title: Instability and acoustics of compressible exponential boundary layer flows
Creators: name: Zhang, Yi
Publisher: name: UNSPECIFIED
Year: 2022

description: In this thesis, inviscid instability and acoustics of compressible exponential boundary layer flows are investigated. Based on the linearised Euler equations (LEEs) and the normal-mode approach, the acoustic wave equation of parallel shear flows, the generalised Pridmore-Brown equation (PBE), is derived. For a boundary layer flow mimicked by an exponential velocity profile, an exact solution to the corresponding PBE is given in terms of the confluent Heun function (CHF). In the stability analyses, the eigenvalue equation for the stability problem based on the exact solution to the PBE is derived, and temporal stability and spatial stability are investigated respectively. For this, asymptotic analyses of the eigenvalue equation are first performed, and analytical solutions for limiting cases are obtained. Then, solutions to the eigenvalue equation are computed, which allow a comprehensive picture of the stability behaviour of the exponential boundary layer. In particular, the first three acoustic modes are computed as a function of the Mach number, the streamwise wavenumber, and the frequency. Unstable modes are found, where the first acoustic mode is always the most unstable one of all acoustic modes. Besides, an acoustic boundary layer thickness (ABLT) is defined, which essentially quantifies how far eigenfunctions reach into the area afar from the boundary layer. Meanwhile, wave angles, which describe the direction of the phase velocity, and eigenfunctions of acoustic modes are displayed. In the end, links between eigenvalues in the temporal stability and spatial stability problems are established. In the study of acoustics of boundary layer flows, the exact solution to the PBE is again employed to derive the reflection coefficient as a function of the Mach number, the streamwise wavenumber, and the incident angle of acoustic waves, and it is computed in wide parameter ranges. It is shown that the over-reflection of acoustic waves arises in boundary layer flows, i.e. the reflected amplitude of acoustic waves is greater than that of incident waves. The phenomenon is validated to be closely related to the critical layer, at which there is an optimal energy exchange from the base flow through the critical layer into the acoustic wave. Meanwhile, a special acoustic phenomenon, the resonant over-reflection, is observed and proved to be caused by the resonant frequency of unstable modes in the temporal stability problem. In addition, the resonant over-reflection also appears at resonant frequencies caused by higher unstable modes, but their over-reflection coefficients are always smaller than that caused by the first unstable mode. In the last part of the present work, the over-reflection of acoustic waves in a supersonic inviscid compressible boundary layer flow is validated by direct numerical simulations (DNS). A wave packet containing plane waves with constant wavelengths and amplitudes is superimposed with the free stream, and the incidence and reflection processes of the wave packet are simulated. In the simulations, the dispersion of the wave packet is observed due to strong shear effects near the wall. Amplification of the amplitude of the reflected waves is determined when the reflected wave eventually returns to the free stream. In particular, there is an exceptionally large over-reflection coefficient when the frequency of the incident wave is close to the resonant frequency, which indicates an occurrence of the resonant over-reflection.