Settling of finite-size particles in an ambient fluid: A Numerical Study

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1 Settling of finite-size particles in an ambient fluid: A Numerical Study Todor Doychev 1, Markus Uhlmann 1 1 Institute for Hydromechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany Keywords: particulate flow, resolved particles, DNS, immersed boundary method, clusters, Voronoї analysis Abstract We have investigated the gravity induced settling of finite-size particles in an ambient fluid by means of direct numerical simulation (DNS). Such configurations are relevant to a large number of applications such as meteorology, mechanical and environmental engineering. For single particle a variety of motion patterns exist, from straight vertical to fully chaotic paths, for which the fluid motion in the near field around the particles and the particle wake play a dominant role. Here, the interface between the dispersed- and carrier-phase was fully resolved by means of an immersed boundary method. Particular care has been taken to meet the respective resolution requirements. We have performed simulations of particles settling with two different Galileo numbers, Ga = 121 and Ga = 178. The settling regimes for a single particle settling with this Galileo numbers are: (a) steady axisymmetric regime and (b) steady oblique regime. We observed, that in the steady oblique regime (Ga = 178) the particles exhibit wake induced clustering, while in the steady axisymmetric regime (Ga = 121) this was not observed. Furthermore, the mean settling velocity of the particles with Ga = 178 was strongly enhanced compared to the velocity of a single settling particle. 1. Introduction Particle-laden flows are found in a large number of environmental natural and technical processes. Examples include pollution dispersion in the atmosphere, raindrop formation in clouds, sediment transport, fluidized bed reactors and combustion devices. Thus, the accurate prediction of such flows is of great importance. This naturally leads to the necessity of reliable theoretical description of the mechanisms that take place in such flows. Despite the progress made in the past, there is still a large scatter of the available data. A recent review of the subject can be found in (Balachandar and Eaton 2010). Here we consider the sedimentation of finite-size particles in an (initially) ambient fluid under the influence of gravity. Under such conditions, the system is characterized by a set of non-dimensional parameters. Given the fluid density ρ f, the kinematic fluid viscosity ν, the vector of gravitational acceleration g on the one hand, and the particle diameter D, particle density ρ p and solid volume fraction Φ s on the other hand, dimensional analysis shows that the problem is determined by three non-dimensional parameters. One has already been mentioned, the solid volume fraction Φ s. The other two can be taken as the density ratio ρ p /ρ f and the Galileo number defined as the ratio between the gravity-buoyancy and the viscosity forces Ga = ( g D 3 ρ p /ρ f 1 ) 1/2 /ν. The particles under consideration in the present work have a particle Galileo (Reynolds) number of the order of O(100). For single particle, the Galileo number and the particle-to-fluid density ratio characterize the regime of particle settling and in particular the particle wake (Jenny et al. 2004). A variety of motion patterns exist, from straight vertical to fully chaotic paths, for which the fluid motion in the near field around the particles play a dominant role (Ern et al. 2012). Therefore, the proper resolution of the flow field in vicinity of the particles is crucial. In the present work the motion of the fluid and the dispersed phase were fully resolved by means of direct numerical simulation and the interface between the dispersed- and carrier-phase was fully resolved by means of an immersed boundary method (Uhlmann 2005). The interaction between the (turbulent) carrier flow and the solid particles can lead to a number of hydro-dynamical coupling phenomena which are most prominently manifested by the following open questions: (a) Do finite-size particles exhibit clustering or not? (b) How does the flow field and/or particle clustering affect the settling rate of the particles? (c) What are the characteristics of the wake-induced turbulence? Particle clustering has been investigated for long time. Majority of the previous studies have been concentrated on the so-called preferential concentration of small particles in turbulent flows (Squires and Eaton 1991; Fessler et al. 1994). Few studies have been devoted to the question, whether finite-size particles exhibit clustering, e.g. (Qureshi et al. 2008; Fiabane et al. 2012). Furthermore, the interaction between the flow filed and the solid particles have been from great interest in the scientific community. Especially the question, whether the particles enhance or attenuate the flow turbulence. Lucci et al have investigated the interaction of finite-size heavy particles with a decaying homogeneous-isotropic turbulence in the absence of gravity. In the present study the analysis will focus primary on: (i) the mean apparent velocity lag; (ii) the flow field fluctuations induced by the particles as they settle through the computational domain as well as the velocity fluctuations of the particles; (iii) the spatial structure of the dispersed phase, i.e. do the particles form cluster and what 1

