Quadrant Analysis of the Reynolds Shear Stress in a Two-Height Canopy

Abstract

We study experimental data from a two-height roughness bed forming a top and bottom canopy sublayer of heights h and h/2, respectively. We focus on the double-averaged profiles of Reynolds stresses and the difference in contributions from sweep and ejection events, $\Delta S_0$. The two-height roughness adds to the typical canopy–air interface at height h another, previously unexplored, inner-canopy interface at height h/2. We apply particle image velocimetry within and above the two-height canopy and obtain the flow statistics over a representative repeating cell area. A quadrant analysis of the turbulent velocity fields is used to explore $\Delta S_0$. Our results show that, like in homogeneous dense canopies, ejections dominate the contribution to the measured shear stress ($\Delta S_0<0$) above 1.5h, while sweeps dominate below ($\Delta S_0>0$). In the two-height canopy roughness, $\Delta S_0$ peaks twice, right below the top and the bottom sublayer heights. We test how well the measured $\Delta S_0$ can be reproduced by the complete and incomplete cumulant expansion methods (CEM and ICEM), and further test a simplified gradient diffusion approach to the third-order velocity moments in the ICEM (ICEM-GD). We demonstrate that CEM and ICEM reproduce the measured $\Delta S_0$ fairly well above the canopy but over-estimate its values inside the canopy. It is also found that ICEM-GD captures the general shape of $\Delta S_0$ at heights dominated by ejections and reproduces the two peaks inside the canopy. But it fails above the canopy in the range $h<z<1.5h$. This failure uncovers the counter-gradient nature of the turbulent energy and shear stress flux associated with dense canopy flows.