论文标题
涡流表几何形状对开尔文 - 霍尔姆尔兹不稳定性的影响
The Influence of Vortex Sheet Geometry on the Kelvin-Helmholtz Instability
论文作者
论文摘要
本文在不可压缩的流体流中重新讨论了尖锐的剪切界面（也称为涡流板）的不稳定性。我们研究了Birkhoff-Rott方程，该方程描述了根据不可压缩的Euler方程在两个维度上的运动。经典的开尔文 - 霍尔姆霍尔茨不稳定性表明，无限的平坦涡流板具有很强的线性不稳定性。我们表明，圆形涡流片并非如此：这种配置具有微妙的线性稳定性，并且是Birkhoff-Rott方程线性稳定解决方案的第一个例子。随后，我们为广义的Birkhoff-Rott内核家族提供了足够的条件，可以使圆形涡流板线性不稳定性，并证明用于数值模拟和分析中的普通正规化核使圆形涡流板不稳定。我们的工作没有破坏稳定的内核正则化，这表明非线性动力学对于理解圆形涡流板不稳定性至关重要，因此开尔文 - 赫尔莫尔兹不稳定性的基本机制取决于全球涡流图的几何形状。正如预期的那样，利用正则内核的非线性数值模拟表现出不稳定的行为。最后，我们显示了实验结果，这些结果在定性上与数值观察到的不稳定性类型相匹配，证明了开尔文 - 霍尔姆霍尔茨不稳定性在实际圆形剪切流中的持续性。
This article revisits the instability of sharp shear interfaces, also called vortex sheets, in incompressible fluid flows. We study the Birkhoff-Rott equation, which describes the motion of vortex sheets according to the incompressible Euler equations in two dimensions. The classical Kelvin-Helmholtz instability demonstrates that an infinite, flat vortex sheet has a strong linear instability. We show that this is not the case for circular vortex sheets: such a configuration has a delicate linear stability, and is the first example of a linearly stable solution to the Birkhoff-Rott equation. We subsequently derive a sufficient condition for linear instability of a circular vortex sheet for a family of generalized Birkhoff-Rott kernels, and prove that a common regularized kernel used in numerical simulation and analysis destabilizes the circular vortex sheet. Absent a destabilizing kernel regularization, our work suggests that the nonlinear dynamics are critical for understanding circular vortex sheet instability, and so the essential mechanism of the Kelvin-Helmholtz instability is dependent on global vortex sheet geometry. As expected, nonlinear numerical simulations utilizing the regularized kernel exhibit unstable behavior. Finally, we show experimental results which qualitatively match the types of instabilities that are observed numerically, demonstrating the persistence of the Kelvin-Helmholtz instability in real circular shear flows.
