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

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.

arXiv commentIn press in the Journal of Fluid Mechanics

References in corpus (26)

• A critical layer model for turbulent pipe flow
• Travelling waves in pipe flow
• The Significance of Simple Invariant Solutions in Turbulent Flows
• Recurrent motions within plane Couette turbulence
• Visualizing the geometry of state space in plane Couette flow
• Turbulence transition and the edge of chaos in pipe flow
• Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
• On a self-sustained process at large scale in the turbulent channel flow
• Spectral analysis of the budget equation in turbulent channel flows at high Re
• Equilibrium and traveling-wave solutions of plane Couette flow
• Periodic orbits near onset of chaos in plane Couette flow
• Self-sustaining process of minimal attached eddies in turbulent channel flow
• Revealing the state space of turbulent pipe flow by symmetry reduction
• Exact coherent states and connections to turbulent dynamics in minimal channel flow
• An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
• Edge states for the turbulence transition in the asymptotic suction boundary layer
• Cause-and-effect of linear mechanisms sustaining wall turbulence
• On the self-sustained nature of large-scale motions in turbulent Couette flow
• Scaling of Lyapunov Exponents in Homogeneous Isotropic Turbulence
• Coarse graining the state space of a turbulent flow using periodic orbits
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Symmetry reduction in high dimensions, illustrated in a turbulent pipe
• Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations
• Sustaining processes from recurrent flows in body-forced turbulence
• Time-periodic inertial range dynamics
• Global energy budgets in turbulent Couette and Poiseuille flows

Cited by in corpus (3)

• Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback
• Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics
• Turbulence suppression in plane Couette flow using reduced-order models