Buaria Dhawal, Pumir Alain, Bodenschatz Eberhard
Max Planck Institute for Dynamics and Self-Organization, Göttingen, 37077, Germany.
Tandon School of Engineering, New York University, New York, 11201, USA.
Nat Commun. 2020 Nov 17;11(1):5852. doi: 10.1038/s41467-020-19530-1.
Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described by the incompressible Navier-Stokes equations. A hallmark of turbulence is spontaneous generation of intense whirls, resulting from amplification of the fluid rotation-rate (vorticity) by its deformation-rate (strain). This interaction, encoded in the non-linearity of Navier-Stokes equations, is non-local, i.e., depends on the entire state of the flow, constituting a serious hindrance in turbulence theory and even establishing regularity of the equations. Here, we unveil a novel aspect of this interaction, by separating strain into local and non-local contributions utilizing the Biot-Savart integral of vorticity in a sphere of radius R. Analyzing highly-resolved numerical turbulent solutions to Navier-Stokes equations, we find that when vorticity becomes very large, the local strain over small R surprisingly counteracts further amplification. This uncovered self-attenuation mechanism is further shown to be connected to local Beltramization of the flow, and could provide a direction in establishing the regularity of Navier-Stokes equations.
湍流在自然界和技术领域中无处不在，并且在数学上由不可压缩的纳维-斯托克斯方程描述。湍流的一个标志是强烈漩涡的自发产生，这是由流体的变形率（应变）对其旋转率（涡度）的放大所导致的。这种相互作用，体现在纳维-斯托克斯方程的非线性中，是非局部的，即取决于流动的整体状态，这在湍流理论中构成了严重阻碍，甚至在确立方程的正则性方面也是如此。在此，我们揭示了这种相互作用的一个新方面，通过利用半径为R的球体中涡度的毕奥-萨伐尔积分将应变分离为局部和非局部贡献。分析纳维-斯托克斯方程的高分辨率数值湍流解，我们发现当涡度变得非常大时，小R范围内的局部应变出人意料地抵消了进一步的放大。这种发现的自衰减机制进一步表明与流动的局部贝特拉米化有关，并且可以为确立纳维-斯托克斯方程的正则性提供一个方向。
