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1 November 2016 Spectral instability of characteristic boundary layer flows
Emmanuel Grenier, Yan Guo, Toan T. Nguyen
Duke Math. J. 165(16): 3085-3146 (1 November 2016). DOI: 10.1215/00127094-3645437


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 proof follows 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.


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Emmanuel Grenier. Yan Guo. Toan T. Nguyen. "Spectral instability of characteristic boundary layer flows." Duke Math. J. 165 (16) 3085 - 3146, 1 November 2016.


Received: 16 March 2015; Revised: 25 November 2015; Published: 1 November 2016
First available in Project Euclid: 31 August 2016

zbMATH: 1359.35129
MathSciNet: MR3566199
Digital Object Identifier: 10.1215/00127094-3645437

Primary: 35Q30
Secondary: 35B25 , 35B35

Keywords: boundary layers , Navier–Stokes equations , spectral instability , Tollmien–Schlichting instability waves

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 16 • 1 November 2016
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