Application of DNS and LES to Dispersed Two-Phase Turbulent Flows

Transcription

1 1 Application of DNS and LES to Dispersed Two-Phase Turbulent Flows Kyle D. Squires and Olivier Simonin Mechanical and Aerospace Engineering Department Arizona State University Tempe, AZ , USA Institut de Mécanique des Fluides UMR 5502 CNRS/INPT/UPS Allée du Professeur Camille SOULA Toulouse, France Abstract An overview and examples of the application of Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) to prediction and the scientific study of dispersed, turbulent two-phase flows is presented. This contribution focuses on Eulerian-Lagrangian treatments in which dispersed phase properties are obtained from discrete particle trajectories. The scope of the approaches considered are on systems in which the ensemble comprising the particulate phase is large enough that direct resolution of the flow in the vicinity of each particle is not feasible and, consequently, models of particle dynamics must be imposed. The advantages and limitations of each technique are first considered, representative applications of both DNS and LES to building block flows are then summarized: statistically stationary particle-laden isotropic turbulence and fully-developed turbulent channel flow. In both flows, the detailed descriptions possible using DNS and LES enable in-depth evaluations of statistical and structural features. The specific properties considered including the role of inter-particle collisions in turbulent channel flow and more recent efforts focused on exploration and analysis of the spatial structure of the particle velocity field. 1. Introduction Numerical simulation continues to evolve as an important tool in the analysis and prediction of dispersed two-phase turbulent flows. Computations are playing an increasingly important role as both a means for study of the fundamental interactions governing a process or flow, as well as forming the backbone for engineering predictions of physical systems. At a practical level, computations for engineering applications continue to rely on solution of a statistically-averaged equation set. Many of the statistical correlations requiring closure in Reynolds-averaged models are often difficult or impossible to measure in experimental investigations of two-phase flows. Computational techniques that directly resolve turbulent eddies are an important component in evaluating closure models, while at the same time offering a useful approach for basic studies of fundamental interactions. The focus of the present contribution is on Eulerian-Lagrangian techniques for prediction and study of turbulent, two-phase flows. Within this broad class, the subset of two-phase flows in which the dispersed phase is comprised of small particles and is present in dilute concentrations is of primary interest, using computational techniques that directly resolve turbulent fluctuations in the carrier phase. The most common approaches to resolving turbulent fluctuations in the carrier flow rely on

2 2 either Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) of the Navier-Stokes equations. It these techniques form the basis for the work summarized in this contribution. From a theoretical perspective, DNS is the most satisfactory approach the equations governing the carrier-phase turbulence are solved without recourse to explicit turbulence modeling at any scale of motion. The restriction of DNS to moderate Reynolds numbers is a well-known limitation that is problematic for prediction of gas-solid flows since it restricts the range of particle parameters that may be realistically considered. LES attempts to circumvent the Reynolds number restriction encountered in DNS by parameterizing with a model the effect of motions smaller than the filter width (linked to the grid sized) on the large, energy-containing scales of motion directly resolved (e.g., see Lesieur and Metais 1996 for a general review). In the majority of previous and current applications of DNS and LES to computation of dilute, turbulent two-phase flows, discrete particle tracking is employed to predict dispersed-phase transport. In these approaches, the trajectories of a large ensemble of particles are considered with dispersed-phase statistics formed by suitably-defined averages. It is important to note that a direct resolution of the flow around individual particles is not attempted, rather particle dynamics are parameterized through use, for example, of a drag law. These approaches necessarily impose restrictions on the dispersedphase, e.g., to point particles with diameters smaller than the smallest turbulent lengthscales in DNS or the filter width in LES. In this manuscript DNS and LES together with Lagrangian particle tracking are first outlined with respect to some of the strengths and limitations when used for prediction of two-phase flows. One of the strongest features is use of both methods for performing well-defined numerical experiments to study a particular effect or phenomena. Two representative applications are considered, particulate transport in isotropic turbulence and fully-developed turbulent channel flow. The applications to these flows enable detailed assessment of theories and also shed new light on the importance of the spatial properties of the particle velocity field, in turn motivating development of new tools for future analysis. 2. Background 2.1 Direct Numerical Simulation True direct numerical simulation of a fluid flow loaded with discrete particles requires solution of the standard single-phase Navier-Stokes equations on the domain occupied by the fluid. The influence of the particles is formally taken into account through the fluid velocity boundary conditions (no-slip conditions) on the particle surface. Simultaneously, particle trajectories are computed using Lagrangian tracking, the force acting on the particle being computed by direct integration of the simulated fluid stress exerted on the particle surface. This approach requires that the velocity field around each particle be accurately resolved, an area of current research in which a variety of approaches are being applied, including the use of overset-grids, immersed-boundary techniques, and using finiteelements (e.g., see Burton and Eaton 2002, Fadlun et al. 2000, Hu et al and references therein). The advantage of fully-resolved solutions is that particle-fluid and particle-particle interactions can be treated without approximation, other than those inherent to the underlying numerical algorithm. Such methods can provide detailed information concerning aspects of, for example, the interactions of large-diameter particles with the fluid flow, interaction of particle wakes, etc. Even with highly op-

