Spectral instability of general symmetric shear flows in a two-dimensional channel

Emmanuel Grenier, Yan Guo, Toan Nguyen

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

Original languageEnglish (US)
Pages (from-to)52-110
Number of pages59
JournalAdvances in Mathematics
Volume292
DOIs
StatePublished - Apr 9 2016

Fingerprint

Shear Flow
Reynolds number
Eigenvalues and Eigenfunctions
Incompressible Navier-Stokes Equations
Operator
Euler Equations
Rayleigh
Asymptotic Expansion
Green's function
Navier-Stokes Equations
Unstable
Singularity
Curve

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

@article{2f2732855072488ab90485889b2a400a,
title = "Spectral instability of general symmetric shear flows in a two-dimensional channel",
abstract = "In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.",
author = "Emmanuel Grenier and Yan Guo and Toan Nguyen",
year = "2016",
month = "4",
day = "9",
doi = "10.1016/j.aim.2016.01.007",
language = "English (US)",
volume = "292",
pages = "52--110",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

Spectral instability of general symmetric shear flows in a two-dimensional channel. / Grenier, Emmanuel; Guo, Yan; Nguyen, Toan.

In: Advances in Mathematics, Vol. 292, 09.04.2016, p. 52-110.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Spectral instability of general symmetric shear flows in a two-dimensional channel

AU - Grenier, Emmanuel

AU - Guo, Yan

AU - Nguyen, Toan

PY - 2016/4/9

Y1 - 2016/4/9

N2 - In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

AB - In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier-Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier-Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R-1/7 and αup(R)≈R-1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

UR - http://www.scopus.com/inward/record.url?scp=84957075471&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957075471&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.01.007

DO - 10.1016/j.aim.2016.01.007

M3 - Article

AN - SCOPUS:84957075471

VL - 292

SP - 52

EP - 110

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -