Low-dimensional exact coherent states in plane Couette flow

Abstract

A new approach to understanding the dynamics of moderate-Reynolds number wall bounded flows has emerged recently, based on the computation of steady (equilibrium) or exactly recurring (periodic) solutions of the Navier-Stokes equations. Such solutions are thought to represent coherent structures, and the connections in state space between these solutions can explain the dynamics and self-sustaining nature of wall-bounded turbulent flows.

Herein new equilibrium solution branches for plane Couette flow are reported which add to the inventory of known solution branches. The exact solutions are found by projecting known equilibria onto the resolvent modes of McKeon and Sharma (2010 to generate approximate solutions that are subsequently used as seeds in a Newto Krylov-hookstep search. Searches initialised with these projections have a convergence rate of 96%. The low-dimensional nature of the resulting equilibria is attributed to the projection generated by the resolvent model; resolvent modes for a given equilibrium solution span a low-dimensional space which the solution approximately inhabits. This property is exploited to generate new branches of equilibria in plane Couette flow, and to jump to known branches. The new branches include bifurcations from previously known bifurcation curves and a disconnected bifurcation curve which displays interesting behaviour when continued in Reynolds number. The unstable manifolds of the new equilibria are computed to expand the known state space structure of plane Couette flow.

Additionally, ten new periodic orbits in plane Couette flow are found by projecting known periodic orbits and quasi-recurrent segments of turbulent flows onto the resolvent modes of McKeon and Sharma (2010) to generate low-order representations of periodic flow, points along which are then used as initial conditions in a Newton-Krylov-hookstep search. Moreover, projections of periodic orbits find one new equilibrium solution. The new periodic solutions are in the neighbourhoods of previously known orbits and reveal the geometry of state space in the region separating the laminar and turbulent attractors. We show that the periodic orbits are appropriate representations of the self-sustaining process proposed by Waleffe (1997) and we examine their effect on turbulent trajectories in state space.

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Identifiers

ORCID for Ati Sharma: orcid.org/0000-0002-7170-1627

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