## 4 Discussion
### Formula of Recirculation Zone Length Behind Alternating Groins  
As previously analyzed in Section 3.1, there are three subzones for the variation of backflow length behind Groin A with the various spacing between groins, i.e., upstream far-field (Cases 1-5), near-field (Cases 6-8) and downstream far-field (Cases 9-13), as a result, the calculation of the backflow length behind Groin A can also be divided into three subzones. When Groin B is on the opposite bank and upstream/downstream far-field of Groin A, the influence of Groin B on the backflow length behind Groin A can be represented by the relative contraction ratio of cross-section (\(\delta_{b}\)), and given as:  
\[\delta_{b}=(\frac{B}{|x|+B}b_{b})/B \tag{5}\]  
The total contraction ratio of the cross-section in front of Groin A due to Groins A and B (\(\delta_{ab}\)) is given as:  
\[\delta_{ab}=(b_{a}+\frac{B}{|x|+B}b_{b})/B \tag{6}\]  
Thus, the passing ratio of main flow through the cross-section is given as:  
\[\delta=1-\delta_{ab}=1-(b_{a}+\frac{B}{|x|+B}b_{b})/B \tag{7}\]  
When Groin B is in the near-field of Groin A, the relative contraction ratio of Groin B to Groin A (\(\delta_{b}\)) can be expressed as:  
\[\delta_{b}=(\frac{B-x}{B}b_{b})/B \tag{8}\]  
The total contraction ratio of the cross-section in front of Groin A from Groins A and B (\(\delta_{ab}\)) becomes:  
\[\delta_{ab}=(b_{a}+\frac{B-x}{B}b_{b})/B \tag{9}\]The current crossing capacity of a river section (\(\delta\)) becomes:  
\[\delta=1-\delta_{ab}=1-(b_{a}+\frac{B-x}{B}b_{b})/B \tag{10}\]  
In Equations (5)-(10), \(B\) is river width, \(b_{a}\) and \(b_{b}\) are the lengths of Groins A and B, respectively, x is the longitudinal spacing between Groins B and A (the \(x\) value is negative and positive responding to the location of groin B upstream and downstream from Groin A, respectively). According to the variation process in Figure 6, a linear relationship between recirculation zone length and spacing of groins can be observed (see Figure 7), indicating that linearity exists with the contraction ratio of a cross-section through groins. The contraction ratio comprehensively reflects the influence of the opposite Groin B on the flow structure around the Groin A, especially the spanwise velocity in the vicinity and rear of Groin A. Hence, the relationship between the backflow length of Groin A under the impact of Groin B (\(L_{a}\)) and the backflow length of a single groin (\(L^{\prime}_{a}\)) can be given as follows:  
\[\frac{L_{a}}{L^{\prime}_{a}}=f(\frac{\delta}{\delta^{\prime}_{a}}) \tag{11}\]  
where  
\[\delta^{\prime}_{a}=\frac{(B-b_{a})}{B} \tag{12}\]  
When Groin B is in the upstream far-field of Groin A, the relationship diagram of \(L_{a}/L^{\prime}_{a}\sim\delta/\delta^{\prime}_{a}\) is drawn by experimental data; the result is shown in Figure 19.  
According to the linear fitting formula in Figure 19, Equation (11) can be written as:  
\[\frac{L_{a}}{L^{\prime}_{a}}=-36.609(\frac{\delta}{\delta^{\prime}_{a}})+37.456 \tag{13}\]  
Furthermore, Equation (4) is substituted into Equation (13), and the length of recirculation zone behind Groin A can be obtained as:  
\[\delta_{ab}=(b_{a}+\frac{B}{|x|+B}b_{b})/B \tag{6}\]0  
When Groin B is in the near-field of Groin A, the relationship diagram of \(L_{a}/L^{\prime}_{a}\sim\delta/\delta^{\prime}_{a}\) is plotted in \[\delta_{ab}=(b_{a}+\frac{B}{|x|+B}b_{b})/B \tag{6}\]1  
\[\delta_{ab}=(b_{a}+\frac{B}{|x|+B}b_{b})/B \tag{6}\]2 The relationship between \(L_{a}/L^{\prime}_{a}\) and \(\delta/\delta^{\prime}_{a}\) when groin B is in the upstream far-field of groin A.