A systematic characterisation of canopy density based on turbulent-structure penetration

Abstract

We investigate turbulent flows over canopies of rigid elements with different geometries, spacings and Reynolds numbers to identify and characterise different canopy density regimes. In the sparse regime, the overlying turbulence penetrates relatively unhindered within the canopy, whereas in the dense regime, this penetration is limited. The frontal density, $\lambda f$ , a common a measure of canopy density, is effective for e.g. conventional vegetation with no preferential orientation, but we observe that it does not fully characterise the density regime for some less conventional topologies, suggesting it may not always capture the underlying physics. To address this, we propose to quantify turbulence penetration directly, from the position and extent of individual turbulent eddies, particularly those associated with intense Reynolds shear stress. We analyse a series of direct simulations for isotropic- and anisotropic-layout canopies with frontal densities $\lambda f\approx 0.01$ – $2.04$ , heights $h^+\approx 44$ – $266$ , element width-to-pitch ratios $w/s\approx 0.06$ – $0.7$ and Reynolds numbers ${\textit{Re}}{\tau} \approx 180$ – $2000$ . For the same $\lambda f$ , canopies with elements closely packed in the streamwise direction but large spanwise gaps result in deeper turbulence penetration, appearing sparser than isotropic or spanwise-packed ones. For the same spanwise gap, turbulence penetration remains similar across canopies independently of their streamwise pitch and gap. As the spanwise gap increases, eddies penetrate deeper and more vigorously into the canopy. Turbulence penetration is also Reynolds-number-dependent: the same canopy can behave as dense at low ${\textit{Re}}{\tau}$ , but increasingly sparse as ${\textit{Re}}{\tau}$ increases. Our results suggest that turbulence penetration depends essentially on the ability of turbulent eddies to fit within the canopy as they travel downstream, and that this can be characterised by an effective spanwise gap, and its ratio to the typical eddy size; turbulence penetration is substantial when this gap is larger than the eddy size, and negligible in the opposite case. A penetration length $d_p$ can then be defined from the effective gap or the eddy size, whichever is smaller. For small $d_p/h$ , the canopy behaves as dense; for moderate $d_p/h$ , as intermediate; and for $d_p/h\approx 1$ , turbulent eddies can penetrate all the way to the canopy bed and the canopy behaves as sparse.