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

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 supports 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.