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Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances

Published online by Cambridge University Press:  08 April 2011

PIERRE RICCO ,
JISHENG LUO  and
XUESONG WU
PIERRE RICCO
Affiliation:
Division of Engineering, King's College London, London WC2R 2LS, UK
JISHENG LUO
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 30072, China
XUESONG WU*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2BZ, UK
*
†Email address for correspondence: x.wu@ic.ac.uk
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Abstract

We investigate the influence of free-stream vortical disturbances on the evolution and instability of an incompressible laminar boundary layer, focusing on components of sufficiently long wavelength, which are known to penetrate into the boundary layer to generate streamwise elongated streaks. The free-stream disturbance is assumed to be sufficiently strong (but still of small amplitude) that the induced streaks acquire an O(1) streamwise velocity in the region where the boundary-layer thickness becomes comparable with the spanwise wavelength of the perturbation. The formation and evolution of the streaks are governed by the nonlinear unsteady boundary-region equations supplemented by appropriate upstream and far-field boundary conditions. This initial-boundary-value problem is solved for the special case where the free-stream disturbance is modelled by a pair of oblique vortical modes with the same frequency but opposite spanwise wavenumbers. Nonlinearity is found to inhibit the response. The nonlinear interaction alters the mean-flow profile appreciably, the shape of which is in quantitative agreement with experimental measurements. Wall-normal inflection points are detected in the instantaneous streamwise velocity profiles. The secondary stability analysis indicates that in the presence of free-stream disturbance with an intensity of 2.8%, the resulting streaky boundary layer becomes inviscidly unstable. The characteristic frequency, phase and group velocities, and growth rate of unstable sinuous modes are found to be in broad agreement with recent experiments. The present theoretical framework allows in principle a quantitative relation to be established between the characteristics of free-stream turbulence and the secondary instability, and this relation may be exploited to develop an efficient and physics-based approach for predicting bypass transition.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability Instability: Transition to turbulence

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Papers
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Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 1 - 38
Copyright
Copyright © Cambridge University Press 2011

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References

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