The critical layer in quadratic flow boundary layers over acoustic linings
The critical layer in quadratic flow boundary layers over acoustic linings
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声学衬里二次流边界层的关键层
DOI：
10.1017/jfm.2022.753
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发表时间：
2021
影响因子：
3.7
通讯作者：
T. Shah
中科院分区：
文献类型：
--
作者：
M. King;E. Brambley;Renan Liupekevicius;Miren Radia;P. Lafourcade;T. Shah
圆柱直管道中除了壁面附近的边界层外，其他地方的轴向平均流都是均匀的，边界层不一定很薄。在该边界层内，平均流量呈抛物线变化。对线性化的欧拉方程进行傅立叶变换，得到Pridmore-Brown方程，并利用Frobenius级数构造了该方程的格林函数。临界层来自连续谱分支截断的非模式贡献，并在某些情况下主导下游压力扰动，特别是对于较厚的边界层。连续谱分支切割也被发现通过将对流不稳定模隐藏在分支切割后面来稳定它们。总体而言，当源位于剪切流区时，临界层的贡献给出了一个中性稳定的非模式波，并且对于位于均匀流区的源，其贡献沿管道以代数形式衰减为$O(x^{-{5}/{2}})$。Frobenius展开式，除了在接近临界层的数值上精确之外，在其他数值方法失去精度的地方，还能够定位隐藏在分支切割后面的模极点，这是其他方法无法找到的；这包括稳定的水动力不稳定性。给出了计算格林函数的MatLab代码。
Abstract A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as $O(x^{-{5}/{2}})$ for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function.