*4.2. Comparison of Vortex Strength*

To further explore the strength evolution of each individual vortex within the evolving vortex system, their normalized circulation is plotted against time in Figure 6a, in which the magnitude of circulation is computed from the vorticity distributions (see left column of Figure 5), as follows,

$$
\Gamma = \iint\limits\_A adA \tag{1}
$$

where *A* is the enclosed area of the vortex. In this study, the vortex center and boundary were determined by using the vortex identification algorithms developed by Graftieaux et al. [20]. The so-obtained coordinates of the vortex centers confirm a reasonable agreemen<sup>t</sup> with those inferred from the visualized vortices shown in the streamline plots (see VC1, VC2, and VC3 in the right column of Figure 5). It is also noted that both V2 and V3 exhibit a clockwise rotation (negative circulation). To compare their relative strength with V1, Figure 6a shows the absolute value of their magnitudes. For easy reference, the temporal development of the maximum scour depth (*d*s,t) is also plotted against time in Figure 6b.

**Figure 6.** Temporal development of (**a**) vortex circulation of V1, V2, and V3; (**b**) maximum scour depth at *X*w = <sup>2</sup>*D*p.

Figure 6a clearly shows that during the initial scouring phase (*t* = 0–0.5 h), V1 undergoes a rapid decrease; meanwhile, V2 and V3 reveal a synchronous increase with a comparable increasing rate. During the developing phase (*t* = 0.5–12 h), V1 and V2 exhibit a similar decreasing trend, whereas V3 still retains a relatively high increasing rate, which is consistent with that of the scour depth development. During the stabilizing phase (*t* = 12–24 h), all three vortices approach an asymptotic state in both size and location. When studying the horseshoe vortex evolution at pier scour, Baker [21] suggested that there exists a constancy of the vortex strength during the scouring process. On the other hand, Muzzammil and Gangadhariah [22] reported that during the scouring process the strength of the horseshoe vortex at a cylinder pier experiences an initial increase, followed with a decreasing trend as the scour hole continues to enlarge. However, these observations are different from that of a propeller-induced scour around a vertical wall. In the present case, Figure 5 reveals that the vortex circulation undergoes a monotonical increase as the scour hole evolves until the asymptotic state is reached around *t* = 24 h. A possible reason may be the fact that pier scour is often caused by multiple horseshoe vortices that formed around the scour hole [23], while in the present study, it is the single primary vortex (V3) that directly shapes and completely fills the scour hole (see Figure 5). As a result, the vortex size exhibits a concurrent growth with the developing scour hole.
