## 5 Analysis of Results  
### Data Assimilation Results  
Initially, the generated sample parameter set is input into the FLUENT commercial software (ANSYS 2021 R1)and analyzed using the RNG k-\(\varepsilon\) turbulence model. The results are subsequently post-processed to derive the initial sample prediction set, as shown in Figure 7. From Figure 7, it is evident that the experimental measurements fall within the range of the predicted sample set. This indicates that the initial sample set captures sufficient information about the flow field, thereby validating the assimilation process. When the EnKF algorithm reaches a converged state, the distribution of all ensemble members closely aligns with the experimental data.  
Figure 6: Distribution Status of Constant Sample Group.  
Subsequently, the initial sample collection and experimental data are integrated using the EnKF algorithm to compute the updated model constants. Table 4 presents both the original and updated constants of the RNG k-\(\epsilon\) turbulence model. The results show that after the EnKF integration, constants \(C_{u}\), \(C_{1\epsilon}\), and \(C_{2\epsilon}\) were modified relative to their initial values, while constant \(Pr_{t}\) remained largely unchanged.  
The Mean Squared Error (MSE) quantifies the deviation between experimental measurements and theoretical values. A smaller MSE indicates higher accuracy of the experimental results. In this study, a lower MSE between the predicted flow rate from the turbulence model and the experimental flow rate reflects greater predictive accuracy of the corresponding model.  
The definition of MSE is as follows:  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}\left(X_{i}-Y_{i}\right)^{2} \tag{23}\]  
\(N\): Sample size  
\(X_{i}\): The \(i\)-th predicted value.  
\(Y_{i}\): The \(i\)-th true value.  
Using the data-assimilated RNG k-\(\epsilon\) model to predict the flow rate of the air-lift plunger under various operating conditions and comparing it with the predictions from the original constant RNG k-\(\epsilon\) model, as shown in Figure 8, it is evident that the accuracy of the assimilated turbulence model has significantly improved. In addition, after data assimilation, the turbulence model exhibits high prediction accuracy for the original rectangular slot plunger case and shows significantly improved accuracy for the circular arc-shaped slot plunger, as illustrated in Figure 9. The mean square errors of the RNG k-\(\epsilon\) model before and after assimilation are detailed in Table 5, showing a decrease from 60.67 and 61.48 to 7.12 and 7.20 for the rectangular and circular arc plungers, respectively. The recalibrated  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{C_{u}}\) & \(\mathbf{C_{1\epsilon}}\) & \(\mathbf{C_{1\epsilon}}\) & \(\mathbf{Pr_{t}}\) \\ \hline RNG k-\(\epsilon\) & 0.0845 & 1.42 & 1.68 & 0.85 \\ ENKF+RNG k-\(\epsilon\) & 0.0828 & 1.4586 & 1.6266 & 0.8503 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Model constants for models before and after data assimilation.  
Figure 7: CFD calculation results.  
model demonstrates robust scalability and effectively predicts the sealing performance of turbulent sealing plungers with various groove types across different operating conditions, thereby significantly enhancing the accuracy of these predictions.