波瓣射流的时间稳定性分析
Temporal stability analysis of jets of lobed geometry
摘要：
对波瓣射流进行了二维时间不可压缩稳定性分析。射流底部流动被假定为平行的涡面型。这个简化的稳定性问题的本征函数展开使用圆形射流的本征函数。原始问题，然后制定为一个创新的矩阵特征值问题，它可以解决一个非常强大和有效的方式。结果表明，波瓣的几何形状改变了对流速度和不稳定波的时间增长率。然而，不同的模式受到不同的影响。特别地，模式0对几何变化不敏感，而高阶模式可以显著地改变。随着叶数N和穿透比$ε$增加，变化变得更加明显。此外，波瓣几何形状可以导致先前退化的本征值（$λ_n = λ_{-n}$）变为非退化的（$λ_n \ne λ_{-n}$），并且导致对应的对称（n）和反对称（-n）模式的稳定性特性的相反变化。它还表明，每个本征模式改变其形状响应的波瓣的涡面，和本征值的简并发生时，涡面有更多的对称平面比相应的模式形状（包括对称和反对称平面）。本文提出的新方法可用于研究其他任意几何形状射流的稳定性特性。
英文摘要：
A 2D temporal incompressible stability analysis is carried out for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet. The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, mode 0 is not sensitive to the geometry changes, while modes of higher-orders can be changed significantly. The changes become more pronounced as the number of lobes N and the penetration ratio $ε$ increase. Moreover, the lobed geometry can cause a previously degenerate eigenvalue ($λ_n = λ_{-n}$) to become non-degenerate ($λ_n \ne λ_{-n}$) and lead to opposite changes to the stability characteristics of the corresponding symmetric (n) and antisymmetric (-n) modes. It is also shown that each eigen-mode changes its shape in response to the lobes of the vortex sheet, and the degeneracy of an eigenvalue occurs when the vortex sheet has more symmetric planes than the corresponding mode shape (including both symmetric and antisymmetric planes). The new approach developed in this paper can be used to study the stability characteristics of jets of other arbitrary geometries in a robust and efficient manner.
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