The effects of near wall corrections to hydrodynamic forces on particle deposition and transport in vertical turbulent boundary layers

Accepted Manuscript

The effects of near wall corrections to hydrodynamic forces on particle deposition and transport in vertical turbulent boundary layers C. Jin, I. Potts, M.W. Reeks PII: DOI: Reference:

S0301-9322(15)00229-3 10.1016/j.ijmultiphaseflow.2015.10.007 IJMF 2297

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

29 November 2013 5 October 2015 21 October 2015

Please cite this article as: C. Jin, I. Potts, M.W. Reeks, The effects of near wall corrections to hydrodynamic forces on particle deposition and transport in vertical turbulent boundary layers, International Journal of Multiphase Flow (2015), doi: 10.1016/j.ijmultiphaseflow.2015.10.007

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Highlights • We study effects of near wall forces on particles in a turbulent boundary

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layer.

• Drag effects reduce small particle deposition, increases large particle deposition.

• Inclusion of lift has a small effect on particle deposition.

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• Corrected drag gives big differences between particle and flow stream-

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wise velocities.

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The effects of near wall corrections to hydrodynamic forces on particle deposition and transport in vertical turbulent boundary layers C. Jin, I. Potts, M. W. Reeks∗

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School of Mechanical & Systems Engineering, Newcastle University, Stephenson Building, Claremont Road, Newcastle upon Tyne, NE1 7RU, UK

Abstract

In a previous paper[1] we presented a model for the transport and deposition of particles in a turbulent boundary layer which involved tracking individual particles via their interactions with a succession of random eddies found

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in each quadrant of the fluid Reynolds shear stress domain. Using drag forces based on unbounded flow, the model in common with other stochastic

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models significantly underestimated the deposition rates for small particles with particle relaxation times (in wall units) St < 5. Here, using the same

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model we consider the influence on the transport and deposition of corrections to the drag and lift force in the near wall region utilizing the composite

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correlations proposed by Zeng et al. [2] for a rigid spherical particle. As far as deposition rates are concerned, the inclusion of near wall effects

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and lift forces are unable to explain the under-estimate in deposition for low inertial particles compared to the benchmark experimental measurements of Liu and Agarwal [3]. More specifically, the overall effect of near-wall ∗

Email:[email protected]

Preprint submitted to International Journal of Multiphase Flow

October 28, 2015

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corrections to the drag is found to decrease deposition rates for small particles and to increase them for large particles. The inclusion of a Saffman lift

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force into the particle equation of motion (for a uniform bounded shear flow) reduced the deposition rates of the particles considered but the change was not significant (unlike previous predictions). The inclusion of a lift force

based on Zeng et al. [2]’s formula which accounts for both wall and Reynolds effects was little different to that based on Saffman’s formula. However in

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both cases the change in deposition rates is relatively small.

In contrast, the corrected drag does yield significant differences between the particle and carrier flow streamwise velocities for large particles ( St ∼ 20) in the near wall region.

Given all the results obtained, whether the wall effects on the hydrody-

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namic forces should be included or not depends on specific applications.

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Keywords: Wall effects, drag coefficient, lift coefficient, particle deposition 1. Introduction

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In a previous paper[1] we presented a novel stochastic quadrant model for the transport and deposition of particles in a turbulent boundary layer

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which involved tracking individual particles via their interactions with a succession of random eddies found in each quadrant of the fluid Reynolds shear

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stress domain. The detailed statistics of each quadrant were based on a quadrant analysis of the wall-normal fluid velocity fluctuations obtained by an LES of a fully developed channel flow and consistent with the skewness of wall-normal fluid velocity fluctuations observed in fully turbulent developed boundary layers. More importantly the model captured the influence 3

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of sweeps and ejections in the near wall region of the boundary layer on the particle deposition process. Whilst the model yielded very good predictions

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of the deposition rate for particle relaxation times (in wall units) St > 5 when compared with benchmark experimental measurements, it significantly

under-estimated the deposition for small particles St < 5. This was a feature

of several other models and suggested that there may be other processes that were ignored or inadequately represented in the model that were contribut-

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ing to the deposition e.g. Brownian motion or that the aerodynamic forces

encountered in the boundary layer were under-predicted. In our model as in other models only the drag forces was accounted for being based on empirical drag coefficients for spherical particles in unbounded flows. No account was taken of the influence of lift forces and the near wall corrections to both

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drag and lift as the particles approached the wall and finally deposited. The neglect of either might explain the under-predicted values of the deposition

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rate for the small particles.

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Numerous expressions for the drag coefficient CD that account for the effect of particle Reynolds number Rep exist (see Lin and Lin [4]). The two

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most frequently used forms are from Schiller and Naumann [5] and Morsi and Alexander [6], respectively. The drag coefficient from Schiller and Naumann

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[5] can be written as CD =


24 1 + 0.15Re0.687 . p Rep

(1)

The above expression reverts to the Stokes relation CD = 24/Rep when

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Rep  1. Morsi and Alexander [6] proposed the following expression c2 c3 + . Rep Re2p

(2)

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C D = c1 +

where c1 , c2 , c3 are known constants. Figure 1 shows that the expression

from Morsi and Alexander [6] exhibits the correct asymptotic behavior at both low and high values of Rep . Compared to the standard experimental

drag-Reynolds number relationship for rigid spherical particles, the above

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two expressions shows no discernible discrepancy.

Experiments Stokes Schiller & Naumann Morsi & Alexander

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102

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100

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CD

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101

10-1

100

101

102

Re10 p

3

104

105

106

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10-1

Figure 1: Drag Coefficient CD as a function of the particle Reynolds number Rep for a

rigid spherical particle

The standard drag correlations from Schiller and Naumann [5] have been 5

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used in numerous studies involving DNS (e.g. [7, 8]) or LES (e.g. [9, 10]) to solve the flow coupled with a Lagrangian particle tracking method to study

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particle motion in a turbulent boundary layer. Chen and McLaughlin [11] first considered the wall effect on the Stokes drag to study particle deposition.

However, the nonlinear drag law was not employed in the work of Chen and McLaughlin [11] as they regarded Rep << 1, which is not necessarily the case for depositing particles (see [12, 13, 14]).

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The lift force can play a part in the process of particle deposition but is one of the most difficult forces to model. The Saffman lift shear force for a rigid spherical particle in an unbounded uniform shear flow, which is the

(3)

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most frequently studied shear-induced lift force, takes the form 1/2  2 1/2 dux fl = 1.615dp (ρµ) (ux − vx ), dy

where ρ and µ denote the fluid density and dynamic viscosity respectively, dp

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is the particle diameter, vx and ux are the particle and fluid velocities respectively at the particle centroid in the x − direction respectively and dux /dy is the gradient of fluid streamwise velocity or the shear rate of the mean flow.

