One-point statistics for turbulent pipe flow up to

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