# Numerical Study of the Flow and Blockage Ratio of Cylindrical Pier Local Scour
## 4 Numerical Results
### Effects of the Channel Width on the Clear-Water Local Scour Pattern  
In a free-stream, the streamlines lateral to an obstacle will contract, resulting in flow acceleration. If the flow acceleration results in a supercritical Shields number, the sediment particles will be set in motion, resulting in scouring. In a narrow channel, the flow is confined between two lateral walls, and this confinement will accentuate the contraction of the streamlines, further accelerating the flow past the obstacle. Under this contraction effect, the dune will be pushed downstream. In clear-water conditions, the region of threshold overshoot has a larger expansion than in live-bed conditions. In this section, the narrow channel case has a blockage ratio of \(D/W=0.1\), and the wide channel has a blockage ratio \(D/W=0.0625\). Figure 15 shows the friction velocity amplification \(u_{\tau}/u_{crit}\), where \(u_{crit}\) is as defined in Equation (16). The figures show the amplification effect of the friction velocity at the sides of the cylinder. [PERSON] et al., 2005 [PERSON] et al. (2005), describes this amplification as being caused by the contraction of the flow streamlines and the horseshoe vortex. The isolines in Figure 15a  
Figure 14: (**a**) Snapshot of the scour in development at time \(t=2.4\) h (Lachausée, 2018 Lachausée, 2018). (**b**) Scour hole at non-dimensional time \(t/T=3500\). Figure 13: (**a**) Topographical map of the experimental results reported in Lachausée, 2018 Lachausée (2018) at time \(t=2.4\) h. (**b**) Topographical map of the state of the sediment bed for the numerical results at non-dimensional time \(t/T=3500\). show that the largest amplification is located at an angle within the interval \(43^{\circ}\leq\varphi\leq 75^{\circ}\). This region is approximately 45 percent wider than the equivalent region in the wide channel case, which is located at \(48^{\circ}\leq\varphi\leq 70^{\circ}\) (Figure 15b). In the narrow channel case, the confinement of the flow results in a larger contracted area relative to the wide channel case, which will result in a larger sediment transport rate at the start of the development. Figure 16 compares the resulting bed topography of the clear-water simulations. It can be seen that both scour holes and dunes present similar characteristics. The horseshoe scour hole is circular without reaching the wake-side of the cylinder. The maximum scour depth is present in both cases at the lee-side of the cylinder, lateral to the stagnation point, at \(|y/D|\approx 0.25D\). The scour hole in the narrow channel is deeper than the scour hole in the wide channel by approximately 15%. Figure 15: Friction velocity amplification distribution for \(\text{Re}_{D}=2000\) in a rigid bed. (**a**) Narrow channel case (\(D/W=0.1\)). (**b**) Wide channel case (\(D/W=6.25\times 10^{-2}\)). Figure 17 compares the time evolution of the normalized scour depth for the three scour simulations performed for the present study. Regarding clear-water, it can be observed that, in both cases, the scour depth evolution is largely monotonically increasing. This continuously growing scour behaviour, although not perfectly monotonic in lab studies, has been documented in experimental studies on clear-water scour, such as [PERSON], 2018 [18], [PERSON] et al., 2007 [48], and [PERSON] et al., 1999 [2]. Moreover, the scour rate decreases over time in both cases. From the the beginning of the scour simulation, both cases presented a similar scour evolution, but the scour rate decreases more rapidly for the wide channel case. This initial similar behavior can be explained by the similar magnitude reached by the friction velocity in the rigid bed simulation (Figure 16). The area where the friction velocity overshoots its critical value is only slightly larger in the narrow channel case, so the faster scour rate does not become apparent in the early time of the simulation.