3 ! 5 3 &!!!! timized numerical algorithms and parallel numerical solution, the computational resources required by methods that fully resolve the flow around individual particles is substantial. Optimistic estimates constrains these approaches to simple domains containing particles and mesh resolutions probably smaller points for the fluid flow, at least for the near future. Nevertheless, one of the key contributions of fully-resolved solutions of the flow around particles in two-phase systems will be providing a means for increasing understanding of some of the basic aspects governing particle-laden flows. Ultimately, simpler working models for complex systems that require more modeling input may evolve from these efforts. For turbulent two-phase flows with a dispersed second phase of particles having diameters comparable to or smaller than the Kolmogorov length scale, fully-resolved computations are not feasible in flows containing many particles, being a typical sample size for current DNS and LES. Prediction of two-phase flows containing large particle ensembles therefore requires parameterizations of particle dynamics. DNS of the carrier phase with two-way coupling, for example, can be currently applied to particle- or droplet-laden flows in regimes in which the solid volumetric fraction is negligible and for particles smaller than the spacing between the computational collocation points (being on the order of the Kolmogorov scale). The modulation effect of the particles on the flow is accounted for by including a term in the instantaneous single-phase Navier-Stokes equations representing the force of the particle on the flow, typically a Dirac distribution (point-force approximation). This approach has been used by numerous investigators, including Squires and Eaton (1990), Elghobashi and Truesdell (1993), Boivin et al. (1998), Sundaram and Collins (1999), Miller and Bellan (1999), Ahmed and Elghobashi (2000) among others. While the point-force approximation can represent the very small-scale perturbations in the fluid velocity field caused by the particle, there are practical constraints on the usefulness of this approach, as discussed below. In addition, neglected in DNS studies is the accompanying sub-particle scale dissipation of these fluid velocity perturbations. In this context, therefore, DNS refers to an accurate numerical solution of the Navier-Stokes equations without the use of explicit turbulence models. Explicit resolution of this additional dissipation rate would be required if one wishes to satisfy the kinetic energy balances for the entire range of fluid velocity fluctuations or the mixture of fluid and particles. For two-way coupling, the governing equations for an incompressible fluid are written in the general form, (1) "!$# % #(' % % % #*) (2) where the velocity is denoted ' +% (, for the fluid,,.- for the particles), the fluid density is, and the pressure&. The term / 102, the dynamic viscosity is particles, implemented in a simulation as, accounts for the effect of the ) ;: =<?>@ 30ACBDEFB (3) where the summation is taken over the particles within the computational domain. The term G 30 6 is the force acting on particle& 0 with center at position 6 <?> at time, and is a three-dimensional

