## 1 Introduction  
Reasonable technical improvements to aircraft engine seals can enhance the component efficiency of an engine and even increase the thrust-to-weight ratio of the entire aircraft [1]. Traditional labyrinth seals with grate teeth suffer from high leakage rates, frequent rotor-stator abrasion issues, and inadequate sealing performance and stability [2,3]. The leakage rate of brush seals is only 1/4 that of grate-tooth labyrinth seals, and the flexible brush bundle exhibits excellent tolerance to rotor vibrations, maintaining a stable low leakage [4, 5, 6]. While single-stage brush seals have a limited ability to withstand differential pressure, three-stage brush seals can endure differential pressures in the range of 0.6-1.0 MPa. However, three-stage brush seals face the issue of the unevenness of pressure drops between stages, which can lead to premature seal failure [7]. Therefore, researching the equilibrium characteristics and sealing performance of three-stage brush seals is of great significance for stabilizing and improving the sealing performance of aircraft engine seals under differential pressures of several megapascals. Prior studies (e.g., [PERSON] [8] or [PERSON] [9]) established foundational models (porous media and tube bundles) to analyze interstage pressure imbalances and leakage flow distributions in multistage brush seals. [PERSON] [10] and [PERSON] [11] further advanced numerical models (porous media and fluid-solid coupling) to quantify the effects of structural parameters (e.g., mounting angles or baffle height) on pressure drop equalization. Experimental studies by [PERSON] [12] and [PERSON] [13] revealed trade-offs between sealing tightness, wear resistance, and temperature gradients in multi-stage brush seals under operating environments. [PERSON] [14] identified geometric parameters (axial/radial dimensions) and operational conditions (pressure ratio and rotational speed) as critical factors influencing leakage and thermal behavior in brush seals. [PERSON] [15] developed a transient fluid-solid coupling model to simulate brush wire dynamics and leakage under varying pressure differentials. [PERSON] [16] generated 20 CFD-based training samples for a two-stage brush seal and trained a neural network to predict 50 sets of leakage and interstage imbalance data under varying brush wire displacements and protective clearances. The study analyzed the coupling effects of these parameters, revealing that brush wire row count impacts leakage inversely with protection clearance reduction. [PERSON] [17] developed a 3D multistage brush seal flow heat transfer model and validated it through numerical and experimental comparisons. The study analyzed how operational parameters affect interstage flow characteristics in both differentiated and uniform multistage brush seal configurations. The study revealed that configurations with an elevated backplane height exhibited the highest leakage, followed by those with a reduced brush bundle thickness. [PERSON] [18] based on the experimental data, modified the porous media model of a three-stage brush seal, and the leakage of the brush seal was accurately simulated by the numerical model through the dynamic adjustment of porosity, and the error of the calculated leakage compared with the experiment was less than 18.12%. [PERSON] [19] conducted a comparative analysis of flow field characteristics across multistage brush seal configurations, examining the impact of stage count on leakage behavior and flow dynamics. The study revealed that under constant pressure ratio conditions, incremental additions of stages (from first to third) progressively reduced both the pressure drop per stage on brush bundles and the flow field's peak velocity. Furthermore, sealing efficiency exhibited a direct correlation with stage quantity under identical pressureratios. [PERSON] [20] conducted flow field analysis on two-stage brush seals, examining how pressure differential and second-stage backplane height impacts on leakage and pressure distribution.