2 is the extension and shape of the clusters in case clustering takes place. The experiments of (Parthasarathy and Faeth 1990a,b) provide one of the most relevant datasets for the present study (Re p =[65,147,262]). They performed measurements of heavy particles settling in an ambient fluid. Their experiments provide measurements of the velocity fields for both phases. Considering the wake-induced clustering of the particles, our parameters are matched most closely by the numerical simulations of Kajishima and Takiguchi (2002) and the experiments of Nishino and Matsuchita (2004). 2. Numerical Method The numerical code used for the present simulation utilizes an efficient and precise formulation of the immersed boundary method for the simulation of particulate flows (Uhlmann 2005). This method employs direct forcing approach by adding a localized volume force term to the momentum equations. The basic idea of the immersed boundary method is to solve the Navier-Stokes equations in the entire computational domain, including the space occupied by the particles. The force term is formulated in such way, as to impose a rigid body motion upon the fluid at the locations of the solid particles and is explicitly computed at each time step as a function of the desired particle positions and velocities, without recurring to a feed-back procedure. Thereby, the stability characteristics of the underlying Navier-Stokes solver are maintained in the presence of particles, allowing for relatively large time steps. The necessary interpolation of variable values from Eulerian grid positions to particle-related Lagrangian positions and vice versa are performed by means of the regularized delta function approach of (Peskin 2002). This procedure yields a smooth temporal variation of the hydrodynamic forces acting on individual particles while these are in arbitrary motion with respect to the fixed grid. The solution of the Navier-Stokes equations is realized in the framework of a standard fractional-step method for incompressible flow. The temporal discretization is semi-implicit, based on the Crank-Nicholson scheme for the viscous terms and a low-storage three-step Runge-Kutta procedure for the non-linear part (Verzicco and Orlandi 1996). The spatial operators are discretized by means of central finite-differences on a staggered grid. The temporal and spatial accuracy of this scheme are of second order. The particle motion is determined by the Runge-Kutta discretized Newton equations for translational and rotational rigid-body motion, which are explicitly coupled to the fluid equations. The hydrodynamic forces acting upon a particle are readily obtained by summing the additional volume forcing term over all discrete forcing points. Thereby, the exchange of momentum between the two phases cancels out identically and no spurious contributions are generated. The analogue procedure is applied for the computation of the hydrodynamic torque driving the angular particle motion. In the case of periodic boundary conditions, the spatial average of the force term needs to be subtracted from the momentum equation for compatibility reasons (Fogelson and Peskin 1988; Höfler and Schwarzer 2000). Since particles are free to visit any point in the computational domain and in order to ensure that the regularized delta function verifies important identities (such as the conservation of the total force and torque during interpolation and spreading), a Cartesian grid with uniform isotropic mesh width is used. For reasons of efficiency, forcing is only applied to the surface of the spheres, leaving the flow field inside the particles to develop freely. During the course of the simulation, particles can approach each other closely. However, very thin inter particle films cannot be resolved by a typical grid and therefore the correct build-up of repulsive pressure is not captured which in turn lead to possible partial overlap of the particle position. In order to prevent such non-physical situations, we use the artificial repulsion potential of (Glowinski at al. 1999), relying upon a short-range repulsion force. For detailed description of the method and for information on the validation tests and grid convergence please refer to (Uhlmann 2005, 2008; García-Villalba et al. 2012; Kidanemariam 2013) and further references therein. 