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.

本文研究了超音速边界层中的无粘性第一和第二模式对局部吹/吸、流向速度扰动和温度扰动形式的时间周期壁面扰动的接受能力，所有这些都是通过平板上的一个小强制槽引入的. 使用直接数值模拟 (DNS)、有限和高雷诺数方法研究了接受性，它们相互补充。有限雷诺数公式与 DNS 一样准确地预测了接收率，但没有对详细的激发过程提供太多洞察，也不能解释相对于吹气/吸入的流向速度和温度扰动的明显较弱的接收效率。为了阐明这些问题，在大雷诺数的极限下进行渐近分析。它表明对所有三种形式的壁扰动的接受度都减少到相同的数学形式：受等效吸入/吹气速度的瑞利方程，可以明确地用物理壁扰动表示。对于每种情况，可以先验地估计激发本征模式的幅度。此外，流动速度和温度扰动的接受效率与吹/吸的接受效率通过简单的比率定量相关，它们是 这可以用物理壁扰动明确表达。对于每种情况，可以先验地估计激发本征模式的幅度。此外，流动速度和温度扰动的接受效率与吹/吸的接受效率通过简单的比率定量相关，它们是 这可以用物理壁扰动明确表达。对于每种情况，可以先验地估计激发本征模式的幅度。此外，流动速度和温度扰动的接受效率与吹/吸的接受效率通过简单的比率定量相关，它们是\(O(R^{-1/2})\)并有简单的表达式，其中R是雷诺数，基于强制槽中心的边界层厚度。对于足够大的雷诺数（大约\(10^4\)），简单的领先阶渐近理论准确地预测了不稳定性和感受性特征，但对于中等雷诺数存在明显的误差。通过使用适当的阻抗条件开发了一种改进的渐近理论，该条件解释了由与壁相邻的斯托克斯层中的粘性运动引起的\(O(R^{-1/2})\)横向速度。改进后的理论预测了中等雷诺数 ( \(R=O(10^3)​​\)) 具有令人满意的准确性。特别是，它很好地捕捉了有限雷诺数效应，包括雷诺数依赖的感受性和发生在所谓同步点附近的强激励。