Complete description of turbulence in terms of hopf functional and LMN hierarchy: New symmetries and invariant solutions

Abstract

This paper deals with two methods for the full statistical description of turbulent field, namely the Lundgren-Monin-Novikov hierarchy (Lundgren, Phys Fluids, 10:969-975 1967, [5]) for the multipoint probability density functions (PDFs) of velocity and Hopf functional equation for turbulence (Hopf, J Ration Mech Anal, 1:87-122 1952, [2]). These equations are invariant under certain transformations of dependent and independent variables, so called symmetry transformation. The importance of these symmetries for the turbulence theory and modelling is discussed.