3. Setup of the Simulations The sedimentation of multiple heavy spherical particles in an otherwise ambient fluid under the influence of gravity was studied. We carried out numerical experiments with two different Galileo numbers, Ga = 121 and Ga = 178. In both cases the particle-to-fluid density ratio and the solid volume fraction were kept constant to ρ p /ρ f = 1.5 and Φ s = 0.5 %. The corresponding particle Reynolds number based on the balance between drag and immersed weight, using the standard drag formula (Clift 1978, p.112) Re p = w clift p D/ν, was calculated to Re p = 141 and Re p = 245. Under this conditions the flow is considered to be dilute, thus dominant effects of inter-particle collisions are avoided. Here and in the following, we will refer to the particles settling with Ga = 121 as case M120 and to the particles settling with Ga = 178 as case M180. Additionally we performed simulations of the settling of a single particle with the exact physical parameters as in case M120 and case M180. The corresponding simulations are denoted by S120 and S180. The computational domain Ω for both cases M120 and M180 is a box elongated in the vertical direction. The domain extends in terms of the particle diameter D to: 68D 68D 341 for case M120 and 85D 85D 170 for case M180. In all three directions periodic boundary conditions are applied. The selected solid volume fraction corresponds then to a total of particles in case M120 and particles in case M180. The particles are initially placed randomly in the computational domain. The initial particle position and the extensions of the computational domains are depicted in figure 1. The flow is resolved by grid points in case M120 and grid points in case M180. Consequently the particle resolution in case M120 results in D/Δx = 15 and in case M180 in D/Δx = 24. The physical and numerical parameters of the performed simulations are summarized in table 1 and table 2. 2

3 defined as the difference of the instantaneous values and the averaged value at that same time e.g. u p (x p (t), t) = u p (x p (t), t) < u p > p (x p (t), t). Similarly, the fluctuations of the fluid velocity field with respect to the average over the volume occupied by the fluid are defined as u f (x (t), t) = u f (x (t), t) < u f > Ωf (x (t), t). The time in the present work is scaled by the gravitational time scale τ g, which is defined as τ g = ( g D ρ p /ρ f 1 ) 1/2. Table 1: Physical parameters for particulate flow with a single and multiple settling particles in an ambient fluid. Solid volume fraction Φ s, density ratio ρ p /ρ f, Galileo number Ga = ( g D 3 ρ p /ρ f 1 ) 1/2 /ν, Reynolds number Re p = w clift p D/ν based on the terminal velocity of a single particle w clift p, particle diameter D and fluid viscosity ν and number of particles N p. The gravitational constant g is applied against the vertical direction z. Φ s ρ p /ρ f Ga Re p N p M M S120 O(10 5 ) S180 O(10 5 ) Figure 1: Dimensions of the computational domain with the initial particle distribution for (a) case M120 and (b) case M180. D denotes the particle diameter. Gravity acts in the negative z-direction. Periodic boundary conditions are applied in all three directions in both cases. The simulations were initialized from a succession of coarse-grained runs with randomly distributed fixed particles. We start with a coarse simulation with fixed randomly distributed particles and evolve the simulation in time until stationary steady state is reached. The results of the coarse simulations are then linearly interpolated on a finer computational grid and the simulations are resumed. This is repeated until the targeted resolution is reached. During this procedure the particles were kept at fixed positions. This ensures that any particle deviations from a random distribution due to insufficient resolution are avoided. Once the desired resolution is reached, the particles are released to move freely and the actual recording of data is started. Here and in the following we arbitrary set the time at which the particles are released to zero, t = 0. During the course of the present document the following nomenclature is applied: the domain is discretized by a Cartesian grid (x, y, z), where z is the vertical component in direction of the gravity g; velocity vectors and their components corresponding to the fluid and the particle phases are distinguished by subscripts f and p respectively, as in u f = (u f, v f, w f ) T and u p = (u p, v p, w p ) T ; particle position vector is denoted as x p = (x p, y p, z p ) T. The fluctuations of particle quantities over time are denoted by a single prime, i.e. u p and are Table 2: Numerical parameters for particulate flow of single and multiple settling particles in an ambient fluid. Particle resolution D/Δx, number of grid nodes N i in the i-th coordinate direction. D/Δx N x N y N z M M S S Results and Discussion 4.1 Settling velocity Figure 2 depicts the temporal evolution of the mean apparent velocity lag w p,lag for both cases M120 and M180. The velocity w p,lag is defined as the difference in the mean streamwise velocity components of the two phases: w p,lag (t) =< w p (t) > p < w f (t) > Ωf, (3.1) where < > Ωf defines a spatial averaging operator over the domain occupied by the fluid and < > p denotes an averaging operator over the particles. The velocities have been scaled as a Reynolds number with the particle diameter and the fluid kinematic viscosity, Re p,lag = w p,lag D/ν. The terminal settling velocity of the single settling particles, case S120 and case S180, is shown as well. As can be seen 3

