Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

Abstract

We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number Re, but the downstream exponent is essentially independent of Re. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…Chaotic sets developed from extended versions of these solutions clearly exhibit wave trains with a varying count of wave periods embedded (Ritter et al , to be published). This will most probably be the case for localised plane Couette and plane Poiseuille solutions too Zammert & Eckhardt 2014). The non-snaking localisation mechanism (Burke & Knobloch 2006) found in double-diffusive convection (Beaume et al 2013b) cannot apply to plane Couette, plane Poiseuille or pipe flows, as the required condition that a nontrivial streamwise-independent state exists is not fulfilled by any of these flows.…”

“…This scenario has been confirmed for extended domains allowing for localisation, albeit with modifications regarding the spatial structure of the underlying solution, which is no longer periodic but localised (Mellibovsky et al 2009;Duguet et al 2009). Fully localised invariant solutions of the Navier-Stokes equations have since been found by restricting the dynamics to appropriate symmetry subspaces in long pipes (Avila et al 2013), and in extended domains (spanwise and/or streamwise) in plane Couette and plane Poiseuille flows (Zammert & Eckhardt 2014).…”

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

“…However, their calculation were strongly truncated and attempts to reproduce their results with better resolution failed [29]. Subsequent studies identified many other three-dimensional exact solutions for PPF [29][30][31][32][33][34]. Waleffe [35] and Nagata & Deguchi [29] studied the dependence of particular solutions on the streamwise and spanwise wavelengths.…”

“…We use a numerical resolution of N x × N y × N z = 32 × 65 × 64 for small domains and increase it to N x × N y × N z = 80 × 65 × 112 for a domain with L x = L z = 4π. For sufficiently large domains the edge state of PPF is a traveling wave [33] that is symmetric with respect to the center-plane,…”

“…In particular, a subcritical long-wavelength instability of T W E at high Reynolds numbers creates a streamwise localized periodic orbit, henceforth referred to as P O E , which is an edge state in long computational domains [33] and appears at low Reynolds numbers in a saddle-node bifurcation. The dependence of the Reynolds number of this saddle-node bifurcation on the spanwise wavelength is shown in figure 7a).The lowest Reynolds number for the turning point is achieved for a spanwise wavelength of 1.75π.…”