Spanwise-localized solutions of planar shear flows

Abstract

We present several new spatially localized equilibrium and traveling-wave solutions of plane Couette and channel flows. The solutions exhibit strikingly concentrated regions of vorticity that are flanked on either side by high-speed streaks. For several traveling-wave solutions of channel flow, the concentrated vortex structures are confined to the nearwall region and form particularly isolated and elemental versions of coherent structures in the near-wall region of shear flows. The solutions are constructed by a variety of methods: application of windowing functions to previously known spatially periodic solutions, continuation from plane Couette to channel flow conditions, and from initial guesses obtained from turbulent simulation data. We show how the symmetries of localized solutions derive from the symmetries of their periodic counterparts, analyze the exponential decay of their tails, examine the scale separation and scaling of their streamwise Fourier modes, and show that they develop critical layers for large Reynolds numbers.

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 mathematical formulation and numerical methods are presented in detail in Gibson & Brand (2014) (GB14); here we present a brief outline. The Reynolds number Re for plane Couette flow is defined in terms of half the relative wall speed, the channel halfheight, and the kinematic viscosity, so that the walls at y = ±1 have velocity ±1 and the laminar flow solution is given by y e x .…”

“…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. Khapko et al (2013) found spanwise-localized relative periodic orbits of the asymptotic suction boundary layer, and Zammert & Eckhardt (2014) found a spanwise-and wall-normal-localized periodic orbit of plane Poiseuille flow.…”

Exact coherent structures in stably stratified plane Couette flow

“…§3.2 then describes the results of edge-tracking in a variety of geometries aimed at uncovering new ECS. Two already known, albeit now stratified, ECS are repeatedly found -EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (2014). In §3.3 these are continued around in (Re, Ri b ) parameter space, including into negative Ri b (unstable stratification), along with Nagata's (1990) first solution in pCf.…”

Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

“…In this sense our use of the word 'localization' is different from that used in recent numerical investigations of fully isolated vortex structures, e.g. Schneider, Gibson & Burke (2010), Deguchi, Hall & Walton (2013) and Gibson & Brand (2014). Similar computations are now presented for larger values of β.…”