# A Large-eddy Simulation Study of Surface Layer Response to Roughness Spatial Heterogeneity

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## Abstract

A rough wall is known to modify the structures of the turbulent boundary layer relative to the boundary layer flow over a smooth wall due to eddies, vortices, and turbulence structures introduced by drag-exerting surface irregularities. The elevated turbulence and mixing due to roughness influence momentum and heat exchange. The roughness also affects flow separation and reattachment and may result in secondary flow. The present study examines surface layer response to roughness heterogeneity in the streamwise and spanwise directions, resulting from various arrangements of drag-exerting tree canopy elements. Previous studies have shown that a predominant streamwise-aligned spanwise heterogeneity results in undulations in streamwise velocity in the transverse-wall-normal plane where low-momentum pathways (LMP) and high-momentum pathways (HMP) are located based on surface roughness characteristics, known as Prandtl’s secondary flow of the second kind. Several numerical and experimental studies reported an HMP over the elevated roughness, whereas others witnessed an LMP over high roughness flanked by counter-rotating vortices corresponding with momentum excess. In addition, flow response to a step change in surface roughness in the flow-wise direction is also widely studied. Through a suite of controlled numerical experiments using large-eddy simulations (LES), we examined the confluence of roughness heterogeneity on the generation and sustenance of secondary flow to address the disagreement above. The roughness is introduced as synthetic trees using the canopy drag model. The results showed that within the transitional roughness regime in the spanwise direction, 1 ≲ δ2/δ ≲ 2, where δ2 is the gap between streamwise aligned parallel rows of canopies and δ is the channel half height, an upwelling of low-momentum fluid is observed over canopy rows for cases with δ1/h ≲ 6 (d-type roughness), where δ1 is the downstream gap between subsequent trees and h is the canopy height from the ground. However, a downwelling of high momentum fluid aloft is found for δ1/h ≳ 6 (k-type roughness) (Raupach and Rajagopalan, 1991; Chung et al., 2021). The observations suggest that the roughness regimes (d and k-type) regulate the polarity of the secondary cells. The flow response learned from simulations categorized the roughness regimes on a δ1 − δ2 parameter space. This suggests that the upwelling and downwelling of low and high-momentum fluid over the elevated roughness depends on the streamwise cavity width. Within the topography regime, δ2/δ ≳ 2, both d- and k-type roughness cases result in the downwelling of high momentum fluid above canopy elements. Later, the transitional roughness regime resulting from δ-scale spanwise roughness heterogeneity is explored to understand the flow response near critical cavity width, δ1/h ≈ 6. We show that the system attains a large-scale non-periodic reversal of the secondary flow near the critical streamwise gap using statistical measures such as joint probability distribution, time series evolution, and probability density functions.