4 4 spatial low-pass filter with a characteristic width of the order of the mesh sizeh. The particular filter <I> depends on the scheme used to project the coupling force to the grid (e.g., see Boivin et al. 2000). It is projection of the coupling force that effectively filters the very small-scale velocity perturbations in the vicinity of the particle. While implicit in the numerical approach used to implement the coupling force, formal application of the filter to the governing equations would in fact yield a subgrid contribution representing the effective stress of the velocity perturbations on the fluid macroscales, a term that is neglected in previous computations. In addition, it is natural then to consider that the other terms in (2) have also been filtered. This again shows that DNS is providing a solution of the macroscales of the flow, and not the entire spectral range that would include resolution of details of the velocity disturbance in the particle wakes. 2.2 Large Eddy Simulation In Large Eddy Simulation the Navier-Stokes equations are spatially filtered (volume averaged) to yield transport equations for the large, energy-containing scales of motion. The influence of the small, subgrid scales of motion on the large eddies is modeled, usually using simple closures that invoke an eddy viscosity hypothesis for part of the subgrid stress. For gas-solid turbulent two-phase flows the fluid momentum equations are written for the filtered variables, similar to the relation (2), also including a contribution from the subgrid-scale stress, JK% % *L L% (4) LNM in which (4) represents the effect of the residual velocity field on the resolved scales ( denotes a filtered variable). Traditionally, the SGS tensor has been written as the sum of a resolved stressok%, the Leonard term, and a residual stress, JK% OP% #RQ TSVUXW L % # TSYUZW % L # [SYUZW TSYUZW %]\ OK% LL % ^L L% (5) where TSYUZW % represents the subgrid-scale velocity. The Leonard term can be computed in terms of the filtered field, following definition of a filter function, while the residual term is unknown. In practice, the whole stress JK% is often modeled together, without a decomposition into resolved and unresolved components. Previous applications of LES to particle-laden flows include homogeneous turbulence by Yeh and Lei (1991), Boivin et al. (2000), and Simonin et al. (1995) as well as turbulent channel flow by Wang and Squires (1996), Fukagata et al. (2001), and Yamamoto et al. (2001). These studies included applications of one- and two-way coupling. Boivin et al. (2000) found in both a priori and a posteriori tests on two-way coupling in computations of isotropic turbulence that LES predictions were substantially more accurate using mixed subgrid models in which the Leonard term was included in the calculation and model coefficients were determined using the dynamic procedure of Germano et al. (1991). Boivin et al. (2000) showed that the dynamic procedure substantially improved predictions of quantities such as the subgrid-scale dissipation rate, even in regimes in which the fluid energy spectrum is non-uniformly modified via momentum exchange with the particulate phase. Also relevant to particle-laden flows are the contributions of the fluid subgrid-scale velocity field to the fluctuating motion of the particles. In gas-solid systems, for particle response times within the range directly resolved on the grid and large compared to the subgrid-scale timescales, the influence

5 5 of subgrid-scale velocity fluctuations on particle motion is not large. In other regimes, e.g., small particle response times, the error resulting from the neglect of the fluid subgrid fluctuations on particle motion will be more substantial. LES used in conjunction with particle tracking approaches for flows in which particle response times are small compared to the fluid timescales requires explicit modeling of transport by subgrid-scale turbulence. Miller and Bellan (2000) have recently analyzed the role of particle transport by subgrid velocity fluctuations in a transitional mixing layer. 3. Recent Applications In gas-solid flows, particle motion is governed by the drag force and any body forces acting on the particle such as gravity. Particle motion can be parameterized in terms of the relaxation timescale, which can be written in the form, J 6 E 6 _F` :a :a # cbd fegihkj]l m 6 (6) where E 6 is the particle diameter. Following Simonin (1991), the mean particle relaxation timejcno is defined as, J no qp J 6sr 6 (7) where t MVu 6 denotes an average with respect to the dispersed phase. Described in this section are application of DNS and LES to particle-laden isotropic turbulence and fully-developed turbulent channel flow. In the first subsection below the use of DNS and LES for measurement and analysis of some basic statistics is described. This discussion, and most current work, focuses on single-point statistical analyses which are inadequate to fully describe some phenomena. Following is a discussion of efforts which motivate and attempt to explain the spatial features of the particulate phase that may be useful for interpreting, and ultimately modeling, effects related to the collective motion of the particulate phase. 3.1 Statistical measures of particle transport Particle-laden isotropic turbulence Fevrier and Simonin (2001) have used DNS and LES of statistically stationary isotropic turbulence together with Lagrangian tracking to study particle interactions with turbulence. A series of calculations were performed over a wide Reynolds number range and for several particle response times. The calculations were performed in the dilute regime for which turbulence modulation by particles is not significant and the effect of inter-particle collisions was neglected. The DNS and LES in this study were computed using resolutions varying fromvw to - _ mesh points and using xzy particles. Shown in Figure 1 are two quantities fundamental to describing the statistical features of turbulent dispersion: the Lagrangian integral timescale viewed by the particles (also referred to as the eddyparticle interaction time) and turbulent dispersion coefficient. The long-time turbulent diffusivity is dependent on the eddy-particle interaction time{} ~ 6, for which Figure 1 shows a local increase for intermediate particle response times, as well as the fluid turbulent kinetic energy sampled along particle trajectories. Fevrier and Simonin (2001) have shown negligible statistical bias in the kinetic

