Symmetries Based Invariant Modeling for Second Moment Turbulence Modeling*

Attempts to develop semi-empirical model turbulence date back to the late 19th century; however, their limitations often stem from missing symmetries, which are axiomatic invariance properties in physics. The key basis for turbulence description, die Navier-Stokes equations, admit the Galilean group extended by scaling. Over the decades modelers have implicitly included increasingly more symmetries into turbulence model. Classical two-equation models eventually included all of them, though have too many and unphysical symmetries, being invariant under constant rotations. Recently, new symmetries of the infinite set of multi-point moment equations (MPME) have been discovered, dubbed statistical symmetries, as they only occur in the statistical descriptions of turbulence such as the MPME and have been linked with intermittency and non-Gaussianity. In PRL 2022 (Oberlack et al.) it was shown that these symmetries are essential for high-moment scaling laws. All turbulence models covered so far are not invariant under these statistical symmetries. We show very how a second moment turbulence model can be developed from all symmetries known so far, and further we present a recently developed model.