In eq 7, β is given by β = 0.5Res/Rep (for 0.005 < β < 0.4). CD is the Stokes coefficient for drag, with CD = (1 + 0.15Re0.687 )·24/ p Rep as Rep ≤ 1000,38 where Rep is the particle Reynolds number and Rep = dp|V − Vp|/ν. The last term on the right-hand side of eq 6 is the slip-shear force, based on the analytical result36,37 and extended for higher particle Reynolds numbers according to Mei.39 Although many forces act on a particle, a number may be neglected without any appreciable loss of accuracy, depending on the particle inertia. The most important is the Stokes drag force, with gravity also significant, depending on the flow orientation. In this work, Stokes drag, gravity, buoyancy, and lift forces were considered. Because of particle−wall impaction, an electrostatic charge can be generated at the particle surface and the duct wall,40,41 although in this work the duct was horizontal so that any electrostatic force is much lower than the gravity 3332

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Figure 2. Profiles of mean streamwise velocity, normalized by the bulk velocity, along the lower wall bisector at different Reynolds numbers: (---) LES; (△) Brundrett and Baines 1964.

and flows along the walls. In the center of the side wall it takes a 90 degree turn toward the duct core. To examine the turbulence statistics, turbulence quantities along the lower wall bisector are plotted in Figure 4. The agreement with other results is again good. The present calculations, and the other results shown in this figure, also indicate that at this Reynolds number there is anisotropy in the turbulent normal stresses along the wall bisector, much as is found in channel flows. This above study confirms that the proposed simulation approach and the numerical solution method adopted faithfully captures the turbulent velocity field within duct flows, including the turbulence-driven secondary motions and turbulent statistics. Therefore, the extension of the simulations to include particles should produce reliable predictions for one-way coupled, particle-laden flows. 3.2. Particle Field Analysis. In studying particle deposition at the wall of a duct, if a particle is within 1.5 particle diameters of the wall then it is considered to be sufficiently close to the solid surface to have effectively deposited. This value is assumed in what follows unless otherwise stated, and has been used so that deposition information on bouncing, floating, and impinging particles is considered in the analysis, in addition to considering particles in contact with a wall. The particle deposition ratio, Nd/N, is then defined as the ratio of the number of deposited particles to the total number of particles. Deposition will be considered in two directions, namely the vertical, x, and horizontal, y, directions. In this flow it takes a short time (t+ < 20 000) for most (>90%) of the largest 500 μm particles to deposit on the floor of the duct, while for the 50 μm particles the deposition rate is still as low as 5% at t+ = 27 563. For this flow the three typical times chosen to study particle deposition, again based on the extent of deposition for the various particle sizes considered, were t+ = 12 326, 27 563, and 46 769. 3.2.1. Particle Deposition In The x-Direction. At the times noted in the previous section, the various deposition ratios for the particle sizes considered were, for the 50μm particles, 2%,

Figure 3. (a) Secondary velocity vectors and (b) streamwise mean velocity contours.

5%, and 12%; for the 100 μm particles, 11%, 30%, and 38%; and for the 500 μm particles, 86%, 99%, and 100%. Figure 5 shows the deposition results in the x-direction for the three sizes of particles in this flow. It is seen that the particle PDFs become more variable as the particle size decreases, particle accumulation near the side walls can be seen at later times for both the 100 μm (Figure 5b) and 500 μm (Figure 5c) particles, while the smallest particles (Figure 5a) have less tendency of accumulation near the side walls. In this flow, and from Table 1, the Stokes numbers of the 50, 100, and 500 μm particles are, respectively, 3.23, 12.38, and 287.26, so that the present results are generally in accord with the findings in ref 45, in which the observed particle deposition was in the duct corners for St ⩾ 30 and at the floor center for St ⩽ 15. The differences between the findings in ref 45 and those of the present study for the case of the 100 μm particles is most likely due to the neglect by the latter authors of the influence of gravity on particle deposition. In Figure 5, the 500 μm particles are seen to accumulate near the side walls at t+ = 12 326, Figure 5c. At large times, however, particle deposition in this flow shows less of a trend for the particles to accumulate at the center of the floor, with a more 3333