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Saffman assumed that the Reynolds numbers, Rep and ReG  1, and that Rep  ReG , where Rep = |ux − vx |dp /ν and ReG = Gd2p /ν; G denotes the

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fluid velocity gradient at the particle centroid and ν the fluid kinematic viscosity. According to Eq: 3, the relative velocity (ux −vx ) between the particle

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and surrounding fluid and the shear rate determine the character of fl that is appropriate for the lift force acting on a rigid spherical particle moving in an unbounded, unidirectional and steady linear shear flow. In addition, it can be inferred that the shear-induced lift force depends strongly on the instantaneous gradient of the fluid streamwise velocity at the particle location. 6

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Kallio and Reeks [12] employed Saffman’s relation [15, 16] to account for the lift force exerted on a rigid spherical particle moving within a fully devel-

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oped turbulent boundary layer. They observed an increase in the deposition rate of heavy particles resulting from the Saffman lift force. McLaughlin [13] included the same form derived by Saffman for the lift force in the par-

ticle equation of motion to study aerosol particle deposition in a turbulent channel flow, in which the incompressible three-dimensional Navier-Stokes

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equations were solved through DNS calculations. McLaughlin [13] found

that the Saffman lift force plays a significant role both in the inertial deposition of particles and in the accumulation of trapped particles within the viscous sublayer where the gradient of the fluid streamwise velocity has the maximum value across the channel. McLaughlin [17] and Mei [18] extended

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Saffman’s expression [15] to the situation when Rep is comparable with or √ larger than ReG . McLaughlin [19], Cherukat and McLaughlin [20] further

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developed expressions for the lift force exerted on a particle in a wall-bounded linear shear flow and for the wall-induced lift force when a particle lies in a

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linear shear flow field near a flat wall. Based on the above research, Chen and McLaughlin [11] considered wall-induced and shear-induced lift in the

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particle equation of motion to study particle deposition coupled with a DNS channel flow.

There have been a number of studies devoted to calculating the influence

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of wall effects on the drag and lift forces. Wang et al. [21] developed an optimum form that combines both the shear-induced and wall-induced parts of the lift force on a particle in a wall-bounded shear flow. Lataste et al. [22] studied the importance of the shear-induced lift force on a particle in a

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turbulent boundary layer. They found that the shear-induced lift force plays a significant role in the near wall region. Furthermore, they observed that

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the formulation of Cherukat and McLaughlin [20] for the lift force gives the best results when compared with experiments. In the DNS calculations of particle-laden channel flows by Arcen et al. [23], the drag force was corrected for the presence of a wall according to the direction of motion of the particle,

i.e. a particle moving parallel or perpendicular to the wall; the lift force was

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the form derived by Cherukat and McLaughlin [20]. They found that even with the inclusion of the most accurate treatment of lift and drag forces due to wall effects there are no significant changes in the statistical properties of the dispersed particle phase, except for the high inertia particles. Using DNS, Bagchi and Balachandar [24] studied the effect of turbulence

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on the drag and lift force acting on a rigid spherical particle suspended in a free stream isotropic turbulent flow. They observed that the standard drag

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correlation from Schiller and Naumann [5] based on the instantaneous or mean relative velocity gives a reasonably accurate prediction of the mean

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drag compared to their DNS calculations. This indicates that the standard drag correlation is applicable to turbulent dispersed particle flows. They also

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demonstrated that the mean drag is insensitive to the fluid velocity measured at the particle center, or acquired by averaging over a fluid volume of the order of the particle size. This confirms that the point particle approach is

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an acceptable approximation for small particles. The objective of this paper is to evaluate the effects of near wall correc-

tions of hydrodynamic forces on particle deposition and transport in a vertical fully developed turbulent boundary layer (i.e.in the absence of gravity).

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The model for the carrier flow in the turbulent boundary layer responsible for the transport of the particles is our stochastic quadrant model which al-

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though being a simple one-dimensional model for the wall normal velocity fluctuations, is based on the statistics of the velocity fluctuation in both the wall normal and streamwise directions. The reasons for using such a model

are two-fold. Firstly computational times for calculating particle deposition

rates are much shorter than using DNS. This is especially noticeable in the

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case of deposition rates for very small particles which ˜ 10−4 of those for

high inertial particles, thus requiring significantly larger numbers particles to achieve the same accuracy. In fact because of the huge computational times involved, accurate DNS predictions for very low values of Stokes number do not exist. See Figure 11 in Soldati and Marchioli [25]. So this model,

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given that it embodies in a simple way the underlying statistics of the turbulent boundary layer appropriate for particle transport, is better able to

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calculate the effect of near wall corrections to the aerodynamic forces acting on the particle. Secondly unlike DNS, this simple stochastic model can

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be used as a part of a RANS formulation for considering particle transport and deposition in a range of complex flows and particle sizes. In view of

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the versatility and potential application of this simple model, it is therefore important to examine how much near wall corrections to the aerodynamic forces acting on the particles have on the transport and particle deposition

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predicted by this simple model. Of course it’s important to compare it with the results of similar DNS calculations recognising the limitations of using DNS for measuring the deposition of very small particles. We do not believe however that the overall conclusions of this study depend critically on the

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stochastic model we have used for the particle transport. In carrying out this assessment of the influence of near wall corrections

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to the aerodynamic forces, we use the recently obtained correlations for CD and CL proposed by Zeng et al. [2] and Lee and Balachandar [26], who used

DNS with the immersed boundary method (IBM) to fully resolve the flow field around a fixed or moving rigid spherical particle in a wall-bounded shear flow. In order to judge the influence of near wall corrections, three

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different cases of simulations are performed. The first case is concerned with the standard drag law CD versus the near wall corrected CD of Zeng et al.

[2]; the second case focuses on the inclusion of the Saffman lift force with the standard drag law CD in the particle equation of motion; the final case considers the effect of the wall corrected CL compared to that based on the

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standard drag law on particle deposition rates, mean streamwise particle velocities and wall-normal r.m.s. velocities.

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The paper is structured as follows. Section 2 contains a brief description of the stochastic quadrant model that forms the basis of our analysis. Section

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3 contains the detailed description of the near wall corrections we apply to the drag force and the addition of a lift force. Section 4 is devoted to the

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modelling methodology and finally the remaining sections 5 and 6 are devoted to the results of our calculation and the conclusions we can draw from them.

