Lower Branch Coherent States in Shear Flows: Transition and Control

Abstract

Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of O(1) streaks, O(R −1 ) streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number R. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggest that the lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

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.…”

“…The physical mechanism supporting the self-sustaining lower branch states was proposed (Waleffe 1997) as a tripartite coupled nonlinear system in which the dominant component is two-dimensional high-and low-speed streaks of O(1) appearing on the underlying streamwise flow; these streaks are driven by weaker two-dimensional rolls, which in turn are driven nonlinearly by the divergence of streamwise-average Reynolds stresses of still weaker three-dimensional waves that are neutrally stable eigenmodes of the streak flow. Waleffe (1997), in dealing with plane Couette flow, supplied no direct means of determining the required amplitudes of the roll and wave components relative to the streak flow, but in Wang et al (2007), where nonlinear equilibria were found directly using Newton's method applied to full Navier-Stokes solutions, it was observed that the strength of the rolls fitted an O(R 1 ) dependence (where Reynolds number R is based on half the relative wall speeds and their semi-distance d) while the strength of the three-dimensional waves fitted O(R 0.9 ). On this basis it was suggested by Wang et al (2007) that an asymptotic theory for the lower branch equilibria appeared feasible.…”

“…Waleffe (1997), in dealing with plane Couette flow, supplied no direct means of determining the required amplitudes of the roll and wave components relative to the streak flow, but in Wang et al (2007), where nonlinear equilibria were found directly using Newton's method applied to full Navier-Stokes solutions, it was observed that the strength of the rolls fitted an O(R 1 ) dependence (where Reynolds number R is based on half the relative wall speeds and their semi-distance d) while the strength of the three-dimensional waves fitted O(R 0.9 ). On this basis it was suggested by Wang et al (2007) that an asymptotic theory for the lower branch equilibria appeared feasible.…”

“…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).…”

“…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). The physical mechanism supporting the self-sustaining lower branch states was proposed (Waleffe 1997) as a tripartite coupled nonlinear system in which the dominant component is two-dimensional high-and low-speed streaks of O(1) appearing on the underlying streamwise flow; these streaks are driven by weaker two-dimensional rolls, which in turn are driven nonlinearly by the divergence of streamwise-average Reynolds stresses of still weaker three-dimensional waves that are neutrally stable eigenmodes of the streak flow.…”