A study of over-reflection in viscous Poiseuille flow

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Abstract
It is shown that viscous Poiseuille flow sustains wave propagation in its centre region for waves whose phase speed is less than the maximum flow speed. When, moreover, there are critical levels, there exists a range of phase speeds for which over-reflection occurs and this range corresponds exactly to the range for which unstable eigenmodes exist. Consistent with the conjecture of Lindzen & Barker (1985), it is found that viscous boundary layers around the critical level and at the wall replace the exponential regions and wave sinks required for inviscid over-reflection.Over-reflection, we find is confined to phase speeds for which these two boundary layers are in close proximity rather than widely separated or substantially overlapping.Over-reflection is inevitably associated with a wave phase tilt opposite in direction to the shear at the critical level. All other cases yield a phase tilt in the direction of the shear. The former is consistent with the condition for the Orr mechanism to produce amplification (Boyd 1983).

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