Minimal transition thresholds in plane Couette flow

Duguet, Yohann; Monokrousos, Antonios; Brandt, Luca; Henningson, Dan S.

doi:10.1063/1.4817328

Cited by 85 publications

(130 citation statements)

References 60 publications

(94 reference statements)

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“…It is noteworthy that such localized minimal states would have an energy scaling E ∼ Re −2.5 in a fixed periodic box, which is close to that obtained by an energy-optimized technique, E ∼ Re −2.7 ; in fact, the optimized states typically have localized structures as seen in Duguet et al (2013).…”

Section: Discussionmentioning

confidence: 81%

“…On the one hand, according to the VWI theory/self-sustaining process, the streaky field must be modulated by an amount O(1) to produce a wave-like instability when the basic flow is inviscidly stable, but on the other hand Duguet et al (2013) showed numerically that the optimized minimal disturbances on the edge of laminar-turbulent attractor have a kinetic energy density E that decreases with Reynolds number: see Hof et al (2004) and Lemoult, Aider & Wesfield (2012) for experimental evidence showing a similar decreasing disturbance amplitude needed to trigger transition. No self-consistent theory has been provided for the Reynolds number scaling of such minimal disturbances.…”

Section: Introductionmentioning

confidence: 99%

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Self-sustained states at Kolmogorov microscale

Deguchi

2015

J. Fluid Mech.

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It is shown theoretically how the scaling of coherent structures in shear flows changes their asymptotic development at large Reynolds number. Based on the theory a family of nonlinear self-sustained states at Kolmogorov microscale is numerically identified on the laminar-turbulent boundary of shear flows. Theoretically and numerically the states connect to known asymptotic states existing at larger scale. The asymptotically very small amplitude of the new states may explain why strongly sheared, linearly stable laminar flows can cause a turbulent transition by small disturbances. The numerically obtained Kolmogorov-scale solutions can be used to describe the theoretically minimal self-sustained structures appearing in various shear flows.

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Section: Discussionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Self-sustained states at Kolmogorov microscale

Deguchi

2015

J. Fluid Mech.

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“…We directly use step size 0.04 < α j < 0.1 if restarting from previously stored fields, which in general are moderately converged with a relatively small r t . This is close to the value of 0.05 used in Duguet et al (2013). We also repeated the optimisations with different choices of α j but nevertheless reached the same optimum.…”

Section: Galerkin Expansions For U B and B †mentioning

confidence: 77%

The optimal kinematic dynamo driven by steady flows in a sphere

Chen

Herreman

et al. 2018

J. Fluid Mech.

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We present a variational optimisation method that can identify the most efficient kinematic dynamo in a sphere, where efficiency is based on the value of a magnetic Reynolds number that uses enstrophy to characterize the inductive effects of the fluid flow. In this large scale optimisation, we restrict the flow to be steady and incompressible, and the boundary of the sphere to be no-slip and electrically insulating. We impose these boundary conditions using a Galerkin method in terms of specifically designed vector field bases. We solve iteratively for the flow field and the accompanying magnetic eigenfunction in order to find the minimal critical magnetic Reynolds number Rm c,min for the onset of a dynamo. Although nonlinear, this iteration procedure converges to a single solution and there is no evidence that this is not a global optimum. We find Rm c,min = 64.45 is at least three times lower than that of any published example of a spherical kinematic dynamo generated by steady flows, and our optimal dynamo clearly operates above the theoretical lower bounds for dynamo action. The corresponding optimal flow has a spatially localized helical structure in the centre of the sphere, and the dominant components are invariant under rotation by π.

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“…In terms of future work, the way is now clear to map out the threshold energy E c for transition as a function of Re just as has been recently done in small-box plane Couette flow 27 (the imposed streamwise periodicity in Ref. 27 is equivalent to L = 2πD here).…”

Section: Discussionmentioning

confidence: 99%

“…27 is equivalent to L = 2πD here). Our results indicate that using small-to-intermediate periodic domains (at least in the streamwise direction) can still yield useful results.…”

Section: Discussionmentioning

confidence: 99%

Fully localised nonlinear energy growth optimals in pipe flow

2015

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A new, fully localised, energy growth optimal is found over large times and in long pipe domains at a given mass flow rate. This optimal emerges at a threshold disturbance energy below which a nonlinear version of the known (streamwise-independent) linear optimal [P. J. Schmid and D. S. Henningson, "Optimal energy density growth in Hagen-Poiseuille flow," J. Fluid Mech. 277, 192-225 (1994)] is selected and appears to remain the optimal up until the critical energy at which transition is triggered. The form of this optimal is similar to that found in short pipes [Pringle et al., "Minimal seeds for shear flow turbulence: Using nonlinear transient growth to touch the edge of chaos," J. Fluid Mech. 702, 415-443 (2012)], but now with full localisation in the streamwise direction. This fully localised optimal perturbation represents the best approximation yet of the minimal seed (the smallest perturbation which is arbitrarily close to states capable of triggering a turbulent episode) for "real" (laboratory) pipe flows. Dependence of the optimal with respect to several parameters has been computed and establishes that the structure is robust. C 2015 AIP Publishing LLC.

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Minimal transition thresholds in plane Couette flow

Cited by 85 publications

References 60 publications

Self-sustained states at Kolmogorov microscale

Self-sustained states at Kolmogorov microscale

The optimal kinematic dynamo driven by steady flows in a sphere

Fully localised nonlinear energy growth optimals in pipe flow

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