## 4 Results and Discussion
### Dimensional Analysis of Effective Mixing Length  
Dimensional analysis can be used to analyze the correlation between the influencing factors and the physical variables, to solve the criterion relationship when problems cannot be described by a differential equation [24]. For clarifying the structural relationship and exploring the significant level of each variable on \(L_{EML}\) in the structural formula, dimensional analysis was used to fit the computational expressions of \(L_{EML}\). Based on a thorough understanding of the research, the main physical quantities affecting the physical process must be correctly determined, to ensure the correct application of dimensional analysis. In this section, a total of eight physical quantities were considered, including effective mixing length (\(L_{EML}\)), main pipe diameter (\(\varphi\)) and cross-flow flux (\(Q\)), water density (\(\rho\)), mixing ratio (\(\delta\)), gravitational acceleration (\(g\)), incident angle (\(\theta\)), and average pressure of the mixture (\(p\)). \[F(\varphi,\rho,g,L_{EML},Q,p,\delta,\theta)=0 \tag{8}\]  
where \(\varphi,g\), and \(\rho\) are the basic physical quantities, \(\delta\), \(\theta\) are the dimensionless quantities. As shown below, \(\pi_{1}\), \(\pi_{2}\), \(\pi_{3}\), \(\pi_{4}\), \(\pi_{5}\), \(\pi_{6}\), \(\pi_{7}\), \(\pi_{8}\) were used to represent eight proportional coefficients. Among them, \(\pi_{1}=\pi_{2}=\pi_{3}\)\(=1\), \(\pi_{7}=\delta\), \(\pi_{8}=\theta\), \(\pi_{4}\), \(\pi_{5}\), \(\pi_{6}\) were formulated. \[\left\{ \begin{array}{l}\pi_{4}=\frac{L_{EML}}{\varphi}\\ \pi_{5}=\frac{Q}{\varphi^{\frac{1}{2}}g^{\frac{1}{2}}}\\ \pi_{6}=\frac{p}{\varphi p\delta}\end{array} \right. \tag{9}\]  
Further, the following relationships were obtained. \[\pi_{4}=\frac{L_{EML}}{\varphi}=f(1,1,1,\pi_{5},\pi_{6},\delta,\theta) \tag{10}\]  
Figure 11: Contours of the distribution of salt solution (**a**) and water velocity (**b**) in the selected region. Substituting Equation (9) into Equation (10), a general equation can be obtained. \[\frac{L_{\text{EML}}}{\varphi}=a\left(\frac{Q}{\varphi^{\frac{5}{2}}g^{\frac{1}{2 }}}\right)^{\!\!b}\left(\frac{p}{\varphi\rho g}\right)^{\!\!c}\delta^{d}\theta^{e} \tag{11}\]  
where \(a\), \(b\), \(c\), \(d\), and \(e\) are coefficients. The coefficients and criterion equation were obtained by means of multivariate nonlinear regression analysis of numerical simulation data. \[\left\{ \begin{array}{l}\frac{L_{EML}}{\varphi}=45.48347*\left(\frac{Q}{ \varphi^{\frac{5}{2}}g^{\frac{1}{2}}}\right)^{0.95026}\left(\frac{p}{\varphi \rho g}\right)^{-0.50847}\delta^{0.00491}\theta^{-0.00297}\\ R^{2}=0.995\end{array} \right. \tag{12}\]  
where \(L_{EML}\) is effective mixing length, \(\varphi\) represents main pipe diameter, \(Q\) is cross-flow flux, \(g\) is gravitational acceleration, \(p\) denotes average pressure of the mixture, \(\rho\) is water density, \(\delta\) signifies mixing ratio, and \(\theta\) is incident angle. Equation (12) can be transformed into the following form. \[L_{EML}=45.48347*Q^{0.95026}\varphi^{-0.86718}g^{0.03334}p^{-0.50847}\rho^{0.50 847}\delta^{0.00491}\theta^{-0.00297} \tag{13}\]  
Through Equation (13) and correlation analysis, it can be seen that \(L_{EML}\) is less affected by \(\delta\) and \(\theta\) which can be ignored, and affected by \(Q\), \(\varphi\), \(p\), \(\rho\), \(g\) dramatically. In order to simplify the calculation process, \(\delta^{0.00491}\) and \(\theta^{-0.00297}\) were assigned a value of 1, and the coefficients were simplified simultaneously. The simplified equation is shown below. \[L_{EML}=45.483*Q^{0.95}\varphi^{-0.867}g^{0.033}p^{-0.508}\rho^{0.508} \tag{14}\]  
Comparison between the simulation results and the calculation results of Equation (14) showed that the maximum relative error was 9.56%, the minimum relative error was 0.09%, and the average error was 4.01%.