Spectral instability of characteristic boundary layer flows

Emmanuel Grenier, Yan Guo, Toan T. Nguyen

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large Reynolds number: R → ∞. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well documented in physics literature, going back to the work of Heisenberg, C. C. Lin, Tollmien, Drazin, and Reid, but a rigorous construction requires delicate mathematical details, involving, for instance, a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of et/√R. The prooffollows the general iterative approach introduced in our companion paper, avoiding having to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators. Unlike in the channel flows, the spatial domain in the boundary layers is unbounded and the iterative scheme is likely to diverge due to the linear growth in the vertical variable. We introduce a new iterative scheme to simultaneously treat the singularity near critical layers and the asymptotic behavior of solutions at infinity. The instability of generic boundary layers obtained in this paper is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

Original languageEnglish (US)
Pages (from-to)3085-3146
Number of pages62
JournalDuke Mathematical Journal
Volume165
Issue number16
DOIs
StatePublished - Jan 1 2016

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Boundary Layer Flow
Boundary Layer
Iterative Scheme
Reynolds number
Singularity
Airy Functions
Singular Solutions
Eigenvalues and Eigenfunctions
Channel Flow
Shear Flow
Asymptotic Behavior of Solutions
Diverge
Turbulent Flow
Rayleigh
Asymptotic Expansion
Green's function
Viscosity
Navier-Stokes Equations
Linearly
Unstable

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Grenier, Emmanuel ; Guo, Yan ; Nguyen, Toan T. / Spectral instability of characteristic boundary layer flows. In: Duke Mathematical Journal. 2016 ; Vol. 165, No. 16. pp. 3085-3146.
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Spectral instability of characteristic boundary layer flows. / Grenier, Emmanuel; Guo, Yan; Nguyen, Toan T.

In: Duke Mathematical Journal, Vol. 165, No. 16, 01.01.2016, p. 3085-3146.

Research output: Contribution to journalArticle

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