Statistical Symmetries in Second Order Turbulence Modeling

Early turbulence models, such as zero- and one-equation models, exhibit a lack of symmetries, which are axiomatic properties of flow physics. Conversely, more comprehensive second moment closure models contain all of the symmetries of the Navier-Stokes equations. Recently, new symmetries in statistical descriptions of turbulence have been discovered, yet current turbulence models fail to account for these symmetries. We show how commonly used turbulence models fail to preserve classical symmetries and propose a second moment closure turbulence model and demonstrate its invariance under statistical symmetries. To achieve this, an auxiliary velocity and pressure field must be defined.