  We study turbulent flows in a smooth straight pipe of circular cross--section
up to $Re_{\tau} \approx 6000$ using direct--numerical-simulation (DNS) of the
Navier--Stokes equations. The DNS results highlight systematic deviations from
Prandtl friction law, amounting to about $2\%$, which would extrapolate to
about $4\%$ at extreme Reynolds numbers. Data fitting of the DNS friction
coefficient yields an estimated von K\'arm\'an constant $k \approx 0.387$,
which nicely fits the mean velocity profile, and which would restore
universality of canonical wall-bounded flows. The same constant also applies to
the pipe centerline velocity, thus providing support for the claim that the
asymptotic state of pipe flow at extreme Reynolds numbers should be plug flow.
At the Reynolds numbers under scrutiny, no evidence for saturation of the
logarithmic growth of the inner peak of the axial velocity variance is found.
Although no outer peak of the velocity variance directly emerges in our DNS, we
provide strong evidence that it should appear at $Re_{\tau} \gtrsim 10^4$, as a
result of turbulence production exceeding dissipation over a large part of the
outer wall layer, thus invalidating the classical equilibrium hypothesis.
