ON THE PREFERENTIAL CONCENTRATION OF INERTIAL PARTICLES DISPERSED IN A TURBULENT BOUNDARY LAYER

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1 3rd International Symposium on Two-Phase Flow Modelling and Experimentation Pisa, -4 September 4 ON THE PREFERENTIAL CONCENTRATION OF INERTIAL PARTICLES DISPERSED IN A TURBULENT BOUNDARY LAYER Maurizio Picciotto (a), Cristian Marchioli (a), Michael W. Reeks (b) and Alfredo Soldati (a) (a) Centro Interdipartimentale di Fluidodinamica e Idraulica and Dipartimento di Energetica e Macchine, University of Udine, v.le delle Scienze, 8, 33 Udine, Italy (b) School of Mechanical and Systems Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne, NE 7RU, UK ABSTRACT The distribution of inertial particles in turbulent flows is strongly non-homogeneous and is driven by the structure of the flow field. In this work, we examine the relationship between particle concentration and flow field in a turbulent channel (Re τ = 5). We use DNS combined with Lagrangian tracking of particles, with Stokes numbers equal to.,, 5 and, to characterize the effect of inertia on particle preferential accumulation. Comparing the Eulerian statistics for fluid and particles, we observe differences which can be explained by the particle preferential sampling of the underlying carrier flow field. This turbulent preferential sampling appears responsible for particle accumulation in the boundary layer. To measure particle preferential accumulation we use the Jacobian, J(t), of the dispersed phase measured along a particle trajectory. This quantity measures the compression of an elemental volume of the dispersed phase along a particle trajectory and is related to the local compressibility/divergence of the particle velocity field itself. Lagrangian statistics of J(t) show that the compressibility increases for increasing particle response times τ + p over the range of particle response times considered. INTRODUCTION In a number of environmental and industrial problems involving turbulent dispersed flows, the information on particle distribution is a crucial issue. In this respect time average / volume averaged concentrations based on point closure models are not sufficient and more sophisticated models are required. Turbulent flow fields in general are of a strongly organized and coherent nature represented by large scale structures. These structures, because of their coherence and persistence have a significant influence on the transport of dispersed particles. Specifically, coherent structures generate preferentially directed, non-random motion of particles leading to non-uniform concentration and to long-term accumulation. The effect of local spatial structures of the flow field on particles is related to their mutual interaction which, in turn, is modulated by inertia [; ; 3] and their action is not captured by engineering models [4; 5; 6]. Preferential accumulation of particles by turbulence coherent structures has been examined previously in a number of theoretical and experimental works [; ; 3; 7; 8; 9]. In the case of homogeneous turbulence [; ; 3; 9], the particle concentration field will be characterized by local particle accumulation in the low vorticity, high strain regions. In the case of inhomogeneous turbulence [7; 8], as for instance the turbulent boundary layer, the local interaction between particles and turbulence structures will lead to a remarkably macroscopic behaviour producing a long term particle accumulation in the viscous sublayer [; ; ]. This effect may be of fundamental significance in applications as particle abatement, flow reactors and control of momentum, heat and mass fluxes at a wall. Corresponding author: soldati@uniud.it In this work we examine precisely the tendency of particles with different inertia to sample preferentially a turbulent channel flow. Several attempts have been made to characterize the regions of particle preferential accumulation. In particular, Rouson & Eaton [3] and more recently Picciotto et al. [4] used the topological classification by Chong et al. [5] and observed that, very near the wall, strong vortical regions are depleted of particles, which accumulate in specific convergence regions. Rouson & Eaton [3] have also shown that, farther away from the wall, the spatial intermittency of particle distribution is not correlated with the topological descriptors even if particle concentration remains highly non uniform suggesting that the characterization of particle clustering and the link with turbulent structure requires deeper analyses. In previous works [; 4] we analyzed the mechanisms leading to particle accumulation with specific focus on the interactions of particles with the local turbulent structures, describing an instantaneous phenomenological picture. In the first part of this work, we aim at providing a statistical framework to give insights about the importance of particle inertial response to the underlying flow field which eventually determines the particle local accumulation. In the second section we propose to quantify and characterize the local preferential accumulation of particles by measuring the compression/expansion of an elemental volume of particles moving along a particle trajectory. This we will call the Jacobian of the particle velocity field [6]. In a fully Lagrangian approach, this Jacobian is the local divergence or local compressibility of the particle velocity field [7] which ultimately results in the accumulation and segregation of the particles sometimes referred to as the demixing of the particles by the turbulence. The novelty of using J consists in the fact that, being measured along the particle trajectory, it provides information on the velocity field that each particle feels during

