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.

Reply to Frewer et al. Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536–1566

“…This symmetry is present also in other methods used to describe stochastic fields, namely the multipoint correlation function approach and the multipoint probability density function approach; see [7,8], where also its physical meaning was discussed.…”

Symmetries and their importance for statistical turbulence theory

“…As it was shown in Wacławczyk et al (2014) scaling of moments according to (234) transforms a PDF of a turbulent signal into a non-continuous PDF with delta function at the origin x = 0. Such function would correspond to a PDF of a turbulent signal interuppted by intervals where U = 0.…”

“…This issue was investigated by Wacławczyk et al (2014), where the form of classical symmetries written in terms of PDF's was derived. Below, we shortly recall the findings.…”

“…It was shown in Wacławczyk et al (2014) that the LMN equations are invariant with respect to the Galilean transformations t * = t, x * i = x i + u 0 t, u * i = u i + u 0 , where u 0 is a constant vector. However, the extended Galilean transformations T * 7 − T * 9 in (181), which read t * = t, x * = x + f (t) and U * = U + f ′ (t), where the vector f is time-dependent, was broken due to the formulation of the pressure term chosen for constructing the LMN hierarchy.…”

“…The statistical symmetries of the LMN hiererchy were first identified in Wacławczyk et al (2014) and their derivation will be shortly recalled below. We discuss here a set of transformations of the PDFs under which the LMN equations (218) turn out to be invariant and which corresponds to the set of statistical symmetries (233) and (234) found for the MPCE, where the moments H are, respectively shifted by a constant and scaled.…”