## 2 Materials and Methods
The working section was located at 16.0 m downstream from the flume entrance with the same width as the flume and was extended for 4.0 m. The flume tailgate was used to maintain a 30.0 cm water depth (\(H_{0}\)) above the original bed level and a 0.247 m/s average cross-section velocity. The average cross-section velocity was selected to provide 95% of the critical bed shear stress. The experiments lasted for 21 h to achieve equilibrium conditions where the variation in the scour depth does not exceed \(D_{50}\) in one hour. Figure 2 shows the contour lines of the developed scour hole at different time steps with the maximum scour depth (\(d_{S}\)) at the concerned time step, until the equilibrium conditions with maximum scour depth (\(d_{Sc}\)) were reached which were selected as the geometric files for the current study. Auto-CAD software was utilized to generate a Stereolithographic (\(STL\)) file for each scour hole geometry in addition to the flat-bed conditions, by approximating the solid surfaces with triangles. Figure 3 illustrates the Cartesian coordinate system used in the description of the results obtained from the CFD simulations. The full-scale geometry model was constructed with the boundary conditions shown in Figure 4.  
The Courant number is one of the key factors which should be considered for a proper numerical simulation since it represents the ratio between the distance traveled by the flow and the grid cell length in the direction of flow through a single numerical simulation time step. In the current study, the Courant number stability criterion equaling to unity was achieved by applying an adjustable variable time step (\(\delta_{t}\)) since the grid size was uniform along with the calculation domain. \[\delta_{t}<min\left(\frac{V_{F}\delta_{x_{i}}}{A_{x}u},\frac{V_{F}\delta_{y_{j }}}{A_{y}v},\frac{V_{F}\delta_{z_{k}}}{A_{z}w}\right) \tag{6}\]  
where (\(\delta x_{i}\), \(\delta y_{j}\), \(\delta z_{k}\)) are the cell sizes in the rectangular Cartesian coordinates. Mesh sensitivity analysis is one of the mandatory CFD steps that enables an acceptable accuracy in the results to be reached with the efficient usage of computational resources. The variation of water depth, lateral velocity, and longitudinal velocity with different grid sizes starting from 2.00 cm to 0.25 cm were evaluated. These hydraulic parameters were calculated at point A, which was selected at 1.0 cm upstream of the pier and 2.0 cm away from the symmetry plane as shown in Figure 5. The calculated values of the hydraulic parameters remained almost constant for the mesh sizes 0.50 and 0.25 cm. Accordingly, a 0.5 cm grid size was selected to generate the mesh for the current study. The selected grid size satisfied the wall function limitations (30 \(<Z^{+}<\) 100) to avoid calculations in the viscous sublayer and to ensure that the law of the wall was applicable. Table 1 shows the variation in the CFD model cell types and numbers while using the selected 0.5 cm grid size in the three Cartesian directions for the seven scour hole stages starting from \(ds/dse\ =\ 0.00\%\) to \(ds/dse\ =\ 100.0\%\). \[Z^{+}\ =\ \frac{u_{*}z}{v} \tag{7}\]  
where (\(Z^{+}\)) is a dimensionless parameter for the wall function, (\(U\)-) is the shear velocity, (\(U\)-) is the distance from the cell centroid to the wall, and (\(v\)) is the kinematic viscosity. \[\frac{\partial u}{\partial t}+\frac{1}{V_{F}}\left(\overset{ \rightarrow}{q}\
abla u\right)\ =\ -\frac{1}{\rho}\frac{\partial P}{\partial x}+G_{x}+f_{x} \tag{2}\]0 Scour hole development, adapted from [46]. \[\frac{\partial u}{\partial t}+\frac{1}{V_{F}}\left(\overset{ \rightarrow}{q}\
abla u\right)\ =\ -\frac{1}{\rho}\frac{\partial P}{\partial x}+G_{x}+f_{x} \tag{2}\]1 Variation of the scour hole bed level at the plane of symmetry during the scour stages. \[\frac{\partial u}{\partial t}+\frac{1}{V_{F}}\left(\overset{ \rightarrow}{q}\
abla u\right)\ =\ -\frac{1}{\rho}\frac{\partial P}{\partial x}+G_{x}+f_{x} \tag{2}\]2 Boundary conditions and generated mesh.