Analytic prediction for planar turbulent boundary layers

TL;DR

This paper presents analytic predictions for the mean velocity profile and development of turbulent boundary layers using symmetry analysis, showing good agreement with empirical data.

Contribution

It introduces a novel symmetry-based analytical framework to predict key turbulent boundary layer quantities.

Findings

Predicted friction velocity matches empirical data.

Derived boundary layer thickness aligns with observations.

Reynolds number relations are validated against experiments.

Abstract

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.