DNS of pipe flow up to Re_τ⁼6000

We present DNS data of turbulent flow in a smooth straight pipe of circular cross-section, up to Re_τ⁼6000. The DNS results highlight mild 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 and heat transfer coefficients yields estimated von Karman constants k ≈ 0.387 for the velocity field, and k_θ≈0.459 for the temperature field, which are well in line with those found in all canonical wall-bounded flows. The same constants also apply to the pipe centerline values,  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_τ ≥ 10⁴. As a result of turbulence production exceeding dissipation over a large part of the outer wall layer, thus invalidating the classical equilibrium hypothesis.