4 the mean apparent lag in case M120 remains over the entire course of the simulation at approximate value of Re p,lag 141. Moreover, the velocity w p,lag in case M120 is well represented by the settling velocity of the single particle from case S120. This implies that, the presence of many freely moving particles in case M120 does not have strong influence upon the mean apparent velocity lag of the particles. In case M180 on the other hand, after the particles are released they begin to accelerate for approximately two hundred gravitational time units and the velocity w p,lag reaches a maximum of approximately Re p,lag = 271. After reaching maximum, the mean settling velocity decelerates and levels up, still exhibiting some fluctuations, at an approximate value of Re p,lag = 260. In contrast to case M120, we found out that the velocity w p,lag in case M180 deviates significantly from the one for single particle in case S180. As will be discussed below, the increase of the mean apparent velocity lag in case M180 in comparison to the settling velocity of a single particle is a direct result of the clustering of the dispersed phase in case M180. (a) (b) Figure 2: Average particle settling velocity for case M120 (black solid line) and case M180 (red solid line) (normalized with the particle diameter and kinematic viscosity) as function of time. The time is normalized with the gravitational time scale τ g = ( g D ρ p /ρ f 1 ) 1/2. Terminal settling velocity of a single particle in ambient fluid for case S120 (black dashed line) and case S180 (red dashed line). 4.2 Velocity fluctuations As mentioned above, the particles are settling in an (initially) ambient fluid. Therefore, any fluctuations of the fluid are induced by the settling of the particles. The relevant question in this context is aimed at determining the intensity of the self-induced fluid motion. This analysis is different (but related to) the study of the modulation of existing (background) turbulence due to the addition of particles. Figure 3a shows the temporal evolution of the root mean square (r.m.s.) values of the fluid velocity components for both cases, M120 and M180. The r.m.s. values are normalized with the terminal settling velocity of the particles from case S120 and case S180 respectively. Figure 3: (a) R.m.s. of the fluid velocity fluctuations for case M120 (black lines), and case M180 (red lines) as function of time. Solid lines show the horizontal components, dashed lines show vertical component. (b) As in (a), but for the particle velocity. The reference velocity w ref denotes the terminal settling velocity w p,lag of case S120 and case S180 respectively. As can be immediately seen, the fluid velocity components in case M180 show higher fluctuations than in case M120. It can be observed, that in both cases the fluid velocity fluctuations of the vertical component are highly dominant. This can be attributed to the wake-induced character of the fluid motion, since fluid motion is primarily caused by the particles moving in streamwise direction. This high level of anisotropy also suggests strong effects of the particle wakes, where the mean particle wake contributes to the fluctuating velocity field. Similar observations were made in the work of Parthasarathy and Faeth (1990a,b). In case M180 the r.m.s. values increase after releasing the particles for about 200τ g time units after which they oscillate around a mean value of 0.27 for the streamwise component and around 0.08 for the lateral component. On the other hand the fluctuations in case M120 do not experience significant increase after particles are released. The r.m.s. values in 4

