Folded Edge of Turbulence in a Pipe

Abstract

The results of an experimental investigation into the threshold boundary between laminar and disordered pipe flow are presented. Complex features have been uncovered using a highly refined experimental approach where an intermediate periodic state forms an integral part of the transition sequence. In accord with the suggestions produced by a numerical investigation, the boundary is found to be folded with a complicated structure. This raises important questions about accepted definitions of threshold amplitudes in this long-standing problem. DOI: 10.1103/PhysRevLett.105.174502 PACS numbers: 47.27.nf, 47.20.Àk, 47.60.Ài The transition to turbulence in pipe flow remains an outstanding challenge to hydrodynamic stability theory although there have been significant advances in recent years [1][2][3]. The central issue is that all numerical and theoretical results indicate that the flow is linearly stable and yet pipe flow is typically turbulent at modest values of the Reynolds number Re. (Here Re ¼ UD= where U is the mean speed, D is the diameter of the pipe and is the kinematic viscosity of the fluid.) It has been recognized since Reynolds' original investigations [4] that finite amplitude disturbances are responsible for transition in practice. More recently, progress has been made in providing experimental estimates for the stability boundary between laminar and turbulent flow [5,6] and various scalings of the amplitude of disturbance required to cause transition have been found [7,8] some of which are in accord with theoretical predictions [9,10]. A reasonable question to ask is whether the boundary is sharp or, as in examples from lowdimensional dynamical systems it is folded or interleaved? In this Letter, the results of an experimental investigation into this issue are reported and the outcomes are interpreted with the aid of numerical results.The possibility that the boundary basin of Poiseuille flow is folded has been investigated numerically where the lifetimes of perturbations are used to determine the boundary between laminar and disordered flow [11]. A dynamical systems approach is taken where the calculation domain is 5D long. Support for the folded stability boundary found in the numerical results lies in experimental observations [5] where a degree of spottiness is found in the transition probability plotted as a function of disturbance amplitude and Re. Precisely mapping out the stability boundary and elucidating any features in an experiment requires the use of a highly reproducible disturbance whose amplitude can be adjusted without changing its form. The experiments [5] were performed using short duration pulses of relatively large amplitude and unknown spatial structure and direct connection with the numerical work is unclear. In general, it is difficult to specify the precise form of a disturbance in an experiment where fluid is either injected and/or withdrawn through small holes [7,8]. Despite this, repeatability of the estimates of thresholds and scaling laws for the dependency of the tr...

Extraction of 3D vortex structures from a turbulent puff in a pipe using two-color illumination and flakes

“…Nevertheless, Peixinho and Mullin 2006 indicated that the lifetime of a turbulent puff approaches infinity around Re = 1750. The work of Tasaka et al 2010, which used the same facility used by Peixinho and Mullin 2006, also confirmed that puffs are persistent for Re values higher than 1,750 ± 10. However, there is a difference between the systems used in these studies to induce the flow in a pipe: the system in Peixinho and Mullin 2006 had a constant mass flux and one in Avila et al 2011 was operated with a constant pressure gradient.…”

Osborne Reynolds pipe flow: Direct simulation from laminar through gradual transition to fully developed turbulence

“…The evolution of Lambda vortex to hairpin packet to turbulent spot discussed in step 5 also occurs naturally (without artificial creation) downstream of the inlet at an irregular and much longer time interval compared with that in ref. 26. Of course, the exponential growth rate in disturbance energy is also different from the algebraic growth in ref.…”

“…6, the perturbation amplitude in ref. 26 was quite weak; only 0.1-0.3% of the mean pipe flow flux was applied through the holes. This feature permits a clear observation of the transition process.…”

“…The present vortex dynamics is distinct from that in ref. 26 for several reasons. Reverse hairpin streets in steps 3 and 4 start far downstream from the inlet where the disturbance is introduced; and the reverse hairpins are not deliberately created through jet-in-cross-flow.…”

Edge states in a boundary layer

“…Similar hairpin structures have been also experimentally recovered on the edge of turbulence in pipe flows. 20 The hairpin vortices are also visualized in Fig. 3 which provides snapshots of the perturbation taken at the abscissa x ¼ 650 for three time instants (the corresponding video shows the whole time sequence from t ¼ 500 to 700).…”