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

非高斯性、湍流标度参数和共形变换的界限 - 使用李对称性分析湍流的 Lundgrenand Hopf 函数方程

• 批准号: 385665358
• 负责人: Professor Dr.-Ing. Martin Oberlack
• 金额: --
• 依托单位: Fachgebiet Strömungsdynamik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/385665358?language=en
• 关键词: Non; Gaussianity; bounds; turbulent; scaling

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.

对称性是所有物理理论的核心，如经典力学、量子力学或相对论，因为它们反映了基础物理学的公理性质。湍流的一些关键性质，特别是非高斯性和非高斯性，最近由本申请人给出了基于高斯性的测量。结果表明，这是一致的，所有三个完整的统计湍流理论，即多点相关方程，Lundel-Novikov-Monin（LMN）的概率密度（PDF）的层次和霍普夫函数的方法。最重要的是，在一系列出版物中，申请人已经表明对称性是所有湍流标度律的公理基础。基于LMN族和Hopf泛函，我们打算推广我们最近的结果，并集中讨论以下关键问题：（i）二维湍流中共形对称的存在性，（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性.由于LMN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变换的扩展对称性方法导出对称性的穷举列表。对于子任务（i），我们在二维湍流涡度的LMN层次结构中寻找共形群，因为它的存在最近被分析数值数据所证实。申请人的初步计算支持这一重要的物理和十年前的猜想。对于子任务（ii），我们将解开的限制群参数和自由函数的对称性，通过调查潜在的代数结构。上述表现例如在kappa的对数定律的壁，这是一个统计和比例组参数的比率，其数值的确定是未知的。我们可以进一步使用PDF的正性，这将对对称参数施加额外的约束。对于子任务（iii），在我们手中的对称性和它的约束，我们可以构建PDF的群不变的解决方案。对于这一点，我们已经看到，某种对称性的组合将导致指数尾和非高斯性，前者与非高斯性密切相关。最后，对于子任务（iv），我们将构造Hopf泛函的群不变解，它提供了无穷序列的时刻，反过来，所有的结构函数，以研究其异常标度性质。对于（iii）和（iv），预期的结果只能在非常特定的对称性组合下观察到。这种特殊性也将使用李代数进行研究，而工作假设是，这些对称性形成一个特定的子代数。根据目前的提议，长期目标是将湍流中一些最紧迫的问题放在合理的位置，这超出了数据匹配和半经验建模的范围。