This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of \(O(\mathrm{{Re}}^{-{1}/{2}})\), and a vorticity deformation layer is also present with thickness \(O(\mathrm{{Re}}^{-{1}/{4}})\). The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness.

本文使用渐近方法研究了在具有粗糙度元素的半无限平板上通过跨音速流动中的自由流湍流产生 Tollmien-Schlichting 波。假设雷诺数（表示为 Re）很大，并且自由流湍流是均匀的，因此可以将其建模为涡度波。靠近板块，形成一个厚度为\(O(\mathrm{{Re}}^{-{1}/{2}})\)的 Blasius 边界层，并且还存在一个涡度变形层，厚度为\(O(\mathrm{{Re}}^{-{1}/{4}})\). 报告表明，涡度波从涡度变形层进入经典边界层的机制不存在；因此，在它们之间引入了一个过渡层，以防止涡量不连续。然后使用跨音速流动的三层模型分析粗糙度元素附近相互作用区域中的流动。一种新颖的渐近扩展用于分析上层甲板，从而可以推导出粘性-无粘性相互作用问题。为了进行分析，假设粗糙度元素的高度很小，由此，我们找到了粗糙度下游远处 Tollmien-Schlichting 波的接收系数的明确公式。