Excitation and evolution of radiating modes in supersonic boundary layers. Part 2. Back effect of spontaneously radiated Mach waves

Abstract

This paper investigates the linear and nonlinear evolution of radiating modes under the influence of the spontaneously emitted Mach waves in a simple set-up of the supersonic boundary layers that develop in the entry region of a channel formed by two parallel semi-infinite flat plates. Two scenarios are considered. The first occurs in the boundary layers having identical wall conditions, where the Mach wave emitted by a radiating mode in one boundary layer influences the instability in the other. The second scenario takes place when the wall temperatures are different, in which case the spontaneously radiated Mach wave is reflected by the other boundary layer back to act on the radiating mode. Appropriate amplitude equations with the acoustic feedback effect being accounted for are derived. In each case, the effect of the spontaneously emitted sound contributes a linear term of delay type to the respective amplitude equation. For the first scenario, analytical and numerical studies of the amplitude equations show that due to the back action of the spontaneously radiated Mach wave, the amplitude exhibits rapid oscillations, and in the case of enhanced feedback effects, its envelope experiences near extinction followed by resurrection. The study of the coupled equations shows that the two modes with different initial amplitudes either undergo oscillations before attenuating, or terminate a finite-distance singularity at different locations. For the second scenario, the acoustic feedback produces similar effects in a broad range of wall temperature. The effects become pronounced, and the dependence on the wall temperature becomes more sensitive when the latter approaches the value corresponding to the resonance. Estimates suggest that such acoustic feedback is likely to be present in typical wind tunnel experiments and models for scramjet combustors.