A multi-layer model for turbulent kinetic energy in pipe flows
  A multi-layer model of an energy length function is developed by employing recent results of the authors. The theory predicts the complete, mean streamwise turbulent kinetic-energy profile (MKP), in good agreement with empirical data for a wide range of Reynolds numbers (Re). In particular, a critical $Re_\tau$ is predicted, beyond which a scaling anomaly appears and MKP develops a second peak. 
Xi Chen, Fazle Hussain, Zhen-Su She
physics.flu-dyn