A weakly nonlinear mechanism for mode selection in swirling jets
A weakly nonlinear mechanism for mode selection in swirling jets
复制标题
旋流射流模式选择的弱非线性机制
DOI：
10.1017/jfm.2012.93
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发表时间：
2012
影响因子：
3.7
通讯作者：
J. Chomaz
中科院分区：
文献类型：
--
作者：
P. Meliga;F. Gallaire;J. Chomaz
摘要采用整体线性和非线性分叉分析方法，对标称轴对称旋转射流的旋涡破裂进行了研究。对于所考虑的参数，稳定性分析从轴对称击穿解中分离出方位波数$m=-1$和$m=-2$的两个不稳定的线性模式。这些模式被解释为缠绕在轴对称气泡周围和后面的螺旋扰动，在时间上与旋流相同的方向旋转，但在空间中以相反的方向缠绕。根据Ruith等人的三维直接数值模拟，讨论了这些模式在涡旋破裂的存在、模式选择和内部结构方面的作用。(《流体机械》，第486卷，2003年，第331-378页)。计算和分析了描述模间前阶非线性相互作用的规范形。它允许与纯单螺旋和双螺旋相对应的两种稳定溶液。在大漩涡中，轴对称解分叉成双螺旋，而双螺旋仍然是唯一稳定的解。在低涡和中涡时，它首先分叉到单螺旋，然后通过一系列在有限雷诺数范围内产生滞后的亚临界分叉到达双螺旋，所估计的分叉阈值与直接数值模拟中观察到的结果很好地吻合。有证据表明，这种选择不应归因于由不稳定模的存在而引起的经典平均流修正，而应归因于谐波之间的一种不平凡的竞争。由于主模的频率接近于强的$2$：$1$共振，因此计算和分析了允许2$模和1$模的一次谐波相互作用的另一种范式。它允许两种稳定的解决方案，即在非共振情况下已经确定的双螺旋，以及与在非共振情况下观察到的不同的单螺旋，只是由于存在受制的、锁相的简谐变形。然而，对于有限偏离$2：$1$共振的情况，从动谐振幅值很低，共振对分叉结构的影响仅限于减小滞后范围。
Abstract Global linear and nonlinear bifurcation analysis is used to revisit the spiral vortex breakdown of nominally axisymmetric swirling jets. For the parameters considered herein, stability analyses single out two unstable linear modes of azimuthal wavenumber $m= \ensuremath{-} 1$ and $m= \ensuremath{-} 2$, bifurcating from the axisymmetric breakdown solution. These modes are interpreted in terms of spiral perturbations wrapped around and behind the axisymmetric bubble, rotating in time in the same direction as the swirling flow but winding in space in the opposite direction. Issues are addressed regarding the role of these modes with respect to the existence, mode selection and internal structure of vortex breakdown, as assessed from the three-dimensional direct numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, pp. 331–378). The normal form describing the leading-order nonlinear interaction between modes is computed and analysed. It admits two stable solutions corresponding to pure single and double helices. At large swirl, the axisymmetric solution bifurcates to the double helix which remains the only stable solution. At low and moderate swirl, it bifurcates first to the single helix, and subsequently to the double helix through a series of subcritical bifurcations yielding hysteresis over a finite range of Reynolds numbers, the estimated bifurcation threshold being in good agreement with that observed in the direct numerical simulations. Evidence is provided that this selection is not to be ascribed to classical mean flow corrections induced by the existence of the unstable modes, but to a non-trivial competition between harmonics. Because the frequencies of the leading modes approach a strong $2$:$1$ resonance, an alternative normal form allowing interactions between the $m= \ensuremath{-} 2$ mode and the first harmonics of the $m= \ensuremath{-} 1$ mode is computed and analysed. It admits two stable solutions, the double helix already identified in the non-resonant case, and a single helix differing from that observed in the non-resonant case only by the presence of a slaved, phase-locked harmonic deformation. On behalf of the finite departure from the $2$:$1$ resonance, the amplitude of the slaved harmonic is however low, and the effect of the resonance on the bifurcation structure is merely limited to a reduction of the hysteresis range.