Spectral instability of characteristic boundary layer flows

By E. Grenier, Yan Guo, Toan T. Nguyen. 2014

142 citations recorded by the source corpus.

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 sufficiently large Reynolds number: $R \to \infty$. 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 $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding 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.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/2b76ce52dc0b8b9e310f50c4937fb067d97be6f1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Well-posedness of the Prandtl equation without any structural assumption (2018, background): “Another example is the so-called Tollmien-Schlichting wave, that is typical of viscous flows at high but finite Reynolds number [5, 15].”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “For instance, in the presence of boundaries it was shown in [31] that linearly spectrally stable Euler flows can become unstable upon the addition of dissipation at large Reynolds number (see also [24]).”
Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows (2016, background): “The best possible value is considered to be k = 1, and the analysis of the Rayleigh equations by [9] indicates that this is indeed the case at least for some specific regime of parameters.”