6 6 - # Figure 1: Effect of the turbulent Reynolds number on the fluid Lagrangian integral time scale viewed by the particles and on the particle dispersion coefficient, from Fevrier and Simonin (2001). ( ) {} ~ 6 { (normalization by the fluid turbulence Lagrangian integral timescale), ( ) ƒ ƒ (normalization by the fluid turbulence diffusivity). gih DNS at _ x ; ˆ gih DNS at gih ; LES at x ee ; gih LES at Š F ; y gih LES at w _F w ; Œ gih LES at fe Š from Deutsch and Simonin (1991); theory of Pismen and Nir (1978). energy following particles, Figure 1 then demonstrates that changes in the dispersion coefficient arise. The figure shows that for small response times, corresponding to particles following nearly all fluid fluctuations, the eddy-particle interaction time and particle diffusivity approach that of the fluid. At the opposite extreme, for large-inertia particles the eddy-particle interaction time approaches the fixed-point Eulerian integral timescale.the interesting feature of Figure 1 is that the inertial bias in particle motion leads to a local increase in{ ~ 6 Ž (between { J"no Ž ), resulting in larger dispersion of dense particles compared to the fluid. The figure shows that this difference is substantial - for intermediate particle inertia the particle dispersion is nearly 30% larger than that of the fluid for the lower turbulent Reynolds numbers. from changes in{ ~ 6 Particle-laden turbulent channel flow Inter-particle collisions are a key feature dictating aspects of dispersed-phase transport in gas-solid turbulent flows. Correlations in particle motion brought about via interactions with the surrounding turbulent flow have important effects on particle-particle collisions. Lavieville et al. (1997) used LES of homogeneous turbulent shear flow to study the effect of inter-particle collisions, testing lowerlevel model predictions against LES results. These investigators developed model approximations that account for the reduction in collision rates due to the correlations developed in particle-particle motion because of entrainment within the same fluid regions. In that work, as well as the present investigations, the method used to account for particle-particle interactions in the LES was based on the detection of all the inter-particle collisions occurring within a timestep and the modification of the final velocities of the colliding particles by considering only binary collisions ( hard-sphere approximation) without energy dissipation nor inter-particle friction. The influence of inter-particle collisions in mono- and bi-disperse mixtures has been examined using LES of turbulent channel flow and discrete particle tracking. The results shown below are from _F, based on the friction velocity gih LES of a low-reynolds number turbulent channel flow ( and channel halfwidth). The fluid flow is resolved on a mesh of vw grid points using the dynamic

7 - 7 # # ( ) c ( ) Figure 2: Mean streamwise velocity in ( ) and inter-particle collision times in ( ) from LES of turbulent channel flow with a bi-disperse mixture of dense particles. The Reynolds number based on the friction velocity and channel halfwidth is 180. ( ) J 6 cbv ef : fluid at the particle position; particle; J 6 qbv F : fluid at the particle position; particle. ( ) J 6 bv e : simulation; model prediction; J 6 - bv F : simulation, model prediction. eddy viscosity model of Germano et al. (1991) to close the subgrid stress. The low Reynolds number of the flow minimizes modeling errors in the subgrid closure, enabling accurate computations on the moderate grids usually needed for the parameter variations considered in two-phase flows. Shown in Figure 2 is the mean streamwise velocity profile of particles and fluid along with the profile of the inter-particle collision time in turbulent channel flow. The profiles shown in Figure 2 are from computations in which the dispersed phase is comprised Fb of a bi-disperse mixture of dense particles. The volume fraction of the particulate phase is Š y š, divided equally between cb a group of fine and coarse particles. The Stokes response time of the fine particles is v ef where and are the channel halfwidth and friction velocity, respectively. The density of the coarse particles is larger by a factor of four than for the fine particles, yielding a Stokes response time of bv œ. The diameter of each particle in the two groups is one viscous unit. The particle equation of motion for the channel-flow calculations contains the drag force, with the particle timescale given by (6). The mean flow profiles shown in Figure 2 show that there is essentially no difference in the mean streamwise fluid velocity viewed by the particles, with in general the particles lagging the mean motion of the fluid in the central region of the channel and leading the fluid close to the wall. Apparent from the figure is the large increase in the mean velocity with response time close to the wall (below about 30 wall units in the figure). This increase is brought about via the effect of particleparticle collisions that cause large changes in wall-normal transport and greater mixing across the entire channel. The figure shows the mean profile for the coarse particles is more uniform, similar to the behavior observed in the number density profile (not shown here). In addition to the mean flow, shown in Figure 2 are wall-normal profiles of the inter-particle collision time (for same particle-type collisions in this bi-disperse mixture). The symbols in the figure correspond to the measured collision times from the computation. The lines are model predictions using a closure proposed by Lavieville et al. (1997). The closure model is derived via introduction of the joint fluid-particle pair distribution function in order to account for reductions in the collision rates

