Analytical study of inviscid instabilities in exponentially modelled compressible boundary layers over porous walls

We investigate the influence of porous walls on the linear temporal stability of compressible boundary layers with adiabatic wall temperature, focusing on effects arising from variations in the porous wall configuration. Assuming an exponential base-flow profile and a regularly structured wall with cylindrical pores as a model assumption allows for an analytic solution of the underlying inviscid compressible Rayleigh equation in terms of the general Heun function, thereby reducing the problem to an algebraic eigenvalue equation. This enables the analysis of eigenmodes across a wide Mach number range and broad variations in the porous-wall parameters, porosity phi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ϕ</mml:mi> </mml:math> $\phi$ and layer thickness h <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>h</mml:mi> </mml:math> $h$ , revealing a novel delineation of parameter regimes associated with different effects on the stability characteristics. We find that porous walls with very small porosities phi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ϕ</mml:mi> </mml:math> $\phi$ allow damping, whereas higher phi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ϕ</mml:mi> </mml:math> $\phi$ lead to destabilisation. The character of destabilisation is largely determined by the layer thickness: for larger h <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>h</mml:mi> </mml:math> $h$ , long-wave, three-dimensional first-mode instabilities dominate, while small h <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>h</mml:mi> </mml:math> $h$ favour short-wave, two-dimensional second-mode instabilities. For intermediate h <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>h</mml:mi> </mml:math> $h$ , the most unstable mode can be a two-dimensional first mode, in contrast to its usual three-dimensional character. Further, we show that the layer thickness h <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>h</mml:mi> </mml:math> $h$ causing strongest destabilisation at a given Mach number decreases with increasing upper M <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" x