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even distribution across the floor apparent and accumulation near the side walls. This is most likely due to the flow properties (Figure 3) since the core of the secondary vortices around the lower corner bisector of the duct tends to approach the duct walls and corners. The increasing proximity of these vortices to those regions where deposition is occurring is therefore likely to have significant effect on the final deposited particle distribution, promoting their accumulation in the corners. Figure 6 shows the particle deposition velocity in the xdirection for this flow. It is seen that the velocity increases with particle size due to the effects of gravity. However, the magnitude of this velocity is evenly distributed across the floor of the duct, despite the preferential accumulation of deposited particles near the side walls in Figure 5b,c. Only for the largest 500 μm particles at t+ = 46 769 (Figure 6c) does the deposition velocity appear slightly higher near the side wall at y+ = 1930. Overall, therefore, this indicates that for this flow the tendency for the larger particles to deposit near the side walls is increased. In absolute terms, the particle deposition velocity in the xdirection looks higher near the side walls. This is likely due to the properties of the flow, from the core of the secondary vortices toward the wall and corner (see Figure 3); increasing turbulent mixing in the x−y plane which would result in higher instantaneous secondary velocities which would in turn enhance the particle deposition velocity. On the other hand, the tendency of particles to deposit near the side walls increases with particle size as noted earlier. As an illustration, at the end stage (t+ = 46 769), for the 500 μm particle (Figure 5c, Nd/N = 100%), the PDF fraction of particles deposited near the side walls is 0.12; for the 100 μm particle (Figure 5b, Nd/N = 38%) the PDF fraction is 0.04, and for the 50 μm particle (Figure 5a, Nd/N = 12%) the PDF fraction is 0−0.04. In terms of the particle deposition velocity, the maximum value for 500 μm particle (Figure 6c) is 0.25; for the 100 μm particle (Figure 6b) the maximum value is 0.09; for the 50 μm particle (Figure 6a) the maximum value is 0.08. In the x-direction, therefore, on the one hand particle deposition, Nd/N, increases with particle size, while on the other hand secondary flow effects lead to an increasing tendency for large particles to deposit near the side walls with a higher deposition velocity.

Figure 4. Profiles of (a) ⟨u′u′⟩ and (b) ⟨v′v′⟩ along the lower wall bisector at different Reynolds numbers: (---) LES; (△) Brundrett and Baines 1964.

Figure 5. PDFs of particle deposition in the x-direction: (a) 50 μm particles (from bottom to top Nd/N = 2%, 5%, 12%); (b) 100 μm particles (from bottom to top Nd/N = 11%, 30%, 38%); (c) 500 μm particles (from bottom to top Nd/N = 86%, 99%, 100%): t+ = () 12 326; (---) 27 563; (-·-·-·) 46 769. 3334

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Figure 6. Particle deposition velocity in the x-direction: (a) 50 μm; (b) 100 μm; (c) 500 μm: t+ = (△) 12 326; (△) 27 563; (○) 46 769.

Figure 7. PDFs of particle deposition in the y-direction: (a) 50 μm particles (from bottom to top Nd/N = 0.04%, 0.3%, 0.4%); (b) 100 μm particles (from bottom to top Nd/N = 0.4%, 2%, 4%); (c) 500 μm particles (from bottom to top Nd/N = 0.5%, 2.5%, 7%): t+ = () 12 326; (---) 27 563; (-·-·-) 46 769).

For this flow, it can be concluded that for deposition in both of the directions considered, the particle PDF location profiles become more variable with decreasing particle size, with peak values of the PDF and deposition velocities generally increasing with particle size. In addition, the medium and large sized particles again have a propensity to accumulate in the corners of the duct, with this tendency increasing with particle size, while the smallest particles have less tendency of accumulation near the duct corners. 3.3. Particle Deposition Mechanism. 3.3.1. Preferential Deposition Mechanism. To investigate the mechanism of particle deposition in the duct flows, particles located in the left lower quadrant of the ducts (x+ < 0, y+ < 0) were analyzed statistically. Figure 10 shows that the points concentrate near the line v+p / u+p = (h/2 − y+p )/(h/2 − x+p ), which is equivalent to (h/2 − y+p )/ v+p = (h/2 − x+p )/u+p or t+py = t+px. This indicates that particle deposition to the walls in both the x- and y-directions requires approximately the same time, suggesting that the particles are located around the corner bisector and tend to deposit in the corners of the duct. This tendency is related to the secondary flows in the duct cross-section, with the predominant effect of these motions being the induced transport of streamwise momentum toward the corner regions from the core of the duct. In this work, particle preferential deposition is therefore likely dominated by the secondary flow in the square duct. For

