Abstract An asymptotic approach is employed to investigate the effects of dispersion on acoustic disturbances propagating through cylindrical ducts, the walls of which may be impermeable or acoustically treated. The disturbances are assumed to be those generated by a rotor-locked pressure field. In the large azimuthal wavenumber (high-frequency) limit, m ≫ 1, linear potential perturbations are found to be weakly dispersive, with the leading order correction occurring at O ( m − 2 / 3 ) . When a liner is present, its additional dispersive and dissipative effects appear at O ( m − 1 ) . The effects of weak nonlinearity, in addition to those of dispersion are then studied. Specifically, on the assumption that the disturbance is comprised of a single radial mode, an evolution equation is derived for the pressure field. Numerical solutions of this equation are carried out for idealized perturbation patterns corresponding to tuned and detuned rotor blades, featuring respectively shocks of uniform and varying strength. In the tuned, untreated case, dispersive effects are found to reduce the dissipation across the shocks without substantially changing the character of the waveform in comparison with the non-dispersive case. The inclusion of the acoustic liner brings about more drastic changes: the periodic shock pattern is transformed into a series of humps with dispersive tails. With the detuned blades, it is found that the untreated duct scenario yields a waveform that differs significantly from the non-dispersive case. In particular, the shock merging that occurs in the latter instance does not transpire. Instead, the shocks exchange amplitudes even as they decay. The presence of the acoustic liner in the detuned case is similar to the tuned case, except with a richer spectrum being generated. Despite the relative simplicity of the models studied, these results exhibit many of the features of previous numerical simulations and point to the need for the inclusion of dispersive effects in designing liners for turbofan applications.

中文翻译：

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圆柱形管道中的声波和冲击波扩散

摘要 采用渐近方法研究色散对通过圆柱形管道传播的声干扰的影响，圆柱形管道的壁可能是不透水的或经过声学处理的。假定扰动是由转子锁定压力场产生的扰动。在大的方位角波数（高频）极限 m ≫ 1 中，发现线性电位扰动是弱色散的，前级校正发生在 O ( m − 2 / 3 ) 处。当存在衬垫时，其额外的色散和耗散效应出现在 O ( m - 1 ) 处。然后研究弱非线性的影响，以及色散的影响。具体而言，假设扰动由单个径向模式组成，推导出压力场的演化方程。该方程的数值解是针对对应于调谐和失谐转子叶片的理想化扰动模式进行的，分别具有均匀和变化强度的冲击。在调谐的、未经处理的情况下，与非色散情况相比，色散效应减少了冲击的耗散，而不会显着改变波形的特征。声学衬垫的加入带来了更剧烈的变化：周期性冲击模式转变为一系列带有分散尾部的驼峰。使用失谐叶片，发现未经处理的管道情况产生的波形与非分散情况有显着差异。特别是，在后一种情况下发生的冲击合并不会发生。反而，即使它们衰减，冲击也会交换振幅。失谐情况下声学衬里的存在与调谐情况类似，只是生成了更丰富的频谱。尽管所研究的模型相对简单，但这些结果展示了先前数值模拟的许多特征，并表明在设计涡轮风扇应用的衬套时需要包含色散效应。