## 1 Introduction  
The considerable quantity of energy conveyed by the flood discharge and the intricate attributes of the high-speed water flow render the flood discharge vibration problem particularly pronounced. The hydraulic steel gate plays a pivotal role in regulating water discharge structures in hydraulic and hydropower engineering [1]. Its safe and flexible operation determines the success of the entire project and the safety of downstream people and property [2]. The development of gates is trending towards larger scales and lighter weights to meet the demands of high water heads, large orifices, and large discharge flows. Fluid-induced vibration (FIV), dynamic stability, and the safety and reliability of hydraulic gate structures under dynamic water action have received increasing attention from the water conservancy engineering community [3]. Due to the complex fluid-structure interaction (FSI), there are cases where the gate does not function properly or even becomes destabilized due to vibration [4]. The intelligent regulation of hydropower stations, which requires the gate to be partially opened in dynamic water, leads to complex operating conditions. FIV caused by flood discharge is the main cause of the dynamic instability of radial gates [5, 6]. The radial gates of discharge structures are primarily installed in spillways with low water heads and bottom outlets with high water heads. The front section of a high-head bottom outlet typically constitutes a pressurized section, where water flow through the pressure slope is regulated by the gate, transitioning from pressurized flow to open channel flow [7]. The submerged radial gate is often chosen as the type of pressurized outlet gate due to its ability to regulate flow stably. Following the pressurized section outlet radial gate, it is typically designed with a bottom drop and sudden lateral enlargement. This design feature helps protect the chute bottom and side walls from cavitation erosion by facilitating appropriate aeration [8; 9]. In this type of configuration, the bottom slope is typically steeper, and the flow velocity increases after passing through the gate [10], resulting in a typical open channel flow condition. This flow condition is significantly different from a submerged hydraulic jump, meaning that the water body downstream of the gate has a negligible impact on the vibration of the gate. Instead, the fluctuating water pressure acting on the gate panel is the direct cause of vibration [11]. The current state of research about gate vibration in high-head bottom outlets can be broadly categorized into three main aspects: prototype testing, hydraulic experiments, and numerical simulation. [PERSON] et al. [12] conducted a prototype test on the high-head and large-dimension gate in the emptying culvert of the Tianshengqiao power station. The variation patterns of flow fluctuating pressure and structural vibration parameters with gate opening indicated that the gate structure should avoid operation in a small gate opening (n = 0\(\sim\)0.1) and large gate opening (n = 0.7\(\sim\)0.9) to prevent cavitation erosion and fatigue damage to the building structure. [PERSON] et al. [13] elucidated the causes and mechanisms of the accompanying vibration of radial gates through prototype observations conducted at the Jinping hydropower station. Their findings suggested that the vibration of the mid-level orifice gate was attributed to the FSI between the gate and the turbulent flow fluctuating pressure. This vibration was subsequently transmitted through the dam body to the surface orifice gate, where it was amplified to a certain extent, resulting in significant vibrations of the surface orifice. [PERSON] et al. [14] conducted hydraulic experiments and three-dimensional finite element simulations to investigate the hydraulic characteristics, static and dynamic characteristics, dynamic stability, and FIV characteristics of submerged radial gates.