Symmetries and turbulence modeling

Abstract

This work applies new insights into turbulent statistics gained by Lie symmetry analysis to the closure problem of turbulence. Founded in the mathematics of partial differential equations, Lie symmetries have helped advances in many fields of modern physics. The main reason for this is their ability to encode important physical principles that are implicitly expressed by governing equations. Newly discovered symmetries of the multi-point correlation equations describing turbulent motion have been shown to encode two central effects of turbulent statistics: intermittency and non-Gaussianity. Moreover, these symmetries play a pivotal role in obtaining turbulent scaling laws such as the logarithmic law of the wall. Evidently, correctly preserving these symmetry properties in a turbulence model would render it capable of accurately predicting important effects of turbulent statistics and turbulent scaling. As these symmetry constraints have so far not been taken into account when devising turbulence models, we present a completely new modeling framework that can yield models fulfilling these conditions. In order to accomplish this, it turns out to be helpful, if not necessary, to introduce an entirely new symmetry-based modeling strategy that allows systematically constructing equations based on symmetry constraints imposed on them. From these considerations, it can be shown that in order to create meaningful turbulence models that fulfill these constraints, it is necessary to introduce a new velocity and pressure field. A possible skeleton of model equations for second moment closure is presented.