Kinetic data-driven approach to turbulence subgrid modeling

TL;DR

The paper tackles the cost of DNS for turbulent flows by introducing a data-driven, kinetic SGS closure within the LBM framework. It learns a collision operator $\Omega^{NN}$ from DNS data to form a Neural LBM (NLBM) that remains local in space and time, achieving stable coarse-grained simulations that reproduce key statistics, including energy spectra and anomalous scaling, and even backscatter. The approach uses unrolled trajectory training and a physics-constrained ANN to deliver a physically interpretable, MRT-like relaxation in moment space, with promising generalization in HIT settings and potential extension to more complex flows and systems. This work lays the groundwork for hybrid data-driven kinetic models capable of capturing the macroscopic dynamics of diverse turbulent and multi-physics systems while reducing computational cost.

Abstract

Numerical simulations of turbulent flows are well known to pose extreme computational challenges due to the huge number of dynamical degrees of freedom required to correctly describe the complex multi-scale statistical correlations of the velocity. On the other hand, kinetic mesoscale approaches based on the Boltzmann equation, have the potential to describe a broad range of flows, stretching well beyond the special case of gases close to equilibrium, which results in the ordinary Navier-Stokes dynamics. Here we demonstrate that, by properly tuning, a kinetic approach can statistically reproduce the quantitative dynamics of the larger scales in turbulence, thereby providing an alternative, computationally efficient and physically rooted approach towards subgrid scale (SGS) modeling in turbulence. More specifically we show that by leveraging on data from fully resolved Direct Numerical Simulation (DNS) we can learn a collision operator for the discretized Boltzmann equation solver (the lattice Boltzmann method), which effectively implies a turbulence subgrid closure model. The mesoscopic nature of our formulation makes the learning problem fully local in both space and time, leading to reduced computational costs and enhanced generalization capabilities. We show that the model offers superior performance compared to traditional methods, such as the Smagorinsky model, being less dissipative and, therefore, being able to more closely capture the intermittency of higher-order velocity correlations. This foundational work lays the basis for extending the proposed framework to different turbulent flow settings and -- most importantly -- to develop new classes of hybrid data-driven kinetic-based models capable of faithfully capturing the complex macroscopic dynamics of diverse physical systems such as emulsions, non-Newtonian fluid and multiphase systems.

Figures (11)

• Figure 1: Snapshots of the vorticity magnitude ($|\omega|$) from 3D simulations of HIT at $\rm{Re} \approx 6000$. In panel a) the upper half of the domain is taken from filtered DNS data with $\rm{cg} = 2$, while the lower half is obtained from a simulation using NBLM, trained and validated on the same flow conditions, albeit with different initial configurations. We show that at a qualitative level the structures generated by NLBM closely resemble those from filtered DNS. This is further highlighted in panel b) and c) where we show 2D slices of $|\omega|$ (cf. red box in panel a) ), respectively at coarse graining factor $\rm{cg} = 2$ and $\rm{cg} = 4$.
• Figure 2: Schematic representation of the training process for the turbulence SGS model. The upper panel corresponds to the DNS simulation on a $L^3$ grid, while in lower panel the Neural LBM (NLBM) operates on a $(L/\rm{cg})^3$ grid, with coarse-graining factor $\rm{cg}$ via subsampling. The mapping between DNS into coarse-grained data is given by the application of a filter: the pre-collision state at a generic time step $t$ at the coarse grained level ( $f_{\rm cg}^{\rm pre}(t)$ ) is obtained by filtering the DNS pre-collision state ( $f_{\rm DNS}^{pre}(t)$ ) (dependence on space has been omitted for conciseness). Similarly, the post-collision data at the coarse-grained level ( $f_{\rm cg}^{\rm post}(t)$ ) is obtained by first filtering the post-collision DNS state at time step $t + \rm{cg} \Delta t$, and then by applying the inverse of the streaming operator. Following this procedure it is possible to create a dataset of arbitrary size for training an ANN to which we assign the task of minimizing the mismatch between $\tilde{f}_{\rm cg}^{\rm post}(t) = \Omega_{\rm NN}(f_{\rm cg}^{\rm pre}(t))$ and $f_{\rm cg}^{\rm post}(t)$ under a given error-metric $\mathcal{L}$.
• Figure 3: Energy spectrum for simulations of HIT at $\rm{Re} \approx 6000$. The results from DNS (blue curve) are compared with NLBM (red) and Smagorinsky (green). For the two SGS models we report the average spectrum from $80$ simulations starting from different initial conditions. The shaded curves corresponds to one standard deviation from the average value.
• Figure 4: Structure functions (cf. Eq. \ref{['eq:structure-fun']}) of order $p$, ranging between $p = 1$ to $p = 6$, versus $S^3$, with in blue data from filtered DNS with $\rm{cg} = 4$, in red data from simulations using NLBM, and in green data from simulations using the Smagorinsky model. The inset shows the deviation for the scaling exponents $\xi_p$ from the K41 scaling $\frac{p}{3}$.
• Figure 5: Probability distribution function (PDF) of the fitted value of the Smagorinsky constant $C^2$ from NLBM data. The inset highlights that the average value $<C^2_{\rm NLBM}> \approx 0.11$ is about a factor two smaller than the one used in simulations with the Smagorinsky SGS model ($C^2 = 0.2$). The presence of a tail with negative values highlights the fact that in NLBM it is possible to capture the inverse transfer of energy from small to large scales.