3.1. Calculation Model
In this study, the FDA process of the Shuibuya Power Station was used as the research object, and a three-dimensional finite element calculation grid was established. The simulated similarity scales are 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, and 1:100 for the FDA process of the Shuibuya Power Station. The calculation grid is shown in Figure 4
. The grid dependency is verified on a 1:1 computing grid, and hexahedral meshes are used. Considering the calculation efficiency and calculation accuracy, the number of calculation grid nodes used is 59,939, and the number of grid elements is 53,244. The calculation grids of other scales are all reduced proportionally on the basis of the 1:1 scale calculation grid. The number of meshes used in numerical simulation with different geometric scales is the same, so the mesh size is different and the Reynolds number is not preserved in small-scale models. The mesh size used in the model with a small geometric scale is smaller. Considering that the size of models with different geometric scales is different and the mesh size adopted by models with different geometric scales cannot be consistent, a consistent mesh number strategy is adopted. It should be noted that the change in mesh size affects the calculation results but does not affect the trend of the calculation results.
The initial boundary conditions are important factors affecting the calculation results. To study the influence of the geometric scale, different models adopt similar boundary conditions, that is, the types of boundary conditions are the same, but the values are different. Taking the 1:1 scale calculation model as an example, the boundary conditions are mainly divided into velocity boundary conditions, pressure boundary conditions, and concentration boundary conditions. The velocity boundary conditions are as follows: at the entrance of the spillway, the velocity in the x-direction below the water surface is known, which equals the discharge divided by the corresponding mesh area, and the velocity of the air above the water surface in the x-, y-, and z-directions is 0; the velocities of water and air at the spillway and the river channel are both 0; the velocities in the y- and z-directions at the downstream exit are 0, and the velocities in the downstream direction are calculated through calculations; and the velocities in the x-, y-, and z-directions at the top of the model are all 0. The discharge of the geometric scale of 1:1 is 3848 m3/s. The pressure boundary conditions are as follows: the model outlet is a known pressure boundary, the pressure below the water surface is calculated by water depth, and the pressure above the water surface is calculated by elevation coordinates; the top of the model is a known pressure boundary; and the pressure of the remaining parts is obtained by calculation. The concentration boundary conditions are as follows: the concentration boundary below the spillway inlet water surface is known, and the concentration in the rest of the area is obtained by calculation. For models with other geometric scales, the speed at the boundary is reduced according to the geometric dimensions, which meets the criterion of equal Froude number, while the pressure boundary conditions and concentration boundary conditions are consistent with the 1:1 model.
The initial conditions of the model calculation for different scales are all kept the same, and the initial velocity in the model calculation domain is 0; the initial concentration of water below the surface of the downstream river is 1, and the concentration of water above the surface is 0.
3.2. Evolution Process of Atomization Wind Speed and Rainfall
By numerically simulating the flood discharge process under different similar scale conditions, the velocity of water vapor movement and the concentration of water mist at a certain place in space at any time can be obtained. We select the node at the 230 m platform (#22 in Figure 10) on the left bank and analyze the changes in wind speed in the x-direction (along the river) component, z-direction (vertical direction) component, and wind speed at this part under different similarity scales. Generally, according to Figure 5
, the x-direction component of atomization wind speed at the 230 m platform on the left bank first increases and then gradually tends to stabilize with the flood discharge process, and the final stable speed is also different under different similarity scales. When the similarity scale is 1, the x-direction component of the atomization wind speed is the largest. As the similarity scale decreases from 1:1 to 1:50, the x-direction component of atomization wind speed gradually changes from positive to negative. When the similarity scale decreases to 1:100, the x-direction component of atomization wind speed changes to positive again. This shows that the position of the air flow gyration formed by the atomized wind on the 230 m platform on the left bank is greatly affected by the geometric scale. At the same position, under different geometric scales, it may be located on the left side of the counterclockwise cyclone center (the x-direction component is positive) or on the right side of the cyclone center (the x-direction component is negative).
shows the components of the atomization wind speed in the z-direction under different similar scale conditions. With the progress of flood discharge, the component of atomization wind speed in the z-direction increases first, then decreases gradually, and tends to be stable. Since the water flows mainly downward, the z-direction component of the atomized wind speed produced mainly moves vertically downward on the 230 m platform on the left bank. The larger the similarity scale, the greater the stable value of the atomizing wind speed component in the z-direction, and the earlier this value starts to change. This shows that under the same initial conditions, the larger the similar scale, the shorter the time for the water to flow down to the 230 m platform on the left bank.
