Equilibrium and travelling-wave solutions of plane Couette flow

Abstract

We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space onto suitably chosen 2-or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. arXiv:0808.3375v2 [physics.flu-dyn]

Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows

“…This theory effectively takes R out of consideration and reduces an unsteady three-dimensional Navier-Stokes problem to a steady two-dimensional Navier-Stokes problem coupled to an advection-diffusion equation and a wave equation. Recognition that the solutions of Nagata (1990), Waleffe (1997), Faisst & Eckhardt (2003), Wedin & Kerswell (2004), Wang et al (2007) and Gibson et al (2009) were finite-R analogues of VWI states stimulated application of the theory to plane Couette flow (Hall & Sherwin 2010). Outcomes of that asymptotic approach agreed remarkably with the 'lower branch' equilibria found in Wang et al (2007) and explicitly provide asymptotic scaling relations R 1 and R 11/12= 0.916 for roll and wave components.…”

“…Two facets of this problem that exist at opposite ends of the Reynolds-number spectrum are the mechanism of † Email address for correspondence: Hugh.Blackburn@monash.edu transition in flows such as Hagen-Poiseuille flow and plane Couette flow, which are linearly stable and whose instability mechanisms have not yielded to analysis with linear tools, and the universal inertial-range scaling of turbulence kinetic energy with wavenumber first proposed by Kolmogorov (1941), which applies at very high Reynolds numbers. In recent years much interest in the dynamics of shear flows has come through the discovery of three-dimensional nonlinear equilibrium states (Nagata 1990;Waleffe 1997Waleffe , 2001Faisst & Eckhardt 2003;Wedin & Kerswell 2004;Wang, Gibson & Waleffe 2007;Gibson, Halcrow & Cvitanovic 2009). These equilibria have been shown to act either as 'edge states' which locally divide the basin of attraction between relaminarized or turbulent outcomes ('lower branch' equilibria) or as 'organizing centres' about which the flow slowly cycles during the approach to turbulence ('upper branch' equilibria) (Wang et al 2007).…”

A doubly localized equilibrium solution of plane Couette flow

“…Avila et al (2013) found a streamwise-localized relative periodic orbit of pipe flow that closely resembles the transient turbulent puffs of Hof et al (2006). Deguchi et al (2013) and Gibson & Brand (2014) independently found spanwise-localized forms of the periodic EQ7/HVS solution of Itano & Generalis (2009);Gibson et al (2009). Gibson & Brand (2014) also presented a number of spanwise-localized and wall-normal-localized traveling waves of channel flow.…”

“…The software and the numerical data for the doubly-localized solution are available at www.channelflow.org Gibson 2014). Initial guesses for the doubly-localized solutions were produced by applying streamwise windowing to the spanwise-localized forms of EQ7 solution from GB14, or twodimensional windowing to the doubly-periodic EQ7 solution from Gibson et al (2009). We used the same tanh-based windowing function as in GB14 equation (2.4), replacing z with x for a streamwise windowing function…”

“…The alignment of the y-averaged flow with the x and z axes also results from symmetry: σ z symmetry requires that u and w are even and odd in z, respectively, about z = 0, and σ xy symmetry requires thatū andw are odd and even in x about x = 0. Periodicity in x and z requires the same symmetries about the x = ±L x /2 and z = ±L z /2 edges of the computational domain, so that the y-averaged flow aligns with these edges as well (see Gibson et al (2009)). …”