# Experimental Investigation in Drag Coefficient of Cubes
## 4 Results and Discussion
### Universal Drag Model for the Cube  
According to the above discussion, the orientation angle will have diverse effects on the drag coefficient at different Reynolds number ranges. Therefore, it is a better way to build the drag model for the cube in different Reynolds number ranges. According to the distribution of the drag coefficient of the spliced cubes, the drag model is built in three Reynolds number ranges, which are 0.4 < \(Re\) < 10, 10 < \(Re\) < 1 \(\times\) 10\({}^{3}\), and 1 \(\times\) 10\({}^{3}\) < \(Re\) < 4.516 \(\times\) 10\({}^{3}\). Checking the several forms of drag coefficient models, the most accurate correlation would be obtained by modifying [PERSON]'s model [12]. The following expression of the drag coefficient in this paper is expressed as follows:  
\[C_{D}=\frac{24}{Re}(1+p_{1}Re^{p_{2}})+\frac{p_{3}Re^{p_{4}}}{Re+p_{5}}. \tag{8}\]  
The parameters \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\), and \(p_{5}\) are constant values in different Reynolds number ranges. Table 4 exhibits the value of parameters of Equation (8) in different Reynolds number ranges. Since the orientation angle has little effect on the drag coefficient of the cube in the ranges of 0.4 < \(Re\) < 10 and 1 \(\times\) 10\({}^{3}\) < \(Re\) < 4.516 \(\times\) 10\({}^{3}\), the drag coefficient of the cube in these two ranges can be expressed by a single expression. The variation trend of the drag coefficients of the cube is quite complex in the range of 10 < \(Re\) < 1 \(\times\) 10\({}^{3}\). However, the drag coefficient of the cube is in direct relation to its orientation angle in this range. Thus, the upper and lower boundary of the drag coefficients of the cube in this Reynolds number range can be obtained. The upper bound of the drag coefficient is that with \(\alpha\) = 45\({}^{\circ}\), while the lower bound of the drag coefficient is that with \(\alpha\) = 0\({}^{\circ}\). The comparisons between the experimental drag coefficients for the cube and the predictions from Equation (8) in the range of 0.4 < \(Re\) < 4.516 \(\times\) 10\({}^{3}\) are shown in Figure 9. As can be seen, the predications based on Equation (8) fit the experimental data well. The RMSE between the prediction and the experimental data are only 0.25 and 0.048 in the ranges of 0.4 < \(Re\) < 10 and 10\({}^{3}\) < \(Re\) < 4.516 \(\times\) 10\({}^{3}\), respec  
Figure 8: Relationship between drag coefficient and Reynolds number of spliced cubes with \(h/a\) = 0.5, 0.67, and 1.0. \(10<Re<1\times 10^{3}\), the RMSE between the prediction and the experimental data with \(\alpha=0^{\circ}\) and \(\alpha=45^{\circ}\) are 0.038 and 0.036, respectively. Figure 10 exhibits the distributions of the drag coefficients predicted by Equation (8) and the drag coefficients of the cube obtained by different investigations [20; 23; 24; 25; 26] as well as that obtained by the uniformed cubes in the current investigations. The results of the drag coefficient correction models brought forward by former investigations [11; 12; 14; 17] are also shown in Figure 10. As can be seen, all the correction models can reflect the variation in the drag coefficient. However, each model has its respective characteristics. The prediction of [PERSON]'s correction [11] is always lower than the measured data, while [PERSON]'s correction [14] is higher than the measured one. In the range of \(0.4<Re<10\), both [PERSON]'s [12] and [PERSON]'s corrections [17] are very close to each other and show good accuracy to the experimental data. However, the prediction of [PERSON]'s correction is higher than the experimental data, while [PERSON]'s correction is lower than the experimental data at a high Reynolds number. The prediction in the current paper shows a better agreement with the drag coefficients of the cubes. When the Reynolds number is smaller than 10, the drag coefficients of the cube show a continuous decreasing trend, and Equation (8) predicates the trend with high accuracy.