## 3 Experimental Results and Analysis
### Hydrodynamic Pressure
#### 3.3.5 Time Domain Characteristics  
The coefficient of association is an important parameter used to analyze the turbulent structure and dimensions of vortexes. The coefficient of association includes two parameters: The coefficient of autocorrelation and the coefficient of cross-correlation. The point of maximum pressure is the critical location at which the coefficient of association, \(R\), should be determined, calculated by  
\[R=\frac{1}{T-\tau}\int_{0}^{T-\tau}[p_{x}(t)-\overline{p_{x}}]\cdot\big{[}p_{y }(t+\tau)-\overline{p_{y}}\big{]}dt \tag{6}\]  
where \(T\) is the entire test duration; \(t\) is the test time and \(\tau\) is an interval of time; \(p_{x}\) is the hydrodynamic pressure at a given test point and \(p_{y}\) is the hydrodynamic pressure at another test point; \(\overline{p_{x}}\) and \(\overline{p_{y}}\) are the average hydrodynamic pressures at test points \(x\) and \(y\), respectively; when \(p_{x}\) and \(p_{y}\) represent a single test point, \(R\) represents the coefficient of auto correlation for \(R_{xx}\), and when \(p_{x}\) and \(p_{y}\) represent two different test points, \(R\) represents the coefficient of cross-correlation for \(R_{xy}\). The coefficient of autocorrelation of the maximum hydrodynamic pressure, and the coefficient of cross-correlation between the point with the maximum pressure and the other points located at 0.1 m intervals downstream, should be calculated. By comparing the coefficients of autocorrelation of the fresh water and water with a sediment concentration of 250 kg/m\({}^{3}\), the results in Figure 13 demonstrate  
Figure 11: Relationships between power spectrum and frequency with an upstream water head of 1.90 m for different sediment concentrations. Figure 12: Relationships of spectral density distribution with concentrations taken by [PERSON] and [PERSON] [37]. that the eddy sizes in sediment-laden flow are larger than those in fresh water. Additionally, the coefficients of cross-correlation of the fresh water and water with a sediment concentration of 250 kg/m\({}^{3}\) indicate that the scale of the vortex increases with the presence of sediment in the flow. The scale of the vortex increases when sediment is present as a result of the changes in turbulent structure and surface tension [47; 48]. Although different objects were studied, and different experimental details were produced, these results correspond well with those in reference [37], as shown in Figure 14. Figure 14, taken from [PERSON] and [PERSON] [37] depicts the relationships of eddy sizes with particle concentrations. The results reveal that the longitudinal sizes of the macroscale eddies increase with the increase in sediment concentration and is mainly decided by the low frequency, large-size eddies. When the frequency of turbulent eddies is reduced, the energy proportion carried by large-size eddies increases. The longitudinal sizes of the microscale eddies increase with an increase in concentration and is mainly decided by small-size eddies with high frequency. For a sediment-laden flow, part of the energy carried by high frequency eddies decreases and the smallest eddies vanish. Hence, the fluctuating pressure is the result of the movement and interaction of different scales of vortexes in the plunge pool, caused by the nappe jet. The results of the experiments and calculations demonstrate that the larger the sediment concentration in the liquid of the jet, the higher its viscosity. This causes a decrease in vortexes of smaller scales and an increase in vortexes of larger scales, leading to an increase in fluctuating pressure and, thus, energy. The scale of vortexes increases with the  
Figure 14: Relationships between eddy sizes and particle concentrations taken by [PERSON] and [PERSON] [37]. Figure 13: Relationships between coefficient of association at peak pressure and time for fresh water and sediment-laden flow of 250 kg/m\({}^{3}\) concentration at an upstream water head of 1.90 m.  
increasing concentrations of sediments.