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2. Stochastic model for particle transport in a fully developed turbulent boundary layer For completeness we present here a brief description of the turbulent

boundary layer particle transport model used in this study. A detailed de10

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scription of the model and its predictions for particle transport and deposition are given in Jin et al. [1]. Its principle feature is the use of a stochastic quad-

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rant model for the wall normal fluid velocity fluctuations which accounts for the ejections and sweeps in the boundary layer through the skewness of the

distribution in each of the 4 quadrants of the Reynolds stress domain. So we begin with the formulation of the quadrant model and the statistics of the

fluid velocity fluctuations that are generated. Then we show how this model

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is implemented within the Lagrangian particle tracking model. 2.1. Formulation of the stochastic quadrant model

The wall normal fluctuating velocity component v 0 can be distinguished as positive or negative according to whether the momentum flux is away from

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0 be a function defined as or towards the wall. Thus let v+

(4)

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  v 0 if v 0 > 0, 0 v+ =  0 if v 0 ≤ 0

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0 and v− defined as

  v 0 if v 0 < 0, 0 v− =  0 if v 0 ≥ 0.

(5)

0 RT 0 0 0 dt It is possible to define the average value of v+ and v− as: v+ = T1+ 0 v+

0 R T 0 and v− = T1− 0 v− dt, where T is the interval of observation time contain-

0 0 ing the fraction of v+ denoted by T+ and the fraction of v− denoted by T− .  

0 0 R T 0 0 We then have v+ + v− = T1+ 0 v+ + TT−+ v− dt. Accordingly,

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1 T

Z

T

0

Thus if T+ < T− ,


0 0 v+ + v− dt = 0.

if T+ > T− ,

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0  0  v+ > v− ,

(6)

(7)

0  0  v+ < v− .

(8)

and

D

0 2 v−

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Similarly, average momentum flux per unit area can be defined as: Z T D E
1 0 2 0 2 v+ = v+ dt T+ 0 1 = T−

Z

T

0

0 v−

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According to Eq: (7), when T+ < T− we have


2

(10)

(11)

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D E D E 02 02 v+ > v− ,

dt.

(9)

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and according to Eq: (8), when T+ > T−

D E D E 02 02 v+ < v− .

(12)

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0 3  0 3  0 3  0 3  > v− when T+ < T− ; whilst v+ < v− It is obvious that v+ when T+ > T− . These two cases mean that the wall normal fluctuating

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component is derived from positively and negatively skewed distributions, respectively. Under the positively skewed distribution, there will be a net upward momentum flux of fluid; whilst under the negatively skewed distribution, there will be a net downward momentum flux of fluid. This imbalance of momentum flux of fluid particle within fully turbulent boundary layers can 12

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play an important role on the transport and deposition of heavy particles. The data from Kim et al.[27] show that the wall normal fluctuating compo-

negative skewness in the range of 10 < y + < 30. 2.2. Statistics of v 0 in each of the four quadrants

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nent is of positive skewness in the range of 0 < y + < 10 and y + > 30 and of

Following the quadrant analysis approach of Willmarth and Lu [28] for

analysing the structure of the Reynolds stresses, we classified the wall nor-

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mal fluctuating velocity and averaged it in the four quadrants according to the instantaneous flow velocity in the quadrant domain. In this sense, the instantaneous velocity of a sufficiently large number of fluid particles at a

specified position may be categorized in terms of the sign of the streamwise

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and wall normal velocity fluctuations. For example, when both u0 and v 0 > 0 the instantaneous velocity signal is allocated to quadrant I (QI ); in the case

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of u0 < 0 and v 0 > 0, it is allocated in quadrant II (QII ); when both u0 and v 0 < 0 , is allocated to the quadrant III (QIII ); finally, if u0 > 0 and v 0 < 0, it is allocated to quadrant IV (QIV ). We note that upward momentum fluxes

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may be associated primarily with the bursting process associated with events in QII , whilst downward momentum fluxes are associated with sweep events

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in QIV . Physically speaking, upward momentum fluxes associated with QII would cause particles to move away from the wall and downward momentum

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fluxes associated with QIV would result in the migration of particles toward the wall. Time averages of vi0 and momentum flux v 02i can be defined for each of

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the four quadrants according to Eq: 7 and 9 as Z 1 T 0 0 hv i i = v i dt; i = I, II, III, IV Ti 0 and

(13)

Z 1 T 02 i = I, II, III, IV, (14) = v i dt; Ti 0 where Ti denotes time spell spent in the quadrant i by v 0 i , and v 0 i is define as

2 v0i

E

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D

(15)

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  v 0 if v 0 satisfies the criterion of quadrant analysis, 0 vi =  0 if not.

A large eddy simulation (LES) of a fully developed channel flow with Reτ = 180 was carried out to obtain the corresponding statistics of vi0 up to y + = 100. The LES was based on a dynamic Smagorinsky sub-grid scale

time-advancement.

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(SGS) model [29] and a generalized fractional-step method[30] for the overall

1/2

hv 02 i

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In Figure 2, the quadrant mean hvi0 i and wall normal flow velocity rms

as a function of y + show that the fluctuating components in the four

quadrants are smaller in magnitude than the wall normal flow velocity rms across the y + range shown. hvi0 i in each of the four quadrants is of

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1/2

hv 02 i

different magnitude, indicating that there is an asymmetry in the wall normal

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fluctuating components. Furthermore, the greatest magnitude of hvi0 i is found in QII across most of the y + range. Figure 3 shows that there is a net upward

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momentum flux resulting from QII for the range of y + > 20. However, this situation is reversed in the range of y + < 20. The asymmetry of hvi0 i and hvi0 2 i in each of the four quadrants is a new feature for modelling velocity

fluctuations encountered by heavy particles, which we believe is particularly useful for measuring particle transport and deposition in the near wall region. 14

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0.8

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0.7 0.6

, hvi0 i

0.5



v 02

1/2

0.4

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0.3

0.0

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0.1

0

20

v 02 hvI0 i

1/2

hvII0 i

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0.2



0 | hvIII i| 0 | hvIV i|

40

60

80

100

y+

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1/2 Figure 2: Profiles of v 02 and hv 0 i i as a function of y + at Reτ = 180 in each of the

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four quadrants.

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0.8

vI0 2

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0 2 vIII

D

0.7 0.6

D

0.5

E

vII0 2

E

0 2 vIV

E

E

0.4

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Mean momentum flux

D

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0.3

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0.2

0.0

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0.1 0

20

40

y+

60

80

100

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Figure 3: Profiles of momentum fluxes as a function of y + at Reτ = 180 in each of the

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four quadrants.

