# Numerical Study of the Flow and Blockage Ratio of Cylindrical Pier Local Scour
## 4 Numerical Results
### Clear-Water Case
#### 4.2.2 Morphodynamic Validation  
This section is devoted to the analysis of the scour around the cylinder. The morphological model is tested for \(\text{Re}_{D}\approx 2000\), in the clear-water regime. Any friction velocity greater than \(1.27\times 10^{-2}\) m/s will result in a Shields number satisfying the condition \(\theta>\theta_{c0}\), and, thus, it will produce sediment transport. Figure 8 illustrates the early evolution of the obtained scour hole at non-dimensional morphological times \(t/T*\) equal to 100, 200, 400, and 800. At these early stages, the development of the scour hole agrees qualitatively with  
Figure 6: Three-dimensional instantaneous flow visualization at \(t/T=100\): (**a**) Detailed view of the vortex structures at the base of the cylinder. (**b**) Front and (**c**) side view depicting stream lines. Figure 7: Friction velocity amplification along the \(x\) direction with \(y=0\) for \(\text{Re}_{D}\approx 2000\). the one presented for live-bed conditions in [PERSON] et al., 2021 [19]. \(T*\) is the morphological time scale defined by [PERSON], 1992 [46] as follows:  
\[T*=\frac{D^{2}}{\sqrt{g(s-1)d^{3}}}, \tag{17}\]  
where \(d\) is the grain size and \(s=\rho_{s}/\rho\) is the relative density of the sediment. In this case, \(T*=5.87\). Figure 9 presents the horizontal profiles of the same results shown in Figure 8, offering a more quantitative insight into the scour hole and the dune. The scour hole upstream of the cylinder presents a consistent angle of approximately \(32^{\circ}\), which is consistent with the prescribed value for the maximum angle of repose in [PERSON] et al., 2018 [16]. By contrast, the dune downstream of the cylinder does not have a consistent shape throughout its formation. While the lee-side angle of the dune is unchanged once it reaches the same slope as the scour hole, the stoss side of the dune presents more inconsistency. This last behavior might result from the wake vortices interacting directly on the stoss side of the dune. Figure 8: Bed elevation near the cylinder at different moments in time, demonstrating the evolution of the scour hole. Additionally, the presence of a horseshoe vortex in the studied range of \(\mathrm{Re}_{D}\) is in agreement with [PERSON], 1979 [47]. The presence of scour induced by the horseshoe vortex (herein referred to as \"horseshoe scour\") in clear-water regime agrees with [PERSON] et al., 2018 [16]. Three measures are compared with their data: the maximum depth \(\xi_{m}\) and the characteristic lengths \(L_{x}\) and \(L_{y}\), illustrated in Figure 10. \(L_{x}\) is defined as the distance between the pier and the point of the bed where its depth is \(\xi_{m}/2\). \(L_{y}\) is also defined as the distance between the pier and the point of the bed where its depth is \(\xi_{m}/2\). Figure 10: Characteristic lengths of the scour hole. Figure 9: Horizontal profile of the bed elevation at \(y=0\), corresponding to the same states shown in Figure 8.  
The time evolution of \(\xi_{m}\), \(L_{x}\), and \(L_{y}\) can be fit to the exponential law  
\[f(t)=A\left(1-B\exp\left(-\frac{t}{T_{a}}\right)-(1-B)\exp\left(-\frac{t}{10T_{a} }\right)\right), \tag{18}\]  
where \(A\), \(B\), and \(T_{a}\) are empirical parameters. This is illustrated in Figure 11a, where the square, triangle, and circle markers are the computed maximum depth, and the upstream and lateral characteristic lengths, respectively. The solid and dashed lines are the adjusted Equation (18) to fit the evolution of \(\xi_{m}\), \(L_{x}\), and \(L_{y}\). Figure 11 presents a comparison between the numerical and experimental results with the model Equation (18). Figure 11a illustrates the fitting of the Equation (18) to the numerical data. The values for the empirical parameters are \(A=-10.3\), \(B=0.6\), and \(T_{a}=1000\) for the time evolution of \(\xi_{m}\); \(A=14.5\), \(B=0.42\), and \(T_{a}=1000\) for the time evolution of \(L_{x}\), and \(A=15\), \(B=0.52\), and \(T_{a}=1000\) for the time evolution of \(L_{y}\).