Metastability of Kolmogorov Flows and Inviscid Damping of Shear Flows

Lin, Zhiwu; Xu, Ming

doi:10.1007/s00205-018-1311-8

Cited by 54 publications

(63 citation statements)

References 21 publications

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“…This contrasts greatly with the nonrotating case = 0, where it was shown in Ref. 28 that for neutral modes in 2 , must be an inflection value of . In the literature, it is common to look for unstable modes near neutral modes.…”

mentioning

confidence: 76%

“…Remark For general flows in class

K^{+}

, when there are no nonzero imaginary eigenvalues for the linearized operator

JL

(defined in ), linear damping can be shown as in Theorems and for

ω false(0 false) \in L^{2}

. For

β = 0

, nonexistence of nonzero imaginary eigenvalues and linear damping is true for flows in class

K^{+}

. Explicit decay estimates of the velocity were obtained for monotone and symmetric flows in Refs.…”

Section: Linear Inviscid Dampingmentioning

confidence: 95%

“…For = 0, it is shown in Ref. 28 that only (i) is true, that is, for all neutral modes in 2 , the phase speed must be an inflection value of .…”

Section: Classification Of Neutral Modes Inmentioning

confidence: 99%

“…Remark When

β = 0

k_{i}^{\leq 0} = 0

and the index formula reduces to

k_{c} + k_{r} = n^{-} false(L_{α} false)

(see Ref. ). When

β \neq 0

, in general, we have

k_{i}^{\leq 0} \neq 0

as seen from Sinus flow in Section 4.…”

Section: Hamiltonian Formulation Index Formula and Instability Critmentioning

confidence: 99%

“…For = 0, nonexistence of nonzero imaginary eigenvalues and linear damping is true for flows in class  + . 28 Explicit decay estimates of the velocity were obtained for monotone and symmetric flows in Refs. 41-43 with more regular initial data (eg, (0) ∈ 1 or 2 ).…”

Section: Lemma 14mentioning

confidence: 99%

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Barotropic instability of shear flows

2020

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We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

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mentioning

confidence: 76%

“…Remark For general flows in class

K^{+}

, when there are no nonzero imaginary eigenvalues for the linearized operator

JL

(defined in ), linear damping can be shown as in Theorems and for

ω false(0 false) \in L^{2}

. For

β = 0

, nonexistence of nonzero imaginary eigenvalues and linear damping is true for flows in class

K^{+}

. Explicit decay estimates of the velocity were obtained for monotone and symmetric flows in Refs.…”

Section: Linear Inviscid Dampingmentioning

confidence: 95%

“…For = 0, it is shown in Ref. 28 that only (i) is true, that is, for all neutral modes in 2 , the phase speed must be an inflection value of .…”

Section: Classification Of Neutral Modes Inmentioning

confidence: 99%

“…Remark When

β = 0

k_{i}^{\leq 0} = 0

and the index formula reduces to

k_{c} + k_{r} = n^{-} false(L_{α} false)

(see Ref. ). When

β \neq 0

, in general, we have

k_{i}^{\leq 0} \neq 0

as seen from Sinus flow in Section 4.…”

Section: Hamiltonian Formulation Index Formula and Instability Critmentioning

confidence: 99%

Section: Lemma 14mentioning

confidence: 99%

See 3 more Smart Citations

Barotropic instability of shear flows

2020

Self Cite

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On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Zelati

Delgadino

Elgindi

2019

Comm Pure Appl Math

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We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation timescales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and an optimization step in the frequency cutoff.Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.

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Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

Wei

Zhang

2019

Comm Pure Appl Math

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In this paper, we study pseudospectral bounds for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [22], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies U 0 k 2 f sin.k f y/; 0; 0 H 2 Ä c 7=4 ( the viscosity coefficient) and k f 2 .0; 1/, then the solution does not transition away from the Kolmogorov flow.

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Metastability of Kolmogorov Flows and Inviscid Damping of Shear Flows

Cited by 54 publications

References 21 publications

Barotropic instability of shear flows

Barotropic instability of shear flows

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

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