We investigate two distinct scenarios of spatial modulation that are candidate mechanisms for streamwise localization of waves in two-dimensional plane Poiseuille flow. The first one stems from a symmetry-breaking bifurcation that disrupts the half-shift and reflect equivariance of Tollmien-Schlichting waves (TSW). A new state, an asymmetric TSW (ATSW), emerges from unstable lower-branch TSWs at subcritical Reynolds number and undergoes subharmonic Hopf bifurcations that lead to branches of asymmetric time-periodic space-modulated waves (MATSW). Streamwise modulation does not evolve into localization within the range of parameters explored. In breaking the last standing remnants of the reflectional symmetry about the channel midplane, ATSW and MATSW admit a bias toward either one of the channel walls, thus bearing a potential for explaining near-wall structures that are typical of developed turbulence. The second scenario follows the fate of a branch of time-periodic space-modulated TSWs (MTSW) initially discovered by Mellibovsky and Meseguer [J. Fluid Mech. 779, R1 (2015)]. We find that these waves can lead to localization but the mechanism is not new, as they do so through their connection, by means of a codimension-2 bifurcation point, with other known localizing MTSWs. The codimension-2 point is, however, responsible for the appearance of MTSWs that exclusively bridge upper-branch TSW-trains of different number of replicas. In this respect, these MTSWs possess all required properties that single them out as possible constituents of the strange saddle that governs domain-filling turbulent dynamics at high Reynolds numbers.

中文翻译：

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二维平面Poiseuille流中的对称破波和时空调制机制

我们研究了两种不同的空间调制方案，它们是二维平面波塞维流中波的流向定位的候选机制。第一个源于打破对称性的分叉，该分叉破坏了半位移并反映了托尔米-施利希廷波（TSW）的等方差。一个新的状态，即非对称TSW（ATSW），从亚临界雷诺数的不稳定下分支TSW出现，并经历了次谐波Hopf分叉，从而导致非对称时间周期空间调制的分支。波浪（MATSW）。在所研究的参数范围内，流向调制不会演变为局部化。在打破关于通道中平面的反射对称性的最后残留部分时，ATSW和MATSW允许朝任一通道壁偏斜，因此具有解释潜在湍流的典型近壁结构的潜力。第二种情况是Mellibovsky和Meseguer最初发现的时间周期空间调制TSW（MTSW）分支的命运。 779，R1（2015）]。我们发现这些波可以导致定位，但是该机制并不新颖，因为它们是通过codimension-2分叉点与其他已知的本地MTSW通过连接来实现的。但是，codimension-2点是MTSW出现的原因，该MTSW仅桥接了不同数量的副本的上分支TSW列。在这方面，这些MTSW具有所有必需的属性，可以将它们选作控制高雷诺数时域填充湍流动力学的奇怪鞍形的可能组成部分。