Lagrangian filtered density function for LES-based stochastic modelling of turbulent particle-laden flows

Abstract

The Eulerian-Lagrangian approach based on Large-Eddy Simulation (LES) is one of the most promising and viable numerical tools to study turbulent dispersed flows when the computational cost of Direct Numerical Simulation (DNS) becomes too expensive. The applicability of this approach is however limited if the effects of the Sub-Grid Scales (SGS) of the flow on particle dynamics are neglected. In this paper, we propose to take these effects into account by means of a Lagrangian stochastic SGS model for the equations of particle motion. The model extends to particle-laden flows the velocity-filtered density function method originally developed for reactive flows. The underlying filtered density function is simulated through a Lagrangian Monte Carlo procedure that solves for a set of Stochastic Differential Equations (SDEs) along individual particle trajectories. The resulting model is tested for the reference case of turbulent channel flow, using a hybrid algorithm in which the fluid velocity field is provided by LES and then used to advance the SDEs in time. The model consistency is assessed in the limit of particles with zero inertia, when "duplicate fields" are available from both the Eulerian LES and the Lagrangian tracking. Tests with inertial particles were performed to examine the capability of the model to capture particle preferential concentration and near-wall segregation. Upon comparison with DNS-based statistics, our results show improved accuracy and considerably reduced errors with respect to the case in which no SGS model is used in the equations of particle motion.2

Large-eddy simulation of turbulent dispersed flows: a review of modelling approaches

“…This is shown in Fig. 17, where the Voronoï diagrams of near-wall particle clusters are shown [51]. The PDFs shown in Fig.…”

“…For a more detailed description of the formalism, the reader is referred to [83,85] and references therein. Figures 15, 16, and 17 show the predictions of the LFMDF approach in terms of concentration and near-wall segregation of inertial particles in turbulent channel flow (at Re τ = 300 based on the half channel height) [51]. A posteriori assessment made against DNS and LES without particle SGS model shows improved predictions of particle statistics, especially at intermediate Stokes numbers as demonstrated in Fig.…”

“…a field from which part of the fluctuations (mainly the highest frequency ones) has been removed by filtering. As discussed also in [72], however, this behaviour is due to the subgrid modelling error in the LES equations for the fluid phase, which leads to an overshoot of the rms of the fluid velocity components (not shown) when very coarse grids are used, as done in [51] where the LES grid is 8 times coarser than the DNS one in each spatial direction. The observed overshoot is a rather well-known behaviour of coarse LES, especially for the RMS of the streamwise component [72].…”

“…When St SG S << 1, particles are SGS-non-inertial with respect to subgrid eddies (here we adopt the terminology of [125]): They are sensitive to the high-frequency fluctuations of subgrid turbulence (which are filtered) while behaving as tracers with respect to the large eddies. In this limit, which is encountered, for instance, in solar-power receivers (where St K ∼ O (1) and St l << 1), a particle SGS model is required to [72,73] (filled squares channel flow at shear Reynolds number Re τ = 150 based on the half channel height); Innocenti et al [51] (open squares, channel flow at Re τ = 300); Fede and Simonin [31] (diamonds HIT at Taylor-microscale Reynolds number Re λ = 34.1); Gobert and Manhart [44] (circles HIT at Re λ = 52); Cernik et al [21] (triangles decaying HIT with initial Re λ = 74). The thick solid line in panel (b) represents the dependence of the parameter St SG S /St K on /η K that can be derived in HIT solve Eq.…”

Modeling the effects of small turbulent scales on the drag force for particles below and above the Kolmogorov scale

“…[24] for a comprehensive review). In most of these approaches, the unresolved scales of the Lagrangian dynamics are reproduced using stochastic models for the flow on subgrid scales [25][26][27][28][29][30][31][32][33][34][35]. Note also the approach of Ref.…”