论文标题
在平面poiseuille流动中的干扰线性演变上
On the linear evolution of disturbances in plane Poiseuille flow
论文作者
论文摘要
通过在傅立叶平面中求解相关的初始边界值问题，计算出频率$ω_0$在频率$ω_0$下的色带振动引起的线性演变，然后进行反转。提出了一种用于识别复杂波数平面中ORR-Sommerfeld方程（OSE）的时间模式的新型算法，这是反转中所需的。与许多先前的研究不同，拉普拉斯积分的表现首先，不仅避免了复杂的因果关系论证和混乱，在定位上游和下游模式时，在文献中很普遍，而且还产生了时空均匀的解决方案。它还表明该解决方案由$ω_0$的时间周期部分组成，与相关的空间模式（Tollmein-Schlichting Wave）和瞬态波装相关，主要与OOSE的鞍点相关联，并且与最陡峭的后裔进行了计算，该方法还可以包含来自Spatial Pole的最陡峭的后裔。这些部分中的哪一个主要取决于雷诺数和ω0。可以看到对这一主导部分的次要稳定性分析来解释Nishioka，Iida＆Ichikawa的开创性实验中观察到的障碍生长（J. Fluid Mech。，Vol.72，1975，P.731）和Nishioka，Iida＆Kanbayashi（NASA TM-tm-75885，1981）。在亚临界雷诺数上的不稳定性re = 5000的不稳定性的阈值幅度是通过结合次级基础状态和增长的浮雕模式的三个维度干扰而获得的。实验中观察到的阈值幅度曲线的最小值是根据这两个基本状态的不稳定性来解释的。对于另一个亚临界（4000）和超临界（6000）雷诺数的计算也通过实验数据进行了验证。
The linear evolution of disturbances due to a ribbon vibrating at frequency $ω_0$ in plane Poiseuille flow is computed by solving the associated initial boundary value problem in the Fourier-Laplace plane, followed by inversion. A novel algorithm for identifying the temporal modes of the Orr-Sommerfeld equation (OSE) in the complex wavenumber plane, which are required in the inversion, is presented. Unlike in many prior studies, the performance of the Laplace integral first, not only avoids complicated causality arguments and confusion, in locating upstream and downstream modes, that is prevalent in literature but also yields a spatio-temporally uniform solution. It also reveals that the solution consists of a time-periodic part at $ω_0$ , associated with the relevant spatial mode (the Tollmein-Schlichting wave) and a transient wavepacket, associated mainly with the saddle points of the OSE and is computed by the method of steepest descents, which can also include contributions from the spatial pole. Which of these parts dominates depends on the Reynolds number and ω0. A secondary stability analysis of this dominant part is seen to explain the disturbance growth observed in the seminal experiments of Nishioka, Iida & Ichikawa (J. Fluid Mech., vol.72 , 1975, p.731) and Nishioka, Iida & Kanbayashi (NASA TM-75885, 1981). Threshold amplitudes for instability at a subcritical Reynolds number Re = 5000 are obtained from the time-averaged three dimensional disturbances, by combining the secondary base states and the growing Floquet modes. The observed minima of the threshold amplitude curves in the experiments are explained in terms of the instabilities of these two base states. Computations, for another subcritical (4000) and a supercritical (6000) Reynolds number, are also validated with the experimental data.
