Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

TL;DR

This study uses a reduced-order model to analyze multi-scale invariant solutions in plane Couette flow at Re=1200, revealing the importance of unsteady solutions for capturing complex turbulent dynamics.

Contribution

The paper demonstrates that incorporating unsteadiness into invariant solutions is crucial for accurately representing multi-scale turbulence in a reduced-order model.

Findings

Most multi-scale physical processes are captured by the model

Equilibrium solutions alone do not reproduce energy balance

Long-period periodic orbits better describe turbulent dynamics

Abstract

Plane Couette flow at Re=1200 (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, {Phys. Rev. Fluids}, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8): i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatiotemporal dynamics associated with the known multi-scale phenomena.