3.2.2. Particle Deposition in the y-Direction. Figure 7 shows particle deposition in the y-direction in this flow. At the three times considered, the particle deposition ratio is 0.04%, 0.3%, and 0.4% for the 50 μm particles; 0.4%, 2%, and 4% for the 100 μm particles; and 0.5%, 2.5%, and 7% for the 500 μm particles. In all cases, the particles are ultimately seen to be deposited near the bottom of the wall at x+ = −1930, with this trend becoming more dominant with increasing particle size due to the effects of gravity. In this flow the deposition PDFs of the 100 μm and 500 μm particles in both the x- and y-directions have high values toward, respectively, the side walls and the floor of the duct, again indicating that these particles preferentially deposit in the duct corners, although this trend is more dominant for the largest particles. These findings are confirmed by the results of Figure 8 which indicate an increasing tendency for particle deposition at the duct corners with increasing particle size. Figure 9 shows the particle deposition velocity in the ydirection. Again, this velocity increases with particle size due to the accompanying increase in the particle inertial force, and decreases with time for all particles. Most notably, for the largest 500 μm particles, Figure 9c, since the particles largely deposit near the bottom of the duct wall at x+ = −1930 (Figure 7c), the deposition velocity is centered at this location over all the times considered. 3335

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in Figure 11), the fluid moves from the center of the duct toward the corner, and close to the corner it turns rapidly with a small radius of curvature and then flows along the wall. In the center of the wall, the secondary stream turns 90° back toward the duct center. Thus, streamwise, counter-rotating vortices are formed in the corner of the duct. Under the secondary flow effect, around the corner bisector region, particles deposit toward the corner with the characteristics of “tpy = tpx” as described in Figure 10. Based on the above analysis, it is known that particle size affects particle deposition. It can be summarized in terms of particle responding time (dimensionless) which is shown in Table 1 for all cases. In this work, both 100 μm and 500 μm, St ranges from 12.38 to 287.26 and most particles deposit near the duct corners, while the smallest particle 50 μm (St = 3.23) has a slight tendency of accumulation near the duct corners. The analysis shows that higher-inertia particles (St > 12.38) tend to deposit efficiently close to the corners while lower-inertia particles (St < 3.23) has less tendency of accumulation near the duct corners. Phares and Sharma45 used DNS of a square duct flow at Reb = 4410 and studied particle deposition for St = 1− 50. They discovered that for the lower-inertia particles (St ≤ 15) deposition is more efficient closer to the center of the walls, whereas for higher-inertia particles (St ≥ 30) deposition is more efficient toward the corners. Winkler et al.21 used LES of a turbulent square flow at Reb = 5810 and studied particle deposition for St = 0.072−256.32, where they obtained that particle deposition is seen to be least likely in the duct corners but most likely in the duct center. The difference between the present work and the former works may lie on two reasons: first the gravity force is considered in the present work but never in Phares and Sharma;45 second the flow Reynolds number in the present work is significantly larger than that in former works. 3.3.2. Dynamic Analysis. To permit further analysis of the results, mean values (averaged over all particles) of the relative slip velocity, that is, the difference between the particle and local fluid velocity, and of the drag and lift forces acting on the particles, were derived. Table 2 shows the results for the slip velocities and forces in both the x- and y-directions, and at the times indicated. 3.3.2.1. Relative Slip Velocity. Table 2 shows some dynamic characteristics of particles in the flow. The relative slip velocity, defined as the difference between the particle and local fluid velocity. In Table 2, it is seen that, mean relative slip velocity mostly increases with particle size. It is reasonable because a small particle can follow the fluid motions and disperse well within the duct while, with increasing particle size, they respond slowly to the fluid motion. It is worthy of note that, in some cases, the above statement is not constant. For example, as large particles (500 μm) mostly deposit in the near-floor region with low velocity, the relative slip velocity in the x-direction is significantly lower than those of smaller particles. At the final stage (t+ = 46 769), the value of |u+ − up+| for large particle (500 μm) is equal to 4.82 × 10−3, which is much lower than that of particle (100 μm) 2.54 × 10−2. This inconsistent case happens because large particles most deposit in the near-wall region, while other smaller particles disperse well in the flow. For all cases regardless of the particle size as shown in Table 2, the slip velocity decreases with time because more particles deposit with time. As particles approach the floor, the slip velocity decreases with a decrease in the particle velocity and fluid velocity.

