Based on the linearized Euler equations and the normal-mode approach, the Pridmore-Brown equation for acoustics and the stability of parallel shear flows is solved for a boundary layer flow with an exponential velocity profile. The general solution is derived in terms of the confluent Heun function, and in turn, using the boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity, the boundary value problem is converted to an algebraic eigenvalue problem. Solutions to the eigenvalue problem allow a comprehensive picture of the stability of compressible boundary layers. In particular, the complex eigenvalues ω are calculated as a function of the streamwise wavenumber α and the Mach number M. We observe that with growing wavenumbers, the number of eigenvalues increases discretely. Only for M > 1, unstable modes exist, and the boundary between neutrally stable and unstable modes is defined by the transonic line ω = α to be calculated from a degenerated eigenvalue equation. For large wall distances, the eigenfunctions converge exponentially toward zero. The spatial decay rate defines an “acoustic boundary layer thickness” δa, which indicates how far outside modes are still audible. For M > 1 and large α, δa diverges exponentially, i.e., in this parameter range, modes are perceptible even far from the boundary layer. A particularly steep rise of δa is observed when M > 2. Thus, there is a wavenumber range in which ωi and δa are both large and thus generates a particularly strong noise impact. From the eigenfunction for the unstable modes, a strong increase in the amplitude of acoustic waves can be identified in the vicinity of the wall, which indicates an accumulation and saturation of energy and thereafter leads to temporal instability and sound radiation in the free stream.
The two-dimensional acoustic wave equation for inviscid compressible boundary layer flows, i.e. the Pridmore-Brown equation with an exponential velocity profile for homentropic flows, is studied for the reflection and over-reflection of acoustic waves based on the exact solution in terms of the confluent Heun function. The reflection coefficient $R$ , which is the ratio of the amplitude of the reflected to that of the incoming acoustic wave, is determined as a function of the streamwise wavenumber $\alpha$ , the Mach number $M$ and the incident angle $\phi$ of the acoustic waves. Over-reflection refers to $R>1$ , i.e. the reflected wave has a larger amplitude than the incident wave. We prove that, in the supersonic context, energy is always transferred from the base flow to the reflected wave, i.e. $R<1$ does not exist. Meanwhile, this fact is intimately linked to the critical layer. We show that the presence of the critical layer leads to an optimal energy exchange from the base flow into the acoustic wave, i.e. the critical layer ensures $R>1$ . In our analysis, we observe a special phenomenon, resonant over-reflection, which is proven to be closely related to resonant frequencies $\omega _r$ of unstable modes of the temporal stability of the base flow. At resonant frequencies of the first unstable mode, the over-reflection coefficient exhibits an unusual peak in an extremely narrow frequency interval. The maximum values of these peaks are largely synchronized with the variation of the growth rate $\omega _i$ of the unstable modes. In addition, resonant over-reflection appears also at resonant frequencies of other higher unstable modes, but their peaks of the over-reflection coefficient are always smaller than that induced by the first unstable mode.
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