Prandtl and von Kármán have developed the famous log-law for the mean velocity profile for turbulent flow over a plate. The log-law has also been applied to turbulent pipe flow, though the wall surface is curved (in span-wise direction) and has finite diameter. Here we discuss the theoretical framework, based on the Navier-Stokes equations, with which one can describe curvature effects and also the well-known finite-size effects in the turbulent mean-velocity profile. When comparing with experimental data we confirm that the turbulent eddy viscosity must contain both curvature and finite-size contributions and that the usual ansatz for the turbulent eddy viscosity as being linear in the wall distance is insufficient, both for small and large wall distances. We analyze the experimental velocity profile in terms of an r-dependent generalized turbulent viscosity ν turb ≡u ∗ ag(ρ/a) (with ρ being the wall distance, a pipe radius, u * shear stress velocity, and g(ρ/a) the nondimensionalized viscosity), which reflects the radially strongly varying radial eddy transport of the axial velocity. After the near wall linear viscous sublayer, which soon sees the pipe wall's curvature, a strong transport (eddy) activity steepens the profile considerably, leading to a maximum in g(ρ/a) at about half radius, then decreasing again towards the pipe center. This reflects the smaller eddy transport effect near the pipe's center, where even in strongly turbulent flow (the so-called “ultimate state”) the profile remains parabolic. The turbulent viscous transport is strongest were the deviations of the profile from parabolic are strongest, and this happens in the range around half radius.