2 its motion without the need to count particles in an Eulerian framework. METHODOLOGY A. DNS of turbulent channel flow The flow into which particles are introduced is a turbulent Poiseuille channel flow of air, assumed incompressible and Newtonian. The reference geometry consists of two infinite flat parallel walls: the origin of the coordinate system is located at the center of the channel and the x, y and z axes point in the streamwise, spanwise and wall-normal directions respectively. Periodic boundary conditions are imposed on the fluid velocity field in both streamwise and spanwise directions, noslip boundary conditions are imposed at the walls. All variables are normalized by the wall friction velocity u τ, the fluid kinematic viscosity ν and the half channel height h. The friction velocity u τ is defined as u τ = (τ w /ρ) /, where τ w is the mean shear stress at the wall and ρ is the fluid density. Thus, the balance equations in dimensionless form are: u i t u i x i =, () u i = u j + u i x j Re x p + δ,i, () j x i where u i is the ith component of the dimensionless velocity vector, p is the fluctuating kinematic pressure, δ,i is the mean dimensionless pressure gradient that drives the flow and Re τ = u τ h/ν is the shear Reynolds number. Eqs. and are solved using a pseudo-spectral method. Details of the numerical method can be found elesewhere [8]. In the present study, we consider air with density ρ =.3 kg m and kinematic viscosity ν = m s. The calculations are performed on a computational domain of wall units in x, y and z discretized with nodes. The shear Reynolds number is Re τ = 5. and the time step used is t + =.45 in wall time units. The statistics of the flow field are consistent with previous simulations [9; ]. B. Lagrangian particle tracking Particles are injected into the flow at concentration low enough to consider dilute system conditions (particle-particle interactions are neglected). Furthermore, particles are assumed to be pointwise, rigid and spherical. The motion of particles is described by a set of ordinary differential equations for particle velocity and position at each time step. For particles much heavier than the fluid (ρ p /ρ, ρ p particle density), Elgobashi and Truesdell [] have shown that the only significant forces are Stokes drag and buoyancy and that Basset force can be neglected being an order of magnitude smaller. In the present work, the effect of gravity has also been neglected. With the above simplifications the following Lagrangian equation for the particle velocity is obtained []: du p dt = 3 ( ) C D ρ u p u (u p u), (3) 4 d p ρ p where u p and u are the particle and fluid velocity vectors, d p is the particle diameter and C D is the drag coefficient given by: C D = 4 Re p ( +.5Re.687 p ), (4) where Re p is the particle Reynolds number (Re p = d p u p u /ν). The correction for C D is necessary because Re p does not necessarily remain small [3], in particular for depositing particles. A Lagrangian particle tracking code coupled with the DNS code was developed to calculate particles paths in the flow field. The code interpolates fluid velocities at discrete grid nodes onto the particle position, and with this velocity the equations of motion of the particle are integrated with time. The interpolation scheme exploits a sixth order Lagrangian polynomials. A fourth order Runge-Kutta scheme is implemented for time integration. The interpolation scheme used was compared both with spectral direct summation and with an hybrid scheme which exploits sixth order Lagrangian polynomials in the streamwise and spanwise directions and Chebychev summation in the wall-normal direction: results showed good agreement between the three schemes. For the simulations presented here, four sets of 5 particles were considered, characterized by different relaxation times, defined as = ρ p d p/8µ where µ is the fluid dynamic viscosity. is made dimensionless using wall variables and the Stokes number for each particle set is obtained. In the present paper we have St =.,,5 and. At the beginning of the simulation, particles are distributed homogeneously over the computational domain and their initial velocity is set equal to that of the fluid in the particle initial position. Periodic boundary conditions are imposed on particles in both streamwise and spanwise directions, elastic reflection is applied when the particle centre is less than a distance d p / from