Complete Description of Turbulence in Terms of Hopf Functional and LMN Hierarchy: New Symmetries and Invariant Solutions

Marta Wacławczyk

Progress in Turbulence VI: Springer Proceedings in Physics

165, 2016, 9-17, 10.1007/978-3-319-29130-7_2

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.