The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet. The proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by the German Aerospace Center in Berlin. A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses.

中文翻译：

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圆形多孔穿孔衬管的阻抗条件

燃气轮机和航空发动机中的声阻尼在很大程度上取决于由空腔和穿孔面板组成的隔音衬里。如今，由于成百上千的小孔重复，通过直接数值模拟来预测衬管的阻抗是不可行的。我们引入了一种程序，以数字方式获得静止的粘性气体的隔音衬垫的瑞利电导率，并由此定义了多孔板的声阻抗。所提出的方法将在不同尺度上占主导地位的影响解耦：（a）在一个孔尺度上的粘性和不可压缩流动，（b）在孔型尺度下的不粘和不可压缩流动，以及（c）粘滞和可压缩流动在波长范围内。通过匹配渐近展开的方法，我们耦合了不同的尺度，并最终获得了宏观尺度上的有效阻抗条件。为此，有效的瑞利电导率是通过对一个孔周围的频域中的平稳斯托克斯问题进行数值求解而得出的，该平稳斯托克斯问题是在距孔无限距离处具有规定压力的情况下围绕一个孔进行的。它取决于孔的形状，频率，平均密度和粘度除以周期性单元的面积。这使我们能够估计耗散损耗和传输特性，并与柏林德国航空航天中心在具有圆形横截面的管道声学测试台中的声学测量结果进行比较。提出了在一个孔的尺度上有效瑞利电导率的精确和合理定义，并通过系统程序得出了宏观压力或速度的阻抗条件。与实验的比较表明，导出的阻抗条件可以很好地预测耗散损耗。