title: Statistical symmetries of the Lundgren-Monin-Novikov hierarchy.
abstract: 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.
authors.affiliation: Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany.
authors.forename: Marta
authors.initials: M
authors.lastname: Wacławczyk
authors.affiliation: Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany.
authors.forename: Nicola
authors.initials: N
authors.lastname: Staffolani
authors.affiliation: Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany and GS Computational Engineering, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany and Center of Smart Interfaces, Alarich-Weiss-Strae 10, 64287 Darmstadt, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Oberlack
authors.affiliation: Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany.
authors.forename: Andreas
authors.initials: A
authors.lastname: Rosteck
authors.affiliation: Institut für Theoretische Physik, Westfälische Wilhelms Universität, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany and Department of Mechanical Engineering, Johns Hopkins University, 3400 N Charles Street, Baltimore, Maryland 21218, USA.
authors.forename: Michael
authors.initials: M
authors.lastname: Wilczek
authors.affiliation: Institut für Theoretische Physik, Westfälische Wilhelms Universität, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany.
authors.forename: Rudolf
authors.initials: R
authors.lastname: Friedrich
mesh_terms: D008962:Models, Theoretical; D055585:Physical Phenomena