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 modelling their impact through the subgrid-scale stress tensor ${\tau }_{\textit{ij}}$ . Traditional models approximate ${\tau }_{\textit{ij}}$ 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 the 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 be 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 the flatness of these distributions are quantified. Our results emphasise that fluctuations are an integral component of the small-scale feedback onto the large-scale dynamics and should be incorporated into subgrid-scale modelling through an appropriate stochastic framework.