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. A logarithmic variance profile is obtained. The peak value of the variance scaled using the friction velocity has a logarithmic dependence on the ratio the wall-normal length of the flow to the thickness of the internal boundary layer induced by the largest attached eddies ($\delta_o$), 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 unbounded at asymptotically large Reynolds numbers. 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 scaled velocity variance there remains finite at asymptotically large Reynolds numbers.