On Continuous Spectra of the Orr-Sommerfeld/Squire Equations and Entrainment of Free-stream Vortical Disturbances in the Blasius Boundary Layer
On Continuous Spectra of the Orr-Sommerfeld/Squire Equations and Entrainment of Free-stream Vortical Disturbances in the Blasius Boundary Layer
复制标题
奥尔-索末菲/斯奎尔方程的连续谱和 Blasius 边界层中自由流涡旋扰动的夹带
DOI：
10.2514/6.2013-2465
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发表时间：
2013
影响因子：
3.7
通讯作者：
M. Dong
中科院分区：
文献类型：
--
作者：
Xue;M. Dong
小振幅扰动由线性化的Navier-Stokes方程控制，对于平行或接近平行的剪切流，通常简化为OrrSommerfeld （O-S）和Squire方程。在本文中，我们考虑了Blasius边界层O-S算子和Squire算子的连续谱（CS），并讨论了连续模态是否以及何时可以表示自由流涡旋扰动及其夹带到剪切层的问题。我们强调了CS的两个特殊性质：(a)连续模态的特征函数同时由两个壁法向波数±k2的分量组成，我们将这种现象称为“傅立叶分量的纠缠”；(b)对于低频扰动，边界层的存在迫使自由流中的顺流速度的振幅比横向速度的振幅大得多。这两个特征似乎都是非物理的，并且对使用CS来表征自由流涡旋扰动及其夹带进入边界层的适当性产生了一些怀疑，这种做法在最近的一些旁路转捩研究中已被采用。给出了连续模和夹带的高雷诺数渐近描述，并证明了纠缠是忽略非并行性的结果，而非并行性对夹带有先导级影响。当考虑到这种效应时，纠缠消失，并且当R ω 1时，边缘层的流向速度被显著放大，其中R为基于局部边界层厚度的雷诺数。
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 OrrSommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius boundary layer, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. We highlight two particular properties of the CS: (a) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers ±k2, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (b) 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 ω 1, where R is the Reynolds number based on the local boundary-layer thickness.