8 ª ( ) ( ) žÿ 1.5 žÿ ( ) ( E ) Figure 3: Instantaneous fluid and particle velocity vectors in planes in isotropic turbulence and fullydeveloped turbulent channel flow. ( ) and ( ) show velocity vectors in DNS of isotropic turbulence. Particle relaxation time E in ( cb+ ) is x{ and in ( ) is - bd Š { where{ is the fluid Lagrangian integral timescale. ( ) and ( ) show cb velocity vectors in LES of fully-developed turbulent channel flow. Particle relaxation time in ( ) is v ef E and in ( ) is - bv œ. Particle positions shown by the circles in the figure, the symbol size is exaggerated for clarity. because of correlations in particle motion. This results in expressions for the inter-particle collision time of the form, J8 J C = X (8) wherej is the collision time computed using kinetic theory and accounts for fluid-particle correlated motion, «ª 6 w ª ª 6 where ª 6 is the trace of the fluid-particle velocity correlation tensor, is the fluid kinetic energy, and ª 6 is the particle kinetic energy. Near the wall, where the effective particle response times are large because of the reduction in turbulence timescales, the model predictions are in reasonable agreement with LES results, though the effect of the correction is weak near the wall. Further from the wall model predictions are below the simulations, indicative of higher collision frequencies predicted than actually occur in the channel. 3.2 The continuum properties of the particle velocity distribution The flows summarized above are representative applications of DNS and LES in which detailed

9 9 statistical analyses are possible using the database generated from these calculations. While useful, most calculations and analyses performed to date have considered, essentially, single-point measures which may not be fully adequate for understanding some of the phenomena of practical importance such as two-way coupling. In addition, DNS and LES as currently practiced are limited to small particles, low volume fractions, etc., and there remains a need for new techniques that will be less restricted in range of available parameters. These issues provide part of the motivation for considering a more detailed investigation of the particulate phase and the limits within which the motion of finite-inertia particles takes place. The first limiting case corresponding to response times that are small enough for particles to follow essentially all fluctuations in the fluid; the second being the large-inertia limit in which particle motion satisfies the assumption of molecular chaos and can be fully described using kinetic theory. For small-inertia particles, neighboring velocities will be spatially correlated through the interactions with the same local fluid turbulence, an effect apparent in the velocity vectors shown in Figure 3a in isotropic turbulence and in Figure 3c in turbulent channel flow. In contrast, for large-inertia particles, with response times that are long compared to the fluid turbulence macroscales, neighboring particle velocities are less correlated since these particles maintain stronger connection to their interactions with distant, and independent, turbulent eddies, an effect shown in Figure 3b and Figure 3d in which velocity vectors of large-inertia particles are shown. While qualitative, the vector plots do provide an indication of the differences in structure of the particle velocity field with changes in particle inertia. Quantitative measures of the particle velocity correlations are plotted in Figure 4. Shown from the DNS of Fevrier and Simonin (2001) is the Lagrangian particle velocity autocorrelation along with the two-point Eulerian spatial correlation of the particle velocities in isotropic turbulence. Figure 4a shows the well-known effect with particles becoming less correlated to their initial velocity with increasing time. The dependence of the Lagrangian velocity correlation on particle inertia is also apparent, i.e., longer memory to the history of the motion as the particle response times increase. The spatial velocity correlations show that for low-inertia particles the shape of the correlation appears as a decaying exponential, approaching that cb œe, the two-point velocity correlations are consis- of the fluid. Figure 4 shows that for J"no { tent with the limiting case of fluid-elements, i.e., for O g 6 6, -ª 6 x. As also true for the fluid, for increasing, the particle velocity correlation decreases monotonically. For increasing inertia, Figure 4 shows that the velocity distribution remains spatially correlated since particles are, sharing the same g fluid velocity field. Nevertheless, one important result is that in the 66 limit O } the condition -±ª 6 x g 66 is not satisfied with instead the two-point correlation remaining smaller than -±ª 6 x. This feature illustrates that a portion of the entire particle kinetic energy corresponds to a velocity distribution which is not spatially correlated. 4. Perspectives DNS and LES treatments of the continuous-phase turbulence together with Lagrangian particle tracking for the dispersed phase will continue to constitute useful tools for developing understanding some of the complex effects governing turbulent two-phase flows. The applications summarized in this manuscript have been useful for validating basic theories governing particle dispersion and developing fundamental insight into the structural interactions influencing particle transport, including the role of inter-particle collisions. Treatments of the particulate phase in the context of DNS and