the wall. Elastic reflection was chosen since it is the most conservative assumption when studying the reasons of particle prefential concentration in a turbulent boundary layer. We will not analyze the effect of the elastic reflection boundary condition on particle behavior, since it is beyond the scope of this paper. In this work we are more interested in the influence of turbulence on particle behaviour than vice versa. We thus employ the one-way coupling approximation under which particles do not feedback on the flow field. RESULTS A. Effect of inertia on particle preferential accumulation: statistical framework Interactions between particles and the turbulent field are influenced by particle relaxation time. In this section, we analyze this influence from a statistical viewpoint. Specifically, we will compare the Eulerian statistics of the fluid velocity field with those of the particle velocity field and with those of the fluid velocity field sampled by particles. Figure shows the mean streamwise velocity profile (averaged in time and along the xy planes) for both particles and fluid as a function of the wall-normal coordinate. Profiles are averaged over the last 5 time instants of the simulation. Differences are readily visible, but there is evidence of an effect of particle inertia. The behaviour of τ + p =. particles is very similar to that of the fluid: differences in the mean velocity profile, between this size particles and the fluid, are negligible. As

3 8 6 3 FLUID + = U 8 rms u FLUID + =. 5.5 (a) Figure. (symbol). Mean streamwise velocity, for the particles (lines) and fluid particle relaxation time increases, the mean particle velocity is seen to lag the mean fluid velocity outside the viscous sublayer, in the region 5 < < 5. The most important differences are observed for the larger particles (τ + p = 5 and τ + p = ) approximately in the region < < 3. Similar results were reported also by van Haarlem et al. [4], by Portela et al. [6] and by Narayanan et al. []. As already observed by van Haarlem et al. [4], this behavior is probably due to the fact that inertial particles in a turbulent flow do not sample the flow field homogeneously. Once in the near wall region, particles tend to avoid areas of high vorticity and prefer areas of lower-thanmean streamwise fluid, in which the strain rate is high. Statistics of the root mean square (RMS) of velocity fluctuations for particles and fluid, shown in figure, confirm this effect. Figure a compares the RMS of particle streamwise velocity with that of the fluid, as function of. Fluctuations of particle streamwise velocity are larger than those of the fluid and this difference becomes more evident as particle relaxation time increases. The RMS profiles of the fluid velocities at the position of the particles (not shown) closely follow the behaviour of particle RMS profiles. Similar results for the streamwise velocity fluctuations were reported by Portela et al. [6] for particles in pipe flow. From a physical viewpoint, the difference in the streamwise values suggests that the gradient in the mean velocity of the fluid can produce significant streamwise particle-velocity fluctuations. Due to particle interaction with near-wall coherent structures, particles move across the channel and perpendicularly to the mean flow, between regions of high and low streamwise-velocity [6]. This seems particularly true in the case of heavy particles, with a large Stokes number and a long memory. As a consequence, particles with streamwise velocities very different from each other can be found confined in the same fluid environment. This is not true for fluid particles, which show no preferential concentration: the local instantaneous streamwise velocity is more homogeneous and the corresponding fluctuation is smaller than that of the particle velocity field. As reported by Portela et al. [6] The gradient in the mean-streamwise velocity is potentially much more important when there exists a net flow of particle towards the wall, as it is in the case with absorbing walls, or not fully developed flows. This is a feature characterizing particle dispersed in shear flows [7]. rms v rms w Figure. (b) (c) Root mean square of the velocity fluctuations for particles (symbols) and fluid (drawn line): streamwise, (a); spanwise, (b); wallnormal, (c). An opposit behavior is observed in the spanwise direction and in the wall-normal direction (figures b and c respectively), where the fluid velocity field has zero mean gradient. The particle-phase turbulence intensity is lower than the fluid due to two mechansims acting in tandem. The first mechanism is preferential concentration of particles in low-speed regions, characterized by lower turbulence intensity. The second is the filtering of high frequency or wavenumber fluctuations by particles due to their inertia: this leads to the suppression of particle fluctuating intensity in the wall-normal direction and in the

