Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509 - Refinement of the logarithmic law of the wall

Available direct numerical simulation of turbulent channel flow at moderately high Reynolds numbers data show that the logarithmic diagnostic function is a linearly decreasing function of the outer-normalized wall distance $\eta=y/\delta$ with a slope proportional to the von Kármán constant, $\kappa=0.4$. The validity of this result for turbulent pipe and boundary layer flows is assessed by comparison with the mean velocity profile from experimental data. The results suggest the existence of a flow-independent logarithmic law $U^+=U/u_\tau=(1/\kappa)\ln(y^\star/a)$, where $y^\star=y\,U_S/\nu$ with $U_S=y\,S(y)$ the local shear velocity and the two flow-independent constants $\kappa=0.4$ and $a=0.36$. The range of its validity extends from the inner-normalized wall distance $y^+=300$ up to half the outer-length scale $\eta=0.5$ for internal flows, and $\eta=0.2$ for zero-pressure-gradient turbulent boundary layers. Likewise, and within the same range, the mean velocity deficit follows a flow-dependent logarithmic law as a function of a local mean-shear based coordinate. Furthermore, it is illustrated how the classical friction laws for smooth pipe and zero-pressure-gradient turbulent boundary layer are recovered from this scaling.
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Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

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Refinement of the logarithmic law of the wall