TY - JOUR

T1 - Spectral instability of characteristic boundary layer flows

AU - Grenier, Emmanuel

AU - Guo, Yan

AU - Nguyen, Toan T.

N1 - Funding Information:
Gou's work is partially supported by National Science Foundation (NSF) grant DMS-1209437 and Chinese NSF grant 10828103. Nguyen's work is partially supported by NSF grants DMS-1108821 and DMS-1405728.
Publisher Copyright:
© 2016.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

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U2 - 10.1215/00127094-3645437

DO - 10.1215/00127094-3645437

M3 - Article

AN - SCOPUS:84994316367

VL - 165

SP - 3085

EP - 3146

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 16

ER -