Through the components of the atomization wind speed in the x-, y-, and z-directions, the change process of the atomization wind speed with time during the flood discharge process can be obtained, as shown in Figure 7
. The change process of atomization wind speed with time is similar to the component of atomization wind speed in the x-direction, and both increase first and then gradually stabilize with the progress of flood discharge. On the whole, the smaller the similarity scale, the smaller the atomization wind speed when the flood discharge is stable, and the more severe the fluctuations. When the similarity scale reaches 1:100, the atomization wind speed fluctuates the most. Figure 6
shows that the geometric similarity scale of the model has a greater impact on the atomization wind speed. From the perspective of energy conversion, the potential energy of the water body is transformed into the kinetic energy of water and air during the flood discharge process. When the potential energy of the water body is greater, the corresponding atomization wind speed is also greater. The atomization wind speed during the flood discharge process is affected by many factors, including the drop between the upstream and downstream, the atmospheric pressure, and the boundary conditions [29
]. When a smaller similar scale is used, the influence of each influencing factor on the atomization wind speed is not negligible.
shows the change process of the water concentration above the 230 m platform on the left bank under different similar scale conditions with time. The water concentration at this location first increases rapidly and then gradually decreases to a stable trend as the flood discharge progresses. When the similarity scale is less than 1:50, the concentration of water at the 230 m platform on the left bank does not have a decreasing stage. The larger the similarity scale, the greater the maximum value that can be reached by the water concentration above the 230 m platform on the left bank, and the greater the concentration that can be reached in a steady state, the earlier the water concentration at the 230 m platform on the left bank changes.
The atomized rainfall intensity is mainly affected by wind speed and water mist concentration (Figure 9
). The variation law of atomized rainfall is similar to that of water mist concentration, which first increases rapidly and then decreases gradually to become stable. The higher the wind speed, the higher the water mist concentration, and the greater the atomization rainfall intensity in this part. When the model scale is less than 1:10, the atomized rainfall intensity measured in the model test is approximately 0. At this time, the rainfall intensity is greatly affected by the surrounding environment, and the accuracy of the results is poor. When the geometric similarity scale of the model is large, the deviation of the predicted prototype rainfall intensity is also large.
3.3. Influence of Geometric Scale on the FDA Wind Field and Rain Field
In 2016, the Shuibuya Hydropower Station carried out a prototype observation test of FDA [31
] in which the discharge flow was 3848 m3
/s. The atomization wind speed at 7 measuring points and the atomization rain intensity at 21 effective measuring points were monitored (Figure 10
]. Generally, we pay more attention to the maximum value and the influence range of atomized rainfall. Therefore, measuring point #22 with the largest rainfall intensity among all monitoring points (left bank
230 m platform) is selected, and the stable atomization wind speed and rainfall intensity obtained under different geometric scale conditions are plotted in Figure 11
and Figure 12
shows that the smaller the geometric scale, the smaller the stable atomization wind speed obtained. When the similarity scale is less than 1:10, the stable atomization wind speed calculated by the model tends to be basically stable, which means that the monitoring data at a small scale are very indistinguishable. From another point of view, the smaller the similar scale, the greater the error caused by the test.
The atomized rainfall is affected by the concentration of water mist and the atomization wind speed. The atomized rain intensity at the 230 m platform on the left bank under different geometric scale conditions obtained through the numerical model is shown in Figure 12
. Similar to the variation law of atomization wind speed, the larger the geometric scale, the greater the atomization rain intensity calculated by the model. In the prototype test, the maximum rainfall intensity of this part can reach 10,000 mm/h. By comparing with the calculation results of the model with a geometric scale of 1:1, it can be seen that the deviation of the model prediction results is −14.5%. When the 1:1 scale is used to predict the FDA of the Shuibuya Hydropower Station, the maximum deviation between the numerical model calculation results and the prototype observation results can reach 14.5%. There are many reasons for this deviation, including computational boundary conditions, model parameters, and numerical solution methods. Generally, the numerical model can be modified through prototype observation data to improve prediction accuracy. However, it is necessary to carry out indoor model tests under the condition of no construction project and a lack of prototype observation data, which can provide a corresponding basis for the modification of numerical calculation models.
