利用映射勒让德函数的谱方法对尾涡进行线性稳定性分析
Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions
AI总结：
本文提出一种基于映射勒让德函数的谱方法，用于尾涡线性稳定性分析，首次识别出粘性临界层谱，并验证其符合$Re^{-1/3}$标度律。
摘要：
本文发展了一种使用带代数映射的连带勒让德函数的谱方法，用于尾涡的线性稳定性分析。这些函数作为伽辽金基函数，能够正确捕捉无界域中涡旋的解析性和边界条件。在旋流上线性化的不可压缩欧拉方程或纳维-斯托克斯方程被转化为关于环向和极向流函数的标准矩阵特征值问题，求解扰动速度特征模态，其复增长率为特征值。这减小了计算问题的规模，并使配置点可调节地聚集在涡核周围。基于该方法，研究了线性扰动波数阶数为1的强旋$q$-涡。在无粘性情况下，与具有临界层奇点的连续特征值谱相关的中性稳定特征模态被成功分辨。无粘临界层特征模态在数值上趋于成对出现，暗示其奇异简并性。在有粘性情况下，与临界层物理正则化相关的谱向一个区域延伸，指的是Mao和Sherwin（2011）发现的具有波包的特征模态。然而，这些潜在特征模态与无粘临界层特征模态没有空间相似性，令人怀疑它们是否真正代表无粘临界层特征模态的粘性残余。相反，数值谱中首次识别出两条不同的连续曲线，命名为粘性临界层谱，其中相似性显著。此外，粘性临界层特征模态符合$Re^{-1/3}$标度律。这两条曲线的出现被认为是由粘性打破奇异简并性所致。
英文摘要：
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.
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