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2.3. Implementation of the stochastic quadrant model The imbalance of hvi0 i within each of the four quadrants will be of differ-

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ing importance to the transport and deposition of heavy particles. Events in quadrant II are mainly associated with violent ejections of low-speed fluid

away from the wall; motions in quadrant IV are primarily associated with an

inrush of high-speed fluid toward the wall, also referred to as the sweeping event. There are no significant structures associated with quadrant I and III.

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The upward momentum flux in quadrant II may be a strongly contributing factor in the transport of particles away from the wall, reducing the depo-

sition rates; whilst the inward momentum flux in quadrant IV may be a strongly contributing factor in the transport of particles towards the wall, tending increase the deposition rates.

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The results on hvi0 i and hvi0 2 i enables us to specify the statistics of wallnormal velocity fluctuations encountered by particles in each eddy along their

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trajectories. In Figure 4 the probability density functions for a half normal distribution for vi0 in each of the four quadrants at y + = 30 are shown to be

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in fair agreement with the original LES results, indicating that a half normal distribution may be used to describe the distribution of vi0 . This probability

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distribution function is given by  √    √2 exp − ω22 if ω ≥ 0, 2σ σ π fΩ (ω; σ) =  0 if ω < 0, where σ is set to equal to the value of

y + location.

pπ 2

1/2

hvi0 2 i

(16)

at the corresponding

The next step is to construct a random process, which models the eddy motions in the four quadrants. Particles will interact with a random succes17

1.2

3.0 LES HND

1.0

pdf

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2.0

0.6 0.4

1.5 1.0

0.2

0.5

0.5

1.0

2.5

1.5 0 vII

LES HND

2.5

1.0 0.5

3.0

0.0 0.0

0.5

1.0

1.4

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1.5

2.0

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0.0 0.0

pdf

LES HND

2.5

pdf

pdf

0.8

2.0

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1.2

1.50 vI

2.0

2.5

3.0

LES HND

1.0 0.8 0.6 0.4 0.2

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0.0 0.0 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0 0 vIII vIV

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Figure 4: Probability density function (pdf) for vi0 obtained by LES and a comparison

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with a half normal distribution (HND)

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sion of eddies resulting from different quadrants. For this, a homogeneous Markov chain was conceived as a model for the evolution of eddy events in

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the four quadrants along the particle trajectories. Particles may interact with an eddy in quadrant I. After this eddy decays, they would then be able to interact with an eddy resulting from any of the four quadrants with a certain transition probability. Figure 5 describes this process.

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p1,2 QII

QI

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p2,3

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p1,3

QIII

p4,1

p2,4 p3,4

QIV

Figure 5: Diagram describing the Markov chain modelling motions in the four quadrants.

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As far as the transition probabilities are concerned, let Qi , i = {I, II, III, IV} be a discrete time Markov chain on {QI , QII , QIII , QIV } with a transition ma

p11 p12 p13 p14



    p21 p22 p23 p24  ,  P =  p31 p32 p33 p34    p41 p42 p43 p44

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trix

(17)

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where (pij : i, j ∈ {1, 2, 3, 4}) denotes the corresponding probability distribution of random eddy events in each quadrant. The transition matrix in X Eq: 17, needs to satisfy the condition pij = 1. For eddy events in the four j

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quadrants, Eq: 17 is reduced to a “degenerate” transition matrix as   P = p11 p22 p33 p44 .

(18)

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Figure 6 shows variations of the relative probability associated with each

of the four quadrants as a function of y + , which are computed in terms of the sign of each individual event within the integrated non-dimensional

Eq: (18).

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time. These probabilities are used as the transition probabilities denoted in

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The time scale of eddies in each of the four quadrants is difficult to es-

timate accurately from the present study, although Luchik and Tiederman

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[31] provided several quantitative techniques to measure time scales associated with bursting events. In the present study, the lifetime of eddies in the four quadrants are assumed to equal to the Lagrangian time scale of fluid particles according to their corresponding y + position. Figure 7 shows the

Lagrangian time scale of fluid particles within turbulent boundary layers. 20

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0.40

0.30

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0.25

0.20

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0.15

0

20

40

y+

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0.10

pI pII pIII pIV

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Relative probability

0.35

60

80

100

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Figure 6: Relative probability of four quadrants as a function of y + .

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This is taken from the curve-fitting of Kallio and Reeks[12]. Furthermore, the random Lagrangian time scale ξ is assumed to obey an exponential dis-

f (ξ, TL ) =

1 − Tξ e L TL

ξ > 0,

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tribution with the following PDF (19)

where TL indicates the integral Lagrangian time scale in wall units at the particle position. Figure 7 also shows the wall-normal rms profile of fluid ve1/2

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locity. hvi0 i in each of the four quadrants is obtained by multiplying hv 02 i

by a scaling factor that is the ratio of the magnitude of the velocity fluctuation in each of the four quadrants to the magnitude of wall-normal velocity fluctuation across the boundary layer. In every eddy generated in the four quadrants, the fluctuating velocity is sampled from a half normal distribution

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with a mean hvi0 i and a variance corresponding to the particular particle y +

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value in the boundary layer.

3. Corrections to hydrodynamic forces

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As stated in [32], in terms of the analytical solutions for flow around rigid and circulating particles, the effect of containing walls is to change

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the boundary conditions for the equations of motion and continuity of the carrier phase. The corrections to CD in the presence of containing walls were

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achieved through analytical methods (see [33, 34]) or numerical simulations (see [2, 26]).

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100

1.0

v

 02 1/2

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0.6

40

20

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0.4

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0.2

0.0

60

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v 02

1/2

0.8

80

0

20

Lagrangian integral time scale, TL

1.2

TL 40

y+

60

80

0 100

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Figure 7: Non-dimensional wall normal fluid velocity and Lagrangian time integral time

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scale as a function of y + within turbulent boundary layers.

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3.1. Near wall correction for the drag force The drag force opposes relative motion between the particle and sur-

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rounding fluid, and for a unit mass particle it is defined as 1 FD = − ρ |v − u| (v − u)Ap CD , 2

(20)

where ρ is the fluid density, u and v represent fluid and particle velocity

respectively, Ap is the projected area of the particle equaling to πd2p /4 for

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a rigid spherical particle, CD is the drag coefficient which is a function of particle Reynolds number Rep . When the particle Reynolds number Rep  1, CD equals to 24/Rep so that Eq: 20 reverts to the Stokes formula. In this

study, CD is also dependent upon the distance of the particle centroid away from the wall and so that it needs wall-effect corrections.

