Abstract
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 $M_\infty$ in the subsonic and moderate supersonic regime so that $0\leqslant M_\infty \leqslant 2$ . 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.
Highlights
It has been known since Kline et al (1967) that transitional and turbulent flows exhibit clear structure within the boundary layer in the form of vortical structures coupled to high- and low-speed streaks in the plane perpendicular to the unperturbed flow
We explore the behaviour of the velocity and thermal streaks as the Reynolds number Re, Mach number M∞ and Prandtl number σ vary
Using the International Standard Atmosphere (International Organization for Standardization 1975) value for temperature at a fixed altitude, we describe in table 2 the range of free-stream velocities u∞ and suction velocities v∞ required to obtain Reynolds numbers in the range 80 000–140 000 and Mach numbers in the subsonic to moderate supersonic range, 0.1 M∞ 2
Summary
It has been known since Kline et al (1967) that transitional and turbulent flows exhibit clear structure within the boundary layer in the form of vortical structures coupled to high- and low-speed streaks in the plane perpendicular to the unperturbed flow Recent understanding of these structures has been aided by the identification of three-dimensional, nonlinear invariant solutions of the Navier–Stokes equations which may take the form of equilibria, periodic orbits or travelling-wave solutions. Travelling-wave-type solutions have been identified in the ASBL by Deguchi & Hall (2014), who found structures localised in the wall-normal direction but periodic in the streamwise and spanwise directions, and by Kreilos, Gibson & Schneider (2016), who found spanwise-localised travelling-wave solutions. Two types of solution were found: a ‘wall mode’ coherent structure with the streaks and vortex structure concentrated near the wall region; and a ‘free-stream’ coherent structure with the streak flow still mainly concentrated in the near-wall region but with the vortical structure residing in the free stream
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