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

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• 作者：Emmanuel Grenier^a ; ^{egrenier@umpa.ens-lyon.fr" class="auth_mail" title="E-mail the corresponding author} ; Yan Guo^b ; ^{Yan_Guo@Brown.edu" class="auth_mail" title="E-mail the corresponding author} ; Toan T. Nguyen^c ; ^{nguyen@math.psu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Channel flows ; Spectral instability ; Viscous destabilization ; Singular perturbation ; Critical layers ; Rayleigh and Airy Green functions ; Orr&ndash ; Sommerfeld equations ; Linearized Navier&ndash ; Stokes ; High Reynolds number limit
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：9 April 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：52-110
• 全文大小：784 K

文摘

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 $e^{t/\sqrt{\alpha R}}$, where R is the sufficiently large Reynolds number and α   is the small spatial frequency that remains between lower and upper marginal stability curves: α_low(R)≈R^−1/7${\alpha }_{\mathrm{low}}(R)\approx R^{-1/7}$ and α_up(R)≈R^−1/11${\alpha }_{\mathrm{up}}(R)\approx R^{-1/11}$. 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.