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The problem of a rigid spherical particle approaching a nearby plane wall in a viscous fluid defines an entire field of research. Brenner [33] established

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an expression for the correction to the Stokes’ drag law necessitated by the presence of a plane boundary at a finite distance from the particle moving

(21)

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normal to the wall. It can be written as   9 dp 24 ∼ CD = 1 + , 8 2L Rep

where L is the distance from the center of the particle to the nearby wall.

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Goldman et al. [34] further analysed a particle moving parallel to a plane wall and proposed the following correction "    3  4  5 #−1 24 9 dp 1 dp 45 dp 1 dp CD = 1 − + − − . 16 2L 8 2L 256 2L 16 2L Rep (22) 24

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Zeng et al. [2] proposed a composite drag coefficient for a particle moving parallel to the plane valid for a wide range of Rep and distance from the wall.

where 0.07 8 = 1.028 − − ln 2 1 + 4δ 15



270δ 135 + 256δ



24 , Rep

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CD0



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It is given by   √ h  √ i [0.687+0.313exp(−2 δ)] CD0 , CD = 1 + 0.15 1 − exp − δ Rep

δ=

(23)

L − 0.5. dp (24)

The above expressions for a particle moving parallel to a planar wall surface are shown in Figure 8 and compared to the standard drag law from Schiller and Naumann [5]. They are plotted as a function of Rep for different

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normalised gaps δ between the particle and wall. It can be seen that the motion of a particle approaching the nearby wall results in an increase of the

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drag coefficient CD . In particular, CD experiences a significant increase when the gap δ is vanishingly small (e.g. δ = 0.005). We note that both expressions for CD of Goldman et al. [34] and of Zeng et al. [2] collapse into the standard

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drag law when the particle moves further away from the wall. It is worth noting that the wall effects would be additionally affected by the shear flow

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in the vicinity of the wall and the rotation of a particle (see Goldman et al. [34], Lee and Balachandar [26]). In this study, the composite correlation for

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CD from Zeng et al. [2] is regarded as the most accurate representation of drag force acting on a particle moving parallel to a wall as it takes into account both particle Reynolds number Rep and the gap from the particle centroid

to the wall surface. This enables us to compare and investigate particle deposition and transport through the inclusion of the composite correlation 25

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proposed by Zeng et al. [2] and the standard drag law. 3.2. Lift force

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In this study, it is assumed that particles attains an equilibrium spin and experience the lift force resulting from vorticity in the underlying carrierphase. Introducing a correction function to account for the Reynolds number

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dependence of the lift fore, Eq: 3 can be extended to 3D as  1/2 1 2 1/2 FL = 1.615dp (ρµ) [(u − v) × ω] f (Rep , ReG ), |ω|

(25)

where ω is the local vorticity of the fluid. In a two-dimensional wall bounded linear shear flow, |ω| equals the shear rate.

Defining the shear Reynolds number Rep of the particle

Eq: 25 can be rewritten as

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ReG =

ρd2p |ω| , µ

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1 4.1126 f (Rep , ReG ). FL = ρAp dp [(u − v) × ω] 1/2 2 ReG

(26)

(27)

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In the light of the definition of CD and from Eq: 27, the Saffman lift coefficient

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CL is given by

CL =

4.1126 1/2

ReG

f (Rep , ReG ).

(28)

1/2

For a creeping flow, the restriction of Rep  ReG  1 was assumed in the

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derivation of Saffman [15, 16]. This condition was relaxed by McLaughlin [17], and Mei [18] proposed the following expression for f (Rep , ReG )  n o
 1 − 0.3314α1/2 exp − Rep + 0.3314α1/2 for 0 < Rep ≤ 40, 10 f (Rep , ReG ) =  0.0524 (αRe )1/2 for Rep > 40, p (29) 26

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9000 8000 7000 6000

CD

5000 4000 δ decreases

3000

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2000

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S&N1933 G1967 δ = 0.005 Z2009 δ = 0.005 G1967 δ = 0.25 Z2009 δ = 0.25 G1967 δ = 0.5 Z2009 δ = 0.5 G1967 δ = 1.5 Z2009 δ = 1.5 G1967 δ = 3.5 Z2009 δ = 3.5

1000 0 10−2

10−1

100

Rep

(a)

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CD

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M

102

δ decreases

S&N1933 G1967 δ = 0.005 Z2009 δ = 0.005 G1967 δ = 0.25 Z2009 δ = 0.25 G1967 δ = 0.5 Z2009 δ = 0.5 G1967 δ = 1.5 Z2009 δ = 1.5 G1967 δ = 3.5 Z2009 δ = 3.5

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CE

101

100

101

102

Rep

(b)

Figure 8: Comparison of corrections for CD (S&N1933 for Schiller and Naumann [5],

G1967 for Goldman et al. [34] and Z2009 for Zeng et al. [2]. ), (a) Rep < 1, (b) Rep > 1.

27

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where α = 0.5

ReG . Rep

(30)

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In the case of a wall-bounded linear shear flow, the direction of the lift force on a particle is determined by the relative slip velocity from Eq: 25,

i.e. for a particle moving in a shear flow parallel to a wall, if the particle leads the surrounding fluid, the lift force points to the wall; conversely if the

particle lags behind the surround fluid, the lift force points away from the

the wall or away from the wall.

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wall. Owing to inertial effects, the lift force causes particles to migrate to

Containing walls have a significant effect on the lift force as well. As far as the shear-induced lift is concerned, Zeng et al. [2] proposed a composite

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correlation for the shear induced lift coefficient CLs . The expression reads (  4/3 )    ReG CLs = CLs,w exp −0.5δ × exp αsL (ReG )δ βsL (ReG ) − λsL (δ, ReG ) , 250 (31) where

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CLs,w =

αsL (ReG ) =

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βsL (ReG ) =

λsL (δ, ReG ) =

         −exp {−0.3 + 0.025 ReG } ,   0.8 + 0.01ReG ,    
5/2  ReG {1 − exp {−δ}} 250 .  3.663 0.22 , (Re2G +0.1173)

(32)

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Eq: 31 is applicable in circumstances when a stationary particle is positioned in a wall-bounded linear shear flow for 1 < ReG < 200 and even when the particle touches the wall (δ → 0). Here, Eq: 31 is extended to the

situation when ReG < 1. This extension seems valid when compared with

numerous earlier research efforts focusing on ReG < 1 (see Zeng et al. [2]). 28

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Figure 9a shows that CLs results from a combination of flow shear and wallinduced effects for 5 values of the particle-wall gap delta. Compared to the

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Saffman lift coefficient with the correction function f (Rep , ReG ) set to 1, CLs is considerably higher than the Saffman lift coefficient when the gap is vanishingly small. However, the value of CLs is pretty close to the value of the Saffman lift coefficient when ReG  1 and the gap δ = 3.5.

