## 3 Results and Discussion  
### Velocity Field Description  
The longitudinal velocity \(U\) decreased at the vertical plane of symmetry as the flow approached the pier before reaching stagnation at the pier's upstream face. The decrease became noticeable approximately at \(x\)/\(D\) = 2.0, as shown in Figure 9. A zone of reversal flow was noticed at \(x\)/\(D\) = 0.2 upstream from the pier at flat-bed conditions, then it moved further upstream from the pier towards the scour hole edge, and downward towards the scour hole bed in conjunction with the progress of the scour hole. These observations indicate that as the scour hole developed, the \(HV\) sunk inside the hole and moved further below the original bed level, which is in agreement with the experimental observations by the author of [14]. 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 transversal velocity \(v\) appeared at \(x\) = 2\(D\) indicating the transformation from a two-dimensional to a three-dimensional flow. The maximum positive lateral velocity was located at a radial distance = 0.05\(D\) measured from the pier surface and formed 45\({}^{\circ}\) on the two sides of the plane of symmetry upstream from the pier, as shown in the lateral velocity distribution at \(Z\)/\(H_{o}\) = 0.5 in Figure 10. The maximum positive values of the lateral velocity remained almost constant in the order of about 0.8\(U_{o}\) during the souring process and reached (0.0) at the bottom level, as shown in the vertical profiles given in Figure 11.  
Figure 12 shows the vertical velocity \(w\) distribution in the plane of symmetry upstream of the pier, through the development of the scouring process. As the flow approached the pier, the vertical velocity components remained approximately negligible until \(X\)/\(D\) = 1.0 then started to appear and subsequently grow considerably in the downstream direction towards the pier. The maximum downward vertical velocity occurred at a distance range of 0.02\(D\):0.05\(D\) which is in agreement with the laboratory observations in [12]. Figure 13 shows the variation of the vertical velocity profiles in the plane of symmetry during the scouring process at a distance of 0.05\(D\) upstream of the pier. 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. This value lies between the \(-0.6U_{o}\) stated by the authors of [16; 17], and the \(-0.8U_{o}\) stated by the author of [14]. The downward velocity decreased with the development of the scour hole from \(d_{S}\)/\(d_{Se}\) = 0.78 to \(d_{S}\)/\(d_{Se}\) = 1.0 until reaching a value of \(-0.57U_{o}\) at \(d_{S}\)/\(d_{Se}\) = 1.00. The increase in the downward velocity until \(d_{S}\)/\(d_{Se}\) = 0.78 then the decrease till \(d_{S}\)/\(d_{Se}\) = 1.00 explains the experimental observation in [46] about the formation of the vertical trench just upstream of the pier at intermediate scour stages and its disappearance in the equilibrium stage. The maximum magnitude of the upward velocity was 0.17\(U_{o}\) at \(Z\)/\(H_{o}\) = 0.95 during all stages. The upward flow near the free surface formed a circulation that is called bow wave, causing the flow depth to increase in front of the pier.  
Figure 9: Simulated longitudinal velocity \(u\).  
Figure 11: Lateral velocity profiles at the location of maximum positive lateral velocity.  
Figure 10: Simulated lateral velocity distribution at \(z/H_{o}=0.5\).  
Figure 12: Vertical velocity distribution at the plane of symmetry.