Free-stream coherent structures in parallel compressible boundary-layer flows at subsonic and moderate supersonic Mach numbers

As a first step towards the description of coherent structures in compressible shear flows, we present an asymptotic description of nonlinear travelling-wave solutions of the Navier-Stokes equations in the compressible asymptotic suction boundary layer (ASBL). We consider free-stream Mach numbers in the subsonic and moderate supersonic regime so that. We extend the large-Reynolds-number asymptotic theory of Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602-625) describing 'free-stream' coherent structures in incompressible ASBL flow to describe a nonlinear interaction in a thin layer situated just below the free stream. Crucially, the nonlinear interaction equations for the velocity field in this layer are identical to those obtained in the incompressible problem, and thus the asymptotic analysis supporting free-stream coherent structures in compressible ASBL is easily deduced from its incompressible counterpart. The nonlinear interaction produces streaky disturbances to both the velocity and temperature fields, which can grow exponentially towards the wall. We complete the description of the growth of the velocity and thermal streaks throughout the flow by solving the compressible boundary-region equations numerically. We show that the velocity and thermal streaks obtain their maximum amplitude in the unperturbed boundary layer. Increasing the free-stream Mach number enhances the thermal streaks and suppresses the velocity streaks, whereas varying the Prandtl number suppresses the velocity streaks, and can either enhance or suppress the thermal streaks depending on whether the flow is in the subsonic or moderate supersonic regime. Such nonlinear equilibrium states have been implicated in shear transition in incompressible flows; therefore, our results indicate that a similar mechanism may also be present in compressible flows.