## 3 Results and Discussion
### Horseshoe Vortex  
In the current study, dense longitudinal and transversal sections with 0.025\(D\) spacing were utilized to investigate the structure and the evolution of the \(HV\) characteristics during the scouring process. The development of the scouring process presented in the _CFD_ model geometry by using seven solidified bed bathymetries was captured experimentally during the progress of the scour hole until the equilibrium scour bed was reached. Vorticity, or circulation per unit area, reflects the tendency for fluid elements to spin. It is essential to know the magnitude of circulation as it implies the strength of the vortex. The out of plane vorticity on the longitudinal sections and the transversal sections are \((\omega_{Y})\), and \((\omega_{X})\), respectively, and are determined as follows:  
\[\omega_{Y}\ =\ \frac{\partial u}{\partial_{Z}}-\frac{\partial w}{\partial_{X}} \tag{8}\]  
\[\omega_{X}\ =\ \frac{\partial w}{\partial_{Y}}-\frac{\partial w}{\partial_{Z}} \tag{9}\]  
Figure 14 shows the contours of Y-vorticity \((\omega_{Y})\) and shows that the flow streamlined upstream from the pier at the plane of symmetry at flat-bed conditions. The interaction between the downward flow and the horizontal flow close to the original flat-bed surface initiates the formation of a primary rotating clockwise \(HV\) vortex. The shape of the vortex in the flat-bed case was almost elliptical in cross-section with a major horizontal axis. The size of the \(HV\) vortex is presented by the mean vortex size \(D_{V}\ =\ a+b\) as proposed by the authors of [49]. In flat-bed conditions, the height (length of the minor axis) of the elliptical vortex at the plane of symmetry is approximately 40% of the length of the major axis, and the size of the vortex is about 0.1\(D\). Figure 15 gives the contour lines of \(y\)-vorticity \((\omega_{Y})\), and x-vorticity \((\omega_{X})\) during the progress of the scour hole starting from the flat-bed until the equilibrium conditions were reached. It is clear from Figure 15 that as the scour hole developed, the \(HV\) vortex sunk into the scour hole and was kept contained within its boundaries. Figure 13: Vertical velocity distribution at X/D = 0.05 during the scouring process. Circulation or vortex strength \((\Gamma)\) at any surface was computed using the out of plane vorticity contours, \((\omega_{i})\) and applying the Stokes theorem as follows:  
\[\Gamma\ =\ \oint_{c}\overline{\
abla}\,d_{dis}\ =\ \iint_{A}\omega_{i}\ dA \tag{10}\]  
where \(\overline{\
abla}\) is the velocity vector, \(d_{dis}\) is the differential displacement vector along with a closed curve \(C\), and \(A\) is the enclosed area. The area-averaged dimensionless circulation \((\Psi_{i})\) was developed to eliminate the area effect in the assessment of the development of the vortex strength. The selected enclosed area for the vortex strength calculation upstream of the pier was chosen to ensure the containment of the \(HV\) vortex during all stages of the scouring process in longitudinal and transversal directions. \[\Psi_{i}\ =\ \Gamma_{i}\,D/(U_{o}\,A) \tag{11}\]  
It is clear from Figures 16 and 17 that the circulation in the \(y\) direction was the main driving factor in the development of the scouring process, since it reached about 4 to 5 times the circulation in the \(x\) direction. The area-averaged dimensionless circulation in both the \(x\) and \(y\) directions decreased with the development of the scour hole, starting from its maximum values at the flat-bed condition. \(\Psi_{Y-\text{flatted}}\) reached its maximum value at the pier centerline where \(y/D\ =\ 0\) and decreased laterally until reaching a constant value of \(0.7\times\Psi_{y-flatted-maximum}\) at \(y/D\ =\ 1.2\).