论文标题
使用映射的legendre函数通过光谱方法对唤醒涡流进行线性稳定性分析
Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions
论文作者
论文摘要
开发了使用带有代数映射的相关Legendre函数的光谱方法，用于对尾流涡流进行线性稳定性分析。这些函数是Galerkin基础函数，捕获了无限域中涡流的正确分析性和边界条件。在旋转流程上线性化的不可压缩的Euler或Navier-Stokes方程将转化为环形和poloidal流的标准矩阵特征值问题，以其复杂的生长速率（作为特征值）求解扰动速度特征速率。这减少了计算的问题大小，并分布了围绕涡流核心的串联搭配点。基于这种方法，检查了具有订单统一的线性扰动波数的强旋转$ q $ - 涡旋。没有粘度，与具有关键层奇异性的连续特征值谱相关的中性稳定本征量已成功解决。数字上的无粘性临界层特征模式倾向于成对出现，这意味着它们的奇异变性。凭借粘度，与临界层的物理正规化有关的光谱延伸到一个区域，指的是Mao＆Sherwin（2011）发现的Wavepackets的潜在特征码。然而，潜在的本征模与无粘性的临界层特征模式没有空间相似性，怀疑它们确实代表了Inviscid关键层的特征型的粘性残余物。取而代之的是，首次确定了数值光谱中的两个不同的连续曲线，称为粘性临界层光谱，其中相似性很明显。此外，粘性的临界层特征模式与$ re^{ - 1/3} $缩放定律一起解决。据信这两条曲线的开始是由粘度破坏奇异变性引起的。
A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier-Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling $q$-vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (2011). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the $Re^{-1/3}$ scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.
