Highly resolved profiles of the mean velocity in turbulent Rayleigh-Bénard convection in air are presented and discussed. The present work extends our recently performed experiments at constant aspect ratio [Phys. Rev. Lett. 99, 234504 (2007)] to variable aspect ratios. The experiments cover a range of Rayleigh numbers 10(9)<Ra<10(12) and aspect ratios 1.13<Gamma<11.3 whereas the Prandtl number is fixed at Pr=0.7. The major finding of the present work is that the profiles of the mean horizontal velocity and its fluctuations are virtually invariant against the variation in Ra or Gamma if the wall distance is scaled by the displacement thickness of the boundary layer. Furthermore we have studied typical length scales of the boundary layer and their scaling with Ra and Gamma. Regarding a potential transition of the heat transport toward the ultimate regime we found that the boundary layer Reynolds number remains below Redelta=250 which is significantly lower than the predicted limit.