## 1 Introduction  
As the core equipment of refrigeration systems, vortex compressors have the advantages of small size, light weight, and low noise, and are widely used in the field of cold chain transportation. When vortex compressors operate under complex conditions, they often require a larger compression ratio. Using traditional uniform thickness and a combined profile for structural design results in more compression chamber numbers and leakage line lengths, leading to increased difficulty in processing the dynamic and static vortex teeth, higher work consumption, and exacerbated gas leakage losses. Additionally, due to the unique structure and motion patterns of vortex compressors, it is difficult to obtain the transient distribution laws of various state fields in the working chamber and leakage characteristics through experimental testing. Therefore, optimizing the vortex profile and using efficient and reliable finite element analysis methods to simulate gas flow and leakage characteristics in the chamber to improve the reliability and volumetric efficiency of vortex compressors has become a recent research hotspot. In the design and optimization of vortex profiles, [PERSON] [1] proposed a vortex tooth profile with gradually decreasing wall thickness from the tooth head to the tail. The distribution of various state fields and the stress-strain situation of the vortex teeth were studied using numerical simulation methods. [PERSON] [2] established two types of variable base circle profile equations based on whether the involute angle starts from the \(x\)-axis, comparing the relationships between the two variable base circle profiles, tooth head modification equations, and their impact on compressor performance. Literature [3]established a comprehensive mathematical model of a variable wall thickness profile combining base circle involute, higher-order curves, and circular arcs, and analyzed the variation laws of state parameters during operation in detail. Literature [4; 5] proposed a vortex compressor profile composed of variable radius base circle involutes, controlling related coefficients through mathematical modeling methods and analyzing the geometric performance of different variable wall thickness vortex compressors. In studying the working fluid flow and leakage characteristics inside the vortex compressor chamber, [PERSON] [6; 7] and others established a three-dimensional model of a dual vortex ring compressor and a thermodynamic model during operation. Computational Fluid Dynamics (CFD) moving grid technology was used to simulate the fluid flow in the working chamber and build a test bench to validate compressor performance parameters. Literature [8; 9; 10] established a three-dimensional flow field model for uniform wall thickness vortex compressors, using a triangular prism unstructured grid for division. Based on CFD moving grid technology, the unsteady flow of the working fluid in the vortex compressor was numerically simulated, revealing the flow mechanism of the working fluid in the chamber. Literature [11] constructed a thermodynamic model of an oil-free compressor, using the Euler method to solve the variation laws of various performance parameters in the compression chamber, developed variable speed performance testing, and analyzed the impact of heat transfer and leakage on performance parameters. Literature [12; 13] established transient performance analysis models and thermodynamic models for leakage areas based on the thermodynamic principles and geometric shapes of vortex compressors. [PERSON] [14] and others focused on the performance and reliability issues of vortex compressors, where a numerical model of the rotor-bearing-frame system was established through the coupling of the Reynolds equation, rotor dynamics equations, and frame flexibility equations, optimizing the profile of the vortex compressor.