# Wind Tunnel Measurements of Surface Shear Stress on an Isolated Dune Downwind a Bridge
## 4 Discussion  
The threshold friction velocity is a key parameter to estimate the deposition and erosion potential of the ground. Some experiments show a higher threshold friction velocity than the calculation results using the [PERSON]'s empirical equation [30]. We calculated the fluid threshold skin friction velocity as:  
\[u_{\texttt{\tiny TT}}\ =\ u_{\texttt{\tiny st}}+2\sigma(u_{\texttt{\tiny T}}) \tag{3}\]  
where \(u_{\texttt{\tiny st}}\) was 0.29 m s\({}^{-1}\) in our experiment [31]. The standard deviation \(\sigma(u_{\texttt{\tiny T}})\) = 0.048 m s\({}^{-1}\) at 15 m/s was determined from skin friction velocity variations measured with Irwin sensors on the smooth wooden floor. As a result, \(u_{\texttt{\tiny TT}}\) in our study was 0.387 m s\({}^{-1}\) and the threshold shear stress \(\tau_{t}\) equaled 0.179 Pa.  
To assess the local dominance of erosion and deposition mechanisms, a threshold parameter of fraction time \(\psi\) was proposed [14], which represents the fraction of time when friction velocity exceeds the threshold parameter. \[\psi(x,y)\ =\ \frac{\Delta t[\tau(t,x,y)>\tau_{t}]}{T} \tag{4}\]  
where \(\Delta t\) is the time period during which the shear stress \(\tau\) is larger than the threshold shear stress \(\tau_{t}\), and \(T\) represents the total time. As defined \(\psi\ =\ 1\) indicates an erosion zone, \(\psi\ =\ 0\) indicates a deposition zone, and \(0<\psi<1\) indicates both erosion and deposition are both possible. Specifically, for \(\psi<0.5\), deposition dominates the local net and for \(\psi>0.5\), erosion dominates. Figure 8 shows the spatial distribution of \(\psi\) at the free-stream wind velocity \(u_{0}\ =\ 15\) m s\({}^{-1}\) in our experiment. In the isolated dune case, the value of \(\psi\) increased from 0 at the windward toe to 0.98 at the windward crest, indicating that the erosion rate increases with height on the windward slope. On the leeward and downwind the sand dune (\(0<\texttt{x}<10\) H), the values of \(\psi\) are less than 0.5, indicating that deposition happened in this region. In the case of S = 1.5 H, the value of \(\psi\) is decreased by the bridge pier and increased by the bridge desk in the lower part of the windward slope. That is, the value of \(\psi\) between x = -5 and -2.5 decreases at y = 0 and increases at y = L/2 compared with the isolated dune case. For instance, the value of \(\psi\) is 0.42 at x = -2.5, y = 0 and the value of \(\psi\) is 0.65 at x = -2.5, y = L/2, decreased and increased compared with the value of 0.5 in the isolated dune case. In the cases of S = 3.0 H, 5.0 H, 10.0 H and 20.0 H, the change of \(\psi\) on the windward slope is not as prominent as in the case of S = 1.5 H. On the leeward slope, \(\psi\) in this region always kept low magnitudes that are lower than 0.3, indicating that the incoming sand particles deposit no matter where the bridge was set. Downstream the sand dune, in the cases of S = 1.5 H, 3.0 H and 5.0 H, \(\psi\) is decreased at y = 0 and increased at y = L/2 compared with the isolated dune case. However, the value of \(\psi\) exceed 0.5 at x = 15 H almost in all cases. That is, between x = 0 and x = 15 H, deposition domains the sand movement and downwind x = 15 H, erosion domains the sand movement. In the cases of S = 10.0 H and 20.0 H, the trend of \(\psi\) is similar to the isolated dune case, especially when S = 20.0 H.  
Many wind tunnel experiments for shear stress measurement on the sand dune and around buildings have been done for engineering applications [32]. However, results on the influence of a building on sand terrain are still lacking. To the authors' knowledge, the study of wind erosion patterns downstream a bridge has not been proposed yet. The results in the present study not only provides predictions on sand dune moving after a bridge which is built upstream, but also confirms the viewpoints that the size of the sand drifts is very sensitive to the frontage of the upwind collecting area, which is accordant with the comment proposed [9].