This paper presents an analytic solution for the sound generated by rotor–stator interaction for aerofoils with small camber and thickness subject to a background flow with small angle of attack. The interaction is modelled as a convected, unsteady vortical or entropic gust incident on an infinite rectilinear cascade of staggered aerofoils in a background flow that is uniform far away from the cascade. Applying rapid distortion theory (RDT) and transforming to an orthogonal coordinate system reduces the cascade of aerofoils to a cascade of flat plates. By seeking a perturbation expansion in terms of the disturbance of the background flow from uniform flow, leading- and first-order governing equations and boundary conditions are obtained for the acoustic potential. The system is then solved analytically using the Wiener–Hopf method. The resulting expression is inverted to give the acoustic potential function in the entire domain, i.e. a solution to the inhomogeneous convected Helmholtz equation with inhomogeneous boundary conditions in a cascade geometry. The solution significantly extends previous analytical work that is restricted to flat plates or only calculates the far-upstream radiation, and as such can give insight into the role played by blade geometry on the acoustic field upstream, downstream and in the important inter-blade region of the cascade. This new solution is validated against solutions that only account for flat plates at zero angle of attack. Various aeroacoustic results, including the scattered pressure, unsteady lift and sound power output, are discussed for a range of geometries and angles of attack.

中文翻译：

---

阵风-级联相互作用噪声的解析解，包括真实翼型几何形状的影响

本文针对具有小弯度和厚度的翼型的转子 - 定子相互作用产生的声音提出了一种解析解，该翼型受到小迎角背景流的影响。相互作用被建模为对流的、不稳定的涡流或熵阵风，入射在远离级联的均匀背景流中的无限直线交错翼型级联上。应用快速畸变理论 (RDT) 并转换为正交坐标系，将机翼级联减少为平板级联。通过从均匀流的背景流扰动方面寻求扰动扩展，获得了声势的领先和一阶控制方程和边界条件。然后使用 Wiener-Hopf 方法对系统进行解析求解。结果表达式被反转以给出整个域中的声势函数，即在级联几何中具有非均匀边界条件的非均匀对流亥姆霍兹方程的解。该解决方案显着扩展了以前仅限于平板或仅计算远上游辐射的分析工作，因此可以深入了解叶片几何形状对上游、下游和重要的叶片间区域的声场所起的作用级联的。这种新解决方案已针对仅考虑零攻角平板的解决方案进行了验证。针对一系列几何形状和攻角讨论了各种气动声学结果，包括散射压力、非定常升力和声功率输出。