Helical Flows - the Transition between 2D and 3D Turbulence

我们对2D和3D湍流的理论理解及其涡拉伸的基本区别揭示了根本的差异。这种差异在大尺度上表现出2D机制的大气流动中是非常统一的，但3D湍流在小尺度上表现出来。本提案的主要目标是了解2D和3D湍流之间的根本差异和过渡。为此目的，螺旋对称流，这在流体物理中无处不在，将被用作一个模型问题。这些流动“存在”于二维流形上，但有三个独立的速度矢量，因此常称为2 1/2维流动。螺旋流分解为沿螺旋线沿着有速度矢量和无速度矢量的流。如果速度矢量不为零，则会引起旋涡拉伸。本提案中的中心问题是围绕确定光谱或结构函数的部分的基本不变量进行分组。例如，在3D湍流和小长度尺度上，根据柯尔莫哥洛夫理论，这是耗散，而在2D湍流中，谱由拟能的积分不变量确定。Kelbin等人（2013）的一个中心发现是螺旋流允许无穷多类守恒律，因此是积分不变量。螺旋湍流的中心新的不变量是广义螺旋度，当涡拉伸，和广义拟能，没有涡拉伸。基本的工作假设是，这两者都是各自光谱的关键决定因素。此外，申请人最近在Deussen等人（2020）中开发了一种螺旋湍流的统计理论，其中包括涡流拉伸。其中的潜力的相关性进行了介绍，将检查其意义的频谱的基础上的模拟数据。相比之下，我们期望螺旋流没有涡流拉伸无限维共形群，申请人可以在Grebenev等人（2017）的2D湍流中首次从理论上证明这一点。工作假设是潜在的联系之间的无限维共形群和无限数量的守恒定律（Casimirs）与拟能是第一个在这一行，并有一个密切的类比广义拟能螺旋流。为了用高度精确的数据支持理论问题，应在非常高的雷诺数下进行大量的螺旋流模拟。该代码是基于不连续Galerkin方法，并已开发的DFG项目OB 96/41-1的背景下，正是为了这一目的。最后，将研究由沿着螺旋线的渐近小速度引起的2D / 3D过渡，从而存在渐近小的涡拉伸。这里的中心问题是上述积分不变量的变化和相关模拟的验证。

Our theoretical understanding of 2D and 3D turbulence and their basic distinguishing feature of vortex stretching reveals fundamental differences. This disparity is remarkably unified in atmospheric flows that exhibit 2D mechanisms on large scales, but 3D turbulence manifests itself on small scales. The main goal of the present proposal is to understand the fundamental differences and transitions between 2D and 3D turbulence. For this purpose, helical-symmetric flows, which occur ubiquitous in fluid physics, will be used as a model problem. These flows "live" on a 2D manifold but have three independent velocity vectors and are therefore often called 2 1/2-dimensional flows. Helical flows decompose into those with and without velocity vectors along the helix. If the velocity vector is non-zero, it induces a vortex stretching. The central questions in the present proposal are grouped around the underlying invariants that determine parts of the spectra or structure functions. E.g. in 3D turbulence and on small length scales this is, according to Kolmogorov's theory, the dissipation, whereas in 2D turbulence the spectrum is determined by the integral invariant of the enstrophy. A central discovery in Kelbin et al. (2013) was that helical flows admit infinite classes of conservation laws and thus integral invariants. The central new invariants for helical turbulence are generalized helicity, when vortex stretching is present, and generalized enstrophy, without vortex stretching. The fundamental working hypothesis is that both of these are key determinants of the respective spectra. In addition, the applicant has recently developed a statistical theory of helical turbulence that includes vortex stretching in Deussen et al. (2020). Therein potentials of the correlations were introduced which are to be examined for their significance of the spectrum on the basis of simulation data. In contrast, we expect for helical flows without vortex stretching the infinite dimensional conformal group, which the applicant could prove theoretically for the first time in 2D turbulence in Grebenev et al. (2017). The working hypothesis is the potential link between the infinite dimensional conformal group and the infinite number of conservation laws (Casimirs) with the enstrophy being the first in this row and which has a close analogy to generalized enstrophy for helical flows. In order to support the theoretical questions with highly accurate data, a large set of simulations for helical flows at very high Reynolds numbers shall be performed. The code is based on the discontinuous Galerkin method and has been developed in the context of the DFG project OB 96/41-1 for exactly this purpose. Finally, the transition 2D / 3D by asymptotically small velocities along the helix will be investigated, so that asymptotically small vortex stretching exist. Central questions here are the changes of the above-mentioned integral invariants and their validation by related simulations.