First and second-moment turbulent scaling laws for wall bounded shear flows - a symmetry approach*

The present work is a fundamental extension of the work of Oberlack (JFM 2001) where presently turbulent scaling laws were extended to second moments using symmetries. The analysis is based on the averaged momentum equation and the multi-point correlation equations. Beside the classical symmetries of Navier-Stokes equations, new statistical symmetries play the key role. They were discovered in Oberlack and Rosteck (2010) and later identified in Waclawczyk et al. (2014) as a measure of intermittency and non-gaussian of the probability density function. On the basis of the above-mentioned symmetries, classical and new turbulent scale laws were derived from first principles, which represent so-called invariant solutions of the above equations. Examples of these solutions are the logarithmic wall law, the law of the wake for BL flows, the classical and rotating Poiseuille flow, which each rotates around one of the three coordinate directions and the Poiseuille flow with wall-transpiration. In all these cases, there is a close connection between the turbulent scaling law of the mean velocity and those of the second moments. Comparisons with experimental and DNS data prove the clear validity of the scaling laws.