## 1 Introduction
This technique pioneers a novel approach to turbulence research, facilitating more precise simulations and predictions of turbulence behavior and offering more reliable guidance for engineering applications. [PERSON] et al. [9] developed a data assimilation method based on the ensemble transformed Kalman filter (ETKF) and successfully predicted the inlet angle of attack and the Mach number for the turbulent flow field around an airfoil. [PERSON] et al. [10] optimized the constants in four turbulence models using an EnKF-based DA technique. Analyzing Particle Image Velocimetry data from a free circular jet flow showed significant improvements in flow field predictions across all models after optimization, with the k-\(\epsilon\) model exhibiting the best performance. [PERSON] et al. [11] conducted an in-depth study on the applicability of the hybrid data assimilation method (EnVar) in turbulence modeling. From the perspective of Bayesian inference, the research optimized RANS simulations for a converging-diverging channel. Experimental results demonstrated that the ensemble-based variational method effectively inferred unknown parameters under both small-sample conditions (D = 20) and large-scale systems (D = 2400). [PERSON] et al. [12] employed a data assimilation approach, utilizing the Ensemble Kalman Filter (EnKF) combined with NASA's PIV experimental data to optimize the constants of the Spalart-Allmaras (SA) turbulence model. The optimized model improved the accuracy of jet flow field predictions, reducing the relative error in the horizontal direction behind the nozzle from 13.04% to 4.6%. [PERSON] et al. [13] used the Ensemble Kalman Filter (EnKF) algorithm to recalibrate the constants of the SST k-\(\omega\) turbulence model, significantly improving the accuracy of flow characteristic predictions for steam valves with filters. The recalibrated model constants are suitable for conditions with similar valve openings but exhibit higher errors when the opening differences are large. The optimized model constants better capture the turbulence characteristics of the internal flow. [PERSON] et al. [14] developed a data-driven framework using a two-step EnKF method to predict compressor cascade flow fields. Validating with a test function, they applied S-A and SST models to the MAN GHH cascade, achieving predictions consistent with experiments. Data assimilation reduced errors by nearly 70%, improved separation bubble predictions, and showed low dependency on turbulence models. [PERSON] et al. [15] utilized the EnKF DA technique to optimize the SST k-\(\omega\) turbulence model constants for enhancing heat transfer predictions in plate heat exchangers. By assimilating experimental data from various operational conditions into the computational model, the study achieved significant reductions in prediction errors. The calibrated model constants enabled more accurate simulations of the complex flow and heat transfer behaviors within plate heat exchangers. This optimization, validated by experimental data, substantially enhanced the accuracy of flow predictions, offering a more precise method for analyzing the flow characteristics of the control valve. [PERSON] et al. [16] optimized S-A turbulence model parameters using the Ensemble Kalman Filter (EnKF) for three wind turbine airfoils (NACA63415, S809, DU97W300) under stall conditions. The optimized parameters reduced simulation errors, improved flow separation predictions, and showed that S809's parameters could be applied to other airfoils with slightly higher errors. Key parameters C\({}_{\text{b1}}\), C\({}_{\text{u1}}\), and \(\sigma\) changed significantly, with C\({}_{\text{b1}}\) being the most critical. Despite extensive research in related fields, studies on data assimilation for air-lift plungers remain scarce, particularly regarding the scalability of assimilated models. This paper employs the Ensemble Kalman Filter (EnKF) algorithm to recalibrate turbulence model constants through data assimilation techniques.