Non-Gaussianity, bounds on turbulent scaling parameter and conformal transformations - analyzing the Lundgrenand Hopf functional equation of turbulence using Lie symmetries

对称性是经典、量子力学或相对论等所有物理理论的核心，因为它们反映了基本物理学的公理性质。湍流的一些关键属性，特别是非高斯性和间歇性，最近由本申请人给出了一种基于对称性的度量。结果表明，这与统计湍流的三个完整理论，即多点关联方程、Lundgren-Novikov-Monin(LMN)概率密度(PDF)层次和Hopf泛函方法是一致的。最重要的是，在一系列的出版物中，申请人证明了对称性是所有湍流标度定律的公理基础。基于LMN族和Hopf泛函，我们打算推广我们的最新结果，并集中在以下关键问题上：(I)二维湍流中共形对称性的存在，(Ii)湍流标度律参数的界，(Iii)速度增量统计的非高斯性，(Iv)任意矩结构函数的反常标度。由于LMN方法和Hopf方法都是非标准类型的方程，我们将首先使用基于Lie-Baecklund变换的扩展对称方法来推导对称性的穷举列表。对于子任务(I)，我们在LMN层级中寻找二维湍流涡度的共形群，因为它的存在最近被数值数据分析所证实。申请人的初步计算支持了这一重要的物理和十年前的猜想。对于子任务(II)，我们将通过研究基本的代数结构来解开对群参数和对称中自由函数的约束。例如，上述表现在墙的对数定律的kappa中，其是统计参数和标度群参数的比率，其数值的确定尚不清楚。我们可以进一步利用PDF的正性，这将对对称参数施加额外的约束。对于既有对称性又有约束的子任务(III)，我们可以构造PDF的群不变解。为此，已经很明显的是，对称性的某种组合将导致指数尾部和非高斯性，前者与间歇性密切相关。最后，对子任务(IV)，我们将构造Hopf泛函的群不变解，该泛函提供无限序列的矩，进而构造所有结构函数，以研究其反常标度性质。对于(III)和(IV)，预期的结果可能只对非常特定的对称性组合才能观察到。这种特殊性也将使用李代数来研究，而工作假设是，这些对称形成一个特定的子代数。通过目前的提议，长期目标是将湍流中一些最紧迫的问题放在基础上，这超出了数据匹配和半经验建模的范畴。

Symmetries lie at the heart of all physical theories such as classical and quantum mechanics or relativity as they mirror axiomatic properties of the underlying physics. Some key properties of turbulence, especially non-gaussianity and intermittency, were recently given a symmetry-based measure by the present applicant. It was shown that this is consistent with all three complete theories of statistical turbulence, i.e. the multi-point correlation equations, the Lundgren-Novikov-Monin (LMN) probability density (PDF) hierarchy and the Hopf functional approach. Most important, in a sequence of publications the applicant has shown that symmetries are the axiomatic basis for all turbulent scaling laws. Based on the LMN hierarchy and the Hopf functional we intend to extend our recent results and focus on the following key questions: (i) presence of the conformal symmetry in 2D turbulence, (ii) bounds on turbulent scaling law parameters, (iii) non-gaussianity of velocity increment statistics, (iv) anomalous scaling of structure functions of arbitrary moments. As both LMN and Hopf approach are non-standard type of equations, we will first derive the exhaustive list of symmetries using extended symmetry methods based on Lie-Baecklund transformations. For subtask (i) we search for the conformal group in the LMN hierarchy for vorticity in 2D turbulence, as its presence was recently confirmed analysing numerical data. Preliminary calculations by the applicant supports this important physical and decade old conjecture. For subtask (ii) we will untangle the constraints on the group parameter and free functions in the symmetries by investigating the underlying algebraic structure. The aforementioned manifests e.g. in kappa of the logarithmic law of the wall, which is the ratio of a statistical and a scaling group parameter, the determination of its numerical value is yet unknown. We may further use the positivity of PDF, which will impose additional constraints on symmetry parameters. For subtask (iii) having both symmetries and its constraints at our hands we may construct group invariant solutions for the PDF. For this it is already visible that a certain combination of symmetries will lead to exponential tails and non-gaussianity, the former being closely related to intermittency. Finally, for subtask (iv), we will construct group invariant solutions of the Hopf functional, which delivers the infinite sequence of moments, and, in turn, all structure functions, to be investigated for its anomalous scaling properties. For both (iii) and (iv) the expected results may only be observed for a very specific combination of symmetries. This specificity will also be investigated using Lie algebra, while the working hypothesis is, that these symmetries form a specific sub-algebra. With the present proposal, the long-term goal is to put some of the most pressing questions in turbulence on grounds, which is beyond data matching and semi-empirical modelling.