Figure 8. Instantaneous deposition of particles along the streamwise direction at t+ = 46 769: (a) 50 μm; (b) 100 μm; and (c) 500 μm particles.

particles arriving from regions further away from the wall, the secondary flow motions project particles to deposit toward the duct corner, as shown in Figure 10. Particle preferential deposition mechanism in a turbulent duct flow can be summarized in the schematic of Figure 11. Most particles are found to accumulate at the duct bottom corner. The factor determines the accumulation region depending on the time needed to deposit on both walls, which are determined by the local particle velocity and the distance from the wall. That is much related to the secondary flow. As stated above, if the time required to deposit on both walls is around the same, particles tend to preferentially deposit at the duct corner. In the square duct, such preferential deposit is basically caused by the specific flow pattern existing in the square duct, namely secondary flow. In the former section (Figure 3), the secondary flow has been well characterized in the square duct. For example, in the lower left quadrant (the dashed line shown 3336

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Figure 9. Particle deposition velocity in the y-direction: (a) 50 μm; (b) 100 μm; (c) 500 μm (t+ = (□) 12 326; (△) 27 563; (○) 46 769); (d) 500 μm (t+ = 27 563); (e) 500 μm (t+ = 46 769).

Figure 11. Preferential particle deposition mechanism in turbulent duct flows.

particle movement in that direction and toward the duct corners. Therefore, the mean relative slip velocity is a useful indicator as to the way in which the particles deposit in the duct flow. 3.3.2.2. Drag Force. From eq 6, it is seen that drag force is closely relevant to the relative slip velocity. On the basis of the mean relative slip velocity listed in Table 2, the drag force can be calculated via eq 6 and the results are listed in Table 2. To compare with gravitational force, all forces calculated are presented in dimensional format. For all cases listed in Table 2, drag force increases with particle size either at the x-direction or at the y-direction, which is regardless of the flow developing

Figure 10. Scatter plot of particle deposition velocity ratio in the region (x+ < 0, y+ < 0) versus particle−wall distance, t+ = 46 769: (●) 500μm, (□) 100 μm, (▲) 50μm.

Comparing the slip velocity in the x- and y-direction for the flows, some significant differences are apparent. For example, at the second and third stages (t+ = 27 563, 46 769), the |u+ − up+| value for the 500 μm particle is significantly less than the corresponding |v+ − vp+| value. Hence particle motion in the ydirection appears to be dominant, which is in accord with 3337

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Table 2. Mean relative slip velocities, and drag and lift forces acting on the particles, in the duct flows mean relative slip velocity time/t+ 12 326

