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.

中文翻译：

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通过慢声波与流向隔离壁粗糙度相互作用在超音速边界层中产生第一麦克模式

本文研究了超音速边界层对特征频率和波长在三层尺度上的慢声波的接受能力，因此相速度渐近小。这些尺度上的声波特别重要，因为它们具有有趣的特性，即自由流中幅度为 $O(\unicode[STIX]{x1D716}_{u})$ 的扰动会产生更大的 $O( \unicode[STIX]{x1D700}_{u}R^{1/8})$ ，边界层内的速度波动，其中 $R$ 是基于到前缘距离的雷诺数。使用三层理论和直接数值模拟 (DNS) 研究了它们与流向局部粗糙度元素的相互作用，导致更强的接受性。接受系数，定义为不稳定模式激发的流动速度幅度与入射自由流声波的流动速度幅度之比，用于表征接收效率。检查其对粗糙度宽度、雷诺数 $R$、自由流马赫数 $M$ 和声波入射角的依赖性。当粗糙度元素位于不稳定性的下分支附近时，假设 $R\gg 1$ 获得的理论预测与 $R$ 中等值下的 DNS 结果在定量上一致。接收率对声波的入射角（或相位速度）敏感，在接近 $\cos ^{-1}(1/M)$ 和 $\unicode[STIX ]{x03C0}+\cos ^{-1}(1/M)$ 分别用于下游和上游传播声波。