Modelling turbulent dispersion with neural stochastic differential equations

In dilute particle-laden turbulent flows, the dynamics of low inertial particles is mainly driven by the fluctuating motion of the carrier fluid phase. To study the particle dynamics in these flows, the Eulerian-Lagrangian (EL) approach is commonly applied where the fluid transport is resolved by the volume-filtered Navier-Stokes equations on an Eulerian grid and each particle is tracked in a Lagrangian manner by solving Newton’s equation of motion. To account for the the carrier phase fluctuations on the particle transport, a stochastic differential equation (SDE) for the fluid velocity ‘seen’ by a particle (us)can be used for the fluid velocity when computing drag force [1]. However, the generalisation of existing models to a range of flow configurations spanning from homogeneous isotropic turbulence (HIT) to aerosol transport in complex geometries such as lung airways is questionable due to manually-tuned empirical constants in the modelling parameters. Therefore, we use neural networks (NNs) to learn modelling terms in the SDE for us (‘neural SDEs’), based on recent neural SDE studies for Brownian motion [2]. We propose a neural SDE for us for isotropic particle-laden flows with a following form: dus = −GNN (τL, ∆) ¯ · (us − uef )dt + BNN (τL, ∆) ¯ · dw. (1)Here, GNN and BNN are the NN modelled return-to-equilibrium and the diffusion terms, respectively. These are functions of the filter/mesh size (∆¯ ), and the subgrid timescale, τL = ksgs/εsgs, where ksgs is the subgrid kinetic energy and εsgsis the subgrid dissipation rate. w is a Wiener process with distribution N (0, dt) and uef is the filtered/resolved fluid velocity. To generate ‘ground truth’ for NN training, we performed direct numerical simulations (DNSs) of decaying particle-laden HIT at Reλ = 20 on a 2563 grid with one-way coupling. We focused on tracer particles and used DNS to generate exact particle trajectories with us = uf , where uf is the fluid velocity. We then applied a top-hat filter at widths∆ = ¯ {3∆DNS, 5∆DNS, 7∆DNS, 9∆DNS} to obtain uef , ksgs and εsgs for modelling GNN and BNN . Each NN term was constructed using a densely connected feed-forward NN with two layers and 25 neurons per-layer. We assessed our neural SDE against DNS results of the kinetic energy decay, k = tr(u ⊗ u) in a priori and a posteriori tests (Open FOAM simulations). The model prediction fits the DNS data well, compared to the case only accounting for the filtered fluid velocity (Figure 1a). A posteriori analysis showed similar improvements in k (Figure 1b). We will extend this model to inertial particles with varying Stokes number, and wall-bounded flows such as turbulent channel flow in a future study.