Acoustic theory of compressible shear layers. Power-series solutions, equivalence transformations and instabilities

We study temporal instabilities of an inviscid, compressible, hyperbolic tangent shear layer. For this, we use a newly developed algorithm to avoid spurious modes based on remapping of the physical transverse coordinate using the analytic flow profile and connecting Frobenius-series solutions in this remapped space. The transformation maps the infinitely many singularities of the Pridmore-Brown equation to four singularities. Our algorithm is combined with symmetry methods that facilitate equivalence transformations, enabling the extension of the two-dimensional (2-D) isothermal eigenvalue problem to the 3-D and non-isothermal one. We provide a comprehensive overview of unstable 2-D solutions to the temporal stability problem, revealing Mach-number regions where only the first or second modes exist, or a combination of both. Extending to the 3-D, non-isothermal case shows how this solution space is constructed from the 2-D propagating modes in an isothermal shear layer. We find that fundamental effects, such as the occurrence of the first and second modes, are preserved but adjusted. The consideration of 3-D propagating modes shows that above Mach number

, all types of modes occur, with some only occurring in a purely 3-D propagation pattern. Above

upper M almost equals 1.3 M ≈ 1.3 $M\approx 1.3$

, the 3-D propagating modes show larger growth rates. Temperature gradients break the fundamental symmetry properties observed in the isothermal case. A temperature gradient enables modes that only existed in the supersonic isothermal case to also exist at subsonic Mach numbers. This work demonstrates how analytical solutions and symmetry methods are combined to solve singular eigenvalue problems, providing an alternative to classical numerical methods.