New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws

Abstract

We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Second, we demonstrate that the recently discovered exponential decay law of isotropic turbulence generated by fractal grids is only admissible with the new statistical scaling symmetry. It may not be borne from the two classical scaling groups implied by the fundamental equations of fluid motion and which has dictated our understanding of turbulence decay since the early thirties implicated by the von-Ḱarḿan- Howarth equation.