Zeng et al. [2] also proposed a composite lift coefficient CLt for translation

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parallel to the nearby plane wall induced lift force, namely (  1.2 ) δ CLt = f (L, Rep )+[CLt,w − f (L = 1/2, Rep )] exp −11 , (33) g(Rep ) where

f (L, Rep ) = f0 (Rep ) CLt,0 (L∗ ) Lf1 (Rep ) ,

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 2 f0 (Rep ) = 1 + 0.329 Rep + 0.00485 Rep ,  f1 (Rep ) = −0.9tanh(0.022 Rep )

(35)

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  (9/8 + 5.78 × 10−6 L∗ ) exp {−0.292 L∗ } for 0 < L∗ < 10, =  8.94 L∗−2.09 for 10 < L∗ < 300, (36)  CLt,w = 0.313 + 0.812 exp −0.125 Rep0.77 (37)

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CLt,0

(34)

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where

 g(Rep ) = 3 exp −0.17 Re0.7 , p L∗ =

LRep . dp

(38)

(39)

Eq: 33 is applicable for 0 < Rep < 100 and 0 < L∗ < 300. Figure 9b shows the curves of CLt for 5 values of the particle-wall gap δ almost collapse into a single curve with a decrease of L∗ . Comparing 9a with 9b, we observe that 29

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103

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δ decreases

101

Saffman1965 Z2009 δ = 0.005 Z2009 δ = 0.25 Z2009 δ = 0.5 Z2009 δ = 1.5 Z2009 δ = 3.5

100

10−1 −2 10

10−1

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CLs

102

ReG

100

101

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(a)

δ decreases

10−1

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CLt

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100

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101

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10−2

10−3 −1 10

Z2009 δ Z2009 δ Z2009 δ Z2009 δ Z2009 δ

= 0.005 = 0.25 = 0.5 = 1.5 = 3.5 100

L∗ ( dLp Rep )

101

102

(b) Figure 9: Comparison of corrections for C30 L , (a) CLs and compared to the Saffman lift coefficient , (b) CLt .

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the value of CLs is much higher than the value of CLt when Rep < 1. In this study, the shear-induced lift force is considered the dominant lift force.

force for comparison with the Saffman lift force. 4. Modelling methodology

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Eq: 31 thus is used to study the effects of near wall corrections to the lift

A vertical fully developed turbulent boundary layer at Reτ = 200 was

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obtained by solving the Reynolds-averaged Navier-Stokes (RANS) equations

with the standard k −  turbulence model and enhanced-wall treatment. As a consequence, the non-dimensional width of the boundary layer is 200 wall units. It is worth pointing out that this equals the half width of a fully developed channel flow. In this way, the computational time is a half that

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for a fully developed turbulent channel flow. We consider the particle size to be sufficiently small so that particle transport to the wall is limited by

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its transport in the boundary layer y + < 200, beyond which particles are assumed to be fully mixed. This avoids tracking particles in the region of

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outside and the need to consider particle tracking in the whole channel, the computation times are significantly reduced as a result. See the seminal paper

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by Kallio and Reeks [12] on this subject. The volume fraction of the particle phase is small enough so that one-way coupling is assumed. Furthermore,

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the density ratio of particle to fluid ρp /ρf  1, so that the non-linear drag force and shear-induced lift force are only considered. Therefore, the particle equation of motion reads dv 1 Rep FL = CD (u − v) + , dt τp 24 mp 31

(40)

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where v, u are the particle velocity and instantaneous fluid velocity at the particle center, τp is the particle response time defined as ρp d2p /18µ, FL

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denotes the shear-induced lift force and mp particle mass. Both the drag coefficient CD and lift coefficient CL are corrected for the wall effect. The

fluid velocity is interpolated at the particle position through a second-order

accurate scheme based on least squares method; the particle equation of motion is integrated in time using an implicit Gear scheme.

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The position xp of particles is obtained from the kinematic relationship dxp = v. dt

(41)

The boundary condition for the above equation is that the particle is captured by the wall when its center is less than its radius away from the

subsequent re-suspension.

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wall. The particle capture is assumed to be perfectly absorbing with no

u = U + u0 ,

(42)

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two parts,

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In RANS calculations, the instantaneous fluid velocity is decomposed into

where U is solved by a RANS calculation, u0 is fluid velocity turbulence

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fluctuations.

In the standard k −  turbulence model, the velocity fluctuations are

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calculated as three equal components on the basis of the isotropic assumption. This is contradictory to the anisotropic nature of turbulent boundary layers. As a consequence, to account appropriately for the turbulence effect of the inhomogeneous and anisotropic turbulent boundary layer on the suspended particles, it is crucial to generate u0 encountered by particles (see [35, 36, 37, 32

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38, 39, 40]). For particles transported in a vertical fully developed turbulent boundary layer studied in this paper, the wall normal velocity fluctuations

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encountered by particles are generated through a stochastic quadrant model reported in [1] 5. Results and discussion

The results presented here are obtained from four sets of particles char-

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acterized by different Stokes number St making use of the standard drag CD ,

near-wall correction for CD from Zeng et al. [2], CL from Saffman [15] and near-wall correction for CL from Zeng et al. [2] in the particle equation of motion, respectively. The particle sets are St = 2, 5, 10, 20 made dimensionless as St = τp /τf , in which τf is defined as ν/u2τ and represents the character-

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istic time scale of the flow. The ratio of particle-to-fluid density ρp /ρ = 770 . Statistics for the dispersed particle phase are based on 5 × 104 particles

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released from the inlet of the boundary layer. The boundary condition of the nearby wall for particles is assumed to be perfectly absorbing when the

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particle centroid is a radius away from the wall surface; the boundary condition of the upper surface of the boundary layer for particles is regarded as

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a mirror reflection.

5.1. Near-wall corrections to CD

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5.1.1. Particle deposition We first studied the effects of the near-wall correction of CD on the depo-

sition rates of a wide range of inertia particles. The results for the deposition rates of fourteen sets of heavy particles are shown and compared with the

33

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experimental measurements by Liu and Agarwal [3] and the curve-fit of McCoy and Hanratty [41] as shown in Figure 10. While the two expressions for

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CD do not yield a significant change in the deposition rates of the particles considered, the overall effect of the near-wall correction is that it predicts a decrease of the deposition rates of small particles and an increases of the

deposition rates of large particles. This may be explained in the following way: when a small particle is transported into within 3.5 times of its own

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diameter, it experiences considerably higher drag as a result of wall-effects and a small Rep (see van Dijk and Swailes [42]); this helps the particle to remain in the vicinity of the near wall with significant longer residence time and reduces deposition. As far as large particles are concerned, the increase of drag force due to the wall-effects may not be strong enough to equilibrate

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their velocity with the local velocity. They then get deposited by their own inertia (free flight) within a relative shorter time compared to small particles.

