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Localised streak solutions for a Blasius boundary layer

Published online by Cambridge University Press:  26 June 2018

Richard E. Hewitt Open the ORCID record for Richard E. Hewitt [Opens in a new window]  and
Peter W. Duck
Richard E. Hewitt*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
Peter W. Duck
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

Streaks are a common feature of disturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of self-sustained structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the classical two-dimensional Libby & Fox (J. Fluid Mech., vol. 17 (3), 1963, pp. 433–449) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes (including the slowest decaying) is fundamentally related to the solutions first presented by Luchini (J. Fluid Mech., vol. 327, 1996, pp. 101–116), as derived for spanwise-periodic disturbances (at small spanwise wavenumber). This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the fully non-parallel downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Encouraging comparisons are made with available experimental data.

JFM classification

Boundary Layers: Boundary Layers Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 885 - 901
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Copyright
© 2018 Cambridge University Press 

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