Logarithmic distribution of mean velocity and turbulent kinetic energy   in a pipe flow
  A Lie-group based similarity theory is developed for both momentum and energy distributions in a turbulent pipe flow, leading to asymptotic logarithmic profiles of mean velocity and turbulent kinetic energy. Both channel and pipe data over a wide range of Re yield 0.45 to be the universal Karman constant. A new spatial invariant characterizing outer dynamics is discovered and validated by reliable experimental data. The theory predicts the mean velocity profile (MVP) with 99% accuracy for high Re experimental data (up to 40 millions), and offers a quantitative explanation for recent observation of logarithmic kinetic energy distribution by Hullmak et al. (Phys. Rev. Lett. 108, 094501). 
Xi Chen, Fazle Hussain, Zhen-Su She
physics.flu-dyn