Streamwise-Localized Solutions at the Onset of Turbulence in Pipe Flow

Avila, Marc; Mellibovsky, Fernando; Roland, Nicolas; Hof, Björn

doi:10.1103/physrevlett.110.224502

Cited by 134 publications

(167 citation statements)

References 43 publications

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“…35,36 The model captures these states in a minimal way. The fixed points (one stable and one unstable) arising as the model transitions to bistability can be viewed as upper and lower branches of spatially extended travelling-wave solutions.…”

Section: Asymptotic Analysismentioning

confidence: 99%

The rise of fully turbulent flow

Barkley

Song

Vasudevan

et al. 2015

Nature

172

222

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This manuscript is the original submission to Nature. The final (published) version can be accessed at http://www.nature.com/nature/journal/v526/n7574/full/nature15701.html 1 arXiv:1510.09143v1 [physics.flu-dyn] Oct 2015Over a century of research into the origin of turbulence in wallbounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. [1][2][3][4][5][6] At slightly higher speeds the situation changes distinctly and the entire flow is turbulent. Neither the origin of the different states encountered during transition, nor their front dynamics, let alone the transformation to full turbulence could be explained to date. Combining experiments, theory and computer simulations here we uncover the bifurcation scenario organising the route to fully turbulent pipe flow and explain the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence 7 and fully turbulent flows. 8,9 The sudden appearance of localised turbulent patches in an otherwise quiescent flow was first observed by Osborne Reynolds for pipe flow 1 and has since been found to be the starting point of turbulence in most shear flows. 2,4,[10][11][12][13][14][15] Curiously, in this regime it is impossible to maintain turbulence over extended regions as it automatically 16,17 reduces to patches of characteristic size, called puffs in pipe flow (see Fig. 1a). Puffs can decay, or else split and thereby multiply. Once the Reynolds number R > 2040 the splitting process outweighs decay, resulting in sustained disordered motion. 7 Although sustained, turbulence at these low R only consists of puffs surrounded by laminar flow (Fig. 1a) and cannot form larger clusters. 17,18 At larger flow rates, the situation is fundamentally different: once triggered, turbulence aggressively expands and eliminates all laminar motion (Fig. 1b). Fully turbulent flow is now the natural state of the system and only then do wallbounded shear flows have characteristic mean properties such as the Blasius or Prandtl-von Karman friction laws. 9 This rise of fully turbulent flow has remained unexplained despite the fact that this transformation occurs in virtually all shear flows and generally dominates the dynamics at sufficiently large Reynolds numbers.A classical diagnostic for the formation of turbulence 2-5, 19, 20 is the propagation speed of the In each case, the flow is initially seeded with localised turbulent patches and the subsequent evolution is visualised via space-time plots in a reference frame co-moving with structures. Colours indicate the value of u 2 r + u 2 θ . Further highlighting the distinction between cases, shown at the top are cross sections of instantaneous flow within the pipe. A 35D section is shown with the vertical direction str...

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Section: Asymptotic Analysismentioning

confidence: 99%

The rise of fully turbulent flow

Barkley

Song

Vasudevan

et al. 2015

Nature

172

222

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“…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.…”

Section: Computation Of Invariant Solutionsmentioning

confidence: 99%

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

2016

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Understanding the origin of large-scale structures in high Reynolds number wall turbulence has been a central issue over a number of years. Recently, Rawat et al. (J. Fluid Mech., 2015, 782, p515) have computed invariant solutions for the large-scale structures in turbulent Couette flow at Re τ ≃ 128 using an over-damped LES with the Smagorinsky model to account for the effect of the surrounding small-scale motions. Here, we extend this approach to an order of magnitude higher Reynolds numbers in turbulent channel flow, towards the regime where the large-scale structures in the form of very-large-scale motions (long streaky motions) and large-scale motions (short vortical structures) energetically emerge. We demonstrate that a set of invariant solutions can be computed from simulations of the self-sustaining large-scale structures in the minimal unit (domain of size L x = 3.0h streamwise and L z = 1.5h spanwise) with midplane reflection symmetry at least up to Re τ ≃ 1000. By approximating the surrounding small scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are found, labelled upper-and lower-branch according to their associated drag. It is shown that the upper-branch equilibrium state is a reasonable proxy for the spatial structure and the turbulent statistics of the self-sustaining large-scale structures.

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“…Schneider et al (2010b) found the first known localized solutions, a pair of spanwise-localized, streamwise-periodic equilibrium and traveling-wave solutions of plane Couette flow, further investigated in Schneider et al (2010a). Avila et al (2013) found a streamwise-localized relative periodic orbit of pipe flow that closely resembles the transient turbulent puffs of Hof et al (2006). Deguchi et al (2013) and Gibson & Brand (2014) independently found spanwise-localized forms of the periodic EQ7/HVS solution of Itano & Generalis (2009);Gibson et al (2009).…”

Section: Introductionmentioning

confidence: 95%

“…Efforts to develop a similar dynamical understanding of transition in extended flows have lead to the computation of a number of localized edge states, but to date these have either been invariant states localized in a single homogeneous direction (Schneider et al 2010b;Avila et al 2013;Khapko et al 2013;Zammert & Eckhardt 2014) or doubly-localized but chaotically wandering states without well-defined stable and unstable manifolds (Schneider et al 2010b;Duguet et al 2012). The doubly-localized invariant solution in this paper thus provides a potential starting point for addressing spatiotemporal transition of extended flows in dynamical terms.…”

Section: Stability and The Evolution Of Unstable Perturbationsmentioning

confidence: 99%

A doubly localized equilibrium solution of plane Couette flow

Brand

Gibson

2014

J. Fluid Mech.

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We present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier-Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

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Streamwise-Localized Solutions at the Onset of Turbulence in Pipe Flow

Cited by 134 publications

References 43 publications

The rise of fully turbulent flow

The rise of fully turbulent flow

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

A doubly localized equilibrium solution of plane Couette flow

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