Receptivity of inviscid modes in supersonic boundary layers to wall perturbations

Abstract

The present paper investigates the receptivity of inviscid first and second modes in a supersonic boundary layer to time-periodic wall disturbances in the form of local blowing/suction, streamwise velocity perturbation and temperature perturbation, all introduced via a small forcing slot on the flat plate. The receptivity is studied using direct numerical simulations (DNS), finite- and high-Reynolds-number approaches, which complement each other. The finite-Reynolds-number formulation predicts the receptivity as accurately as DNS, but does not give much insight to the detailed excitation process, nor can it explain the significantly weaker receptivity efficiency of the streamwise velocity and temperature perturbations relative to the blowing/suction. In order to shed light on these issues, an asymptotic analysis was performed in the limit of large Reynolds number. It shows that the receptivity to all three forms of wall perturbations is reduced to the same mathematical form: the Rayleigh equation subject to an equivalent suction/blowing velocity, which can be expressed explicitly in terms of the physical wall perturbations. Estimates of the magnitude of the excited eigenmode can be made a priori for each case. Furthermore, the receptivity efficiencies for the streamwise velocity and temperature perturbations are quantitatively related to that for the blowing/suction by simple ratios, which are of $$O(R^{-1/2})$$ O ( R - 1 / 2 ) and have simple expressions, where R is the Reynolds number based on the boundary-layer thickness at the centre of the forcing slot. The simple leading-order asymptotic theory predicts the instability and receptivity characteristics accurately for sufficiently large Reynolds numbers (about $$10^4$$ 10 4 ), but appreciable error exists for moderate Reynolds numbers. An improved asymptotic theory is developed by using the appropriate impedance condition that accounts for the $$O(R^{-1/2})$$ O ( R - 1 / 2 ) transverse velocity induced by the viscous motion in the Stokes layer adjacent to the wall. The improved theory predicts both the instability and receptivity at moderate Reynolds numbers ($$R=O(10^3)$$ R = O ( 10 3 ) ) with satisfactory accuracy. In particular, it captures well the finite-Reynolds-number effects, including the Reynolds-number dependence of the receptivity and the strong excitation occurring near the so-called synchronisation point.