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Boundary layer collapses described by the two-dimensional intermediate long-wave equation

Oloo, J.O.; Shrira, Victor

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Authors

J.O. Oloo



Abstract

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width s, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, s) for various values of D. The amplitude threshold a increases as D and s decrease and tends to infinity at D ? 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

Journal Article Type Article
Acceptance Date Nov 7, 2020
Publication Date Apr 30, 2020
Journal Theoretical and Mathematical Physics
Print ISSN 0040-5779
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 203
Pages 512 - 523
DOI https://doi.org/10.1134/S0040577920040078
Keywords boundary layer instabilitynonlinear evolution equationcollapsesingularity formationlaminar-turbulent transition
Publisher URL https://link.springer.com/article/10.1134%2FS0040577920040078#article-info

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