Using the above models and simulation methods, we first simulated the falling distance of the leaves of the Pagoda. Figure 1 shows the relationship between falling leaf distance and wind speed with the tree height of 5m, 10m, 15m, and 20m, respectively. The scattered points in the figure are the test measurement data under different wind speeds, and the broken line is the average value calculated by Monte Carlo method many times. It can be seen from the data in the figure that the fluctuation range of the test data is large. Even for the given tree height and wind speed, the difference in the falling leaf distance obtained in different test measurements is large, which is mainly related to the shape of the leaves of the Pagoda. The leaves are flat, the leaf area is large, and they are sensitive to various factors in the process of falling, so the results of different tests are different. The results of the numerical simulation were in good agreement with the test data on the whole, i.e., for the specific wind speed, the simulated leaf falling distance was basically in the middle of the test measurement results, thus showing that the chosen model could well reflect the relationship between the falling distance of different types of leaves and the changes in tree height and wind speed. In addition, as shown in Fig. 1, for all tree heights, the falling distance of leaves increased with the increase in wind speed, and the overall wind speed was not large. However, when the wind speed was high enough, the falling distance of leaves decreased because for flat leaves like locusts, when the wind speed is too high, the leaves tend to rotate during the falling process, which diminishes their falling distance.
In order to better compare the experimental results with the numerical simulation results, we also calculated the average variance of the experimental measurement results, i.e., the average square difference of each velocity parameter group, which represents the degree of sample dispersion. At the same time, the average variance of the numerical simulation results was calculated, i.e., the variance of each speed parameter group of the simulation was balanced, indicating the degree of dispersion of the simulation operation. The formula for average variance is:
$${\text{average variance}}=\frac{{\sum\limits_{{i=1}}^{n} {{\text{varianc}}{{\text{e}}_i}} }}{n}$$
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The average variance of the test and simulation results is shown in Table 1.
As shown in Table 1, because the leaves of Amorpha japonica are flat and large in area, they are sensitive to various factors in the process of falling. The average variance of both test results and simulation results is large, and the average variance will further increase with the increase in tree height.
Next, we simulated the falling distance of poplar leaves. Figure 2 shows the relationship between leaf falling distance and wind speed when the tree height is 5m, 10m, 15m, and 20m, respectively. The scattered points in the figure are the test measurement data under different wind speeds, and the broken line is the average value calculated by Monte Carlo method many times. It can be seen from the data in the figure that the fluctuation range of the test data is large. Even for the given tree height and wind speed, the difference in the falling leaf distance obtained in different test measurements was large, which is similar to that of locusts. The reason is that the leaves of poplar are flat, large in area, and sensitive to various factors during falling, which leads to different test outcomes. The results of the numerical simulation were in good agreement with the overall test data, i.e., for the specific wind speed, the simulated leaf falling distance was basically in the middle of the test measurement results, thus showing that the chosen model well reflected the relationship between the falling distance of different types of leaves and the change in tree height and wind speed. Compared with the results of acacia japonica and poplar, the falling distance of poplar leaves was smaller than that of acacia japonica because the leaves of poplar are generally larger than that of acacia japonica.
As shown in Table 2, we also calculated the average variance of the experimental measurement results and the average variance of the numerical simulation results for the falling distance of poplar leaves. As mentioned above, because the overall falling distance of poplar leaves was smaller than that of acacia, the fluctuation range of falling distance of poplar leaves under given conditions was smaller, and the average variance of test data was relatively small. Moreover, we could see that the average variance of the numerical simulation results was smaller than the average variance of the test measurement results, indicating better convergence of the simulation results. If more test data could be measured, the average variance of the test measurement data could be further reduced.
The simulation results of the leaf falling distance of Juglans mandshurica are shown in Fig. 3. The four sub-graphs show the relationship between the leaf falling distance and the wind speed for the tree height of 5m, 10m, 15m, and 20m, respectively. The scattered points in the figure are the test measurement data under different wind speeds, and the broken line is the average value calculated by Monte Carlo method many times. It can be seen from the data in the figure that the leaf shape of Juglans mandshurica is similar to that of poplar and locust, so the results of both experimental data and numerical simulation data are similar.