5 streamwise direction seem to have a large period oscillating behaviour. In the following we assume the flow to be in statistically steady state after 200τ g. For case M180, after reaching statistically stationary state, the fluid velocity fluctuations exhibit somehow more prominent peaks in the r.m.s. values than in case M120 and they are more distinctive for the streamwise component than for the lateral velocity component. The corresponding fluctuations of the velocity components of the dispersed phase are depicted in figure 3b. The fluctuations of the dispersed phase experience similar evolution over time as for the fluid velocity: the fluctuations in case M180 are larger then in case M120 and the streamwise component is approximately 2.5 times larger then the cross-stream component of the particle velocity. The intensity of the fluctuations of the vertical velocity component can be attributed entirely to the collective effects, since a single particle at both Galileo numbers is in steady motion (Jenny et al. 2004). By comparison of the velocity fluctuations of both phases, we found out that, both phases experience comparable values. Similar values for the cross-stream velocity component indicate that turbulent dispersion plays an important role in the present cases. As mentioned earlier the flow is dominated by the wakes generated from the particles as they settle through the domain. The particles do not settle on straight vertical trajectories, rather they settle on curved or oblique paths. As result the particle wakes are not oriented exactly in vertical direction, which causes momentum along the wake axis to be deposited into the lateral direction. Figure 4: Temporal evolution of the relative turbulence intensity for case M120 (black lines) and case M180 (red lines). Solid lines: I r = (< u f,i >/3) 1/2 /w p,lag. Dashed lines I rv = (< w f >) 1/2 /w p,lag. A measure of the influence of the fluid turbulence on the particles is the relative turbulence intensity, defined as the ratio between the intensity of the incoming fluid flow fluctuations and the apparent slip velocity. In homogeneous flows the definition of relative turbulence intensity is often based upon the three-component turbulent intensities, viz. I r = (< u f,i >/3) 1/2 /w p,lag. In cases with unidirectional mean flow (z-coordinate direction) the following definition is commonly employed, I rv = (< w f >) 1/2 /w p,lag. The temporal evolution of the relative turbulence intensity I r (I rv ) is depicted in figure 4. Although, the flow in the present simulations is not considered to be turbulent in the general sense, the relative turbulence intensities in the present simulations are showed to be comparable to the turbulence levels often considered in studies of the influence of background turbulence upon the particle motion, e.g. (Wu and Faeth 1994; Bagchi and Balachandar 2004; Yang and Shy 2005; Legendre et al 2006; Poelma et al. 2007; Snyder et al 2008; Amoura et al. 2010). For the present cases we calculated the relative turbulence intensity to I rv = 0.2 (0.24) for case M120 (M180). This indicates that the particles in the present cases generated substantial fluid turbulence as they settle through the domain. 4.3 Spatial structure of the dispersed phase As aforementioned, an interesting feature of particulate flows is the ability of the particles to form clusters. As already mentioned all the fluctuations of the flow field are induced by the settling particles. Moreover, we saw that the particles react to the flow field fluctuations. This reaction was manifested in the fluctuations of the particle velocity field. For the considered Galileo (Reynolds) numbers in this study, the wakes are important even at considerably large distances behind the particles. Thus, the particles can interact with each other over large distances through the particle wakes. The most prominent example of the particle wake interaction is the drafting-kissing-tumbling motion of pairs of trailing particles (Fortes et al. 1987; Wu and Manasseh 1998). Since the flow field is homogeneous in all three directions any deviation of the particles from the random distribution can be attributed to the wake character of the flow. The particle distribution for the present simulations can be visually examined in figure 5 (case M120) and figure 6 (case M180) where the position of the particles is projected on the 2-D horizontal plane. While in case M120 no significant difference in the distribution of the particles at the beginning and at later time of the simulation can be observed, it is clearly observable that the particle distribution in case M180 at later time deviates significantly from the random distribution at the beginning of the simulation. The distribution of the particles shows regions with high number of particles (high particle concentration) and regions with small number of particles (low particle concentration), or even void regions with complete absence of particles. Hereafter, regions with high particle concentration are referred as clusters and regions with low particle concentration as voids. The clusters and the voids in case M180 (figure 6b) appear to extend throughout the entire height of the computational domain. This is in line with previous experimental and numerical findings (Kajishima and Takiguchi 2002, Kajishima 2004a) where similar columnar particle accumulation (Nishino and Matsushita 2004) was observed. Time sequences of such visualization (not shown here) shows that these structures are quite robust and they persist over long time intervals. More quantitative information on the particle clustering can be obtained by performing a Voronoї analysis (Monchaux et al. 2010). The Voronoї tessellation is a decomposition of the space into independent cells, which have the property that 5

