Analyses of Reynolds and Mach number effects on Tollmien–Schlichting wave–bump interaction in subsonic flows

Abstract

We investigated the effects of two-dimensional sharp-edged rectangular bumps on Tollmien–Schlichting (TS) wave evolution using direct numerical simulation. The bump height, $h$ , ranged from 5 % to 40 % of the local displacement thickness, $\delta ^*$ . Behind the bump, a recirculating flow region could be formed whose length increased nonlinearly with $h$ . The bump height effect on the TS wave, which was the dominant, scaled super-exponentially with $h$ . We also showed a substantial effect of the $\delta ^*$ -based Reynolds number, ${\textit {Re}} _{\delta ^*}$ . Firstly, the bump wake extended with ${\textit {Re}} _{\delta ^*}$ , promoting larger TS wave growth rates. The second effect is related to proximity to the upper branch of the instability loop, accounting for the influence of the TS frequency, as well. It dictates the bump impact increases as it gets closer to transition, either by the bump moving downstream or the transition moving upstream. For a 40 % high bump, for example, changing the ${\textit {Re}} _{\delta ^*}$ at the bump location from 1500 to 2000 increased $\Delta N$ by a factor of 2 ( $\Delta N$ represents a measure of a surface irregularity effect on the smooth plate N-factor). We also found that $(\Delta N)_{max}$ increases linearly with ${\textit {Re}} _{hh}$ . Results in the subsonic regime showed that the bump impact attenuates with Mach number up to 0.7 but above it, stabilisation is surpassed by the destabilising effect caused by the recirculation lengthening. This is mostly associated with the bump wake that extends with the pressure gradient which increases substantially towards the sonic speed. This is enhanced if the surface is adiabatic rather than isothermal.