## 1 Introduction  
High-speed railways have become an important part of transportation, with advantages of safety, comfort, and convenience. When a train passes through environments such as mountains, rivers, and buildings, aerodynamic problems such as train wind, pressure wave, noise, and micro-pressure waves will be induced. The micro-pressure wave (MPW) is generated by the compression wave propagating outward from the tunnel exit, and is formed when a train enters the tunnel. Serious problems such as structural vibration and noise near the line will be caused by the MPW, and even a sonic boom can be induced, which may affect the health of residents surrounding the tunnel exit [1]. The micro pressure wave value at the exit of the tunnel is directly proportional to the pressure gradient inside the tunnel, and the pressure gradient value directly depends on the train speed [2]. With the development of high-speed rail technology, the running speed of trains gradually increases, and the problem of micro-pressure waves generated by trains entering the tunnel becomes more serious. Therefore, the MPW is a critical environmental problem that must be solved in the construction of high-speed railways. Its formation can be divided into three stages: compression wave generation, propagation and evolution, and micro-pressure formation. Extensive studies have been carried out on the mitigation of the MPW according to its formation process. The air in the tunnel is compressed violently when a train enters a tunnel at high speed, resulting in a sudden change in pressure, and a compression wave is formed. The intensity of the MPW is related to the pressure gradient of the compression wave, and various hoods set at the tunnel entrance have been found to be effective in reducing the gradient of the compression wave [3; 4]. [PERSON] [5] studied the effects of the blind hood with a section on the generation of compression waves through reduced scale experiments; the results showed that a unique optimum hood exists for a fixed train/tunnel blockage ratio. [PERSON] [6] conducted a model experiment to investigate the effective composition of an unvented hood, and the optimum cross-section area and length of the hood were obtained at 350 km/h. [PERSON] [7] investigated the effect of the cross-section area of hoods using axisymmetric models; they found that a hood with multi-step cross-sections is more effective than those with a constant cross-section. [PERSON] [8; 9; 10] studied the effects of different tunnel portals on tunnel aerodynamics using a moving model test; the results indicated that the hat oblique portal combined with a buffer structure with top holes is particularly effective. [PERSON] [11] revealed that opening holes on the hood can delay the rising time of the compression wave further by splitting the wavefront into multiple stages. [PERSON] [12] carried out a parametric investigation using a model experimental facility to study the effects of different portals, which indicated that the progressive entrance has a relieving impact on the gradient of the compression wave. [PERSON] [13; 14] proposed an entrance hood with air slits, which are designed to bio-mimic the \"ram ventilation\" technique, and a subsequent reduction in the MPW was observed in the reduced scale model test. [PERSON] [15; 16; 17; 18; 19] proposed a compression wave analysis method based on potential flow theory, and the influence of a portal on the initial thickness of the compression wave was studied. Numerical simulation is also widely used in the study of the tunnel portal. [PERSON] [20] studied the aerodynamic effects of a train passing through tunnels without and with a hood by numerical method, and an enlarged hood with an arch lattice-shell was proposed. [PERSON] [21] studied the formation of compressed wavefronts for different hoods based on the three-dimensional compressible Euler equation. When compressed waves propagate in tunnels, they tend to steepen gradually, influenced by inertia.