## 4 Discussion
### Flow Analysis for Daniel 3400  
In order to perform comparisons among velocity profiles along the same-type paths under different \(Q\), the flow velocity \(U\) as shown in Figure 9 was first nondimensionalized using the corresponding mean axial flow velocity U\({}_{0}\) and weighted by the corresponding w, and the location variable x was also nondimensionalized using the length of the corresponding path length L. As shown in equation 1, the values of the weight are 0.1382 along paths A and D (\(w_{\text{A}}=w_{\text{D}}=0.1382\)) and 0.3618 along paths B and C (\(w_{\text{B}}=w_{\text{C}}=0.3618\)), respectively. Note that the tested UFMs' structure is symmetrical at the origin, and each of the paths form a 60\({}^{\circ}\) with the axial flow; however, the velocity along the paths also approximately distributes according to Prandtl-von Karman logarithmic law, which is a common velocity distribution for fully developed turbulent pipe flow [14]. Despite this, there were pronounced differences between the velocity profiles along paths A and D (denoted as U-Profiles A and D) and that along paths B and C (denoted as U-Profiles B and C), because paths A and D were closer to the pipe wall with shorter L, and could be more sensitive to the boundary layer. To be more specific, U-Profiles A and D tend to be an inverted V shape, whereas their counterparts along paths B and C display an inverted \(U\) shape. As we can see from Figure 9, when \(n\) increases, the difference in U-Profiles of each path line also begins to increase, which is manifested as the overall deviation for U-Profiles A and D and the deviation of the main flow area for U-Profiles B and C. Further, we compare the U-Profiles A and B of three tested UFMs at the same flow rate (Figure 10). We find that the velocity profile of the main flow area hardly changed when \(n=3\), while the velocity profile of the near boundary layer area has obvious dispersion phenomenon. Obviously, the negative power-law \(E\)-\(Q\) relationships can be partly derived from these results about the U-Profile at different \(Q\) and the larger value of \(w_{\text{B}}\) and \(w_{\text{c}}\) (weighted coefficients). Figure 8: Factors associated with velocity profile as a function of Q: (**a**) swirl ratio for DN250; (**b**) swirl ratio for DN150. Figure 9: Velocity profiles along the same type of path under different \(Q\) for three tested UFMs (DN150): (**a**) A path; (**b**) B path. As noted above, more tested UFMs would account for poorer performance of SVUFMST and we sought to explain this finding in accordance with the inlet flow condition of each UFM, especially the secondary flow, in the following discussion. In most practical engineering applications, the effect of the entry length on turbulent flow becomes insignificant beyond a pipe length of 10 times the diameter, as imposed by \"JIG1030-2007.\" However, our detailed numerical results showed that the entry length would increase as Re increased, which was in line with the previous observations reported by [PERSON] et al. (2014), and hence we had ensured that all the entry lengths in our numerical simulations were larger than 50D. However, the entry velocity profiles of the UFMs at different streamwise locations can still be affected to some extent by the flow distortion caused by the transducer recesses, which can be treated as large-scale roughness elements. The mass-conservation law was satisfied by reducing the residual of the continuity equation by less than \(10^{-6}\), and the relative deviation between the area-averaged inlet velocity at UFM1 and that at other UFMs was less than \(10^{-3}\). The significant differences among the transverse velocity fields at the inlet of UFMs at different locations demonstrate occurrences of distinguishable secondary flows and remarkably diverse inlet flow conditions of each tested UFM, as shown in Figures 11 and 12. In particular, the phenomenon of secondary flows near the boundary is very obvious and it becomes more severe with the increase of the number of installations.