turbulence的功能重整化组
Functional renormalisation group for turbulence
摘要：
湍流是一种复杂的非线性多尺度现象。尽管基本的纳维-斯托克斯方程已经被人们知道了两个世纪，但是从它们中提取湍流的统计特性仍然是极具挑战性的。因此，为了实际的目的，一直致力于获得一些有效的描述湍流，我们可以称之为湍流模型，或湍流的统计理论。在这方面，重正化群（RG）似乎是一个工具的选择，因为它的目的是提供有效的理论，从基本方程进行系统的方式平均波动。然而，对于Navier-Stokes湍流，一个合适的框架的RG，特别是允许非微扰近似，已经失踪，这已经挫败了长RG应用程序。这个框架是由RG的现代公式提供的，称为功能性重整化组。FRG的使用在均匀和各向同性湍流的理论理解方面取得了重要进展。主要是从纳维-斯托克斯方程出发，严格推导了任意欧拉多点多时间关联函数的解析表达式，该表达式在大波数范围内是精确的。在这篇{ \it JFM观点}中，我们提出了一个湍流FRG方法的综述。我们提供了一个基本的介绍联邦德国，并强调如何场理论框架允许系统和深刻地利用对称性。然后，我们表明，FRG使一个描述湍流被迫在大尺度上，这是不可访问的微扰手段。我们阐述了推导的时空行为的$n$点关联函数，并在很大程度上说明这些结果通过分析数据从实验和直接数值模拟。
英文摘要：
Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying Navier-Stokes equations have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purpose, a sustained effort has been devoted to obtaining some effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the Renormalisation Group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier-Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, have been missing, which has thwarted for long RG applications. This framework is provided by the modern formulation of the RG called functional renormalisation group. The use of the FRG has rooted important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier-Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this {\it JFM Perspectives} a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We then show that the FRG enables one to describe turbulence forced at large scale, which was not accessible by perturbative means. We expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.
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