The fundamental difference between continuous modes of the Orr-Sommerfeld/Squire
equations and the entrainment of free-stream vortical disturbances (FSVD) into the
boundary layer has been investigated in a recent paper (Dong & Wu 2013, J. Fluid
Mech.). It was shown there that the non-parallel-flow effect plays a leading-order role in
the entrainment, and neglecting it at outset, as is done in the continuous-mode formulation,
leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy.
The analysis, which was for incompressible boundary layers and for FSVD with a characteristic
wavelength of the order of the local boundary-layer thickness, is extended in
this paper to compressible boundary layers and FSVD with even shorter wavelengths,
which are comparable with the width of the so-called edge layer. Non-parallelism remains
a leading-order effect in the present scaling, which turns out to be more general in that
the equations and solutions in the previous paper are recovered in the appropriate limit.
Appropriate asymptotic solutions in the main and edge layers are obtained to characterize
the entrainment. It is found that when the Prandtl number Pr < 1, free-stream
vortical disturbances of relatively low frequency generate very strong temperature fluctuations
within the edge layer, leading to formation of thermal streaks. A composite
solution, uniformly valid across the entire boundary layer, is constructed, and it can be
used in receptivity studies and as inlet conditions for direct numerical simulations of bypass
transition. For compressible boundary layers, continuous spectra of the disturbance
equations linearised about a parallel base flow exhibit entanglement between vortical and
entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in
the free stream and vice versa, and this amounts to a further nonphysical behaviour.
High-Reynolds-number asymptotic analysis yields the relations between the amplitudes
of entangled modes.