27 563

46 769

dp/μm 50 100 500 50 100 500 50 100 500

+

|u − 1.94 3.54 4.01 1.42 2.93 8.32 1.51 2.54 4.82

× × × × × × × × ×

up+| −2

10 10−2 10−2 10−2 10−2 10−3 10−3 10−2 10−3

+

|v − 1.61 × 2.41 × 0.102 7.53 × 2.22 × 5.34 × 1.32 × 1.46 × 2.87 ×

vp+| −2

10 10−2 10−3 10−2 10−2 10−3 10−2 10−2

drag force/N

lift force/N

|F̅Dx| 9.12 3.12 2.12 6.42 3.13 3.86 5.15 2.66 2.32

× × × × × × × × ×

|F̅Dy| −10

10 10−9 10−8 10−10 10−9 10−9 10−11 10−9 10−9

7.15 2.54 8.02 3.72 2.56 3.54 5.43 1.43 1.78

× × × × × × × × ×

|F̅Lx| −10

10 10−9 10−8 10−10 10−9 10−8 10−11 10−9 10−8

2.43 3.43 1.05 3.21 3.43 8.04 1.76 3.22 4.98

× × × × × × × × ×

|F̅Ly| −10

10 10−9 10−7 10−10 10−9 10−9 10−14 10−9 10−9

1.01 1.22 7.05 5.12 6.45 3.77 0.0 1.56 2.07

× × × × × ×

10−10 10−9 10−7 10−11 10−10 10−7

× 10−9 × 10−7

Figure 12. Lift force effect on particle deposition in the x direction: (a) 5, (b) 50, (c) 100, and (d) 500 μm particles [() no lift and (----) lift].

stage. Such a conclusion is different from that for relative slip velocity since, for drag force, the particle size and drag coefficient are the other two important factors affecting its magnitude. With increasing time, the drag force from the flow acting on a particle generally decreases, mainly because of the

accompanying decease in the relative slip velocity, as noted in the previous section, caused by the gradual deposition of the particles into near-wall regions. In the x-direction, the gravity force also acts on the particles. From Tables 1 and 2, the drag force (9.12 × 10−10 N at t+ = 12 3338

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326; 6.42 × 10 N at t+ = 27 563; 5.15 × 10−11 N at t+ = 46 769) acting on the 50 μm particle is lower than the gravity force (1.60 × 10−9 N) at all times, indicating that in this direction particle motion is governed by the gravity. For the 100 μm particle, the drag force (3.12 × 10−9 N at t+ = 12 326; 3.13 × 10−9 N at t+ = 27 563; 2.66 × 10−9 N at t+ = 46 769) is less than the gravity force (1.28 × 10−8 N) by 1 order of magnitude, indicating that the particle motion is governed by gravity also. For the 500 μm particles, the drag force (2.12 × 10−8 N at t+ = 12 326; 3.86 × 10−9 N at t+ = 27 563; 2.32 × 10−9 N at t+ = 46 769) is significantly less than the gravity force (1.60 × 10−6 N), by at least 2 orders of magnitude, therefore the motion of these particles is dominated by the effects of gravity which rapidly cause deposition. Therefore, in the x-direction the drag force is always lower than that of gravity by between 1 and 3 orders of magnitude. It suggests that in the flow the gravitational force plays a significant role in determining particle deposition in the x-direction for all sizes of particles considered. This finding is similar to that in ref 14 for the aerosol particle transport and deposition in horizontal turbulent channel flows. Comparing the drag force in the x- and y-direction, it is seen that for most cases listed in Table 2, the two drag forces are at a similar level which is related to the fact that the secondary flow is symmetrically distributed around the corner bisector in the x−y plane. However, in some cases the drag force in the ydirection is significantly higher than that in the x-direction. As an example, for a large particle 500 μm at the second and final stages, the drag force in the y-direction is higher than that in the x-direction by a magnitude of 1 order. This means that at that moment the drag force from the fluid acts on particles stronger in the y-direction than that in the x-direction. This indicates a tendency for the particles to move in the y-direction, in line with their accumulation in the duct corners. From this dynamic analysis, it can therefore be concluded that in the x-direction gravity controls particle deposition. In the y-direction, the drag force is of the same magnitude as that in the x-direction. However, in the near-wall region, the drag force in the y-direction becomes so high that it dominates the particles’ deposition process and dictates their final accumulation in the duct corners. 3.3.2.3. Lift Force. Turning to a consideration of the influence of the lift force, Figure 12 presents profiles of the particle deposition rate in the vertical direction determined both with and without the lift force accounted. In the figure the lower half of the duct is separated equally into five regions in the x direction, with the deposition rate profiled in each. It is seen that the lift force has only a slight effect on the smaller particles, that is, the 5 and 50 μm particles shown in Figure 12 panels a and b, while a more significant influence is evident for the larger 100 and 500 μm particles considered in Figure 12 panels c and d. This could be found in Table 2 as well. Table 2 presents the mean lift force based on all particles presented in the computation system in the x- and y- directions, where the mean lift force (absolute value) was calculated for all particles considered in the whole duct. This finding is reasonable since the lift force arises due to particle inertia and so is most important for large particles. This finding is in line with the conclusions of Zhang and Ahmadi14 who considered aerosol particle transport and deposition in vertical and horizontal turbulent duct flows. In addition, Figure 12 shows that the effect of the lift force increases with decreasing height in the duct, with this being particularly evident in the results of Figure 12 panels c and d. This is likely due to the occurrence of larger

