Lie symmetry analysis of the Lundgren–Monin–Novikov equations for multi-point probability density functions of turbulent flow

Abstract

The problem of turbulence statistics described by the Lundgren–Monin–Novikov (LMN) hierarchy of integro-differential equations is studied in terms of its group properties. For this we perform a Lie group analysis of a truncated LMN chain which presents the first two equations in an infinite set of integro-differential equations for the multi-point probability density functions (pdf’s) of velocity. A complete set of point transformations is derived for the one-point pdf’s and the independent variables: sample space of velocity, space and time. For this purpose we use a direct method based on the canonical Lie–Bäcklund operator. Due to the one-way coupling of correlation equations, the present results are complete in the sense that no additional symmetries exist for the first leading equation, even if the full infinite hierarchy is considered.

A critical examination of the conformal invariance in the statistical equations of 2D turbulent scalar fields

“…which Wac lawczyk et al refer to as the intermittency symmetry [20,21]. 1 But this invariance is not realizable, i.e., the transformed moments cannot be realized by the defining stochastic equation (4.1), no matter which starting configuration in the parameters or which mapping in the variables is chosen.…”

Comment on 'Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow'

“…They definitely may not be treated as exogenous conditions like, for example, initial or boundary conditions which under certain asymptotic assumptions can be neglected or ignored within the search for symmetries of an underlying dynamical equation. In particular, due to not including all these internal LMN constraints into their symmetry analysis in a strict systematic manner, it has been overlooked in Wac lawczyk et al (2017) that the two "new" symmetries X * (Eq. [47]) and X * * (Eq.…”

“…where a s is a group parameter of the infinitesimal X * , constrained according to the LMN constraints (see Sec. [4] in Wac lawczyk et al (2017)). Instead of the invariant result…”

“…In this regard, it is also to be noted that X * is not commuting with X * * , as mistakenly presented in Tab. [1] in Wac lawczyk et al (2017).…”

“…Now, the two new symmetries Eqs. [47][48] in Wac lawczyk et al (2017) are pure statistical transformations that clearly change the statistical fields from f n to f * n (see (1.2) and (1.3), respectively), and hence, in each case, this change should have a cause on the instantaneous level, where, of course, the cause itself need not to be a symmetry. Furthermore, since these two statistical transformations (1.2) and (1.3) are so simple and trivial, it is very easy to find a unique cause for the change of the lowest velocity moment, the mean velocity (see e.g.…”