The results of the average variance of the experimental measurement results and the average variance of the numerical simulation results of the falling distance of Juglans mandshurica leaves are shown in Table 3. Similar to poplar leaves, the average variance of the test data of the falling distance of Juglans mandshurica leaves was relatively small. Only when the tree height was 20m the average variance became larger. The average variance of the numerical simulation results was smaller than that of the experimental measurement results, indicating better convergence of the simulation results. Similarly, when the tree height was 20m, the average variance of the numerical simulation results significantly increased, indicating that when the tree height was higher, the uncertainty of the numerical simulation results also rapidly increased. However, it was still smaller than the average variance of the test results.
Table 3
Average variance of experimental measurement results and numerical simulation results of the falling distance of Juglans mandshurica leaves under different tree heights
| Tree height 5m | Tree height 10m | Tree height 15m | Tree height 20m |
Average variance of experimental results | 5.4970 | 4.9729 | 5.3602 | 12.6990 |
Average variance of numerical simulation results | 2.1207 | 3.0387 | 2.6201 | 7.5510 |
The simulation results of spruce leaf falling distance are shown in Fig. 4, including the relationship between leaf falling distance and wind speed with the tree height of 5m, 10m, 15m, and 20m. The scattered points in the figure are the test measurement data under different wind speeds, and the broken line is the average value calculated by Monte Carlo method many times. Compared with the results of Pagoda, poplar and Juglans mandshurica, the fluctuation range of the falling distance of the spruce leaves was significantly smaller because the spruce leaf shape is different from the previous several types. Spruce leaves are needle-shaped, with a relatively stable falling process, so the data deviation was not large when different measurements were made. Moreover, the falling distance of spruce leaves increased with the increase in wind speed, which is also due to the needle shape of spruce leaves, and is not affected by wind speed. Accordingly, when the wind speed is high, the falling distance of spruce leaves does not fluctuate significantly.
The results of the mean-variance of the experimental measurement results and the mean-variance of the numerical simulation results of the falling distance of spruce leaves are shown in Table 4. As the above analysis shows, the average spruce variance was very small in experimental data and numerical simulation results, indicating that the falling distance of needle-shaped leaves was relatively stable. Also, the numerical simulation results and the test data had different characteristics. As for the numerical simulation results, because they were obtained from the same model, but only the parameter values changed, the mean square error of the numerical simulation results had apparent regularity. This suggests that with the increase of the tree height, the mean square error of the numerical simulation results increases monotonically, which was obviously different for the experimental data. Because the falling of needle leaves is relatively stable and is not affected by the tree's height, the average variance of the experimental measurement results did not show obvious characteristics. The average variance of the tree height of 20m was even lower than that of the tree height of 5m.
Table 4
Average variance of experimental measurement results and numerical simulation results of the falling distance of spruce leaves under different tree heights
| Tree height 5m | Tree height 10m | Tree height 15m | Tree height 20m |
Average variance of experimental results | 2.1352 | 0.848 | 2.368 | 1.4531 |
Average variance of numerical simulation results | 0.5211 | 0.8474 | 1.285 | 2.942 |
Finally, we compared the relationship between the simulation results of falling leaf distance of locust, poplar, Juglans mandshurica and spruce with the wind speed, respectively, for the trees 5m, 10m, 15m, and 20m high, as shown in Fig. 5. As shown in the figure, the falling leaf distance of acacia japonica was the farthest. Nonetheless, there was an exception with the tree height of 10m, where the leaf falling distance of poplar exceeded that of acacia japonica when the wind speed was not very high. Because the leaves of sophora japonica, poplar and Juglans mandshurica are flat, their falling distance was relatively large, and it increased in all of them with the increase in wind speed. However, they demonstrated obvious fluctuations when the wind speed was high. In contrast, because spruce is a needle-shaped leaf, its leaf falling distance was relatively small, and it was not significantly affected by the wind speed, nor did it show obvious fluctuations when the wind speed was high.