velocity gradients as the duct floor is approached, which gives rise to greater differences between the particle and fluid velocities, which in turn results in significant shear-induced lift forces. This observation is in agreement with earlier findings where the lift force was noted to be especially important for particles in the diffusion impaction region close to a wall. Therefore, it can be concluded that the lift force acting on the particles increases with the particle size, with its effect increasing as the floor of the duct is approached. A look at the x- and y-directions in Table 2shows that there is a variation of lift force responses to that of mean relative slip velocity at both directions. For example, at the final stage of particle 500 μm deposition, the mean relative slip velocity in the y-direction (|v+ − vp+| = 2.87 × 10−2) is higher than that in the x-direction (|u+ − up+| = 4.82 × 10−3). Correspondingly, the lift force in y-direction (2.07 × 10−7) is higher than that in the x-direction (4.98 × 10−9). Such a rule is consistent for all cases. This finding is reasonable because the velocity gradient, that is, the differences between the particle and local fluid velocities, directly affects the shear-induced lift force. If there is an increase in the velocity gradient such that the differences between the particle and fluid velocities are greater, a significant shear-induced lift force results. A comparison of the lift force in the x-direction and ydirection shows, for most cases, that they are at a similar level. However, in some cases, lift force in the y-direction is significantly higher than that in the x-direction. As an example, for particle 500 μm at the second stage, the lift force in the ydirection (3.77 × 10−7 N) is significantly larger than that in the x-direction (8.04 × 10−9 N) by a magnitude of 2 orders. A similar difference can be found for the same particle at the third stage. This result is in line with the fact that such particles accumulate in the duct corners. At both stages, there are particles with high PDFs (shown in Figure 7) near the side wall due to the secondary flow effect; at the same time these particles deposit toward the floor under a gravity effect. In this case, the shear-induced lift force generated tends to promote particle deposit on the side wall. This is most clearly evident from accumulation at the corner and for the large particles (Figure 8c). Such an explanation is in agreement with those of earlier studies14,46 which demonstrated that gravity affects the deposition rate in vertical ducts via the shear-induced lift force. This force was therefore found to enhance the particle deposition rate when gravity acted in the same direction as the flow, and reduced the deposition rate when gravity opposed the flow. It should be noted that lift forces in both cases (particle 500 μm at second and third stages) are significantly higher than drag forces by a magnitude of one order. Therefore, it is reasonable to believe that, lift force may be another important factor causing particles to accumulate at the corner.

4. CONCLUSIONS The occurrence of turbulence-driven secondary flows within straight, square duct flows plays an important role in the particle deposition process. For particle deposition in the vertical direction, most particles independent of particle size tend to accumulate in the duct corners. In the horizontal direction, particle deposition increases with particle size. Generally, in both the vertical and horizontal directions, the deposition profile, in terms of the PDF of deposited particle locations, is more variable for small particles in comparison 3339

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ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (No. 51376153); Science Foundation of China University of Petroleum, Beijing (No. 2462013YJRC030) and EPSRC (No. EP/C549465/1) of the United Kingdom. The authors would also like to express their gratitude to Prof W.P. Jones for providing the BOFFIN LES code and for many helpful discussions on its use.