4 W = Figure 3. Mean wall-normal velocity; particles, and fluid at particle position. fluid at + =. 5 spanwise direction. Similar filtering effect has been observed in homogeneous turbulence [7]. It must be noted that preferential concentration is related to pattern formation of particle ensembles whereas inertial filtering refers to the incomplete response of a single particle to its fluctuating fluid environment. A further difference is that preferential concentration has a complex dependence on particle inertia whereas the inertial filtering effect increases monotonically with particle inertia []. Smaller particles (τ + p =.,) are more sensitive to the fluid velocity fluctuations and show spanwise and wall-normal RMS profile very close to the fluid RMS velocity profiles. Larger particles (τ + p = 5,) are less sensitive and their velocity fluctuations are smaller than those of the fluid [8]. The inertial filtering in the wall-normal direction will cause a drift leading to a net wallward velocity of the particle-phase. [8; ] This drift velocity is influenced by inertia [8] and may have significant implications in a number of technological and environmental applications since it also determines the non-uniform particle distribution in the wall-normal direction. In figure 3 we show the time-averaged (5 instants) wall-normal velocity profiles for the different particle sets, plotted against the fluid wall-normal velocity at particle position. Recall that W is directed toward the wall if negative and away from the wall if positive. Particle velocity profiles (symbols) drop to zero at the wall and at the channel centerline, whereas their modulus reaches a maximum in the region < < 4 roughly. This maximum increases as particle inertia increases: a difference of more than one order of magnitude is observed between the maximum velocities of τ + p =. and τ + p = particles. The non-zero value of particle wall-normal velocity occurs for the observed increase of particle number concentration at the wall [8]. Our results confirm that for particle relaxation times up to τ + p =, increasing the inertia will correspond to increasing wall-normal turbophoretic accumulation. If we consider now the fluid velocity at particle position (lines) we can observe a behaviour which is monotonic with particle inertia: i.e., the fluid wall-normal velocities at particle position are different from zero on average and their maxima and minima increase with particle inertia. In the near-wall region ( < < 4), the fluid wall-normal velocity at particle position is positive (i.e. directed away from the wall) reaching W a local maximum at = roughly. Thus, particles in this region sample preferentially ejection-like environments. The fluid wall-normal velocity at particle position becomes negative (i.e. directed toward the wall) outside the near wall region reaching a local maximum at 7-8 : the cross-over point is located around 45. It is interesting to observe that the cross-over position is not much influenced by particle inertia. The sampling of regions with positive wall-normal fluid velocity by particles can be interpreted as a sort of continuity effect of the fluid velocity field for which sweep events, characterized by negative wall normal fluid velocity (directed toward the wall), are more intense and spatially concentrated than ejection events, characterized by positive wall-normal fluid velocity (directed away from the wall). In order to examine in more detail the numerical values of the profiles shown in figure 3, in figure 4 we plotted vis-a-vis the particle wall-normal velocity against the fluid wall-normal velocity, at particle position, for each particle set with the appropriate velocity axis scale. For τ + p =., particle and fluid profiles (figure 4a), at < + z < 5, are similar confirming that far away from the wall the small inertia particles follow promptly the fluid whereas approaching the wall region, <, particles seem to lag the fluid. Moreover, the fluid wall-normal velocity for such particles, after reaching a local negative peak at 8, becomes positive in the near wall region ( < 5) exhibiting a peak near = while particle velocity, though small, remains negative. Increasing particle inertia the particle wall-normal velocity profiles show more clearly a wall directed mean wall-normal velocity. The profiles for τ + p = and τ + p = 5 (figures 4b and c) are similar, though the different in magnitude: for 5 < < 5, these particle sample a wallward fluid wall normal velocity whose modulus is greater than that of the particle wall-normal velocity. Near the wall for < 5 the fluid velocity becomes positive whereas the particle wall-normal velocity reach a peak which is approximately equal to.4 and.5 for τ + p = and τ + p = 5 respectively. τ + p = particles (figure 4d) have profiles which scale similarly with respect to τ + p = and τ + p = 5 except for the region corresponding to 5 < < 5 for which they exhibit a wallward velocity greater than the underlying flow ve- W =. fluid at + =. (a) (c) Figure 4. + =5 fluid at + = = fluid at + = + = fluid at + = (b) (d) Mean wall-normal velocities; (a)-(d), comparison for each particle set between particle and fluid at particle position wall-normal velocity.