6 each point in the cell is closer to the cell's site than to any other cell's site. As a consequence, the inverse of a Voronoї cell's volume is proportional to the local particle concentration. In order to quantify preferential concentration we compared the probability distribution function (p.d.f.) of the normalized Voronoї volumes with the p.d.f. of randomly distributed particles (Monchaux et al. 2010, 2012; Fiabane et al. 2012). The Voronoї cell volumes are normalized to be of unit mean. This normalization allows a qualitative comparison of the p.d.f.s for different particle number densities, since the so normalized p.d.f.s are independent of the particle number density (Ferenc and Neda 2007). The distribution of particles experiencing clustering is expected to be more intermittent than the distribution of randomly distributed particles. This implies that regions with higher particle concentration are more probable than for random distribution. Respectively, void regions with low particle concentration are also more probable. Figures 7a and 8a depict the p.d.f. of the normalized Voronoї volumes for case M120 and case M180. As can be observed, the p.d.f. for both cases at the time when the particles were released in the computational domain is well represented by the theoretical Gamma distribution for random particle fields (Ferenc and Neda 2007), confirming that the initial particle distribution was indeed random. At later times the p.d.f.s in case M120 (figure 7a) show that the extremes for the present data are less probable than in the case of randomly distributed particles. This indicates that the particles in case M120 become more ordered than randomly distributed particles. On the other hand, the p.d.f.s in case M180 (figure 8a) deviate significantly from the distribution function for the random distributed particles. The distribution function of the present data exhibits tails with higher probability of finding Voronoї cells with very large or very small volumes than in the random case with uniform probability, indicating that cluster formation takes place in case M180. Figure 5: (a) Top view of the particle position at the begin of the simulation t/τ g = 0 for case M120. (b) The same, but at t/τ g = Figure 6: (a) Top view of the particle position at the begin of the simulation t/τ g = 0 for case M180. (b) The same, but at t/τ g =

7 Further, we have analysed the shape of the regions with high particle concentration by computing the aspect ratio of each Voronoї cell, defined as the ratio of the largest cross-stream extension l x,vi to the largest streamwise extension l z,vi of the Voronoї cell, A V = l x,vi /l z,vi. The aspect ratio A v provides a qualitative measure of the anisotropy of the particle clusters and can be seen as measure for the stretching of the cells. Figures 7b and 8b show the p.d.f.s of the Voronoї cells aspect ratio A V for case M120 and case M180. It can be immediately seen that the p.d.f.s in case M120 do not show any deviation from the p.d.f. of the randomly distributed particles, while in case M180 an appreciable difference is observed. This indicates, that the majority of the Voronoї cells are squeezed/stretched in the vertical/horizontal direction and that the particle structures in case M180 are more likely to be aligned in vertical direction. This confirms the observations made in figure 6b. An alternative way of characterizing the spatial structure of the dispersed phase is by performing nearest-neighbour analysis (Kajishima 2004b). Figure 9 depicts the time evolution of the average distance to the nearest particle neighbour d min. The distance d min is calculated as: N p d min = 1 N p min i=1 j=1,n p j i d i,j, (4.2) where d i,j = x p,i x p,j is the distance between the centers of particles i and j. The distance d min has been normalized by its value for a homogeneous distribution with the same solid volume fraction, d hom min = ( Ω /N p ) 1/3. The hom lower limit of d min corresponds to the minimum value of the function, when all particles are in contact with a neighbouring particle. Figure 7: Case M120: Probability density function of (a) the normalized Voronoї cell volumes and (b) the aspect ratios A V. Different lines represent data assembled over different time intervals. The initial random distribution is represented by black solid line. t/τ g = [279,307] (red). t/τ g = [559,587] (blue); t/τ g = [1216,1244] (green); t/τ g = [1496,1524] (magenta). The dashed black line represents an analytical Gamma function fit (Ferenc and Neda 2007). Figure 8: Case M180: Probability density function of (a) the normalized Voronoї cell volumes and (b) the aspect ratios A V. Different lines represent data assembled over different time intervals. Initial random distribution is represented by black solid line. t/τ g = [199,213] (red). t/τ g = [345,359] (blue); t/τ g = [572,576] (green); t/τ g = [794,810] (magenta). The dashed black line represents an analytical Gamma function fit (Ferenc and Neda 2007). 7

