Andreas Rosteck

LIE ALGEBRA OF THE SYMMETRIES OF THE MULTI-POINT EQUATIONS IN STATISTICAL TURBULENCE THEORY

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The latter are denoted statistical symmetries and have no direct counterpart in classical mechanics. Finally, we considerably extend the set of symmetries by Lie algebra methods and give the corresponding commutator tables. Due to the infinite dimensionality of the multi-point correlation equations completeness of its symmetries is not proven yet and is still an open question.

New Statistical Symmetries of the Two-Point Correlation Equations for Turbulent Flows

We briefly introduce the two-point correlation equations based on the Navier-Stokes equations for an incompressible fluid. For this special case we determine the set of Lie symmetries, which can be calculated from the classical symmetries of the Navier-Stokes equations and further we present new symmetries, so called statistical symmetries. Finally we give examples where these symmetries can be used, e.g. for wall bounded turbulence and decaying turbulence scaling laws.

Applications of the new symmetries of the multi-point correlation equations

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. Additional to the symmetries stemming from the Navier-Stokes equations 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. Exemplarily, we consider one of the key foundations of statistical turbulence theory, the universal law of the wall, and show that the log-law fundamentally relies on one of the new translational groups. Furthermore, we present rotating channel flows, where different rotational axes result in very different scaling laws.