Three-dimensional eigensolutions of the unsteady compressible boundary-region equations

A previous paper of the authors (Duck \& Stephen (2021) \lq Eigensolutions of the unsteady boundary-layer equations revisited (with extensions to three-dimensional modes)', J. Fluid Mech. {\bf{917}}) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the {\it{boundary-region equations}} methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-~) classification into entropy and non-entropy modes. The former, according to this analysis are spatially damped, with one caveat as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.