8 hom The upper limit of the function d min /d min has a value of unity and arises for homogeneously distributed particles, i.e. particles are positioned on a regular lattice. It can be observed, that the average distance to the nearest neighbour at initial time is very close to the value for randomly distributed particles. As time evolves, the value of hom d min /d min in case M120 increases quickly and reaches a statistically steady state value of approximately This finding is in line with the findings of the Voronoї analysis that the particles in case M120 are more ordered than randomly distributed particles. Contrarily, the value of hom d min /d min in case M180 initially decreases for approximately 200τ g and undulates around approximate value of 0.5. This again corroborates the results obtained by the Voronoї analysis that the particles in case M180 tend to form agglomerations. It was found that the flow field was considerably anisotropic with the vertical direction being the dominant direction. The results show that, independent of clustering, considerable turbulence levels were induced by the settling particles, with relative turbulence intensities reaching values of 0.2 to Moreover, the velocity fluctuations of both phases were comparable indicating that the particles respond strongly to the flow field. In the future more detailed analysis of the flow will be performed, e.g.: (i) The effect of clustering upon the flow statistics will be more deeply analysed; (ii) Lagrangian analysis of the data; (iii) Statistics of conditionally averaged data. The existence of such strong differences in the spatial distribution of the particles between the two different settling regimes is quite interesting and the role of the settling regimes on the spatial distribution of the particles will be further analysed. Next step will be the simulation of the interaction between finite size particles and forced homogeneous turbulence. Acknowledgements Support through a research grant from DFG (UH 242/1-1) is thankfully acknowledged. The authors also want to also acknowledge the computer resources, technical expertise and assistance provided by the Leibniz Supercomputing Center (LRZ), Jülich Supercomputing Center (JSC) as well as by the Steinbruch Center for Computing (SCC) at KIT. References Figure 9: Temporal evolution of the average distance to the nearest neighbor for cases M120 and M180, normalized by the value for a homogeneous distribution on a regular cubical lattice. Case M120 (black solid line). Case M180 (red solid line). Random particle distribution with the same volume fraction (black dashed line). 5. Conclusions We have simulated the settling of spherical particles at moderate Reynolds numbers and low solid volume fractions. The solid/fluid interfaces were fully resolved by means of an immersed boundary method. The particles were released to move freely in an initially ambient fluid. The settling in two different single particle regimes was investigated: steady axisymmetric regime with Galileo number Ga = 121 (Re p,lag = 141) and steady oblique regime with Ga = 178 (Re p,lag = 250). Voronoї analysis of the particle spatial distribution was performed. Our analysis revealed that the particles in the steady oblique regime exhibit significant clustering, while in the steady axisymmetric regime no clustering of the dispersed phase was observed. It was observed that, the clustering of the particles led to a significant increase of the mean apparent velocity lag. Furthermore, the shape of the clusters was investigated and it was found that the cluster structures have strong anisotropic shape, where particles were aggregated in a column like structures which extended throughout the entire height of the computational domain. Amoura, Z., Roig, V., Risso, F., Billet, A.M., Attenuation of the wake of a sphere in an intense incident turbulence with large length scales. Physics of Fluids 22, Bagchi, P., Balachandar, S., Response of the wake of an isolated particle to an isotropic turbulent flow. Journal of Fluid Mechanics 518, Balachandar, S., Eaton, J.K., Turbulent Dispersed Multiphase Flow. Annual Review of Fluid Mechanics 42, Clift, R., Grace, J., Weber, M., Bubbles, drops, and particles. Academic Press. Ern, P., Risso, F., Fabre, D., Magnaudet, J., Wake-Induced Oscillatory Paths of Bodies Freely Rising or Falling in Fluids. Annual Review of Fluid Mechanics 44, Ferenc, J.S., Néda, Z., On the size distribution of Poisson Voronoi cells. Physica A: Statistical Mechanics and its Applications 385, Fessler, J.R., Kulick, J.D., Eaton, J.K., Preferential concentration of heavy particles in a turbulent channel flow. Physics of Fluids 6,

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