Transition in pipe flow: the saddle structure on the boundary of turbulence

Abstract

The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on Σ are organized around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally attracting travelling waves embedded within Σ. Four new types of travelling waves were found using this approach.

Invariant solutions of minimal large-scale structures in turbulent channel flow for up to 1000

“…Skufca et al 2006), we start by computing the edge state for the S550 simulation in the given subspace, using the standard bisection technique to obtain a good initial guess for the Newton iteration (e.g. Duguet et al 2008;Avila et al 2013). Several instantaneous flow fields on the edge state are given for initial guess of the Newton solver, and we found an invariant solution propagating downstream with a constant speed (i.e.…”

Exact coherent structures in stably stratified plane Couette flow

“…These states, which start to populate phase space at some critical Reynolds number and quickly proliferate as the Reynolds number increases, eventually produce a sufficiently tangled structure of stable and unstable manifolds in phase space to act as a scaffold for the observed complex dynamics (Kerswell 2005;Eckhardt et al 2007;Gibson and Cvitanović 2010;Kawahara et al 2012). These states (variously referred to as simple invariant solutions or Exact Coherent Structures (ECS) ), can either be embedded in the laminar-turbulent boundary (Wang et al 2007;Duguet et al 2008) or sit in the basin of attraction of the turbulent state with some buried in the turbulent attractor itself (Kerswell & Tutty 2007;Gibson et al 2008). As a consequence, properties of the ECS have something to say about how transition is triggered (Viswanath & Cvitanović 2009;Duguet et al 2010;Pringle et al 2012), the subsequent transitional process (Itano & Toh 2001;Skufca et al 2006;Schneider et al 2007;Mellibovsky et al 2009) and features of the turbulent state itself (e.g.…”

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…Jimenez 1990;Hamilton et al 1995), have been extensively used as a proof-of-concept. The saddle solution, together with its stable manifold, has been shown to reside in the laminar-turbulent separatrix and to govern the transitional dynamics (Duguet et al 2008;Schneider et al 2008;Itano & Toh 2000;Kreilos et al 2013). Meanwhile, the nodal solution, initially presenting little or no time complexity, undergoes a bifurcation cascade that eventually results in a chaotic saddle that is capable of sustaining turbulent dynamics of transient nature (Mellibovsky & Eckhardt 2012).…”