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. 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.