10 10 µ ½¼ ¼½Á ¼½À ¼½ ¼½¾ ¼½¼²³² ¹+º8» Å Æ Ç Å Äà à ÄÊ Ã ÄÉ Ã ÄÈ Ã ÄÆ Ã Äà à ÄÃ Ë ÌPÍÏÎ Figure 4: Influence of particle inertia on the Lagrangian temporal velocity correlation (in ) and Eulerian spatial velocity correlation (in ) of non-colliding particles in DNS isotropic turbulence, gih from Fevrier and Simonin (2001). The turbulence Reynolds number is F (based onñð and O ). J no { cb œe # ; J no { cbx ; J no { Fbw Š ;,J no { x bw ;,J no { w b _ x LES considered here do not fully resolve details of the very local perturbations in the fluid velocity near the particle. DNS and LES studies have been useful for understanding some of the basic mechanisms governing particle-laden flows, showing for example the strong role of inter-particle collisions. As shown above, models for predicting collision rates based on presumed forms of the fluid-particle joint pdf can account for some of the effects observed in turbulent wall-bounded flows. Extension of the methods to account more accurately for effects of, for example, the anisotropy of the velocity component fluctuations will improve the overall accuracy of these modeling approaches. While models and analyses of the type described in this manuscript have been useful for studying some aspects of particle-laden flows, different approaches are probably required for both improved understanding of some phenomena and eventually development of new schemes for modeling and simulation. These and other considerations motivate study of the spatial structure of the particle velocity distribution. The velocity vector plots in both isotropic turbulence and turbulent channel flow show the apparent differences in correlation of particle velocities, indicative of the effective partitioning of the particle velocity by inertia. The two-point spatial correlations from Fevrier and Simonin (2001) make quantitative these differences, showing the reduction in integral scales with inertia and the discontinuous nature of the two-point correlation for large particle inertia in the limit of zero separation. This feature shows that the particle velocity distribution can be decomposed into two contributions, the first a spatially-continuous velocity field representing the collective effects of the entire particulate distribution and a second contribution which is random. The random part, referred to as the Quasi-Brownian Velocity Distribution by Fevrier and Simonin (2001), is associated with each particle of the system. These features of the particle velocity field have important implications for both modeling and the development of new simulation techniques for turbulent two-phase flows. For modeling, the continuous part of the particle velocity distribution will allow accounting of the collective, large-scale effects of the particulate phase such as those observed in studies of two-way coupling, showing the entire spectral range is nonuniformly modified via interactions with the particles (e.g., see Squires and

