We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Second, we demonstrate that the recently discovered exponential decay law of isotropic turbulence generated by fractal grids is only admissible with the new statistical scaling symmetry. It may not be borne from the two classical scaling groups implied by the fundamental equations of fluid motion and which has dictated our understanding of turbulence decay since the early thirties implicated by the von-Kármán-Howarth equation.

It was shown by Oberlack and Rosteck [Discr. Cont. Dyn. Sys. S, 3, 451 2010] that the infinite set of multipoint correlation (MPC) equations of turbulence admits a considerable extended set of Lie point symmetries compared to the Galilean group, which is implied by the original set of equations of fluid mechanics. Specifically, a new scaling group and an infinite set of translational groups of all multipoint correlation tensors have been discovered. These new statistical groups have important consequences for our understanding of turbulent scaling laws as they are essential ingredients of, e.g., the logarithmic law of the wall and other scaling laws, which in turn are exact solutions of the MPC equations. In this paper we first show that the infinite set of translational groups of all multipoint correlation tensors corresponds to an infinite dimensional set of translations under which the Lundgren-Monin-Novikov (LMN) hierarchy of equations for the probability density functions (PDF) are left invariant. Second, we derive a symmetry for the LMN hierarchy which is analogous to the scaling group of the MPC equations. Most importantly, we show that this symmetry is a measure of the intermittency of the velocity signal and the transformed functions represent PDFs of an intermittent (i.e., turbulent or nonturbulent) flow. Interesting enough, the positivity of the PDF puts a constraint on the group parameters of both shape and intermittency symmetry, leading to two conclusions. First, the latter symmetries may no longer be Lie group as under certain conditions group properties are violated, but still they are symmetries of the LMN equations. Second, as the latter two symmetries in its MPC versions are ingredients of many scaling laws such as the log law, the above constraints implicitly put weak conditions on the scaling parameter such as von Karman constant κ as they are functions of the group parameters. Finally, let us note that these kind of statistical symmetries are of much more general type, i.e., not limited to MPC or PDF equations emerging from Navier-Stokes, but instead they are admitted by other nonlinear partial differential equations like, for example, the Burgers equation when in conservative form and if the nonlinearity is quadratic.

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...

No abstract

We briefly derive the infinite set of multi-point correlation equations based on the Navier-Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The latter are denoted statistical symmetries and have no direct counterpart in classical mechanics. Finally, we considerably extend the set of symmetries by Lie algebra methods and give the corresponding commutator tables. Due to the infinite dimensionality of the multi-point correlation equations completeness of its symmetries is not proven yet and is still an open question.