Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence

TL;DR

This paper introduces a symmetry reduction method for fluid flow data that enhances the analysis of complex turbulent dynamics by removing translation drifts, enabling clearer identification of invariant solutions and low-dimensional structures.

Contribution

The study develops a continuous symmetry reduction technique for fluid flows in rectangular channels, improving dynamic mode decomposition by eliminating translation symmetries.

Findings

SRDMD captures dynamics near invariant solutions like travelling waves.

The method reveals episodes of turbulence approximable by low-dimensional linear models.

Demonstrated on plane-Couette and plane-Poiseuille flow simulations.

Abstract

Data-driven dimensionality reduction methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.