5 C p.. Figure 5. Mean particle concentration profiles as function of the wallnormal coordinate ( ) for all particle sets at time t + =. locity. The above results confirm: (i) the tendency of particles to exhibit a wallward drift which enhances with the inertia and (ii) that, when approaching the wall region ( < 5), particles sample preferentially flow regions with wall-normal velocity directed toward the outer layer. To explain how, on an average, particles can travel towards the wall despite an environment of adverse flow, let us recall the interaction between particles and sweep events. Particles are trapped by the sweeps in the outer region, where they find a favorable fluid velocity field (on average) and gain sufficient momentum to approach the wall (turbophoretic drift). Around the cross-over point of the averaged wall-normal fluid velocity (located at = 4 45 roughly, in figures 3 and 4), particles moving toward the wall start to feel an adverse velocity field which acts to dump their turbophoretic drift. As a result of the adverse flow, particle wall-normal velocity (profiles with symbols in figure 3) starts decreasing after the cross-over. Again, this decrease is more evident for particles with larger Stokes number and a longer memory, the averaged wall-normal velocity of the smaller particles being much smaller in magnitude. The statistical framework shows the importance of the inertial sampling on determining the intensity of particle preferential flux toward the wall which eventually is responsible for particle accumulation in the near wall region. In figure 5, we plot the instantaneous space average particle concentration (taken at time t + = ) for all particle sets. The concentration represents particle number density and was calculated by dividing the channel into wall-parallel bins following the Chebychev collocation method: for each bin we counted the number of particles, normalized by the total number of particles and then multiplied by a proper shape factor to account for the non uniform spacing of the bins. A log-log scale is used to capture the different behaviour of the profiles in the proximity of the wall and to underline the different magnitude of particle concentration for each relaxation time. The variation of particle concentration is rather different for each particle set. τ + p =. particles seem to be weakly involved in such accumulation process. As the particle relaxation time increases, the accumulation process takes up more quickly. Concentration profiles for the τ + p = and the τ + p = 5 particles reach a peak value near = which is about one and two orders of magnitude higher than that observed for the smaller particles respectively. The concentration profile for the τ + p = particles develops a sharp peak well into the viscous sublayer at.4.5: at time t + = the normalized particle concentration in this region is O( ) times larger than the initial value. Recall that the dimensionless value of particle diameter d + are:.648,.53,.34 and.765 for τ + p =.,, 5 and respectively. The tendency of particle accumulation to increase with the relaxation time has been already observed [4; 8] and is correlated with the different intensity of particle wall-normal velocities observed in figures 3 and 4. We remark here that the process of particle accumulation has not reached a steady state within the time span covered by the simulation (t + = 9): the peak values of particle concentration in the near wall region are still increasing with time and the process shows no tendency toward saturation, as might be expected at later stages of the simulation [6]. B. Integral characterization of particle preferential accumulation The Eulerian statistics reported in the previous section provided useful quantitative information about particle preferential sampling of specific flow regions, which causes their drift toward the wall and determines the local accumulation. However, these statistics can not be used to characterize the local flow field experienced by the particles. To this aim, we calculate the Jacobian J(t) of the dispersed phase, which provides an integral characterization of particle accumulation with respect to the underlying flow structure. In doing so, the present work goes beyond previous attempts to quantify the non-random distribution of particles: see for example the works by Fessler et al. [9], who employed the correlation dimension introduced by Grassberger & Procaccia [3], and by Rouson & Eaton [3], who used the topological description of turbulent flows based on the classification scheme