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}) 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), 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}) 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)) 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.
Highlights
Transition prediction of boundary layer flows from a laminar to turbulent state is of critical technological importance, especially for the development of super/hypersonic vehicles that are to be used for deep space access [1,2]
We have investigated the receptivity of inviscid first and second modes in a supersonic boundary layer to time-periodic streamwise localised wall perturbations, namely, blowing/suction, streamwise velocity perturbation and temperature perturbation, all introduced by a forcing slot on the flat plate
Three complementary approaches have been employed, including a finite-Reynolds-number theory [37,38,57], and the large-Reynolds-number asymptotic theory and direct numerical simulations (DNS), which is performed in order to assess the accuracy of the former two approaches
Summary
Transition prediction of boundary layer flows from a laminar to turbulent state is of critical technological importance, especially for the development of super/hypersonic vehicles that are to be used for deep space access [1,2]. The value of the asymptotic methods has not been diminished This is because in the O–S type of equations, the instability modes are, explicitly or implicitly, taken to have wavelengths and phase speeds comparable with the boundary-layer thickness and the base-flow velocity, respectively. An incident sound wave impinges on the rapidly changing boundary layer flow near the leading edge to excite first the so-called Lam–Rott asymptotic eigenmode The latter undergoes wavelength shortening to evolve into a lower-branch viscous T–S instability mode [27]. By using the asymptotic techniques, the slow and fast acoustic waves in the free stream, which propagate upstream and downstream relative to the background flow, respectively, were found to match with the slow and fast boundary-layer eigenmodes, respectively, in the leading-edge region, exciting the latter directly [43,44,45]. The le√ading-edge receptivity mechanisms for strongly three-dimensional (viscous) modes with Θ > M2 − 1 and Θ ≈ M2 − 1 were discussed by [53] and [54], respectively
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