Small scale exact coherent structures at large Reynolds numbers in plane Couette flow

Abstract

Abstract. The transition to turbulence in plane Couette flow and several other shear flows is connected with saddle node bifurcations in which fully 3-d, nonlinear solutions, so-called exact coherent states (ECS), to the Navier-Stokes equation appear. As the Reynolds number increases, the states undergo secondary bifurcations and their timeevolution becomes increasingly more complex. Their spatial complexity, in contrast, remains limited so that these states cannot contribute to the spatial complexity and cascade to smaller scales expected for higher Reynolds numbers. We here present families of scaling ECS that exist on ever smaller scales as the Reynolds number is increased. We focus in particular on two such families for plane Couette flow, one centered near the midplane and the other close to a wall. We discuss their scaling and localization properties and the bifurcation diagrams. All solutions are localized in the wall-normal direction. In the spanwise and downstream direction, they are either periodic or localized as well. The family of scaling ECS localized near a wall is reminiscent of attached eddies, and indicates how self-similar ECS can contribute to the formation of boundary layer profiles.

A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows

“…At large Reynolds number, the nonlinear self-interaction of the Rayleigh mode is confined to a critical layer where the ECS phase speed matches the streak velocity. Blackburn, Hall & Sherwin (2013) and Eckhardt & Zammert (2018) have demonstrated that VWI states can exist on ever smaller spatial scales as the Reynolds number is increased. Specifically, when the streamwise and spanwise wavenumbers are increased such that their ratio remains fixed, the ECS adopt a self-similar form in which the coherent structure becomes localized in the wall-normal direction within a 'production' layer having a thickness comparable to the inverse spanwise (or streamwise) wavenumber.…”

Shear stress-driven flow: the state space of near-wall turbulence as

“…The stability boundary of the laminar flow, which separates initial conditions that relaminarise from those that become fully turbulent, is referred to as the edge (Skufca et al 2006;Schneider et al 2008Schneider et al , 2007 and plays a fundamental role in structuring the state space of turbulence. The computation of invariant solutions of the Navier-Stokes equations has allowed for a simplified analysis of a number of physical processes, including an equilibrium selfsustaining process (Waleffe 1998), the self-similarity of equilibria localised in the wallnormal direction (Eckhardt & Zammert 2018) and the high-Re inner-scaling of wallattached equilibria (Yang et al 2019).…”

Self-similar invariant solution in the near-wall region of a turbulent boundary layer at asymptotically high Reynolds numbers

“…The value of the Couette control parameter Re P CF formally depends on the arbitrarily chosen H + reflecting the fact that a solution localised at the wall, only depends on the shear rate at the wall while the distance of the second upper wall and thereby the value of Re P CF is irrelevant. Thus, at asymptotically high Re, all state space structures representing wall attached flow fields in the MFU of ASBL have counterparts in high Reynolds number PCF, such as those identified by Eckhardt & Zammert (2018). This suggests that the relevant statespace structures for near-wall turbulence that we identified in ASBL are universal in that they are not only independent of Re when expressed in inner units but also do not depend on the specific shear flow system considered.…”

“…Deguchi (2015) identifies a solution which scales in inner units at high Reynolds numbers but is not localised at the wall. More recently Eckhardt & Zammert (2018) present two solutions in plane Couette flow, one localized in the center of the channel and one attached to the wall. The solutions are followed up to a Couette Reynolds number of Re = 100, 000, and become approximately Reynolds number independent when rescaled by the inner length scale.…”

“…Likewise Yang et al (2019) follow a wall-attached solution in fixed flux channel flow up to Re τ = 268 and show that the solution approximately scales in inner units. Both Eckhardt & Zammert (2018) and Yang et al (2019) use inner units corresponding to the wall-drag of the solution itself, which differs from the mean wall drag of turbulence at the same controlled relative plate velocity in Couette or controlled flux in channel flow.…”