On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances
On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances
复制标题
DOI：
10.1017/jfm.2013.421
复制
发表时间：
2013-09
影响因子：
3.7
通讯作者：
M. Dong;Xuesong Wu
中科院分区：
文献类型：
--
作者：
M. Dong;Xuesong Wu
摘要小振幅扰动由线性化的N-S方程控制，对于平行或近平行的剪切流，线性化的N-S方程通常简化为O-索末菲(O-S)方程和斯奎尔方程。在本文中，我们考虑了Blasius边界层和渐近吸力边界层中O-S算子和Squire算子的连续谱，并讨论了连续模是否以及何时可以表示自由流涡扰动及其对剪切层的卷吸。对于Blasius边界层，我们强调了CS的两个特殊性质：(I)连续模式的本征函数同时由两个壁法向波数为$\pm{k}_(2)$的分量组成，这种现象我们称之为‘傅立叶分量纠缠’；(Ii)对于低频扰动，边界层的存在迫使自由气流中的流向速度比横向速度的振幅大得多。这两个特征似乎都是非物理的，并使人们对使用CS来表征自由流涡扰动及其进入边界层的适宜性产生了一些怀疑，这一做法已经在最近的一些绕过转折研究中得到了采用。给出了连续模和纠缠的高雷诺数渐近描述，结果表明纠缠是忽略非平行度的结果，非平行度对纠缠具有前序效应。当包含这一效应时，纠缠消失，并且当${R}^{-1}\ll\omega\ll 1$时，边界层中的流向速度显著放大，其中$R$是基于局部边界层厚度的雷诺数。对于渐近吸力边界层，这是一种完全平行的流动，时间和空间CS都可以在数学上定义。然而，在有限的$R$下，它们都不代表自由流涡扰动穿透边界层的物理过程。后者的特征必须是一种特殊类型的连续模式，其本征函数随离墙的距离呈指数增长。在极限值$R\gg1$中，三种CS在前导阶上都是相同的，因此可以用来表示自由流涡扰动及其卷吸。发现低频扰动会在边界层中产生很大幅度的流向速度，这让人联想到纵向条纹。
Abstract Small-amplitude perturbations are governed by the linearized Navier–Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the Orr–Sommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius and asymptotic suction boundary layers, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. For the Blasius boundary layer, we highlight two particular properties of the CS: (i) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers $\pm {k}_{2} $ , a phenomenon which we refer to as ‘entanglement of Fourier components’; and (ii) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when ${R}^{- 1} \ll \omega \ll 1$ , where $R$ is the Reynolds number based on the local boundary-layer thickness. For the asymptotic suction boundary layer, which is an exactly parallel flow, both temporal and spatial CS may be defined mathematically. However, at a finite $R$ neither of them represents the physical process of free-stream vortical disturbances penetrating into the boundary layer. The latter must instead be characterized by a peculiar type of continuous modes whose eigenfunctions increase exponentially with the distance from the wall. In the limit $R\gg 1$ , all three types of CS are identical at leading order, and hence can be used to represent free-stream vortical disturbances and their entrainment. Low-frequency disturbances are found to generate a large-amplitude streamwise velocity in the boundary layer, which is reminiscent of longitudinal streaks.