Reynolds-number dependence of streamwise velocity variance in wall-bounded turbulent flows

We propose a model for the streamwise velocity variance in wall-bounded turbulent flows. It hypothesizes that the wall-parallel motions of the attached eddies induce internal turbulent boundary layers. The streamwise momentum balance in both outer and inner layers of the internal boundary layers is analysed to obtain the model equations. A logarithmic variance profile is obtained. The peak value of the variance scaled using the friction velocity has a logarithmic dependence on the ratio of the wall-normal length of the flow to the thickness of the internal boundary layer induced by the largest attached eddies, the latter having a dependence on the friction Reynolds number in the form of a Lambert W function. Both the peak and the length ratio are predicted to be unbounded at asymptotically large Reynolds numbers. The model explains the data from the Princeton Superpipe well. The model also predicts that the streamwise velocity fluctuations induced by the attached eddies near the viscous layer scale with the friction velocity; therefore the contributions from the attached eddies to the scaled velocity variance there remains finite at asymptotically large Reynolds numbers.