Exact coherent structures in an asymptotically reduced description of parallel shear flows

Beaume, Cédric; Knobloch, Edgar; Chini, Gregory P.; Julien, Keith

doi:10.1088/0169-5983/47/1/015504

Cited by 7 publications

(12 citation statements)

References 27 publications

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“…The nonlinear self-interaction of the wavy streaks drives the streamwise rolls, thus closing the cycle. The mechanism is similar to that proposed by Hall and co-workers [25][26][27] and by Beaume and co-workers [28][29][30]. Experimental evidence for the SSP in plane boundary layer and channel flow has been reported by Wesfreid and colleagues in [31,32].…”

supporting

confidence: 87%

Self-sustaining process in Taylor-Couette flow

et al. 2018

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The transition from Tayor vortex flow to wavy-vortex flow is revisited. The Self-Sustaining Process (SSP) of Waleffe [Phys. Fluids 9, 883-900 (1997)] proposes that a key ingredient in transition to turbulence in wall-bounded shear flows is a three-step process involving rolls advecting streamwise velocity, leading to streaks which become unstable to a wavy perturbation whose nonlinear interaction with itself feeds the rolls. We investigate this process in Taylor-Couette flow. The instability of Taylor-vortex flow to wavy-vortex flow, a process which is the inspiration for the second phase of the SSP, is shown to be caused by the streaks, with the rolls playing a negligible role, as predicted by Jones [J. Fluid Mech. 157, 135-162 (1985)] and demonstrated by Martinand et al. [Phys. Fluids 26, 094102 (2014)]. In the third phase of the SSP, the nonlinear interaction of the waves with themselves reinforces the rolls. We show this both quantitatively and qualitatively, identifying physical regions in which this reinforcement is strongest, and also demonstrate that this nonlinear interaction depletes the streaks.

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confidence: 87%

Self-sustaining process in Taylor-Couette flow

et al. 2018

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“…However, the computation of the upper branch is more delicate. As observed by Beaume et al [21], upper-branch solutions and their critical layer have a different spatial structure which dramatically increases the computational cost. To continue these solutions, we used a 64 × 128 mesh grid and adjusted the preconditioner as necessary.…”

Section: B Continuation In Reynolds Numbermentioning

confidence: 93%

“…Finally, and somewhat remarkably, we demonstrate in Sec. IV that our asymptotically reduced partial differential equation (PDE) model admits both lower-branch and upper-branch solutions [21]: in spite of the large Reynolds number scaling incorporated into the theory, the approach proves sufficiently robust to capture the saddle-node bifurcation at which the lower-and upper-branch ECS are born. Thus, our reduced PDEs should prove useful for a variety of further studies of parallel shear flows that aim, for example, to investigate streamwise and spanwise localization.…”

Section: Introductionmentioning

confidence: 99%

Reduced description of exact coherent states in parallel shear flows

et al. 2015

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A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower-and upper-branch states are characterized in detail.

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“…The quantities with an overline represent streamwise-invariant quantities while primed quantities represent fluctuations about this mean state. Note that streamwise-invariant quantities do not vary with the (fast) time t [5].…”

Section: Reduced Modelmentioning

confidence: 99%

“…These are complemented with rolls, a streamwise-invariant vorticity/streamfunction mode whose amplitude decays like Re −1 , and fluctuations, i.e., the streamwise dependent part of the solution that decays roughly like Re −0.9 . In this paper, we use a reduced model based on this scaling derived by Beaume et al [2,5] to calculate spatially extended ECS in the spanwise direction. In the next section, we introduce the reduced model, followed in section 3 by a description of the new ECS computed in a moderately large domain.…”

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confidence: 99%

Modulated patterns in a reduced model of a transitional shear flow

et al. 2016

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We use a reduced model of such flows constructed elsewhere to compute stationary exact coherent structures of Waleffe flow in periodic domains with a large spanwise period. The computations reveal the emergence of stationary states exhibiting strong amplitude and wavelength modulation in the spanwise direction. These modulated 1 states lie on branches exhibiting complex dependence on the Reynolds number but no homoclinic snaking.

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Exact coherent structures in an asymptotically reduced description of parallel shear flows

Cited by 7 publications

References 27 publications

Self-sustaining process in Taylor-Couette flow

Self-sustaining process in Taylor-Couette flow

Reduced description of exact coherent states in parallel shear flows

Modulated patterns in a reduced model of a transitional shear flow

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