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Response of a compressible laminar boundary layer to free-stream vortical disturbances

Published online by Cambridge University Press:  31 August 2007

PIERRE RICCO  and
XUESONG WU
PIERRE RICCO
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2BZ, UK
XUESONG WU
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2BZ, UK
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Abstract

As a first step towards understanding the role of free-stream turbulence in laminar–turbulent transition, we calculate the fluctuations induced by free-stream vortical disturbances in a compressible laminar boundary layer. As with the incompressible case investigated by Leib et al. (J. Fluid Mech. vol. 380, 1999, p. 169), attention is focused on components with long streamwise wavelength. The boundary-layer response is governed by the linearized unsteady boundary-region equations in the typical streamwise region where the local boundary-layer thickness δ* iscomparable with the spanwise length scale Λ of the disturbances. The compressible boundary-region equations are solved numerically for a single Fourier component to obtain the boundary-layer signature. The root-mean-square of the velocity and mass-flux fluctuations induced by a continuous spectrum of free-stream disturbances are computed by an appropriate superposition of the individual Fourier components.

Low-frequency vortical disturbances penetrate into the boundary layer to form slowly modulating streamwise-elongated velocity streaks. In the compressible regime, vortical disturbances are found to induce substantial temperature fluctuations so that ‘thermalstreaks’ also form. They may have a significant effect on the secondary instability. The calculations indicate that for a vortical disturbance with a relatively large Λ, the induced boundary-layer fluctuation ultimately evolves into an amplifying wave. This is due to a receptivity mechanism, in which a vortical disturbance first excites a decaying quasi-three-dimensional Lam–Rott eigensolution. The latter then undergoes wavelength shortening to generate a spanwise pressure gradient, which eventually converts the Lam–Rott mode into an exponentially growing mode. The latter is recognized to bea highly oblique Tollmien–Schlichting wave. A parametric study suggests that this receptivity mechanism could be significant when the free-stream Mach number is larger than 0.8.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 587 , 25 September 2007 , pp. 97 - 138
Copyright
Copyright © Cambridge University Press 2007

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