## 3 Results
### Numerical Simulation Verification by Experiments for 1-3 Tested DN 250 UFMs
Therefore, once the decrease in \(Q\) decreases Re to a certain extent, the transition from a flattened velocity profile to a more convex velocity profile may occur, resulting in the evolution of the velocity distribution and the variance in \(C\). Here, we only took \(Q\) (equivalent to Re) and \(n\) into consideration for our simulations, and other factors would be analyzed according to the difference \(\delta\) between the experimental and simulated E, whereas  
Figure 5: Relative measurement error \(E\) as a function of flowrate \(Q\) for 1–3 tested flowmeters: (**a**) a single tested UFM; (**b**) two tested UFMs; (**c**) three tested UFMs. previous studies had suggested that \(E\) depends on Re, pipe roughness, inlet flow conditions, distance from an inlet, etc. [12; 15; 16; 17]. \(\delta\) can be calculated by  
\[\delta=E_{\mathrm{t}}-E_{\mathrm{s}}, \tag{4}\]  
where the subscript t indicates the test results and s indicates the simulated results. The maximum deviation \(\delta\)max between the experimental and simulated \(E\) is about 0.06% for a single UFM at \(Q\) = 500 m\({}^{3}\)/h and 2000 m\({}^{3}\)/h, 0.18% for UFM2 of the two tested UFMs at \(Q\) = 500 m\({}^{3}\)/h, and 0.18% for UFM3 of the three tested UFMs at \(Q\) = 500 m\({}^{3}\)/h. Compared with the maximum permissible error of the UFM (Daniel-3400) of 1.0%, the value of \(\delta\) as discussed above can be acceptable, and also confirms the feasibility of implementing CFD simulations in UFMs' internal flow and then calculating the measurement errors. It was clear that \(\delta\)max always occurred in the flow transition regime with \(Q\) approximately ranging from 500 m\({}^{3}\)/h to 1000 m\({}^{3}\)/h. As illustrated in Figure 5, the deviation \(\delta_{\mathrm{UFM}}\) between the upstream and downstream UFMs' \(E\) (adjacent or nonadjacent) is smaller than 0.15% with the only one exception (\(\delta_{\mathrm{UFM}}\) = 0.3%) that can be found at \(Q\) = \(Q_{\mathrm{min}}\) for three tested UFMs. It is noted that \(\delta_{\mathrm{UFM}}\) is different from \(\delta\), and is given as follows:  
\[\delta_{\mathrm{UMF}}=E_{i+j}-E_{i}\;(i=1,2,3,\ldots,n-1;\;j=1,2,3,\ldots,n-i), \tag{5}\]  
where a larger subscript number indicates a greater distance from the flow inlet at the test bench and \(n\) is the number of the tested UFMs. Overall, the simulated \(|\delta_{\mathrm{UFM}}|\) was smaller than the experimental \(|\delta_{\mathrm{UFM}}|\) because only the flow field effect on \(E\) was taken into consideration for the simulated results, whereas there were other factors introduced previously, such as UFMs' manufacturer and pipe roughness, affecting experimental data and then affecting \(\delta_{\mathrm{UFM}}\). Thus, it can be observed that the magnitude of \(\delta_{\mathrm{UFM}}\) is similar to that of the standard facilities' uncertainty, which is equivalent to 0.29% (k = 2) and can be regarded as an important inlet flow condition. Therefore, the uncertainty is large enough to change \(\delta_{\mathrm{UFM}}\) from negative to positive values. In other words, it is ambiguous whether \(E\) of the upstream UFM is larger than \(E\) of the downstream UFM, as shown in Figure 5. From two tested UFMs to three tested UFMs, the absolute values of the measured \(\delta_{\mathrm{UFM}}\) hardly changed, whereas those calculated from simulated flow fields increased obviously with increases in \(n\). For three tested UFMs, the simulated \(E\)-\(Q\) curves were concentrated in two nonlinear striped regions and the \(E\)-\(Q\) curve of UFM1, which exclusively occupied the lower striped region, showed a clear shift to a smaller relative error, while \(E\) of both downstream UFM2 and UFM3 were very close to each other. The two concentrated striped regions hereinafter can also be found in flow through more than three tested UFMs. It is of interest to note that the increase in \(n\) can induce an overall shift in the \(E\)-\(Q\) curve and the two concentrated striped regions are consequently built on the abrupt shift in an \(E\)-\(Q\) curve associated with \(n\).