Validating the Boltzmann approach to the Large-Eddy simulations of forced homogeneous incompressible turbulence

TL;DR

This work tests whether a Smagorinsky-augmented lattice Boltzmann MRT framework for forced homogeneous isotropic turbulence operates in a kinetic or hydrodynamic regime. By comparing a high-resolution DNS ($800^3$) with a coarser MRT-LBM-LES ($100^3$) at $Re=2\times10^4$, the authors compute the turbulent Knudsen number $K_t$ and analyze spectra and higher-order statistics. They find $\tau_e-\tfrac{1}{2}= \mathcal{O}(10^{-3})$ with no negative excursions, indicating a strictly hydrodynamic regime and preserved locality, while standard LES behavior is recovered in spectra and intermittency. The results validate the use of LBM-Smagorinsky LES as a conventional hydrodynamic LES, retaining LBM’s computational advantages and locality for FHIT.

Abstract

The simulation of turbulent flows remains a central challenge, as even our most powerful computers cannot resolve the finest scales of motion in many flows of practical interest. As a result, the effects of unresolved scales on large eddies must be modelled via closures and coarse-graining procedures. Large-eddy simulation (LES) traditionally coarse-grains Navier-Stokes equations using Smagorinsky's effective viscosity model. This has the merit of simplicity but fails to account for strong non-equilibrium effects, as they typically arise in most flows in the vicinity of solid walls, the reason being that the notion of eddy viscosity assumes scale separation between small and large eddies, an assumption that fails for high-Reynolds flows far from equilibrium. The lattice Boltzmann method (LBM) offers an alternative by coarse-graining at the kinetic level, potentially capturing non-equilibrium effects beyond reach of hydrodynamic closures. This paper addresses whether LBM-Smagorinsky LES of forced homogeneous isotropic turbulence (FHIT) exhibits kinetic behavior. We test whether the turbulent Knudsen number $K_t$, measuring scale separation, reaches order one (kinetic regime) or remains asymptotically small (hydrodynamic regime). Using reference DNS ($800^3$) and iso-Reynolds LES ($100^3$) at $Re = 2 \times 10^4$, we quantify $K_t$ via spatial maps, temporal statistics, energy spectra, and higher-order moments. Results show $K_t \sim O(10^{-3})$, strictly positive without negative excursions, with spectra and flatness following canonical LES behavior. We conclude that despite its kinetic formulation, LBM-Smagorinsky LES operates in the hydrodynamic regime, with small FHIT eddies remaining in local equilibrium with larger ones, validating Smagorinsky viscosity and confirming that LBM-LES functions as conventional hydrodynamic LES while preserving LBM efficiency and locality.

Figures (13)

• Figure 1: Temporal evolution of the time-averaged relative error for different grid resolutions. Faint lines show raw instantaneous values, while solid lines represent the 20-point rolling average. The vertical dashed line at $t/t_0 = 200$ marks the start of the statistically stationary period used for turbulence post-processing.
• Figure 2: Assessment of statistical isotropy. (a) Absolute deviation of directional energy fractions from the isotropic value $1/3$; (b) Anisotropy invariants on the Lumley triangle showing the DNS trajectory approaching the isotropic vertex. X-axis: $\xi = (III_b/2)^{1/3}$ (transformed third invariant), Y-axis: $\eta = (II_b/3)^{1/2}$ (transformed second invariant). Statistical isotropy is validated for the $256^3$ DNS at ${Re}_\lambda \approx 180$, $\eta^+ \approx 1.0018$.
• Figure 3: Validation of DNS energy spectra against Pope model (Eq. \ref{['eq:pope_model']}) using normalized wavenumber $k\eta$ and Kolmogorov-scaled energy $E(k)/(\nu^5\varepsilon)^{1/4}$. Collapse across resolutions and $Re_\lambda$ confirms grid independence and $k^{-5/3}$ scaling.
• Figure 4: Energy spectra $E(k)$ at $Re = 5000$ comparing reference DNS (blue dashed line) with LES on a $128^3$ grid using varying Smagorinsky constants $C$. The plot illustrates the sensitivity to sub-grid dissipation: the $C=0$ case (green) collapses to the DNS profile. Increasing the constant to $C=0.1$ (purple) and $C=0.16$ (orange) yields intermediate dissipation levels, whereas $C=0.25$ (brown) results in significant suppression of small scales (over-dissipation). The theoretical Kolmogorov scaling $k^{-5/3}$ is shown for reference.
• Figure 5: Turbulence intermittency quantification. (a) ESS scaling $S_p \sim S_3^{\zeta_p}$ reveals anomalous exponents $\zeta_p$ (inset: comparison with She-Leveque model (SL94) and experimental anomalies of Benzi et al. (B93)). (b) Flatness $F_4(r)$ exceeding the Gaussian value of 3 indicates small-scale intermittency.