Exact coherent structures in channel flow

Waleffe, Fabian

doi:10.1017/s0022112001004189

Cited by 272 publications

(302 citation statements)

References 20 publications

(32 reference statements)

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“…This feature is consistent with that of the invariant solutions in e.g. Waleffe (2001). The invariant solutions here, however, are obtained by modelling the surrounding smallscale structures with an eddy viscosity.…”

Section: Spatial Structure Of the Invariant Solutionssupporting

confidence: 90%

“…An examination of the energy spectra reveals that these numbers of grid points are also found to provide good spatial resolution for the computed invariant solutions at least for the elevated C s . We also note that the resolution of the invariant solution for Re m = 38133 is finer than that in Waleffe (2001). A reference simulation is first performed to check turbulence statistics of the halfchannel LES simulation at Re τ ≃ 950, with the value C s = 0.05 (F 950) known to provide the best statistical fit to the full DNS result (Hwang & Cossu 2010b).…”

Section: Methodsmentioning

confidence: 99%

“…Nagata 1990;Waleffe 2001;Faisst & Eckhardt 2003;Wedin & Kerswell 2004;Hall & Sherwin 2010;Park & Graham 2016, and many others). The simplest non-trivial exact solutions of the Navier-Stokes equations are typically in the form of a stationary or travelling wave, being equilibria or relative equilibria in phase space.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Spatial Structure Of the Invariant Solutionssupporting

confidence: 90%

Section: Methodsmentioning

confidence: 99%

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

2016

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“…Two facets of this problem that exist at opposite ends of the Reynolds-number spectrum are the mechanism of † Email address for correspondence: Hugh.Blackburn@monash.edu transition in flows such as Hagen-Poiseuille flow and plane Couette flow, which are linearly stable and whose instability mechanisms have not yielded to analysis with linear tools, and the universal inertial-range scaling of turbulence kinetic energy with wavenumber first proposed by Kolmogorov (1941), which applies at very high Reynolds numbers. In recent years much interest in the dynamics of shear flows has come through the discovery of three-dimensional nonlinear equilibrium states (Nagata 1990;Waleffe 1997Waleffe , 2001Faisst & Eckhardt 2003;Wedin & Kerswell 2004;Wang, Gibson & Waleffe 2007;Gibson, Halcrow & Cvitanovic 2009). These equilibria have been shown to act either as 'edge states' which locally divide the basin of attraction between relaminarized or turbulent outcomes ('lower branch' equilibria) or as 'organizing centres' about which the flow slowly cycles during the approach to turbulence ('upper branch' equilibria) (Wang et al 2007).…”

Section: Introductionmentioning

confidence: 99%

Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows

2013

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We consider the development of nonlinear three-dimensional vortex-wave interaction equilibria of laminar plane Couette flow for a range of spanwise wavenumbers. The results are computed using a hybrid approach that captures the required asymptotic structure while at the same time providing a direct link with full numerical calculations of equilibrium states. Each equilibrium state consists of a streak flow, a roll flow and a wave propagating on the streak. Direct numerical simulations at finite Reynolds numbers using initial conditions constructed from these parts confirm that the scheme generates equilibrium solutions of the Navier-Stokes equations. Consideration of the form of the vortex-wave interaction equations in the highspanwise-wavenumber limit predicts that for small wavelengths the equilibria take on a self-similar structure confined near the centre of the flow. These states feel no influence from the walls, and lead to a class of canonical states relevant to arbitrary shear flows. These predictions are supported by an analysis of computational results at increasing values of the spanwise wavenumber. For the wave part of these new canonical states, it is shown that the mass-specific kinetic energy density per unit wavenumber scales with the 5/3 power of the wavenumber.

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“…Travelling wave solutions attracted recently much attention. For plane channel flow they were found by Itano & Toh (2001) and Waleffe (2001Waleffe ( , 2003, while the most recent attempt to find travelling waves in the velocity fields produced by direct numerical simulation was performed by Kerswell & Tutty (2007), where further references can be found. Note that as the Reynolds number and the size of the computational domain increase the likelihood of finding a travelling wave occupying the entire computational domain decreases.…”

Section: Travelling Wavesmentioning

confidence: 99%

Master-Mode Set for 3D Turbulent Channel Flow

Chernyshenko¹,

Bondarenko²

2007

Springer Proceedings Physics

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Turbulent flow fields can be expanded into a series in a set of basic functions. The terms of such series are often called modes. A master-(or determining) mode set is a subset of these modes, the time history of which uniquely determines the time history of the entire turbulent flow provided that this flow is developed. In the present work the existence of the master-mode-set is demonstrated numerically for turbulent channel flow. The minimal size of a master-mode set and the rate of the process of the recovery of the entire flow from the master-mode set history are estimated. The velocity field corresponding to the minimal master-mode set is found to be a good approximation for mean velocity in the entire flow field. Mean characteristics involving velocity derivatives deviate in a very close vicinity to the wall, while master-mode two-point correlations exhibit unrealistic oscillations. This can be improved by using a larger than minimal master-mode set. The near-wall streaks are found to be contained in the velocity field corresponding to the minimal master-mode set, and the same is true at least for the large-scale part of the longitudinal vorticity structure. A database containing the time history of a master-mode set is demonstrated to be an efficient tool for investigating rare events in turbulent flows. In particular, a travelling-wave-like object was identified on the basis of the analysis of the database. Two mastermode-set databases of the time history of a turbulent channel flow are made available online at http://www.dnsdata.afm.ses.soton.ac.uk/. The services provided include the facility for the code uploaded by a user to be run on the server with an access to the data.

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Exact coherent structures in channel flow

Cited by 272 publications

References 20 publications

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

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

Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows

Master-Mode Set for 3D Turbulent Channel Flow

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