When the reduced scale model is used to predict the FDA wind speed and rainfall, it is necessary to establish the conversion relationship between the model and the prototype. Generally, this relationship can be expressed by an exponential function with a geometric scale as the base. For example, the atomized rain intensity scale
can be expressed as a function of the geometric scale
, where the value of n
is affected by flood discharge hydraulic conditions, flow Weber number, geometric scale, energy dissipation shape, and other factors. Using the atomization wind speed and rainfall intensity at the 230 m platform on the left bank, the relationship between the rain intensity scale
, the wind speed scale
, and the geometric scale
can be obtained, as shown in Table 1
As shown in Table 1
, under similar boundary conditions and calculation parameters, by changing only the size of the geometric grid, the index n
of the atomization rain intensity scale obtained by simulation is between 1.019 and 1.734, the conversion coefficient of fog–rain intensity is between 3.33 and 179.47, and the smaller the geometric scale, the greater the conversion value of fog–rain intensity. Wu Shiqiang et al. [18
] recommended that the atomized rain intensity scale be converted to the 1.53 power of the geometric scale, which can be used as the outer envelope control line of the atomized rain intensity. This value is within the range of the conversion coefficient of the atomized rain intensity scale in Table 1
. The index n of the atomization wind speed scale is between 0.572 and 1.43, and there is no detectable monotonic relationship between different geometric scales. The conversion value of atomization wind speed also fluctuates greatly, ranging from 2.1 to 16.81. When the physical model test is used to predict the FDA wind speed and rainfall, the smaller the geometric scale, the larger the rain intensity scale and the wind speed scale, but a larger rain intensity scale causes a larger prediction deviation. Under different combined scale conditions, the prediction deviation of the atomization wind speed and rain intensity at each measurement point is shown in Figure 13
According to Figure 13
, when models with different geometric scales are used to predict atomization wind speed and rainfall intensity at different measuring points, the prediction deviations of models with different geometric scales are different for each measuring point of the prototype. Generally, the prediction deviation of the atomization fraction of models with different geometric scales is −76~+54%, while the prediction deviation of atomization rainfall is −55~+47%. When determining the conversion relationship between the model and the prototype, the coefficient n
is calculated from the values of the maximum measuring points of wind speed and rainfall. When the conversion relationship is applied to other measuring points, the deviation of the predicted value is large. It should be noted that the monitoring data compared here are the prototype observation data of the Shuibuya Hydropower Station, not the data from the reduced geometric scale model experiment.
3.4. Influence of Atmospheric Pressure on the Model Test
Through the research in Section 3.3
, it is found that when the geometric similarity is satisfied, the coefficient n
in the conversion relationship between the prototype and the model is solved through the prototype observation data, and then the coefficient is used to predict the atomization wind speed and rainfall intensity of other measurement points. The calculation model with a small geometric scale still causes a large deviation. In fact, the occurrence of atomization during flood discharge depends not only on the turbulence of the water flow itself but also on its ability to eliminate surface tension. The surface tension of water is essentially caused by the imbalance of pressure on the interface, so the atmospheric pressure where the flood discharge is located also has a greater impact on atomization. Liu Haitao (2019) et al. [30
] studied the influence of altitude on FDA and reported that as the altitude increases, the area of atomization downstream of the flood discharge tunnel tends to increase. Considering that most physical model tests are carried out under normal temperature and pressure, the atmospheric pressure between the prototype and model does not meet similar conditions. In this section, on the premise that the calculation model meets the gravity similarity, the atmospheric pressure of the model is changed to make the environmental pressure of flood discharge meet similar conditions to explore the impact of atmospheric pressure on FDA. The atmospheric pressure of the calculation model with a geometric scale of 1:1 is standard atmospheric pressure, and the pressure of the other calculation models is reduced on the basis of standard atmospheric pressure according to the geometric scale. The other calculation conditions are the same as those in Section 3.3
shows the relationship curve between the atomization wind speed and the atomization rain intensity at the 230 m platform on the left bank with different geometric scales and geometric scales under the condition of similar pressure. It can be seen that the smaller the similarity scale, the smaller the atomization wind speed and atomization rainfall calculated by the model. As the geometric scale decreases, the atomization wind speed and rainfall decrease in a power function relationship with the geometric scale as the base. The atomization wind speed calculated by the models of different geometric scales shows a power function relationship with an exponent of 0.529, and the correlation reaches 0.99, while the atomized rainfall obtained by calculation models of different geometric scales presents a power function relationship with an exponent of 0.554, and the correlation can reach approximately 0.97. From the perspective of numerical simulation, when the calculation model meets geometric similarity and atmospheric pressure similarity, the conversion relationship between different geometric scale models is relatively fixed, and the power function index is approximately 0.5, which approximately meets the gravity similarity criterion.
The conversion relationship coefficient n between the model and prototype with different geometric scale values is solved by using the prototype observation values of the maximum atomization wind speed and rainfall measurement points, as shown in Table 2
According to Table 2
, when the models with different geometric scales meet similar pressure conditions, the variation range of the conversion relationship coefficient n between the atomization wind speed and rainfall intensity and the prototype observation results is significantly reduced. The conversion coefficient n obtained in Table 2
is used to predict the atomization wind speed and rainfall intensity at other observation points. The deviation of the prediction results is shown in Figure 15
shows the comparison between the predicted value of the calculation model of different geometric scales and the observed value of the prototype. Under the condition of similar atmospheric pressure, the deviation between the atomization wind speed obtained by the geometric models with different scales and the measured wind speed is smaller than that of the models that did not meet similar atmospheric pressure, and the deviation between the predicted value of the model and the measured value is ±30%. After considering similar atmospheric pressure conditions, the prediction accuracy of the prototype’s atomization wind speed and rainfall intensity is greatly improved by the calculation model.