Visualizing the geometry of state space in plane Couette flow

Gibson, John; Halcrow, Jonathan; Cvitanović, Predrag

doi:10.1017/s002211200800267x

Cited by 301 publications

(388 citation statements)

References 49 publications

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“…The solution trajectories of these simulations are maintained roughly around 20I l and 20D l in the I-D plane (figure 5b), the values an order of magnitude larger than those at low Reynolds numbers (e.g. Kawahara & Kida 2001;Gibson et al 2008;Willis et al 2013Willis et al , 2016. This is essentially due to the high Reynolds number considered in the present study, which leads to the energy input to the system, substantially larger than that at low Reynolds numbers.…”

Section: Spatial Structure Of the Invariant Solutionsmentioning

confidence: 62%

“…Kerswell & Tutty 2007;Schneider et al 2007;Park & Graham 2016). As these authors indicate, this observation suggests that computation of unstable time periodic orbits would be a more promising way to represent the dynamics of coherent structures in the S950 simulation (Kawahara & Kida 2001;Gibson et al 2008;Willis et al 2013). Also as expected, the statistics of the lower-branch solution L950 do not show such a level of agreement -the lower-branch solution sits on the edge state of S950 simulation.…”

Section: Spatial Structure Of the Invariant Solutionsmentioning

confidence: 66%

“…Together with unstable periodic and/or relative periodic orbits (e.g. Kawahara & Kida 2001), these invariant solutions have been shown to form a skeleton for solution trajectories in phase space (Gibson et al 2008;Willis et al 2013Willis et al , 2016, and their understanding from a dynamical systems viewpoint has been at the heart of recent advancement in the understanding of bypass transition and low-Reynolds-number turbulence.…”

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 Solutionsmentioning

confidence: 62%

Section: Spatial Structure Of the Invariant Solutionsmentioning

confidence: 66%

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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“…For plane Couette flow with downstream wavenumbers of O(1) we know that the lower branch states of HS2 or equivalently those of Wang et al (2007) are visited in unsteady Navier-Stokes simulations of turbulence (see Gibson et al 2008). The nature of the localized solutions we have uncovered suggests that the interaction problem we have formulated has streamwise-localized states which might well be weakly unstable saddle-like points for trajectories of the unsteady Navier-Stokes equations.…”

Section: Resultsmentioning

confidence: 84%

The emergence of localized vortex–wave interaction states in plane Couette flow

2013

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The recently understood relationship between high Reynolds number vortex-wave interaction theory and computationally-generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex-wave interaction theory, which we now refer to as VWI, is used in the long streamwise wavelength limit to continue the development found at order one wavelengths by Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokes equations to the so-called boundary-region equations for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from O(h) to O(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave-vortex states. The solutions we calculate have an asymptotic error proportional to R −2 when compared to the full Navier-Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel so that the roll part of the flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling which when approached causes the flow to localise and take on similarities with computationally-generated or experimentally-observed turbulent spots. In effect the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

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“…Numerous non-linear equilibrium solutions have already been identified in plane Couette, plane Poiseuille and pipe flows [3,4,5], and their role in the transition process as well as their relevance to the statistics of turbulent flow have been investigated [6,7,8]. No non-linear travelling-wave solutions for the flow through a square duct have been published to date.…”

Section: Introductionmentioning

confidence: 99%