Laminar–turbulent transition in pipe flow for 
Newtonian and non-Newtonian fluids

Recently discovered traveling-wave solutions to the Navier-Stokes equations in plane shear geometries provide model flows for the study of turbulent drag reduction by polymer additives. These solutions, or "exact coherent states" (ECS), qualitatively capture the dominant structure of the near-wall buffer region of shear turbulence, i.e., counter-rotating pairs of streamwise-aligned vortices flanking a low-speed streak in the streamwise velocity. The optimum length scales for the ECS match well the length scales of the turbulent coherent structures and evidence suggests that the ECS underlie the dynamics of these structures. We study here the effect of viscoelasticity on these states. The changes to the velocity field for the viscoelastic ECS, where the FENE-P model calculates the polymer stress, mirror the modifications seen in experiments of fully turbulent flows of polymer solutions at low to moderate levels of drag reduction: drag is reduced, streamwise velocity fluctuations increase while wall-normal fluctuations decrease, and smaller wavelength structures are suppressed. These modifications to the ECS are due to the suppression of the streamwise vortices. The polymer molecules become highly stretched in the wavy, streamwise streaks, where the flow is predominately elongational, then relax as they move from the streaks into and around the streamwise vortices, where the flow is predominately rotational. This relaxation of the polymer molecules produces a force that directly opposes the fluid motion in the vortices, weakening them. Since the pressure fluctuations have their greatest magnitude (i.e., they are most negative) in the cores of the vortices, a reduction in vortex strength leads to a decrease in the magnitude of the pressure fluctuations. The pressure fluctuations redistribute energy from the streamwise velocity fluctuations to the Reynolds shear stress, so a decrease in their magnitude leads to a reduction in turbulent drag. For the viscoelastic ECS, we also find that after the onset of drag reduction (at Weissenberg number, WeϷ 7) there is a dramatic increase in the critical wall-normal length scale at which the ECS can exist. This sharp increase in length scale mirrors experimental observations and is also consistent with the observed shift to higher Reynolds numbers of the transition to turbulence in polymer solutions.

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“…8,14 Evidence suggests that the existence of ECS is a prerequisite for transition (cf. the Introduction).…”

Section: A Existence Of the Ecsmentioning

confidence: 99%

“…Most notably, the wall-normal thickness of the buffer region increases, 1 the coherent structures in this region shift to larger length scales, 5,[12][13][14] and the bursting rate decreases. 5 These structural changes are accompanied by changes in the root-mean-square (rms) velocity fluctuations and Reynolds stresses.…”

Section: Introductionmentioning

confidence: 99%

Polymer drag reduction in exact coherent structures of plane shear flow

et al. 2004

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“…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?…”

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confidence: 94%

Folded Edge of Turbulence in a Pipe

2010

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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...

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“…The working fluid was water and the flow was driven by a pump. A smooth contraction in combination with several screens and meshes at the pipe entrance ensured that the flow could be kept laminar up to Re 60 000 [23]. The laminar flow was perturbed 450 pipe diameters downstream of the inlet, where the laminar profile was fully developed.…”

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Turbulence Regeneration in Pipe Flow at Moderate Reynolds Numbers

et al. 2005

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We present the results of an experimental investigation into the nature and structure of turbulent pipe flow at moderate Reynolds numbers. A turbulence regeneration mechanism is identified which sustains a symmetric traveling wave within the flow. The periodicity of the mechanism allows comparison to the wavelength of numerically observed exact traveling wave solutions and close agreement is found. The advection speed of the upstream turbulence laminar interface in the experimental flow is observed to form a lower bound on the phase velocities of the exact traveling wave solutions. Overall our observations suggest that the dynamics of the turbulent flow at moderate Reynolds numbers are governed by unstable nonlinear traveling waves. A process which is believed to be of relevance for transition in shear flows is the so called lift-up mechanism [6]. Here streamwise vortices of relatively small magnitude transport low momentum fluid away from the wall and high momentum fluid towards the wall creating strong anomalies in the velocity profile, so-called low-and high-speed streaks (elongated regions of low or high streamwise velocity with respect to their surroundings). During this process the initial perturbation amplitude can grow substantially. Since the mechanism is of a linear nature, in linearly stable flows eventually all perturbations will decay. However it is assumed that once the perturbation has reached a sufficiently large amplitude nonlinear effects will become important and lead to secondary instabilities. The mathematical reason for the transient growth of initial perturbations is the non-normality of the linearized NavierStokes operator and this mechanism has attracted considerable attention [7][8][9].In more recent studies the relevance of nonlinear effects and, in particular, the role of nonlinear solutions has been discussed [10,11]. Such solutions have been calculated for Couette and Poiseuille flow [12 -16] and more recently also for pipe flow [17,18]. These new flow states arise at finite values of the control parameter (Re) and in the case of pipe flow they take the form of traveling waves (TWs), which are coherent flow structures propagating at a constant phase speed in the streamwise direction. Typically these TW solutions are unstable so that the laminar flow state remains the only stable solution. In analogy to observations in low dimensional models [19,20] it has been suggested that as the Reynolds number is increased further the unstable states undergo secondary bifurcations and form an attracting region in phase space which gives rise to turbulent dynamics. The basin of attraction of the turbulent state grows with increasing Reynolds number, so that smaller and smaller perturbations are sufficient to destabilize the laminar flow.Confirmation of this picture has been provided by recent experiments in pipe flow where transients of unstable TWs have been observed that are in close agreement with the numerical solutions [21]. Further experimental investigations have demonstrated that the sta...

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Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids

Cited by 170 publications

References 49 publications

Polymer drag reduction in exact coherent structures of plane shear flow

Polymer drag reduction in exact coherent structures of plane shear flow

Folded Edge of Turbulence in a Pipe

Turbulence Regeneration in Pipe Flow at Moderate Reynolds Numbers

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