Cancellation of Tollmien-Schlichting waves with surface heating

Two-dimensional boundary layer flows in quiet disturbance environments are known to become unstable to Tollmien–Schlichting waves. The experimental work of Liepmann et al. (J Fluid Mech 118:187–200, 1982), Liepmann and Nosenchuck (J Fluid Mech 118:201–204, 1982) showed how it is possible to control and reduce unstable Tollmien–Schlichting wave amplitudes using unsteady surface heating. We consider the problem of an oncoming planar compressible subsonic boundary layer flow with a three-dimensional vibrator mounted on a flat plate, and with surface heating present. It is shown using asymptotic methods based on triple-deck theory that it is possible to choose an unsteady surface heating distribution to cancel out the response due to the vibrator. An approximation based on the exact formula is used successfully in numerical computations to confirm the findings. The results presented here are a generalisation of the analogous results for the two-dimensional problem in Brennan et al. (J Fluid Mech 909:A16-1, 2020).