How far does the influence of the free surface extend in turbulent open channel flow?

Abstract

Turbulent open channel flow is known to feature a multi-layer structure near the free surface. In the present work we employ direct numerical simulations considering Reynolds numbers up to \mathrm{Re}_\tau=900$ and domain sizes large enough ($L_x=12 \pi h$, $L_z=4 \pi h$) to faithfully capture the effect of very-large-scale motions in order to test the proposed scaling laws an ultimately answer the question: How far does the influence of the free surface extend? In the region near the free surface, where fluctuation intensities of velocity and vorticity become highly anisotropic, we observe the previously documented triple-layer structure, consisting of a wall-normal velocity damping layer that scales with the channel height $h$, and two sublayers that scale with the near-surface viscous length scale $\ell_\mathrm{V}=\mathrm{Re}_\mathrm{b}^{-1/2}h$ and with the Kolmogorov length scale $\ell_\mathrm{K}=\mathrm{Re}_\mathrm{b}^{-3/4}h$, respectively. The scaling laws previously proposed by Calmet and Magnaudet [J. Fluid. Mech. \textbf{474}, 355--378 (2003)] are found to hold with the following exceptions. The thin layer, where the intensity of surface-parallel components of the vorticity rapidly decreases to zero, is here found to scale with the Kolmogorov length scale $\ell_\mathrm{K}$ rather than with the near-surface viscous scale $\ell_\mathrm{V}$. In addition, we argue that the Kolmogorov length scale is the relevant scale for the mean velocity gradient near the free surface. Both the mean velocity gradient and the fluctuation intensity of the surface-parallel component of vorticity decay to zero in the Kolmogorov sublayer $\delta_\mathrm{K} \approx 20 \ell_\mathrm{K}$. On the other hand, the layer, where the wall-normal turbulence intensity decreases linearly to zero near the free surface, scales with $\ell_\mathrm{V}$ rather than $\ell_\mathrm{K}$ as suggested by Calmet and Magnaudet. The corresponding near-surface viscous sublayer measures $\delta_\mathrm{V} \approx \ell_\mathrm{V}$. Importantly, the streamwise turbulence intensity profile for $\mathrm{Re}_\tau \ge 400$ suggests that the influence of the free-slip boundary penetrates essentially all the way down to the solid wall through the appearance of enhanced very-large-scale motions ($ delta_{\mathrm{SIL}}\approx h$). In contrast, the layer where the surface-normal turbulence intensity is damped to zero is restricted to the free surface ($\delta_{\mathrm{NVD}}\approx 0.3h$). As a consequence, the partitioning of the surface-influenced region has to be expanded to a four-layer structure that spans the entire channel height $h$.