with larger ones. The particle deposition velocity also increases with particle size. The preferential deposition of particles in turbulent duct flows has been found to be basically caused by the specific flow pattern that exists in the square duct, namely the secondary flow. One place was found where most particles, independent of particle size, accumulate, the duct bottom corner. The factor determines the accumulation region depending on the time needed to deposit on both walls, which is dictated by the local particle velocity and the distance from the wall. The drag force acting on a particle was found to increase with particle size in both the vertical and horizontal directions. In both directions, drag forces are at a similar level to each other, which is relevant to the fact of secondary flow symmetrically distributed around the corner bisector in the x−y plane. However, in the near-wall region, the drag force in the horizontal direction was found to be the dominant factor. In addition, in the vertical direction, gravity dominates particle deposition. As large particles 500 μm deposit near the wall region, the drag force in the vertical direction dominates the particles deposition location. The lift force increases with particle size and becomes significant as a particle approaches the floor. The variation of lift force may be related to that of particle mean relative slip velocity at both directions. In addition, gravity affects the deposition rate in the horizontal direction via the shear-induced lift force. Hence, lift force is another important factor causing the accumulation of particles in the duct corners. In this work, because of computation based on a large amount of particles, some assumptions have been made, for example, no particle−wall interaction, dilute particle-laden flow, and one-way coupling. Although particle deposition in the duct is demonstrated by the main flow pattern wholly occupied in the duct, secondary flow, that is, accumulation at the duct corners, and the above simplifications may cause a little difference between the current study and real status. For example, as particle deposition concentrates at the near-wall region, dense particle distribution may cause local flow turbulence modulation by the presence of wakes behind the particles.47 The mechanism of particle transport in the wakes mostly depends on the interactions between vortex structures,35 which induces more complicated particle behaviors in the nearwall region, such as rolling, resuspension, mixing, and so on. Therefore some work for future improvement is suggested in the following: First, apply a physical particle−wall interaction model to consider particle−wall collisions. Second, in the nearwall region, particle−particle interaction will be considered. Third, in the near-wall region, a high density of particles affects local flow turbulence, two-way coupling is then suggested to clarify the problem. In addition, further work will be carried out to compare particle deposition in turbulent duct flows at various Reynolds number including considering particle−particle interaction and particle−flow two-way coupling.

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NOMENCLATURE CD = Stokes coefficient, dimensionless cls = ratio of the extended lift force to the Saffman force, dimensionless dp = particle diameter, m f = a spatial filter function, dimensionless Fls = extended lift force, N FDx, FDy = drag force components in (x, y) directions, N FLx, FLy = lift force components in (x, y) directions, N Fls,Saff = Saffman force, N g = gravity, N kg‑1 G = a filter function, dimensionless h = width of square duct, m Nd = number of deposited particles, dimensionless N = total number of particles, dimensionless p = pressure, N/m2 Re = flow Reynolds number, dimensionless Reb = Reynolds number based on flow bulk velocity, dimensionless Rep = particle Reynolds number, dimensionless Res = particle Reynolds number of the shear flow, dimensionless Reτ = Reynolds number based on flow friction velocity, dimensionless St = particle Stokes number, dimensionless t = time, s t+ = time in wall units, dimensionless t+px= deposition time (in wall units) to the wall in the xdirection, dimensionless t+py = deposition time (in wall units) to the wall in the ydirection, dimensionless u, v, w = velocity components in (x, y, z) directions, m s‑1 u+, v+, w+ = flow velocity components in (x+,y+,z+) directions, dimensionless up, vp, wp = particle velocity components in (x, y, z) directions, m s‑1 up+,νp+,wp+ = particle velocity components in (x+,y+,z+) directions, dimensionless uτ = shear velocity, m s‑1 V = fluid velocity (u,v,w), m s‑1 Vp = particle velocity, m s‑1 wb = bulk flow velocity in streamwise direction, m s‑1 x, y, z = Cartesian coordinate system, m x+, y+, z+ = Cartesian coordinate system in wall units, dimensionless

Greek Letters

ρ = fluid density, kg m−3 ρp = particle density, kg m−3 τp = particle relaxation time, s τw = wall shear stress, N m−2 ν = kinematic viscosity, m2 s‑1 ω = fluid rotation, dimensionless τij = subgrid scale stress, dimensionless

AUTHOR INFORMATION

Corresponding Author

*Tel. +86 5925952782. Fax: +86 5922188053. E-mail: yaojun@ xmu.edu.cn. Notes

The authors declare no competing financial interest. 3340

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νsgs = subgrid scale viscosity, dimensionless Ω = entire flow domain, dimensionless λ = a characteristic width of filter function, dimensionless

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