With respect to (A), we have run both, individual high-Reynolds number simulations to obtain reference data sets as well as ensembles of smaller simulations to model high-Re flows. In particular, we have conducted a multi-scale analysis of turbulence focusing on the velocity gradients conditional on various variables related to the velocity gradient filtered at different scales.
As part of (B), we are currently investigating ensembles of Gaussian fields with turbulence-like statistics. We have meanwhile developed a good understanding of many of the non-Gaussian features of these field ensembles. We also studied dynamical models of small-scale turbulence for which ensemble decomposition is exact and the corresponding statistical field theory is analytically tractable. This provides us with useful information for further theoretical investigations of Navier-Stokes turbulence.
(C) aims at developing ensemble simulations (EnSims). We first investigated the impact of large-scale intermittency on small-scale statistics, before progressing to internal, small-scale intermittency. Our results on large-scale flow variations show significant statistical differences in small-scale properties depending on the large-scale driving.
Based on this, we currently investigate EnSims for small-scale turbulence. We have obtained first very promising results on modeling high-Reynolds-number flows with comparably small, affordable simulations. We are currently systematically exploring how computationally affordable the ensemble simulation can be made while maintaining a reasonable approximation to a high-Re reference simulation.
As part of (D), we have already made a significant step toward the reduced-order modeling of small-scale turbulence by developing a dynamical model for Lagrangian velocity gradients that can accurately reproduce statistics of a reference high-Re dataset. This physics-informed machine learning model represents a significant reduction of computational cost while delivering high-fidelity turbulence data. With respect to future work, this enables a combination with EnSims to model high-Re flows with computationally affordable reduced-order models.