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On the other hand, the increased drag may prevent larger particles escaping from the near wall region. For instance, for a particle experiencing an

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ejection event in quadrant II (see Marchioli and Soldati [7]), the increased drag may reduce the probablity of a particle moving away. Consequently,

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the near-wall correction for CD increases deposition rates of heavy particles. 5.1.2. Mean streamwise velocity profiles

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Figures 11 and 12 show a comparison of mean streamwise velocity profiles

for the four sets of particles as a function of y + . The results were obtained

from the standard drag law and the composite drag correlation by Zeng et al. [2] for CD , which are also compared with the mean streamwise fluid velocities. It can be observed that there is no discernible difference between the standard 34

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Exp of Liu & Agarwal 1974 McCoy & Hanranty 1977 Stochastic quadrant CD Stochastic quadrant CD Z2009

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10−1

10−2

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10−3

10−4

10−5

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+ Dimensionless Deposition Velocity Vdep

100

100

101

102

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Dimensionless Particle Ralaxation Time τ + (St)

Figure 10: Deposition results obtained with the standard drag law CD from Schiller and Naumann [5] (denoted as Stochastic quadrant CD ) and near-wall corrections from Zeng

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et al. [2] (denoted as Stochastic quadrant CD Z2009) and compared with data from Liu

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and Agarwal [3] and McCoy and Hanratty [41]

35

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drag law CD and corrected CD for the particle sets St = 2, 5, 10. However, the mean streamwise particle velocities for the particle set St = 20 acquired

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from the corrected CD diverges significantly from the corresponding fluid velocities in contrast to the standard CD , particularly in the very near wall

region. These results are similar to the experimental data of Kulick et al. [43] and the numerical simulations of Wang and Squires [9]. Wang and Squires

[9] considered the inclusion of gravity in the particle equation of motion

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but did not consider the influence of wall effects on the drag coefficient CD . Intriguingly, van Dijk and Swailes [42] also show that the low inertia particles closely follow the mean streamwise fluid velocity and large particles maintain

a relatively high streamwise velocity in the near wall region. These results are based on the numerical solution of a PDF equation for PDF models with an

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absorbing-wall boundary condition for depositing particles. This deviating streamwise particle velocity profile is different from the numerical results of

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Arcen et al. [23], which shows that large particles still follow the local fluid perfectly with a wall-corrected drag force in their DNS calculations. However,

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the boundary condition when particles contacting the wall surface studied by Wang and Squires [9] and Arcen et al. [23] was that for elastic collisions,

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which is not the case considered here. For small particle sets St = 2, 5, 10, the mean streamwise velocities of small particles (St = 2, 5) are lower than the counterpart of the fluid in the near wall region. This might result from

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the fact that we account only for the wall-normal fluid velocity fluctuations seen by particles.

36

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20 St = 2

Fluid Particle CD

+

U

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Particle CD Z2009

15

10

0 10−1

100

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5

101

102

101

102

y+

20 St = 5

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(a)

Fluid Particle CD

ED

10

Particle CD Z2009

PT

U

+

15

0 10−1

AC

CE

5

100

y+

(b)

Figure 11: Comparison of mean streamwise particle velocities with the standard drag law CD and near-wall corrections from Zeng et al. [2] and compared to the fluid velocities (a)

St = 2, (b) St = 5.

37

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20 St = 10

Fluid Particle CD

+

U

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Particle CD Z2009

15

10

0 10−1

100

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5

101

102

101

102

y+

20 St = 20

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(a)

Fluid Particle CD

ED

10

Particle CD Z2009

PT

U

+

15

0 10−1

AC

CE

5

100

y+

(b)

Figure 12: Comparison of mean streamwise particle velocities with the standard drag law CD and near-wall corrections from Zeng et al. [2] and compared to the fluid velocities (a)

St = 10, (b) St = 20.

38

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5.2. Near-wall corrections to CL 5.2.1. Particle deposition

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From the deposition rates shown in Figure 13, it is evident that the inclusion of lift force into the particle equation of motion does not result in significant change in the deposition rates of particles considered. The two

expressions from Saffman [15, 16] and Zeng et al. [2] for CL yield reduced

deposition rates for the three sets of particles St = 2, 5, 10. This is in con-

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trast to the previous reported results on the effects of lift force upon particle

deposition (see Kallio and Reeks [12], Wang et al. [21]). From previous discussions, these three sets of particles move slower than fluid particles in the very near-wall region. According to the definition of the lift force expressed in Eq 27, the positive velocity difference between particles and surrounding

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fluids results in a lift force directed away from the wall. This lift force reduces the deposition rates of small particles. As far as particles with St > 10 are

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concerned, it can be observed that the inclusion of lift force increases the deposition rates. This is quite reasonable since for the large particles more

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particles may experience a wall-ward lift force, which causes an increase of particle deposition. However, there is no discernible difference in deposition

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rates from the Saffman [15, 16] and Zeng et al. [2] for CL . In addition, comparing the effects of corrected drag force on particle depo-

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sition rates shown in Figure 10, it can be observed that the changes resulting from corrected drag coefficient CD are more evident than those resulting from the corrected lift coefficient CL . As can be observed in Figures 8a and 9a, the correction due to wall-effects yields considerable differences to the values of the drag and lift coefficients for the particle Reynolds number Rep << 1 39

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and δ = 0.005. Consequently, it can be inferred that a change of deposition

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rates may result from these differences.

Exp of Liu & Agarwal 1974 McCoy & Hanranty 1977 Stochastic quadrant CD 10−1

Stochastic quadrant CD CL Saffman Stochastic quadrant CD CL Zeng

10−2

10−3

10−4

10−5

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100

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+ Dimensionless Deposition Velocity Vdep

100

101

102

Dimensionless Particle Ralaxation Time τ + (St)

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Figure 13: Deposition results obtained with the standard drag law CD from Schiller and Naumann [5] (denoted as Stochastic quadrant CD ), standard drag law coupled with Saffman lift coefficient CL Saffman [15] (denoted as Stochastic quadrant CD CL Saffman

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), standard drag law coupled with near-wall corrected CL from Zeng et al. [2] (denoted as Stochastic quadrant CD CL Zeng). The results are compared with data from Liu and

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Agarwal [3] and McCoy and Hanratty [41].

