The role of the in-plane solidity on canopy flows

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Turbulent open channel flows developing above submerged canopies made of slender cylinders mounted perpendicular to the channel bed are known to be largely governed by the solidity parameter$\lambda =dh/\Delta S^2$($d$and$h$being the diameter and height of the filament, and$\Delta S$the average spacing between filaments). When the filaments are sufficiently slender, the ratio between the height of the stems and the spacing sets the hydrodynamic regime developing inside and outside the canopy. This ratio also establishes the conditions leading to the transition from a dense to a sparse canopy flow regime (Nepf,Annu. Rev. Fluid Mech., vol. 44, 2012, pp. 123–142). In a previous, companion numerical investigation, Montiet al.(J. Fluid Mech., vol. 891, 2020, A9) used large eddy simulation (LES) to study the influence of the canopy height on the onset of the different regimes without modifying the average spacing$\Delta S$between the stems. In that LES study, we were looking at the complementary situation in which the height of the stems is constant while the filaments’ number density of the canopy is changed. It was found that for low values of$\lambda$(i.e. sparse or moderately dense canopies:$\lambda \lessapprox 0.26$), the flows sharing the value of the solidity obtained by either varying$h$or$\Delta S$are very similar. Differently, for higher values of$\lambda$(i.e. in denser canopies), the effects of$h$and$\Delta S$start to diverge although sharing the same nominal value of$\lambda$. In this paper, we analyse the different physical mechanisms that come into play for dense configurations obtained by varying either$\Delta S$or$h$. In particular, we focus on the most relevant length scales and carry out a detailed analysis of the flows using a triple decomposition approach. We show that the inner region of dense canopy flows, characterised by tall stems, is dominated by wall-normal sweeps delivering high momentum in the wall vicinity. Here, the impenetrability condition of the bed redistributes the available momentum in the wall-parallel directions re-energising an otherwise stagnating flow. Differently, in densely packed canopies, the penetration of the outer jet and the momentum transfer from the external flow are limited by the decreasing value of the wall-parallel permeabilities leading to different behaviours, including a reduction of the total drag offered by the canopy.