旋转弹形体轴对称尾流的整体稳定性分析
Global stability analysis of the axisymmetric wake past a spinning bullet-shaped body
摘要：
本文分析了旋转弹形体轴对称绕流的全局线性稳定性，它是雷诺数$Re=w_{\infty}D/ν$和旋转参数$Ω=ωD/(2 w_{\infty})$的函数，范围为$Re<450$和$0\leqΩ\leq 1$。这里，$w_{\infty}$和$ω$分别是自由流和主体旋转速度，并且$ν$是流体运动粘度。得到的谱和本征函数使我们能够解释不同的分叉从以前的数值研究中观察到的轴对称状态。结果表明，在$Re-Ω$参数平面的不同区域，三个全局本征模（低频、中频和高频）变得不稳定.我们提供了精确的计算相应的中性曲线，划分为四个不同的区域的$Re-Ω$平面：稳定的轴对称流占上风足够小的值的$Re$和$Ω$，而三个不同的冻结状态，其中的尾流结构与身体在不同的角速度共同旋转，发生的LF，MF和HF模式的不稳定的后果。几个直接的数值模拟的非线性状态相关的MF模式，确定在这里的第一次，也报道补充线性稳定性的结果。最后，我们指出了一个重要的事实，即由于轴对称基流是$SO(2)$对称的，等变分岔理论意味着接近临界的弱非线性区必然是行波态。这些状态，以前被称为冻结唤醒在文献中，因此，从基流对称性的结果。
英文摘要：
We analyze the global linear stability of the axisymmetric flow around a spinning bullet-shaped body as a function of the Reynolds number, $Re=w_{\infty}D/ν$, and of the rotation parameter $Ω=ωD/(2 w_{\infty})$, in the ranges $Re<450$ and $0\leqΩ\leq 1$. Here, $w_{\infty}$ and $ω$ are the free-stream and the body rotation velocities respectively, and $ν$ is the fluid kinematic viscosity. The spectrum and the eigenfunctions obtained allow us to explain the different bifurcations from the axisymmetric state observed in previous numerical studies. Our results reveal that three global eigenmodes, denoted Low-Frequency (LF), Medium-Frequency (MF) and High-Frequency (HF) modes, become unstable in different regions of the $Re-Ω$ parameter plane. We provide precise computations of the corresponding neutral curves, that divide the $Re-Ω$ plane into four different regions: the stable axisymmetric flow prevails for small enough values of $Re$ and $Ω$, while three different frozen states, where the wake structures co-rotate with the body at different angular velocities, take place as a consequence of the destabilization of the LF, MF and HF modes. Several direct numerical simulations of the nonlinear state associated to the MF mode, identified here for the first time, are also reported to complement the linear stability results. Finally, we point out the important fact that, since the axisymmetric base flow is $SO(2)$-symmetric, the theory of equivariant bifurcations implies that the weakly non-linear regimes that emerge close to criticality must necessarily take the form of rotating-wave states. These states, previously referred to as frozen wakes in the literature, are thus shown to result from the base-flow symmetry.
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