11 11 Eaton 1990, Elghobashi and Truesdell 1993, Sundaram and Collins 1999). The random part of the velocity distribution, on the other hand, will dominate the mechanisms responsible for inter-particle collisions. Though not shown here, preliminary evaluations of the inter-particle collision times in turbulent channel flow are accurately evaluated using closures developed in terms of the random part of the velocity distribution. References [1] Ahmed, A.M. and Elghobashi, S., 2000, On the mechanisms of modifying the structure of turbulent homogeneous shear flows by dispersed particles, Phys. Fluids, 12, pp [2] Boivin, M., Simonin, O. and Squires, K.D., 1998, Direct numerical simulation of turbulence modulation by particles in isotropic turbulence, J. Fluid Mech., 375, pp [3] Boivin, M., Simonin, O. and Squires, K.D., 2000, On the prediction of gas-solid flows with two-way coupling using Large Eddy Simulation, Phys. Fluids, 12, pp [4] Burton, T. and Eaton, J.K., 2002, Analysis of a fractional-step method on overset grids, J. Comp. Physics, 177, pp [5] Deutsch, E. and Simonin, O., 1991, Large eddy simulation applied to the motion of particles in stationary homogeneous fluid turbulence, in Turbulence Modification in Multiphase Flow, ASME-FED vol. 110, pp [6] Elghobashi, S. and Truesdell, G.C., 1993, On the two-way interaction between homogeneous turbulence and dispersed solid particles. I. Turbulence modification, Phys. Fluids, 5, pp [7] Germano, M., Piomelli, U., Moin, P. and Cabot, W.H., 1991, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids, 3, pp [8] Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yusof, J., 2000, Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comp. Physics, 161, pp [9] Fevrier, P. and Simonin, O., 2001, On the spatial distribution of heavy particle velocities in turbulent flow: from continuous field to particulate chaos, in Turbulence and Shear Flow Phenomena, Second Intnl. Symp., KTH, Stockholm [10] Fukagata, K., Zahari, S., Kondo, S. and Bark, F.H., 2001, Anomalous velocity fluctuations in particulate turbulent channel flow, Intnl. J. Multiphase Flow, 27, pp [11] Hu, H.H., Patankar, N.A. and Zhu, M.-Y., 2001, Direct numerical simulations of fluid-solid systems using Arbitrary-Lagrangian-Eulerian technique, J. Comp. Physics, 169, pp [12] Laviéville, J., Simonin, O., Berlemont, A., Chang, Z., 1997, Validation of inter-particle collision models based on Large-Eddy Simulation in gas-solid turbulent homogeneous shear flow, Proc. 7th Intnl. Symp. on Gas-Particle Flows, ASME Fluids Engineering Division Summer Meeting, FEDSM [13] Lesieur, M. and Metais, O., 1996, New trends in Large-Eddy Simulations of turbulence, Ann. Rev. Fluid Mech., 28, pp

12 12 [14] Miller, R.S. and Bellan, J., 1999, Direct numerical simulation of a confined three-dimensional gas mixing layer with one evaporating hydrocarbon-droplet-laden stream, 384, pp [15] Miller, R.S. and Bellan, J., 2000, Direct numerical simulation and subgrid analysis of a transitional droplet laden mixing layer, Phys. Fluids, 12, pp [16] Pismen, L.M. and Nir, A., 1978, On the motion of suspended particles in stationary homogeneous turbulence, J. Fluid Mech., 84, pp [17] Reeks, M.W., 1977, On the dispersion of small particles suspended in an isotropic turbulent field, J. Fluid Mech., 83, pp [18] Simonin, O., 1991, Second-moment prediction of dispersed-phase turbulence, in Proc. 8th Symp. on Turbulent Shear Flows, 1, pp [19] Simonin, O., Deutsch, E. and Boivin, M., 1995, Comparison of Large-Eddy Simulation and second-moment closure of particle fluctuating motion in two-phase turbulent shear flows, Turbulence and Shear Flows 9, pp , Springer-Verlag. [20] Squires, K.D. and Eaton, J.K., 1990, Particle response and turbulence modification in isotropic turbulence, Phys. Fluids, 2, p [21] Sundaram, S. and Collins, L.R., 1999, A numerical study of the modulation of isotropic turbulence by suspended particles, 379, pp [22] Wang, Q. and Squires, K.D., 1996, Large eddy simulation of particle-laden turbulent channel flow, Phys. Fluids, 8, pp [23] Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, T. and Tsuji, Y., 2001, Large-eddy simulation of turbulent gas-particle flow in a vertical channel: effect of considering inter-particle collisions, J. Fluid Mech., 442, pp [24] Yeh, F. and Lei, U., 1991, On the motion of small particles in a homogeneous turbulent shear flow, Phys. Fluids A, 3, pp