Analytic prediction for planar turbulent boundary layers
  Analytic predictions of mean velocity profile (MVP) and streamwise ($x$) development of related integral quantities are presented for flows in channel and turbulent boundary layer (TBL), based on a symmetry analysis of eddy length and total stress. Specific predictions are the friction velocity $u_\tau$: ${ U_e/u_\tau }\approx 2.22\ln Re_x+2.86-3.83\ln(\ln Re_x)$; the boundary layer thickness $\delta_e$: $x/\delta_e \approx 7.27\ln Re_x-5.18-12.52\ln(\ln Re_x)$; the momentum thickness Reynolds number: $Re_x/Re_\theta=4.94[{(\ln {{\mathop{\rm Re}\nolimits} _{\theta} } + 1.88)^2} + 1]$, all in good agreement with empirical data. 
Xi Chen, Zhen-Su She
physics.flu-dyn