## 4 Discussion
### Formula of Recirculation Zone Length Behind a Single Groin  
Based on the energy equation, [PERSON] et al. [61] considered the contraction ratio of the cross-section through a single groin and used the momentum conservation equation of separating flow. They proposed a hypothesis for the surface gradient of the mainstream and the turbulence shear stress of the boundary between mainstream and backflow. The formula for evaluating the recirculation zone length behind a single groin arranged perpendicular to the river bank was derived through integration of the governing equation, such that,  
\[\ln\frac{B}{B-b_{x}}=-[A_{1}(1-\frac{x}{L})^{2}+A_{2}(1-\frac{x}{L})-\frac{L}{ C_{0}^{2}H}(1-\frac{x}{L})-\frac{2A_{3}}{3}(1-\frac{3x}{2L}+\frac{x^{3}}{2L^{3}}) \frac{L}{b_{a}} \tag{3}\]  
where \(C_{0}\) is the dimensionless Chezy formula and \(C_{0}=\frac{C}{\sqrt{8}}\); \(H\) is the water depth, which does not change much generally, and therefore the main water depth or the tailgate water depth is not affected by the groin and can be taken as constant; \(A_{1}\), \(A_{2}\) and \(A_{3}\) are variable coefficients. The detailed definition of each parameter in Equation (3) is shown in Figure 18.  
Although the backflow pattern of Groin A is affected by Groin B on the opposite bank, the recirculation zone remained intact as a whole due to the thickness of short groins being  
Figure 18: Schematic diagram of the recirculation zone behind a single groin.  
less than a third of the flume width. The scale of the downstream recirculation zone was influenced by both the constriction ratio of the cross-section through groins and spanwise velocity in front of the groin tip. According to the statistical results listed in Table 1, the length of recirculation zone for a single Groin A (Case 14), which is 0.25 m, is close to that of symmetrical groins (Case 7) and Groin B deployed 0.55 m downstream (Case 13). In addition, the averaged spanwise velocity of observed points distributed nearest to the groin tip are also similar to each other in the above three cases. Hence, based on the empirical formula regarding the recirculation length of a single groin [61], using the experimental data, a re-calibration was conducted for the relevant variables. The parametric relationship between two alternate groins and one single groin was established for the groin-induced recirculation zone.  
In the parameter pre-calibration, three stochastic points on the boundary line of the recirculation zone behind a single groin were chosen, extracting \(x\) and \(y\) coordinates, respectively. The coordinates of each point were substituted into Equation (3), which established three equations correspondingly. \(A_{1}\), \(A_{2}\) and \(A_{3}\) were determined by solving the equations. Meanwhile, one more point was extracted from the boundary line of the recirculation zone, and then the x coordinate of the point and solved parameters \(A_{1}\), \(A_{2}\) and \(A_{3}\) were substituted into Equation (4). Accordingly, the basic formula of downstream recirculation length (\(L^{\prime}_{a}\)) for a single groin was obtained as follows:  
\[L^{\prime}_{a}=\frac{C_{0}^{2}Hb_{a}}{0.223C_{0}^{2}H+b_{a}}(1+\ln\frac{B}{B-b_ {a}}) \tag{4}\]