PCM-CMM-2015 – 3rd Polish Congress of Mechanics & 21st Computer Methods in Mechanics September 8th–11th 2015, Gda ´ nsk, Poland PCM-CMM-2015 – 3rd Polish Congress of Mechanics & 21st Computer Methods in Mechanics September 8th–11th 2015, Gda ´ nsk, Poland PCM-CMM-2015 – 3rd Polish Congress of Mechanics & 21st Computer Methods in Mechanics September 8th–11th 2015, Gda ´ nsk, Poland A stochastic model for Lagrangian particle tracking in large-eddy simulation velocity fields A. Innocenti 1 , S. Chibbaro 2 , M.V. Salvetti 3 , C. Marchioli 4 , A. Soldati 5 1, 3 DICI, University of Pisa, Pisa (Italy) e-mail: alessioinnocenti@yahoo.it 1 , mv.salvetti@ing.unipi.it 3 2 D’Alembert Institute, Pierre and Marie Curie University and CNRS, Paris (France) e-mail: chibbaro@ida.upmc.fr 4, 5 DIEG, University of Udine & Department of Fluid Mechanics, CISM, Udine (Italy) e-mail: marchioli@uniud.it 4 , soldati@uniud.it 5 Abstract A Lagrangian stochastic model is proposed for tracking the inertial particles in fluid velocity fields obtained in large-eddy simulations; the model is formulated for the fluid velocity seen by the particles along their trajectory. The behaviour of the model is first investigated for tracer particles in a turbulent channel flow. It is checked that in this limit case similar statistics to these given by a fluid phase LES are obtained. Next, stochastic model is used for Lagrangian tracking of particles of different inertia. The results will be shown in the final presentation and will be compared to particle statistics and concentration obtained in DNS and in LES with no model for particle equations. Keywords: inertial particles, Lagrangian tracking, large-eddy simulation, stochastic subgrid scale modeling 1. Introduction The dispersion of small inertial particles in inhomogeneous turbulent flows is important in a number of industrial applica- tions and environmental phenomena, such as, mixing, combus- tion, depulveration, spray dynamics, pollutant dispersion or cloud dynamics. Direct Numerical Simulations (DNS) of turbulence coupled with Lagrangian Particle Tracking (LPT) demonstrated their capability to capture the mechanisms characterizing parti- cle dynamics in turbulent flows. Due to the computational re- quirements of DNS, however, analysis of problems characterized by complex geometries and high Reynolds numbers demands al- ternative approaches; Large-Eddy Simulation (LES) is increas- ingly gaining popularity, especially for cases where the large flow scales control particle motion. LES is based on a filtering ap- proach of the fluid phase governing equations; thus, only the fil- tered fluid velocity is available for particle tracking and particles are prevented from interacting with the small (unresolved) Sub- Grid Scales (SGS) of turbulence. This may strongly influence clustering of inertial particles and lead to significant underesti- mation of particle preferential concentration and deposition rates ( see e.g. [1]). Hence, there is currently a general consensus about the need to model the effect of SGS turbulence on particle dynamics. Different kinds of SGS models for particle motion equations were proposed in the literature, e.g. filtering inversion or approx- imate deconvolution [2, 3], fractal interpolation [2], stochastic modeling [4, 5] or mixed models [6]. Previous studies [7, 8, 9], focusing on the error purely due to filtering of the fluid velocity field seen by the particle along its trajectory, showed that this er- ror is stochastic and may exhibit a non-Gaussian and intermittent nature. The work is aimed at developing and appraising a new stochastic model for subgrid scales of large eddy simulation of turbulent polydispersed two-phase flows. The model is based on the formalism for the filtered density function (FDF) approach in LES simulations. Contrary to the FDF used for turbulent reactive single-phase flows, the present formalism is based on Lagrangian quantities and, in particular, on the Lagrangian filtered mass den- sity function (LFMDF) as the central concept [10]. A first example of Langevin model constructed within the above formalism is proposed considering isotropic sub-grid fluc- tuations, but taking into account crossing-trajectory effects and paying attention to the consistency of the model with the fluid limit case. 2. Physical Problem, Numerical Methodology and Model- ing The physical problem considered in this study is the disper- sion of inertial particles in turbulent channel flow. The refer- ence geometry consists of two infinite flat parallel plates sepa- rated by a distance 2h. The origin of the coordinate system is located at the center of the channel with the x, y and z axes pointing in the streamwise, spanwise, and wall-normal direc- tions, respectively. Periodic boundary conditions are imposed on the fluid velocity field in the homogeneous directions (stream- wise, x, and spanwise, y), no-slip boundary conditions are im- posed at the walls. The size of the computational domain is L x ×L y ×L z =4πh ×2πh ×2h. The shear Reynolds number is Re ∗ = u ∗ h/ν = 300, where u ∗ = √ τ w /ρ is the shear velocity based on the mean wall shear stress. The flow solver is based on a Fourier-Chebyshev pseudo- spectral method to discretize the LES equations. The SGS models considered for the fluid phase are the classical and the dynamic Smagorinsky model. The Lagrangian tracking is based on the following equation of motion:        dx p (t) dt = U p (t) dU p (t) dt = 1 τ p (U s (t) − U p (t))(1 + 0.15Re 0.687 p ) (1) In these equations, U s (t)= U(t, x p (t)) is the fluid velocity “seen”, i.e. the fluid velocity sampled along the particle trajec- tory x p (t), τ p is the particle relaxation times and Re p the par- 521