Fluid flows are intrinsically characterized via the topology and dynamics of underlying vortex lines. Turbulence in common fluids like water and air, mathematically described by the incompressible Navier-Stokes equations (INSE), engenders spontaneous self-stretching and twisting of vortex lines, generating a complex hierarchy of structures. While the INSE are routinely used to describe turbulence, their regularity remains unproven; the implicit assumption being that the self-stretching is ultimately regularized by viscosity, preventing any singularities. Here, we uncover an inviscid regularizing mechanism stemming from self-stretching itself, by analyzing the flow topology as perceived by an observer aligned with the vorticity vector undergoing amplification. While, initially, vorticity amplification occurs via increasing twisting of vortex lines, a regularizing anti-twist spontaneously emerges to prevent unbounded growth. By isolating a vortex, we additionally demonstrate the genericity of this self-regularizing anti-twist. Our work, directly linking dynamics of vortices to turbulence statistics, reveals how the Navier-Stokes dynamics avoids the development of singularities even without the aid of viscosity.

中文翻译：

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扭曲的涡旋线使 Navier-Stokes 湍流正则化


流体流动本质上是通过底层涡流线的拓扑和动力学来表征的。水和空气等常见流体中的湍流由不可压缩的纳维-斯托克斯方程 （INSE） 数学描述，会产生涡线的自发自拉伸和扭曲，从而产生复杂的结构层次结构。虽然 INSE 经常被用来描述湍流，但它们的规律性仍未得到证实;隐含的假设是自拉伸最终被粘度正则化，从而防止任何奇点。在这里，我们通过分析与正在放大的涡度向量对齐的观察者感知的流动拓扑，揭示了源于自拉伸本身的无粘正则化机制。虽然最初，涡度放大是通过涡旋线的增加扭曲发生的，但会自发地出现正则化的反扭曲以防止无限生长。通过分离涡旋，我们还证明了这种自正则化抗扭曲的通用性。我们的工作将涡流动力学与湍流统计直接联系起来，揭示了纳维-斯托克斯动力学如何在没有粘度帮助的情况下避免奇点的发展。