## 4 Conclusions  
A three dimensional \(CFD\) model was used to simulate the flow pattern around circular bridge piers during the development of the scour hole with a uniform mesh size of \(0.025D\). Seven geometries of the scour hole were captured experimentally during the development of the scouring process [46]. The seven geometries were utilized to prepare the geometric file for each \(CFD\) simulation to represent scour development. The simulated scour stages were 0.0%, 30%, 47%, 69%,78%, 92%, and 100% of the equilibrium scour hole. The numerical results were verified againsts the experimental findings of the authors of [12; 14; 30; 49]. Three turbulence models were evaluated in the current study. The \(k-\omega\) turbulence model captured the flow oscillations at the downstream side of the pier with economical computational resources. The \(k-\omega\) turbulence model was selected to simulate the turbulence in the \(CFD\) numerical simulations for the current study due to its proven accuracy in  
Figure 20: The shear stress distribution during scour stages. capturing flow oscillations at the downstream side of the pier with relatively economical computational resources. A detailed description of the flow field was introduced, and it was concluded that:  
* The reversal longitudinal flow reached a maximum value of \(-0.55U_{o}\) at \(d_{s}/d_{se}~{}=~{}0.92\) and reduced to \(-0.25U_{o}\) at the equilibrium scour stage;
* The maximum positive values of the lateral velocity remained almost constant in the order of about \(0.80U_{o}\) during the souring process and reached \(0.0\) at the bottom level;
* The downward vertical velocity increased with the progress of the scour hole until reaching its maximum value of \(-0.67U_{o}\) at \(d_{s}/d_{se}~{}=~{}0.78\);
* As the scour hole developed, the HV vortex sunk into the scour hole and was kept contained within its boundaries. The area-averaged dimensionless circulation \(\Psi_{i}\) was developed to allow for the assessment of the development of the vortex strength through the scouring process by eliminating the calculation area effect. It was concluded that the circulation in the \(Y\) direction was the main driving factor in the development of the scouring process more than the circulation in the \(X\) direction. The relationship between the relative mean vortex size \(D_{v}/D\) and the relative scour depth \(d_{s}/D\) was developed as \(D_{V}/D~{}=~{}0.67(d_{s}/D)^{2}+0.24(d_{s}/D)+0.1\). As a simplification for the previous equation, the ratio between the HV vortex mean size and the scoring depth \(d_{v}/d_{s}\) and \(d_{s}/d_{se}\) remained constant with a value of \(0.8\) during the development. The \(0.8\) ratio was valid for \(d_{s}/d_{se}\geq 0.3\) until the equilibrium scouring stage was reached. The maximum values of bed shear stress were located at the scour hole's downstream edges within the range \((\pm 13^{\circ})\) to \((\pm 40^{\circ})\) and at the location where the scour started \((\pm 100^{\circ})\). These locations remained steady for \(d_{s}/d_{se}>0.3\) until the equilibrium conditions were reached. The current \(CFD\) numerical simulation overestimated the values of bed shear stress. This overestimation is reported in many \(CFD\) simulations. Conceptualization, methodology, software, validation, A.M.H. ; investigation, resources, writing, original draft preparation, A.H.S. ; writing--review and editing, visualization, A.M.H. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Data Availability Statement: Data are contained within the article. The authors declare no conflict of interest. The following symbols are used in this paper: