Finite-Amplitude Instability of the Compressible Laminar Wake. Strongly Amplified Disturbances

Abstract

The interaction between mean flow and finite‐amplitude disturbances in certain experimentally observed unstable, compressible laminar wakes is considered theoretically without explicitly assuming small amplification rates. Boundary‐layer form of the two‐dimensional mean‐flow momentum, kinetic energy and thermal energy equations and the time‐averaged kinetic energy equation of spatially growing disturbances are recast into their respective von Kármán integral form which show the over‐all physical coupling. The Reynolds shear stresses couple the mean flow and disturbance kinetic energies through the conversion mechanism familiar in low‐speed flows. Both the mean flow and disturbance kinetic energies are coupled to the mean‐flow thermal energy through their respective viscous dissipation. The work done by the disturbance pressure gradients gives rise to an additional coupling between the disturbance kinetic energy and the mean‐flow thermal energy. The compressibility transformation suggested by work on turbulent shear flows is not applicable to this problem because of the accompanying ad hoc assumptions about the disturbance behavior. The disturbances of a discrete frequency which corresponds to the most unstable fundamental component, are first evaluated locally. Subsequent mean‐flow and disturbance profile‐shape assumptions are made in terms of a mean‐flow‐density Howarth variable. The compressibility transformation, which cannot convert this problem into a form identical to the low‐speed problem of Ko, Kubota, and Lees because of the compressible disturbance quantities, nevertheless, yields a much simplified description of the mean flow.

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