by Chong et al. [5]. To compute J(t), we refer to Osiptsov s method [6]. Osiptsov s method is a Full Lagrangian method for calculating the particle concentration and the particle velocity field in dilute gas-particle flows. This method (i) is based on integrating equations for the particle concentration along individual pathlines, (ii) has great potential for dramatic reductions in computational time and improvements in accuracy compared with the standard Eulerian-Lagrangian approach, (iii) can deal with steady and unsteady flows without change of algorithm, and (iv) the mathematical formulation provides a sound framework for interpreting the physics of the particle behavior: specifically, the method can handle certain types of singularities where the particle concentration becomes infinite, in a mathematical sense. In Osiptsov s method, J(t) is computed by integration along particle pathlines and particle concentration is then obtained algebraically from the Lagrangian form of the particle continuity equation by computing the change in volume of an element of particle fluid along its trajectory. J(t) measures precisely this change in volume, i.e. the compression or expansion of a particle element volume moving along a particle trajectory. According to Osiptsov [6], the Jacobian J(t) is defined as: ( ) Xi J(t) = det, (5) X j in which X(X,t) is the particle position at time t. Thus, J(t)

6 can be either positive or negative. From a physical viewpoint, J(t) represents a mapping of the particle field - instantaneously sampled for a specific pathline - with respect to a given initial condition in which we suppose uniform particle elements, namely J(t = ) =. Note that J(t) can be related to the Eulerian divergence of the particle velocity field, which also quantifies the local compressibility of the particle velocity field, through the following equation [7]: dln J(t) dt = u p. (6). + =. 5 Our objective here is to use J(t) to characterize the effect of inertia on the local compressibility of the particle velocity field. This quantity is a measure of particle accumulation since it influences the drift towards the wall in the same way that it influences the settling velocity in homogeneous isotropic turbulence [] except that in this case the drift is driven by the inhomogeneity in the near wall turbulence and not by gravity. In Lagrangian form the conservation equations of mass and momentum for a particle element read: du p dt C p = C p J(t) (7) = (u u p). (8) neglecting the effect of particle Reynolds number. C p is the initial particle phase concentration. C p, u and u p are functions of two Lagrangian independent variables: the time, t, and the initial position, X p, which tags the specific particle pathline. Referring to the Osiptsov method [6] and recalling that dx/dt = u p, we can take the derivative of eq.8 and write: d u pi (X,t) = dt X o j Applying the initial conditions: [ ui (X,t) X k (X,t) u ] pi(x,t), X k (X,t) X o j X o j (9) d X i (X,t) = u pi(x,t). () dt X o j X o j X i (X,t = ) X o j = δ i j, () u pi (X,t = ) X o j = u i(x,t = ) X j, () Equations 9 and can be integrated numerically, along each particle pathline. Once the unknowns X i / X j are calculated, J(t) is computed from eq. 5. Figure 6 shows the PDF of particles as function of ln J(t) at time t + =. For the smaller particles (τ + p =.) the PDF is narrow, roughly centered around zero and skewed towards the ln J(t) > side. As the particle relaxation time increases, the PDF has a larger spreading and appears skewed towards the ln J(t) < side. Thus, larger particles preferentially sample ln J(t) Figure 6. PDF of particles as function of the Jacobian J(t) at time t + =. flow regions where J(t) <. Following eq. 7, we can conclude that the local particle concentration C p in such convergence regions is higher than the initial value C p. Also, we observe that C p is more focused for larger particles: figure 6 shows that sampled regions are characterized by values of J(t) which become smaller as particle inertia increases. In figure 7 the time evolution of the PDFs is plotted as function of ln J(t). This figure confirms that the tendency of particles to cluster and converge in specific flow regions is dominated by inertia. The PDF of τ + p =. particles (fig. 7a) remains centered around the initial value of J(t), showing that expansions and compressions of the dispersed phase are weak. Conversely, profiles for the larger particles shift towards smaller values of J(t) continously during the simulation. This is particularly evident for the τ + p = particles (fig. 7d). Recalling eq. 6, a decrease of J(t) in time implies that large inertia particle are more likely to converge and that their velocity field is preferentially compressing, even if the underlying flow field is incompressible. CONCLUDING REMARKS This paper addresses the issue of particle preferential concentration in a fully developed turbulent boundary layer with specific reference to the influence of particle inertial response to the underlying flow field. Statistical analysis of particle and fluid velocity fields has shown the crucial effect of inertia in determining particle sampling of specific flow regions. As a consequence of the preferential sampling, particles cluster and accumulate in the near-wall region. We quantified the tendency of particles to cluster by means of the Jacobian, J(t), of the dispersed phase. Lagrangian statistics of J(t) show that the particle velocity field along particle trajectory is more likely to be compressible and that inertial particles sample preferentially convergence flow regions in which their local concentration is higher with respect to the initial conditions. An increase in particle inertia appears to enhance this trend. The overall picture resulting from this statistical analysis shows the following features: (i) the preferential sampling by inertia determines a particle drift toward the wall that increases with particle inertia (ii) the near wall regions in which particles accumulate are regions of outflow, (iii) the regions into which

7 6 5 (a) t + = particles preferentially accumulate are convergence regions in which the particle velocity field results to be compressed. NOMENCLATURE t + = 45 4 (b) t + = 45 9 (c) (d) -4-4 ln J(t) t + = Figure 7. PDF of particles as function of the ln J(t) for different times for particles with τ + p =. a), b), 5 c) and d). Kinematic and time variables as well as coordinates are made non dimensional by wall variables (apex +): u τ, ν, h. C D particle drag coefficient [-] C p particle concentration [-] C p particle initial concentration [-] d p particle diameter [m] h half channel height [m] J Jacobian of Eulerian-Lagrangian transformation [-] U mean streamwise velocity [m/s] u instantaneous streamwise velocity [m/s] u τ fluid friction velocity [m/s] u i i-th component of fluid instantaneous velocity [m/s] u ip i-th component of particle instantaneous velocity [m/s] v instantaneous spanwise velocity [m/s] w instantaneous wall-normal velocity [m/s] W mean wall-normal velocity [m/s] X particle initial position [m] X particle instantaneous position [m] ρ fluid density [kg/m 3 ] ρ p particle density [kg/m 3 ] particle relaxation time [s] REFERENCES [] M.W. Reeks, On the dispersion of small particles suspended in an isotropic turbulent fluid. J.Fluid Mech., vol. 83, pp [] M.R. Maxey, The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields, J. Fluid Mech., vol. 74, pp , 987. [3] L.P. Wang & M.R. Maxey, Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence, J. Fluid Mech., vol. 6, pp. 7-68, 993. [4] S.E. Elgobashi & T.W. Abou Arab, A two equation turbulence model for two-phase flows, Phys. Fluids, vol. 6, pp , 983. [5] J. Young & A. Leeming, A theory of particle deposition in turbulent pipe flow, J. Fluid Mech., vol. 34, pp. 9-59, 997. [6] S., Cerbelli, A., Giusti & A. Soldati, ADE approach to predicting dispersion of heavy particles in wall bounded turbulence, Int. J. Multiphase Flow, vol. 7, pp [7] M. Caporaloni, F. Tampieri, F. Trombetti & O. Vittori, Transfer of particles in nonisotropic air turbulence, J. Atmos. Sci., vol. 3, pp , 975. [8] M.W. Reeks, The transport of discrete particles in inhomogeneous turbulence, J. Aerosol Sci., vol. 3, pp , 983. [9] J.K. Eaton & J.R. Fessler, Preferential concentration of particles by turbulence, Intl J. Multiphase Flow, vol., pp. 69-9, 994. [] C. Marchioli & A. Soldati, Mechanisms for particle transfer and segregation in turbulent boundary layer, J. Fluid Mech., vol. 468, pp ,. [] C. Marchioli, A. Giusti, M.V. Salvetti & A. Soldati, Direct numerical simulation of particle wall transfer and deposition in upward turbulent pipe flow, Int.J.Multiphase Flow, vol. 9, pp. 7-38, 3. [] C. Narayanan, D. Lakehal L. Botto & A. Soldati, Mechanisms of particle deposition in a fully developed turbulent channel flow, Phys. Fluids A, vol. 5, pp , 3. [3] D.W. Rouson & J.K. Eaton, On the preferential concentration of solid particles in turbulent, J. Fluid Mech., vol. 48, pp ,. [4] M. Picciotto, C. Marchioli & A. Soldati, Remarks on the distribu-

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