Color of turbulence

TL;DR

This paper develops a data-driven framework to model turbulent flow statistics using low-complexity stochastic linearized Navier-Stokes models with colored-in-time forcing, improving accuracy and tractability.

Contribution

It introduces a maximum entropy-based method for completing second-order statistics with colored forcing, enhancing turbulence modeling from first principles.

Findings

Colored-in-time forcing models turbulence more accurately.

The approach allows low-rank modifications to linearized dynamics.

Models are suitable for analysis, optimization, and control.

Abstract

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier-Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for colored-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that colored-in-time excitation of the Navier-Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization, and control design.