Periodic Reporting for period 2 - UniTED (Unraveling Turbulence through Ensemble Decomposition)
Reporting period: 2023-07-01 to 2024-12-31
The goal of UniTED is to combine numerical simulations and theory to better understand and model the statistics of high-Reynolds (high-Re) number small-scale turbulence. The proposed framework for this is “ensemble decomposition,” i.e. the idea that the comparably complex dynamics of turbulence can be decomposed into sub-ensembles displaying simpler statistics. This idea has been repeatedly employed in the field, e.g. in the framework of superstatistics and multifractal modeling. In preliminary work, we developed the concept for Lagrangian turbulence, which ultimately led up to this proposal.
Within UniTED, we want to generalize the idea to the full spatial complexity of Eulerian turbulence, both theoretically as well as computationally through four objectives:
(A) Investigating the multi-scale nature of turbulence by means of high-resolution direct numerical simulations (DNS) of turbulence
(B) Exploring the idea of ensemble decomposition in the framework of statistical field theories of turbulence
(C) Develop ensemble simulations (EnSims) of turbulence that are capable of capturing small-scale features of high-Reynolds number turbulence
(D) Develop computationally affordable reduced-order models that leverage the results of objectives (A)-(C)
Within UniTED, we want to generalize the idea to the full spatial complexity of Eulerian turbulence, both theoretically as well as computationally through four objectives:
(A) Investigating the multi-scale nature of turbulence by means of high-resolution direct numerical simulations (DNS) of turbulence
(B) Exploring the idea of ensemble decomposition in the framework of statistical field theories of turbulence
(C) Develop ensemble simulations (EnSims) of turbulence that are capable of capturing small-scale features of high-Reynolds number turbulence
(D) Develop computationally affordable reduced-order models that leverage the results of objectives (A)-(C)
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.
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.
We are currently in the process of publishing the results achieved so far, see Results page. For the second half of the project, we will continue our work on the four objectives. On the theoretical side, we aim to apply the results obtained so far to three-dimensional turbulence. On the computational side, we expect to bring EnSims to maturity, perform detailed comparisons to standard direct numerical simulations, and explore EnSims in application scenarios.