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The laminar boundary layer on a flat plate in periodic sideslip

Published online by Cambridge University Press:  21 April 2006

L. Bernstein  and
M. S. Ishaq
L. Bernstein
Affiliation:
Department of Aeronautical Engineering, Queen Mary College, University of London. Mile End Road, London E1 4NS, UK
M. S. Ishaq
Affiliation:
Department of Aeronautical Engineering, Queen Mary College, University of London. Mile End Road, London E1 4NS, UK
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Abstract

A theoretical study has been made of the laminar boundary layer on a semi-infinite flat plate parallel to a stream consisting of a uniform steady component U∞ normal to its leading edge and a periodic sideslip component of the travelling-wave type, where the wave travels with velocity Q in the direction of the steady component. It is found that the longitudinal flow (that in planes perpendicular to the leading edge) is independent of the transverse flow, and satisfies the well-known Blasius equations. The transverse flow is governed by a linear partial differential equation which may be approximated in different ways for high and low values of the ‘reduced’ frequency $\overline{\omega}$. A series-expansion solution for small $\overline{\omega}$ appears to be valid up to about $\overline{\omega} = 2$; the solution for large $\overline{\omega}$ is applicable down to $\overline{\omega} \approx 10$. A third approximation has been developed which joins the others smoothly. Numerical solutions of the equations for the transverse flow are presented for $0 \leqslant\overline{\omega}\leqslant 40$ and Q/U∞ = 0.6 (the value appropriate to the Queen Mary College (QMC) ‘gust-tunnels’) and for $\overline{\omega} = 10$ and 0.4 [les ] Q/U∞ [les ] ∞. The value of Q/U∞ has a profound influence: for values less than about one there are large phase lags within the boundary layer; for large values there are phase leads throughout most of the layer. For Q/U∞ < 1 the amplitude of the oscillation within the boundary layer exceeds that of the external driving oscillation, this ‘overshoot’ increasing as the wave-speed ratio diminishes. At Q/U∞ = 0.6 peak velocities more than 3 times those outside appear within the viscous layer.

As $\overline{\omega} \rightarrow \infty $, the transverse viscous layer becomes thinner; the oscillatory boundary layer, here transverse, becomes a ‘Stokes layer’ and is virtually uncoupled from the longitudinal flow. Far downstream the amplitude of the transverse skin-friction grows as x½ and becomes comparable with the streamwise component even for moderate values of the sideslip amplitude.

Experiments were conducted in one of the QMC gust-tunnels for values of $\overline{\omega}$ up to 2.0. Measurements of the transverse velocity amplitude and phase profiles confirm the ‘low frequency theory’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 175 , February 1987 , pp. 511 - 535
Copyright
© 1987 Cambridge University Press

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