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 Karman 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.

Recent experimental observations appear to indicate that turbulent mean flow velocity profiles with points of inflection are dynamically unstable with respect to traveling wavy disturbances. As an illustration, the description of the nonlinear development of an instability wave in the turbulent wake behind a thin body is presented on the basis of integrals of the equations of mean flow momentum, kinetic energy, and the time-averaged disturbance kinetic energy, with the turbulent dissipation integrals relegated to phenomenological considerations. The over-all physical mechanisms leading to the streamwise distribution of the mean flow decay and disturbance energy amplification and decay are explained through the disturbance energy production integral and the mean flow and disturbance turbulent-dissipation integrals. The growth of the disturbance wave is limited by (1) the rapidity of turbulent mean flow decay which renders the production integral less spectacular and (2) the prominant role of the turbulent dissipation integral which draws energy away from the disturbance wave to the turbulence. These limitations and the rapidity with which events take place in the streamwise direction essentially distinguishes the instability wave in a turbulent mean flow with inflectional velocity profile from that in a corresponding laminar flow.