  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.
