Fluctuations around Turbulence Models

Numerical simulations of turbulent flows at realistic Reynolds numbers generally rely on filtering out small scales from the Navier Stokes equations and modeling their impact through the Reynolds stress tensor $τ_{ij}$. Traditional models approximate $τ_{ij}$ solely as a function of the filtered velocity gradient, leading to deterministic subgrid scale closures. However, small scale fluctuations can locally exhibit instantaneous values whose deviation from the mean can have a significant influence on flow dynamics. In this work, we investigate these effects by employing direct numerical simulations combined with Gaussian filtering to quantify subgrid scale effects and evaluating the local energy flux in both space and time. The mean performance of the canonical Clark model is assessed by conditioning the energy flux distributions on the invariants of the filtered velocity gradient tensor, $Q$ and $R$. The Clark model captures to a good degree the mean energy flux. However, the fluctuations around these mean values for given ($Q,R$) are of the order of the mean displaying fat tailed distributions. To become more precise, we examine the joint distributions of true energy flux and the predictions from both the Clark and the Smagorinsky models. This approach mirrors the strategy adopted in early stochastic subgrid scale models. Clear non Gaussian characteristics emerge from the obtained distributions, particularly through the appearance of heavy tails. The mean, the variance, the skewness and flatness of these distributions are quantified. Our results emphasize that fluctuations are an integral component of the small scale feedback onto large scale dynamics and should be incorporated into subgrid scale modeling through an appropriate stochastic framework.

💡 Research Summary

This paper investigates the role of small‑scale fluctuations in large‑eddy simulation (LES) modelling by analysing direct numerical simulation (DNS) data filtered with a Gaussian kernel. The authors compute the sub‑grid stress tensor τₗᵢⱼ = uᵢuⱼₗ − uᵢₗuⱼₗ and the associated local energy flux Πₗ = −τₗᵢⱼ ∂ⱼuᵢₗ at a filter scale ℓ that lies well inside the inertial range (k ≈ 16). They condition the flux on the two invariants of the filtered velocity‑gradient tensor, Q = −½ Tr(∇u·∇u) and R = −⅓ Tr(∇u·∇u·∇u), thereby constructing conditional means ⟨Π|Q,R⟩ and conditional probability density functions (PDFs).

The first part of the analysis compares the DNS‑based conditional mean with the predictions of two classic deterministic sub‑grid models: the Clark model (τ ≈ ℓ² ∂ₖuᵢ ∂ₖuⱼ) and the Smagorinsky model (τ ≈ −ℓ²|S|S). In the (Q,R) diagram, the DNS shows forward cascade (positive Π) mainly in the region of vortex stretching (R > 0) and backscatter (negative Π) confined to the upper‑right quadrant associated with vortex compression. The Clark model reproduces the qualitative layout of the mean flux, but systematically underestimates its magnitude, especially in regions of strong forward transfer. The Smagorinsky model, by construction, yields non‑negative fluxes and therefore completely misses the backscatter observed in DNS.

The second, more detailed part examines the full PDFs of Π for a set of five representative (Q,R) points (the origin and one point in each quadrant). While the Clark model captures the overall shape of the distribution, the DNS PDFs display fluctuations of the same order as the mean and possess pronounced heavy tails. The positive tail (large forward transfer events) is especially under‑predicted by the Clark model, and the skewness and flatness (kurtosis) differ markedly. This discrepancy is traced back to the fact that the Clark closure retains only the leading ℓ² term of an infinite series expansion of the exact sub‑grid stress; higher‑order non‑local contributions (e.g., ℓ⁴ ∂⁴u terms) are omitted and are responsible for the extreme events seen in the DNS. The Smagorinsky model shows similar deficiencies, with its PDFs being narrower and lacking any negative excursions.

Joint PDFs of the true flux and the model predictions further highlight the non‑Gaussian character of the sub‑grid energy transfer. The probability of large positive fluxes is severely underestimated, while the probability of negative fluxes is essentially zero for Smagorinsky. These findings demonstrate that deterministic closures, even when calibrated to reproduce the mean cascade, fail to represent the stochastic nature of the small‑scale feedback onto the resolved scales.

In conclusion, the authors argue that fluctuations around the mean sub‑grid energy flux are an intrinsic and essential component of turbulent dynamics. Accurate LES therefore requires a stochastic sub‑grid model that can reproduce the observed heavy‑tailed PDFs, the correct skewness, and the occasional backscatter events. The paper suggests that future work should incorporate stochastic terms derived from the higher‑order terms of the exact stress expansion, possibly using data‑driven techniques to calibrate the noise statistics, thereby moving beyond purely deterministic eddy‑viscosity closures. This approach could improve the fidelity of LES in predicting both average and extreme turbulent events across a wide range of applications.