Symmetries and their importance for statistical turbulence theory

Abstract

PrefaceThe special importance of turbulence maybe comprehended by its ubiquity in innumerable natural and technical systems. Examples for natural turbulent flows are the atmospheric flow and the oceanic current which to calculate is a crucial point in climate research. Classical engineering application involving turbulence are the flows around airplanes or cars or turbulence within jet or reciprocating engines. Though supposed for more than hundred years, only with the advent of super computers it became apparent that the Navier-Stokes equations provide an excellent continuum mechanical model for turbulent flows. Still, the exclusive and direct application of the Navier-Stokes equations to practical flow problems at very high Reynolds numbers without invoking any additional assumptions is still several decades away.However, in most applications it is not at all necessary to know all the detailed fluctuations of velocity and pressure present in turbulent flows but for the most part statistical measures are sufficient. This was in fact the key idea of O. Reynolds who was the first to suggest a statistical description of turbulence. The Navier-Stokes equations, however, constitute a non-linear and, due to the pressure Poisson equation, a non-local set of equations. As an immediate consequence of this the equations for the mean or expectation values for velocity and pressure lead to an infinite set of statistical equations, or, if truncated at some level of statistics, an un-closed system is generated. Received 4 March 2015 AbstractThe present article is intended to give a broad overview and present details on the Lie symmetry induced statistical turbulence theory put forward by the authors and various other collaborators over the last twenty years. For this is crucial to understand that our present text-book knowledge proclaims that Lie symmetries such as Galilean transformation lie at the heart of fluid dynamics. These important properties also carry over to the statistical description of turbulence, i.e. to the Reynolds stress transport equations and its generalization, the multi-point correlation equations (MPCE). Interesting enough, the MPCE admit a much larger set of symmetries, in fact infinite dimensional, subsequently named statistical symmetries. Apart from the MPCE also the two other known complete theories of turbulence, the Lundgren-Monin-Novikov (LMN) hierarchy of probability density functions and the Hopf functional theory, share this property of admitting both classical mechanical and statistical Lie symmetries. As the Galilean transformation illuminates fundamental properties of classical mechanics, the new statistical symmetries mirror key properties of turbulence such as intermittency and non-gaussianity. After an introduction to Lie symmetries have been given, these facts will be detailed for all three turbulence approaches i.e. MPCE, LMN and Hopf approach. From a practical point of view, these new symmetries have important consequences for our understanding of turbulent scaling laws. The symmet...

Lie Symmetry Analysis of the Hopf Functional-Differential Equation

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“…Relative to wall turbulence, the contributions of Oberlack and co-workers (Oberlack 2001; Oberlack & Rosteck 2010; Oberlack et al. 2015) are perhaps most notable, as their analyses have revealed symmetry properties that are admitted by the full RANS equations. Their findings indicate that under a maximal symmetry hypothesis these equations admit a logarithmic mean profile solution, as well as other solution properties such as distance-from-the-wall scaling (Oberlack 2001).…”