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We study turbulent flows in a smooth straight pipe of circular cross--sectionup to $Re_{\tau} \approx 6000$ using direct--numerical-simulation (DNS) of theNavier--Stokes equations. The DNS results highlight systematic deviations fromPrandtl friction law, amounting to about $2\%$, which would extrapolate toabout $4\%$ at extreme Reynolds numbers. Data fitting of the DNS frictioncoefficient yields an estimated von K\'arm\'an constant $k \approx 0.387$,which nicely fits the mean velocity profile, and which supports universality ofcanonical wall-bounded flows. The same constant also applies to the pipecenterline velocity, thus providing support for the claim that the asymptoticstate of pipe flow at extreme Reynolds numbers should be plug flow. At theReynolds numbers under scrutiny, no evidence for saturation of the logarithmicgrowth of the inner peak of the axial velocity variance is found. Although noouter peak of the velocity variance directly emerges in our DNS, we providestrong evidence that it should appear at $Re_{\tau} \gtrsim 10^4$, as a resultof turbulence production exceeding dissipation over a large part of the outerwall layer, thus invalidating the classical equilibrium hypothesis.

One-point statistics for turbulent pipe flow up to $Re_{\tau} \approx 6000$