On stability of streamwise streaks and transition thresholds in plane channel flows

Reddy, Satish C.; Schmid, Peter J.; Baggett, Jeffrey S.; Henningson, Dan S.

doi:10.1017/s0022112098001323

Cited by 267 publications

(346 citation statements)

References 54 publications

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“…This question is analogous to a well-known set of problems in hydrodynamic stability: that of estimating the "transition threshold" of an equilibrium, such as a shear flow (see e.g. [53,3,47,7,8,9,14] and the references therein). The goal there is to estimate the basin of nonlinear stability of a shear flow (in a certain sense) as a scaling law in high Reynolds numbers.…”

Section: Msc: 35b35 35b34 35b40 35q83 35q84mentioning

confidence: 99%

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Bedrossian

2017

Ann. PDE

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In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-FokkerPlanck equation with nonlinear collisions in the phase-space (x, v) ∈ T n x × R n v . The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. We give an estimate for the scaling law between Knudsen number and the maximal size of the perturbation necessary for linear theory to be accurate in Sobolev regularity. We conjecture this scaling to be sharp (up to logarithmic corrections) due to potential nonlinear echoes in the collisionless model.

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Section: Msc: 35b35 35b34 35b40 35q83 35q84mentioning

confidence: 99%

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Bedrossian

2017

Ann. PDE

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“…Finite amplitude perturbations of the form (72) are of special interest in the nonlinear stability analysis of shear flows. Streamwise vortices are particularly efficient in triggering transition due to the usually termed lift-up effect, advecting slow axial flow to high speed regions and vice versa [12,21,23,26]. This mechanism modulates the axial parabolic flow in a new transient profile, usually termed streak, which contains saddle points, thus being potentially unstable with respect to three-dimensional infinitesimal disturbances [6,7,23].…”

Section: Convergence Analysismentioning

confidence: 99%

“…This mechanism is just one possible scenario of transition to turbulence in shear flows and it is usually referred to as streak breakdown [16,21,23,26]. In order to obtain a streak breakdown, three-dimensional disturbances of a suitable axial periodicity must be added to the two-dimensional perturbation.…”

Section: Transition To Turbulencementioning

confidence: 99%

On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow

Meseguer

Mellibovsky

2007

Applied Numerical Mathematics

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“…those exhibiting the largest transient growth, are considered in the optimisation. We focus on two different transition scenarios involving departure from either oblique waves -with wavenumbers (1, ±1)-or streamwise vortices with wavenumber (0, 2)-, see also [5]. (The integers refers to multiples of the fundamental streamwise and spanwise wavenumbers, respectively) For the range of parameters tested, the oblique wave scenario seems to be optimal, consistently with threshold studies in other shear flows [2].…”

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confidence: 58%

Nonlinear optimal perturbations in plane Couette flow

Brandt¹,

Duguet²,

Larsson³

2009

Springer Proceedings in Physics

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Subcritical transition to turbulence can occur in a variety of wall-bounded shear flows when the laminar base state is not subject to linear instability. In this case, perturbations to the base state with a finite but very small amplitude can be amplified by non-normal effects, up to a level where nonlinear interactions come into play. Since transition is often undesired, leading to a dramatic increase in wall drag, it is important to know which kind of (weak) perturbations are susceptible to trigger it if we wish to delay it. A vast amount of litterature has described the linear mechanisms responsible for disturbance amplification. Optimisation methods have yielded linear optimal disturbances[1, 2], the disturbances which exhibit the largest linear growth. Yet the focus has been on linear amplification rather than on actual transition, which requires full nonlinearity to be taken into account. In this study, we go back to the original question of which perturbation is most likely to trigger transition to turbulence. We are hence investigating nonlinear optimal disturbances, those with smallest initial energy which effectively lead to a highly disordered flow. In the recent years, progress in the understanding of subcritical transition was made using the concept of 'edge states', originating from dynamical systems theory. 'Edge state' refers to the flow regime reached asymptotically by critical trajectories at the exact onset of transition. It is an unstable flow state and cannot be observed in experiments, but its stable manifold determines the basin of attraction of the base state. Direct numerical simulation in minimal domains of plane Couette flow has shown that the flow at the onset of transition asymptotically approaches an unstable finite-amplitude steady state solution, characterised by wavy streaks and streamwise rolls [3]. Using a shooting method, approach to such steady states is the way to characterise whether a transitional trajectory in phase space is actually an 'edge' trajectory.In our investigation, we also focus on plane Couette flow in a computational box of size 4πh × 2h × 2πh. The Reynolds number (based on the half-gap h) used for most of the computations presented here is 400. The numerical simulations are performed with a pseudospectral code using 48×33×48 grid points in

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On stability of streamwise streaks and transition thresholds in plane channel flows

Cited by 267 publications

References 54 publications

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow

Nonlinear optimal perturbations in plane Couette flow

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