2 results
Receptivity of inviscid modes in supersonic boundary layers due to scattering of free-stream sound by localised wall roughness
- Ming Dong, Yinhui Liu, Xuesong Wu
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- Journal:
- Journal of Fluid Mechanics / Volume 896 / 10 August 2020
- Published online by Cambridge University Press:
- 04 June 2020, A23
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The present paper investigates the receptivity of inviscid first and second modes in super/hypersonic boundary layers due to the interaction between a weak free-stream acoustic wave and a small isolated surface roughness element. The large-Reynolds-number asymptotic analysis reveals the detailed processes of the excitation. The distortion of the acoustic signature by the curved wall contributes to the leading-order receptivity, producing an eigenmode of $O({\mathcal{E}}h)$ amplitude, where ${\mathcal{E}}\ll 1$ is the magnitude of the acoustic wave and $h\ll 1$ the roughness height normalised by the local boundary-layer thickness $\unicode[STIX]{x1D6FF}$. The interactions between the roughness-induced mean-flow distortion and the acoustic signature contribute to the second-order receptivity, which is of $O({\mathcal{E}}hR^{-1/3})$ with $R\gg 1$ being the Reynolds number based on $\unicode[STIX]{x1D6FF}$. Interestingly, the leading-order receptivity is equivalent to a canonic receptivity problem, the excitation by time-periodic blowing and suction through a local slot on the wall, and the effective periodic outflux velocity forced from the underneath Stokes layer can be determined explicitly in terms of the roughness shape function. This equivalence holds when $h=O(R^{-1/3})$, for which the roughness-induced mean-flow distortion becomes nonlinear. A systematic parametric study is carried out for the excitation of the first and second modes by both fast and slow free-stream acoustic waves, and the dependence of the receptivity efficiency on the relevant parameters is provided. Interestingly, the second-order receptivity could become dominant (e.g. in the case of slow acoustic waves with low frequencies and small incident angles), but the present mathematical theory remains valid. In order to check the accuracy of the asymptotic predictions, we have carried out direct numerical simulations (DNS) and also extended the existing finite-Reynolds-number theory to the supersonic regime. The asymptotic solutions agree with the results given by the finite-Reynolds-number calculations and DNS when $R$ is sufficiently large (typically $R=O(10^{5})$). An improved large-Reynolds-number approach is developed by replacing the non-penetration boundary condition by an unsteady outflux, which accounts for the $O(R^{-1/2})$ viscous correction by the Stokes layer. With this modification, the accuracy of the receptivity calculation at moderate Reynolds numbers (approximately a few thousands) is improved remarkably.
Generation of first Mack modes in supersonic boundary layers by slow acoustic waves interacting with streamwise isolated wall roughness
- Yinhui Liu, Ming Dong, Xuesong Wu
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- Journal:
- Journal of Fluid Mechanics / Volume 888 / 10 April 2020
- Published online by Cambridge University Press:
- 06 February 2020, A10
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This paper investigates the receptivity of a supersonic boundary layer to slow acoustic waves whose characteristic frequency and wavelength are on the triple-deck scales, and the phase speed is thus asymptotically small. Acoustic waves on these scales are of special importance as they have the interesting property that a perturbation with a magnitude of $O(\unicode[STIX]{x1D716}_{u})$ in the free stream generates much larger, $O(\unicode[STIX]{x1D700}_{u}R^{1/8})$, velocity fluctuations inside the boundary layer, where $R$ is the Reynolds number based on the distance to the leading edge. Their interaction with streamwise localized roughness elements, leading to stronger receptivity, is studied using triple-deck theory and direct numerical simulations (DNS). The receptivity coefficient, defined as the ratio of the streamwise-velocity amplitude of the instability mode excited to that of the incident free-stream acoustic wave, serves to characterize receptivity efficiency. Its dependence on the roughness width, the Reynolds number $R$, the free-stream Mach number $M$ and the incident angle of the acoustic wave is examined. The theoretical predictions, obtained assuming $R\gg 1$, are found to be in quantitative agreement with the DNS results at moderate values of $R$ when the roughness elements are located near the lower branch of the instability. The receptivity is sensitive to the incident angle (or the phase speed) of the acoustic wave, being highly effective within a small range of angles close to $\cos ^{-1}(1/M)$ and $\unicode[STIX]{x03C0}+\cos ^{-1}(1/M)$ for downstream and upstream propagating sound waves, respectively. The amplitude of the instability mode excited is proportional to the streamwise-velocity amplitude of the acoustic signature inside the boundary layer, and scales with the roughness height $h^{\ast }$ as $(h^{\ast }/\unicode[STIX]{x1D6FF}^{\ast })R^{1/4}$, where $\unicode[STIX]{x1D6FF}^{\ast }$ is the boundary-layer thickness.