## 3 Results  
### Numerical Simulation Verification by Experiments for 1-3 Tested DN 250 UFMs  
For one, two, and three UFMs in tandem arrangement, the relative measurement errors \(E\) are plotted in Figure 5a-c as a function of flowrate \(Q\), respectively. Both the experimental data (distributed symbols) and numerical results (continuous lines), can be fully described by a negative power-law relationship with two assigned constant coefficients \(A\) and \(B\), which can be expressed as  
\[\ln(E+k)=A\ln(Q)+B, \tag{3}\]  
The values of \(A\) and \(B\) for different numbers of UFMs and goodness of fit \(R^{2}\) are listed in Table 4. An \(R^{2}\) value very close to 1 indicates high goodness of fit of the simulated results. In addition, we add a standardized residual \(\delta^{*}\) to observe the regression fit degree of the experimental value and the simulation curve. Once \(\delta^{*}\) is in the range of \((-2,2)\), we think the fitting is almost satisfactory. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\)-UFM\(n\) & \(A\) & \(B\) & \(k\) (Intercept) & **Numerical R\({}^{2}\)** &
\begin{tabular}{c} **Experimental Standardized** \\ **Residual \$*** \\ \end{tabular} \\ \hline
1-UFM1 & \(-0.427\) & 2.037 & 0 & 0.967 & (\(-1.150\)–\(0.732\)) \\
2-UFM1 & \(-0.432\) & 2.219 & 0 & 0.962 & (\(-0.781\)–\(0.813\)) \\
2-UFM2 & \(-0.403\) & 2.040 & 0 & 0.959 & (\(-1.531\)–\(1.138\)) \\
3-UFM1 & \(-0.325\) & 1.416 & 0 & 0.996 & (\(-0.356\)–\(0.342\)) \\
3-UFM2 & \(-0.261\) & 1.166 & 0 & 0.951 & (\(-0.907\)–\(1.661\)) \\
3-UFM3 & \(-0.288\) & 1.393 & 0 & 0.994 & (\(-0.636\)–\(1.180\)) \\
4-UFM1 & \(-0.307\) & 1.584 & 0 & 0.978 & \\
4-UFM2 & \(-0.349\) & 1.880 & 0 & 0.934 & \\
4-UFM3 & \(-0.558\) & 3.037 & 0 & 0.853 & \\
4-UFM4 & \(-0.520\) & 2.680 & 0 & 0.910 & \\
5-UFM1 & \(-0.277\) & 1.428 & 0 & 0.978 & \\
5-UFM2 & \(-0.305\) & 1.602 & 0 & 0.968 & \\
5-UFM3 & \(-0.367\) & 1.984 & 0 & 0.929 & \\
5-UFM4 & \(-0.408\) & 2.210 & 0 & 0.918 & \\
5-UFM5 & \(-0.309\) & 1.599 & 1.713 & 0.996 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Constants in the fitting functions modeling the negative power-law \(E\)–\(Q\) curves. It is clear that \(E\) decreases nonlinearly and monotonically with increasing \(Q\) and the attenuation ratios for \(Q<\) 600 m\({}^{3}\)/h are significantly larger than those for \(Q>\) 2000 m\({}^{3}\)/h. Similarly, the approximately negative power-law relationship between K\({}_{\rm S}\) and \(Q\) can be found in Figure 5 and this phenomenon shows that the swirl secondary flow would be reduced effectively as \(Q\) (or Re) increased. The above discussions indicate that all the available experimental data agree remarkably well with the numerical results and especially the overall \(E\)-\(Q\) power-law distribution for a single tested UFM is in good accordance with the previous results [12, 15, 16]. Because the volumetric flowrate measured using UFMs is calculated as the product of the mean axial flow velocity U\({}_{0}\) and the area \(S\) over the cross section, the velocity distribution correction factor \(C\) (also known as velocity profile factor), which is used to evaluate U\({}_{0}\) corresponding to the uniform velocity profile (fully developed and usually turbulent), must be known [15]. The negative power-law relationship between \(E\) and \(Q\) for \(1\leq n\leq 3\) in geometric tandem arrangement can be mainly attributed to the change in \(C\) as a function of Reynolds number Re, which can be calculated approximately for the fully developed velocity distribution in the axisymmetric non-swirling flow. Re can be calculated using Re = UD/\(\
u\), where \(D\) is the pipe or UFMs internal diameter and \(\
u\) is the kinematic viscosity of the fluid. For a constant internal diameter of UFM and fluid viscosity, Re should increase linearly with increasing flowrate.