## 3 Results
### Vorticity Distribution and Its Influence on Recirculating Flow  
Based on the measured data, the vertical vorticity (i.e., z-component of vorticity, \(\Omega_{z}\)) component is calculated using the following equation,  
\[\Omega_{z}=\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y} \tag{1}\]  
where \(v_{x}\) is the velocity in the streamwise, \(x\)-direction; \(v_{y}\) is the velocity in the spanwise, \(y\)-direction; and \(\Omega_{z}\) is vertical rotational strength of the recirculating flow. The vorticity distribution plays a leading role in the research of recirculating flow, particularly the flow behind some rigid structures such as groins [36]. Figure 13 shows the vertical vorticity (magnitude indicated by colors) distributed in the horizontal measured plane for one single groin and two symmetrical groins. The maximum vertical vorticity occurred near the groin tip, as illustrated in Figure 13a. The observed vertical vorticity pattern of a single groin is consistent with the previous numerical results from [PERSON] [15]. The vertical vorticity \(\Omega_{z}\) generated by a single groin is manifested as the negative vertical vorticity \(\Omega_{z}\) downstream, which originates from the groin tip and is concentrated within the mixing layer zone. Interestingly, as Groin B was symmetrically placed at the opposite bank to Groin A, the maximum positive vertical vorticity was induced by Groin B, where flow in the vicinity of the Groin A tip was dominated by high negative vertical vorticity along the recirculation zone, see Figure 13b. Since the propagation of negative vertical vorticity induced by Groin A was blocked at the transverse direction, two high-vorticity zones with similar size but opposite orientation symmetrically showed along the mixing layer zone behind Groins A and B. The velocity gradient in the mixing layer greatly increased due to the contraction of main flow by two symmetrical groins; as a result, the high-vorticity zones at both banks were extended further longitudinally compared to that of a single groin. Figure 14 shows the vertical vorticity distribution of two alternating groins with various spacing between groins, downstream from Groin A. The vorticity fields also display distributions similar to the recirculation zone behind Groin A shown in Figure 5 for different test cases. High vorticity appeared at the mixing layer at the interface between the mainstream and recirculation zones. When Groin B was upstream of Groin A, the extent of high-vorticity was larger compared to that of Groin B downstream of Groin A. In addition, when Groin B was downstream of Groin A, the advection and diffusion of vorticity induced  
Figure 12: Relationship between the length and width of the recirculation zone behind Groin A.  
by Groin A was hindered by the three-dimensional flow patterns over Groin B. Therefore, a smaller high-vorticity zone was formed behind Groin A and it skewed more to the flume wall, which impeded the formation of large-scale recirculation downstream of Groin A. Furthermore, the vertical vorticity flux (\(E_{z}\)) is introduced to describe the spatially integrated scalar value of \(\Omega_{z}\) and the bulk effect of vertical vorticity on the downstream recirculation zone, defined as:  
\[E_{z}=\iint\Omega_{z}dxdy=\iint(\frac{\partial v_{y}}{\partial x}-\frac{ \partial v_{x}}{\partial y})dxdy \tag{2}\]  
where the integration area is the whole horizontal plane measured by the PIV system. The vertical vorticity flux through the whole horizontal plane measured by the PIV is calculated from Equation (2) and plotted in Figure 15, which also shows the recirculation zone length for different cases. The vertical vorticity flux \(E_{z}\) vs. recirculation zone length displays a consistent variation trend for different test cases. The vertical vorticity flux \(E_{z}\) increases with increasing recirculation zone length, resulting in more dissipation of flow energy.