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

• Applicant: Professor Dr.-Ing. Martin Oberlack
• Subject Area: Fluid Mechanics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 385665358

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.

• DFG Programme: Research Grants