## 1 Introduction  
A gas turbine is a fluid machine designed based on the Brayton cycle, and is primarily used as an engine for generators or propulsion systems, therefore, its efficiency is a critical factor affecting its performance. The efficiency of a gas turbine can be improved through various methods, but the most effective approach is to increase the turbine inlet temperature, which enhances the cycle efficiency [1]. However, material limitations impose constraints on raising the turbine inlet temperature. To overcome these constraints, cooling techniques are employed to supply cooling fluid to both the interior and exterior of the turbine blades. However, the cooling fluid is delivered to the blades through secondary flow paths [2, 3], where gaps inevitably form between rotating and stationary components to enable rotation. These gaps can lead to leakage of the cooling fluid, necessitating appropriate sealing measures. Labyrinth seals have been widely used in sealing applications, however, in recent years, flexible and durable brush seals have been increasingly adopted. Compared with traditional labyrinth seals, brush seals can reduce leakage by up to 80% [4] and are better equipped to handle the temperature and pressure fluctuations that occur under high-temperature and high-pressure operating conditions. Thus, brush seals significantly contribute to improving the performance and lifespan of turbines [5] and are now commonly used in modern gas turbines [6]. In aerospace applications, brush seals offer distinct advantages over traditional sealing methods, particularly because of their ability to reduce leakage by up to 50%, which directly improves engine efficiency and reduces fuel consumption. Their design allows them to maintain tight clearances, even under challenging conditionssuch as thermal variations and stop/start operation, where traditional labyrinth seals tend to degrade. Their durability and consistent performance render brush seals especially valuable in modern aerospace engines, where precise clearance control is essential for optimizing propulsion and minimizing energy losses [7]. As shown in Figure 1, a brush seal consists of three primary components: the bristle pack, the front plate, and the backing plate. The bristle pack, which is composed of fine wires, is fixed at an angle between the front plate and the backing plate. Positioned above the surface of the rotor, the brush seal prevents leakage flow from the high-pressure side to the low-pressure side. The backing plate supports the bristle pack in withstanding pressure loads and prevents small-diameter bristles from escaping outward [8]. As shown in Figure 1a, the fence height refers to the radial distance between the back plate bore and the bristle bore. It plays a key role in determining the pressure limit of a brush seal; a greater fence height reduces the pressure limit due to increased stress on the bristles [9]. The bristles in the brush seal form a flexible barrier between the high-pressure (upstream) and low-pressure (downstream) regions by making contact with the rotor. The front plate ensures the bristle pack's stiffness, while the back plate counters pressure forces [10]. To minimize wear and allow the bristles to adjust to the rotor movement, they are inclined at an angle, known as the lay angle (\(\phi\)), of 30\({}^{\circ}\) to 60\({}^{\circ}\) with respect to the radial direction, as depicted in Figure 1b [11]. The performance characteristics of brush seals under varying flow conditions and different geometric configurations have been primarily investigated through experiments. [PERSON] et al. [12] reported that the initial wear of brush seals varies significantly, depending on the operating conditions. Additionally, they confirmed that brush seals can adapt to a range of operating conditions while maintaining stable performance over time. [PERSON] et al. [13] demonstrated that the performance of brush seals is influenced by mechanical strength and thermal stability.