A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage
A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage
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DOI：
10.1017/s0022112094003939
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发表时间：
1994-11
影响因子：
3.7
通讯作者：
K. Shariff;R. Verzicco;P. Orlandi
中科院分区：
文献类型：
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作者：
K. Shariff;R. Verzicco;P. Orlandi
本文用随机扰动和单模扰动的双精度差分方法研究了涡环的三维不稳定性。目前对这一问题的理解的基础是启发式的无粘模型（Widnall，布利斯& Tsai 1974）和预测薄核均匀涡环增长率的严格理论（Widnall & Tsai 1977）。在足够高的雷诺数的结果定性对应于那些预测的启发式模型：多个波段的波数被放大，每个波段有一个独特的径向结构。然而，对于峰值无粘增长率，我们发现了一个粘性修正因子，它可以用第一项1 - α1（β）/Res很好地描述，其中Res是Saffman（1978）定义的雷诺数，它包含了曲率引起的应变率。发现这是适当的选择，因为α1（β）随芯厚度β的变化很小。三个最非线性放大的模式是一个平均方位角速度的形式相对流，一个n = 1模式（n是方位角波数），这是从两个第二模式弯曲波和谐波的初级第二模式的相互作用产生的。当单个波被激发时，高次谐波开始以与n成比例的非线性增长率连续增长。修正后的平均流具有双峰的方位涡度。由于曲率诱导的应变并不完全是驻点流动，因此倾向于向环的后部伸长：不稳定波的外部结构形成由n个发夹涡组成的长尾流，其波度相对于核心波度相移π/n。而最放大的线性模式具有三个径向层的结构，较高的径向模式具有更多层的径向结构（发夹堆在发夹）被激发时，初始扰动是大的，让人想起可视化实验的湍流环的形成在发电机。
Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α1(β)/Res, for a large range of Res. Here Res is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α1(β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.