# Numerical Study of the Flow and Blockage Ratio of Cylindrical Pier Local Scour
## 1 Introduction
The scour equilibrium depth at the upstream face of the cylinder is frequently reported in studies, while the dune downstream and the kinematics of scour hole evolution are rarely described ([PERSON] et al., 2022 [17]). Furthermore, while a few studies have been performed in computational domains that could be considered narrow channels [26; 28], the effects of the channel width on sediment transport tend to be dismissed. This is because the traditional approach seeks to emulate an open channel by using virtual lateral walls boundaries. In a fluid-structure problem tested between two lateral walls, the term \"narrow channel\" is used when the lateral walls affect the flow pattern in the vicinity of the structure and at its wake. A number of conditions will determine whether the wall effect is negligible or not. [PERSON] et al., 1995 [32], observed a delay in the transition to vortex shedding, compared with the \"free-stream\" case, for blockage ratios (i.e., the ratio between the diameter \(D\) of the obstacle and the width \(W\) of the channel) \(D/W\geq 0.1\). Furthermore, [PERSON] and [PERSON], 2010 [33] describe three factors in which the presence of the side walls will alter the flow structure: (i) the incoming flow will be sheared, (ii) the flow through the gap directly above the obstacle will be higher than the \"free-stream\" case to maintain the mass continuity, and (iii) the overall vortex structure will be altered due to the interaction between the wall vortices and the vortices from the obstacle. Regarding the effect of the blockage ratio over the scour dynamics, [PERSON] et al., 2015 [34], state that the blockage ratio will not influence the scouring process when the obstacle is separated from the side walls by a fully developed flow with a supercritical Froude number. [PERSON], 2018 [18], noted that the critical Shields number for the presence of horseshoe vortex-induced scour will also deviate from the \"free-stream\" case for blockage ratios \(D/W\geq 0.1\)In the present study, we propose a numerical investigation of the combined effects of the velocity and the blockage ratio over the scouring process around a vertical cylinder. The numerical model uses a second-order finite-volume method in \(\sigma\)-coordinates, with a Large Eddy Simulation (LES) method to model the turbulence near the sediment bed and the cylinder. This model is used alongside the Exner-Polya equation with the bedload function of [PERSON] and [PERSON], 1976 [35], similar to the one proposed by [PERSON] et al., 2021 [19]. The narrow channel is emulated using a large blockage ratio and non-slip lateral boundaries, and the wide channel is emulated by a small blockage ratio alongside slip lateral boundaries. Table 1 highlights the position of the present study by presenting an overview of different configurations used to simulate the scour process in the literature. To the best of our knowledge, the present study presents a novel study of the effect of the blockage ratio and the flow velocity over the scour evolution at the base of a circular pier. The present study is structured as follows. Section 2 covers the mathematical model, including the fluid and sediment models. Section 3 focuses on the numerical methods implemented for the solution of the mathematical model. Section 4 first reproduces the scour depth evolution of a live-bed experiment in a wide channel in [PERSON] et al., 2005 [4]. Then, the model is validated with a clear-water case in [PERSON], 2018 [18]. Finally, simulations with the same flow conditions of the clear-water case in two channels with different blockage ratios are presented to assess the effect of the flow confinement in the scour evolution. Section 5 gives a discussion of the numerical results and the conclusions of the work, as well as some perspectives on the combined effects of the velocity, blockage, and relative coarseness, or the shape of the obstacle, over the scour depth.