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5.2.2. Mean streamwise and normal r.m.s velocity profiles Figures 14 and 15 suggests that the inclusion of the lift force in the

particle equation does not yield a noticeable effect on the streamwise mean velocity profiles for the four sets of particles St = 2, 5, 10, 20, even with

the corrected lift coefficient CL . In contrast to the significant influence of 40

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the wall corrected drag coefficient CD on the mean particle velocity profile shown in Figure 12, the wall corrected lift coefficient CL by has no noticeable

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influence on the particle mean velocity profile as shown in Figure 15(at least within the statistical accuracy of the simulation and for reasons of lack of

statistical accuracy we have not included similar comparisons for the particle rms velocity).

It should be pointed out that these results and conclusions are influenced

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by the significant statistical error imposed by measuring the behaviour of

particles at extremely low concentrations in the boundary layer. Establishing conditions starting from fully mixed conditions necessarily impose long computing times starting from fully and increasing numbers of particles to compensate for the low near wall concentrations. This highlights the limi-

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tations of using DNS in such circumstances compared to perfectly reflecting walls when the near wall particle concentration is actually greater that its

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fully mixed value where we have pointed out earler that the streamwise mean velocity profile has a different behaviour [23].

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6. Concluding remarks

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We have investigated the effect of near wall corrections to the drag and lift coefficient CD and CL of our stochastic model [1] predicaions of the trans-

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port and deposition of heavy particles in a vertical fully developed turbulent boundary layers. Compared to the standard drag coefficient CD , the overall effect of the near-wall corrected drag coefficient CD is to decrease deposition rates of small particles and to increase those of large particles; however, the correction has no significant effect on particle deposition. The wall-effect 41

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20 St = 2

Fluid Particle CD

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Particle CD + CL S1965

15

U

+

Particle CD + CL Z2009

10

0 10−1

100

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5

101

102

101

102

y+

(a)

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20

St = 5

Fluid Particle CD

Particle CD + CL S1965

ED

15

10

PT

U

+

Particle CD + CL Z2009

0 10−1

AC

CE

5

100

y+

(b)

Figure 14: Comparison of mean streamwise particle velocities with the standard drag law CD (denoted as Particle CD ), the standard drag law CD with CL from Saffman [15] (denoted as Particle CD + CL S1965) and the standard drag law CD coupled with a near-wall corrected CL from Zeng et al. [2] (denoted as Particle CD + CL Z1965) and

42

compared to the fluid velocities (the same legends assumed in the following figures) (a) St = 2, (b) St = 5.

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20 St = 10

Fluid Particle CD

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Particle CD + CL S1965

15

U

+

Particle CD + CL Z2009

10

0 10−1

100

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5

101

102

101

102

y+

(a)

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20

St = 20

Fluid Particle CD

Particle CD + CL S1965

ED

15

10

PT

U

+

Particle CD + CL Z2009

AC

CE

5

0 10−1

100

y+

(b)

Figure 15: Comparison of mean streamwise particle velocities with the standard drag law CD , with CL and inclusion of near-wall corrections on CL from Zeng et al. [2] and

compared to the fluid velocities (a) St = 10, (b) St = 20.

43

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corrected CD yields a significant deviation of the mean streamwise velocity profile of the large particle set (St = 20) compared to that of the carrier

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flow in the near wall region. The predicted deposition rates obtained from the inclusion of Saffman lift force and wall-effect corrected lift force yield reduced deposition rates for the small particle sets St = 2, 5, 10, in contrast to previous reported results. On the other hand, the inclusion of a near wall corrected lift force has a moderate effect on particle deposition and transport

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in the near wall region for the large particle sets St > 10. Given all the results presented, whether the wall effects on the hydrodynamic forces should be included or not depends on specific applications. For instance, making

use of the wall-effect corrected drag coefficient CD to study the dynamics of large particles (i.e. St > 10) with containing walls may provide better

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particle dispersion characteristics. Similar conclusions can be drawn for the inclusion of the near wall corrected lift force in the particle equation of mo-

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tion. Although the inclusion of lift and corrections of near-wall effects have a noticeable influence on particle deposition, it is not sufficient to explain

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the under-estimate in the predicted deposition rates for low inertial particles compared to the corresponding experimental benchmark measurements of

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Liu and Agarwal[3].

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7. Acknowledgments We wish to acknowledge the support of British Energy (Part of EDF).

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References [1] C. Jin, I. Potts, M. W. Reeks, A simple stochastic quadrant model for

Physics of Fluids 27 (2015) 053305.

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the transport and deposition of particles in turbulent boundary layers,

[2] L. Zeng, F. Najjar, S. Balachandar, P. Fischer, Forces on a finite-sized particle located close to a wall in a linear shear flow, Physics of Fluids

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21 (2009) 033302.

[3] B. Y. H. Liu, J. K. Agarwal, Experimental observation of aerosol deposition in turbulent flow, Journal of Aerosol Science 5 (1974) 145–148, IN1–IN2, 149–155.

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[4] S.-Y. Lin, J.-F. Lin, Numerical investigation of lubrication force on a spherical particle moving to a plane wall at finite reynolds numbers,

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International Journal of Multiphase Flow 53 (2013) 40 – 53. [5] L. Schiller, A. Naumann, A drag coefficient correlation, Vdi Zeitung 77

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(1933) 318–320.

[6] S. A. Morsi, A. J. Alexander, An investigation of particle trajectories in

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two-phase flow systems, Journal of Fluid Mechanics 55 (1972) 193–208.

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[7] C. Marchioli, A. Soldati, Mechanisms for particle transfer and segregation in a turbulent boundary layer, Journal of Fluid Mechanics 468 (2002) 283–315.

[8] C. Marchioli, A. Soldati, J. G. M. Kuerten, B. Arcen, A. Tani`ere, G. Goldensoph, K. D. Squires, M. F. Cargnelutti, L. M. Portela, Statis45

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tics of particle dispersion in direct numerical simulations of wall-bounded turbulence: Results of an international collaborative benchmark test,

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International Journal of Multiphase Flow 34 (2008) 879–893. [9] Q. Z. Wang, K. D. Squires, Large eddy simulation of particle-laden turbulent channel flow, Physics of Fluids 8 (1996) 1207–1223.

[10] Q. Wang, K. D. Squires, Large eddy simulation of particle deposition in

Flow 22 (1996) 667–682.

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a vertical turbulent channel flow, International Journal of Multiphase

[11] M. Chen, J. B. McLaughlin, A new correlation for the aerosol deposition rate in vertical ducts, Journal of Colloid and Interface Science 169 (1995)

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437–455.

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