Excitation and evolution of radiating modes in supersonic boundary layers. Part 1. Fundamental resonance with impinging sound waves

Abstract

This paper investigates the linear and nonlinear evolution of radiating modes in supersonic boundary layers under the influence of impinging sound waves. It is found that the ensuing boundary-layer response is extraordinarily large for a subset of the sound frequency and incident angle, and the resonant over-reflection, corresponding to the reflection coefficient becoming infinite, occurs at a particular pairing of frequency and incident angle. At this point, the reflected wave coincides with a locally neutral radiating mode, which emits spontaneously sound in the form of Mach waves. A fundamental resonance takes place between the incident wave and the radiating mode. Viewed in a developing boundary layer, the response is rendered finite by introducing non-parallelism and nonlinear effects near the neutral location of the radiating mode, where the sound wave directly excites the radiating mode and/or acts on the pre-existing radiating mode. Inhomogeneous amplitude equations are derived to describe the excitation as well as the nonlinear development of the radiating mode in the two regimes where non-equilibrium and non-parallelism play a leading-order role, respectively. A composite amplitude equation is then constructed to take into account both non-parallelism and non-equilibrium effects. This amplitude equation is, with an appropriate initial condition, solved to quantify the impact of the impinging sound wave on the linear and nonlinear instability characteristics of the radiating mode. The far-field analysis shows that the Mach wave field of the radiating mode is changed significantly due to the incident sound.