Inviscid instability of compressible exponential boundary layer flows

Abstract

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.

Over-reflection of acoustic waves by supersonic exponential boundary layer flows

“…In this range, the over-reflection coefficient is not influenced by resonant frequencies, i.e. the unstable eigenvalues ω emerging due to acoustic instabilities of the exponential boundary layer profile (see Zhang & Oberlack 2021). We refer to those over-reflections that are not affected by resonant frequencies as non-resonant over-reflections.…”

“…The PBE (2.9) not only constitutes an eigenvalue problem for modes to investigate stability problems, as was detailed in Zhang & Oberlack (2021), but also describes acoustic wave propagation (non-resonant spectra) in boundary layer flows, which is the main topic of the present work.…”

“…In Zhang & Oberlack (2021), they deduce an eigenvalue problem for the temporal stability of the exponential boundary layer profile, and this is essentially based on solution (2.12). The imaginary part of the eigenvalue derived there, i.e.…”

“…By studying the amplitude of acoustic waves, they noticed an attenuation of the wave amplitude adjacent to the critical layer. Recently, Zhang & Oberlack (2021) gave an exact solution to the PBE for an exponential velocity profile in terms of the confluent Heun function (CHF) (Ronveaux & Arscott 1995). Based on this, they investigated the temporal stability of exponential boundary layer flows and succeeded in giving unstable acoustic modes and the mechanism of the instability.…”

“…This gap triggered our interest. Applying the exact solution in Zhang & Oberlack (2021) to the PBE for an exponential velocity profile, it is straightforward to obtain an explicit expression for the reflection coefficient of acoustic waves. Note that the approach to establishing a link between the over-reflection and instability in the present study differs from most previous work in GFD.…”