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

TL;DR

This work addresses the inadequacy of frontal density $λ_f$ alone to predict canopy-density regimes by introducing a turbulence-penetration metric based on intense $u'v'$ eddies that penetrate the canopy. Using direct numerical simulations with immersed-boundary canopies, the authors isolate background eddies via spectral filtering and quantify their penetration with metrics such as centroid locations, percentile extents, and a penetration depth $d_p^+$, linked to a relative volume $V_r$ near the canopy tip. A key finding is that the spanwise gap $g_z^+$ is the dominant geometric control on penetration: many cases obey $d_p^+ \approx g_z^+$, and dense, intermediate, and sparse regimes collapse onto a $d_p/h$ vs $g_z/h$ map, with dense regimes yielding small $d_p/h$ and sparse regimes approaching unity. The results also show that canopy topology (streamwise vs spanwise packing) and Reynolds-number effects modulate penetration, and that staggered layouts hint at extending the canyon-gap idea to more complex geometries. Overall, the paper provides a physically motivated, eddy-based framework for canopy density that improves predictive capability for turbulence penetration and related drag characteristics, with implications for vegetation flows, heat transfer canopies, and urban microflows.

Abstract

Turbulent flows over canopies of rigid elements with different geometries and Reynolds numbers (Re) are investigated to identify and characterise different canopy density regimes. In the sparse regime, turbulence penetrates relatively unhindered within the canopy, whereas in the dense regime, the penetration is limited. A common measure of canopy density is the ratio of frontal to bed area, the frontal density $λ_f$. This is effective for canopies with no preferential orientation, but we observe that it does not accurately predict the density regime for less conventional ones, so it may not encapsulate the governing physics. Instead, we propose density metrics based on the position and extent of eddies of intense Reynolds shear stress. We analyse a series of direct simulations for isotropic and anisotropic layouts, across a range of $λ_f$, height, element width-to-pitch ratio and Re. Canopies with streamwise-packed elements but large spanwise gaps allow significant turbulence penetration, and appear sparse compared to isotropic or spanwise-packed canopies with the same $λ_f$. Turbulence penetration depends essentially on the spanwise gap, and increases with it, but depends also on Re. A canopy can behave as dense at low Re, but as sparser as Re increases. This suggests that turbulence penetration depends on the size of the spanwise gap relative to the typical width of the overlying eddies. Turbulence penetrates easily when the spanwise gap is larger than the eddy size, and is essentially precluded from penetrating in the opposite case. A penetration length can then be defined that is of the order of the spanwise gap or the eddy size, whichever is smaller. If the penetration length is small compared to the canopy height, the canopy behaves as dense; if it is comparable, as intermediate; and if it is roughly equal or larger, as sparse.

Figures (30)

• Figure 1: Graphical illustration of the flow regimes over (a) dense and (b) sparse canopies. The dark blue, orange and green arrows represent the background-turbulence eddies, mixing-layer-like eddies and element-coherent eddies, respectively. Adapted from poggi2004effect and sharma2020turbulent.
• Figure 2: Schematic representation of (a) a canopy element and (b) a layout of canopy elements. In (a), $A_f$ is the frontal area and $A_b$ is the bed area occupied by each element. In (b), $h$ is the element height; $w_x$, $g_x$ and $s_x$ are the element width, gap and spacing (pitch) in the streamwise direction, respectively, and $w_z$, $g_z$ and $s_z$ are those in the spanwise direction.
• Figure 3: Instantaneous realisations of the $u'v'$ structures over and within canopies with the same number of elements per area and frontal density $\lambda_f\approx0.91$: (a) streamwise-packed canopy $\mathrm{C_{S27\times108}}$ with $s_x^+\approx27$, $s_z^+\approx108$ and $h^+\approx110$, and (b) spanwise-packed canopy $\mathrm{F_{S108\times27}}$ with $s_x^+\approx108$, $s_z^+\approx27$ and $h^+\approx110$. The structures are coloured by distance to the floor, and consist of ejections ($u'<0, v'>0$ blue to green), sweeps ($u'>0, v'<0$ red to yellow), and outward and inward interactions ($u'v'>0$ grey to white).
• Figure 4: Schematics of the numerical channel for (a) spanwise-packed canopy $\mathrm{F_{S216\times54}}$, with streamwise spacing $s_x^+\approx216$, spanwise spacing $s_z^+\approx54$ and canopy height $h^+\approx110$, all in viscous units, and (b) streamwise-packed canopy $\mathrm{C_{S54\times216}}$, with $s_x^+\approx54$, $s_z^+\approx216$ and $h^+\approx110$. The channel half-height is $\delta$. Details of the canopy geometries are presented in table \ref{['tab:canopy_param']}. An instantaneous realisation of the streamwise velocity is shown in axis-orthogonal planes, from dark to clear $u^+=0$ to $u^+=15$ in (a) and $u^+=0$ to $u^+=19$ in (b).
• Figure 5: Instantaneous realisations of $u'v'$ structures for (a, c) case $\mathrm{I_{S216\times216}}$ and (b, d) case $\mathrm{F_{S144\times36}}$. Structures are coloured by distance to the floor, ejections in blue to green, sweeps in red to yellow, and outward and inward interactions in grey to white. (a, b), raw flow fields; (c, d), flow fields spectrally filtered to remove the element-coherent flow.