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

Wacławczyk M., Grebenev V.N., Oberlack M.

Journal of Physics A - Mathematical and Theoretical

50, 2017, 175501, 10.1088/1751-8121/aa62f4

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.