Complexity of the laminar-turbulent boundary in pipe flow

Abstract

Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry-reduction, which we formulate here for the first time for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.

Upper edge of chaos and the energetics of transition in pipe flow

“…Fig. 3(a) shows the [P/D] ratio as a function of radius for the traveling waves S 1 and S 1N , which are embedded in the laminar-turbulent boundary [15]. For the traveling waves S 1 and S 1N , the [P/D](r) curve peaks at r = 0.69R and r = 0.73R respectively, which, is quite different from the turbulent case.…”

“…We utilize the Openpipeflow solver [24] to simulate pipe flow in an axially-periodic computational domain of length L = 2π/0.625 ≈ 10R, where R is the pipe radius. The Reynolds number is set to Re = U c R/ν = 3000, where U c is the centerline velocity, ν is the kinematic viscosity, and the resolution is identical to the one used in Budanur & Hof [15]. In results, the time scale is chosen to be advective unit 4R/U c .…”

“…Moreover, refs. [29] and [15] reported that edge tracking trajectories approach traveling waves of the pipe flow, which are by definition time-invariant. Fig.…”

“…In most of the current literature, irrespective of the type of their time-dependence, such asymptotic states at the laminarturbulent boundary are referred to as the "edge states"; and the bisection-based methods that probe edge states are called "edge tracking". In the pipe flow setting, which we are going to consider in the current study, the edge state appears to be chaotic with transient visits to traveling wave solutions [15].…”

Inferring symbolic dynamics of chaotic flows from persistence

“…However, there has been considerable development in recent years following the introduction of "first Fourier mode slice" by Budanur et al 36 It is a straightforward method for reducing the SO(2)-symmetry due to translation-equivariance and periodic boundary conditions. Since then, this method was adapted to the two-dimensional Kolmogorov flow, 37 three-dimensional pipe flow, 38,39 one-dimensional Korteweg-de Vries equation, 40 and pilot-wave hydrodynamics. 41 For a pedagogical introduction to the first Fourier mode slice, we refer the reader to Ref.…”