## 1 Introduction  
Marine biofouling, which is caused by the settlement and accumulation of marine organisms on hull surface during sailing, can significantly impact the navigation efficiency and energy efficiency of ships [1; 2]. The additional fuel consumption caused by biofouling on global vessels can result in an estimated annual cost of USD 22-204 billion [3]. Moreover, biofouling can result in species invasion [4; 5; 6] and increased maintenance costs associated with artificial structures [7; 8]. Consequently, it is imperative to investigate cleaning technologies capable of eliminating biofouling. Cavitating jet technology emerges as a viable solution for in situ removal of biofouling without damaging the hull surface paint [9; 10], rendering it a preferable choice for practical applications. Cavitation is a phenomenon occurring when the local pressure falls below the saturated vapor pressure, vapor bubbles arise in water. The bubbles break at a solid surface in a submerged environment, generate high-pressure micro-shock jets, causing damage of the solid surface [11]. Thus, cavitation jets can be utilized for biofouling cleaning. Generally, the cavitation jet is produced in water, especially for biofouling cleaning, so the submerged cavitation jet was mainly researched in this paper. A cavitating nozzle is the main actuator for generating a cavitating jet in biofouling cleaning. The nozzle geometry has a significant impact on the intensity of the cavitation jet [12]. A lot of work has been performed on the design of cavitation nozzles to optimize the cavitation characteristics. [PERSON] et al. [13] proposed a cavitating nozzle optimization algorithm to improve axial vapor volume fraction and a 9.41% increment was realized. [PERSON] et al. [14] found that increasing the length and diameter of the resonator of the organ-pipe self-resonating cavitating nozzles contributed to the occurrence of cavitation. [PERSON] et al. [15] experimentally studied the influence of chamber dimensions on aggressive intensity of the cavitating jet. [PERSON] et al. [16]proposed a dual cavitating waterjet, which proved to be more efficient than the traditional cavitation nozzle. [PERSON] et al. [17] investigated a double-hole nozzle and the aggressive ability of cavitating jets was significantly improved in contrast to the single-hole nozzle. [PERSON] et al. [18; 19; 20] extensively researched the cavitation characteristics of a self-resonance pulsed organ pipe nozzle. They examined the influence of nozzle geometry, including the nozzle lip and upstream and downstream contraction ratios, on these characteristics. [PERSON] et al. [21] further conducted experimental studies on the impact of the downstream contraction ratio of the nozzle on its resonant chamber. Recent studies [22; 23; 24; 25; 26] have also investigated the erosion behavior and flow regime of jets generated by double-hole nozzles, revealing a significant enhancement in aggressiveness due to interactions between adjacent jets. In addition, surface roughness [27; 28], standoff [29; 30], and pressure [31] can also influence the cavitation characteristics. Computational fluid dynamics (CFD) and erosion test are two common methods for studying cavitating jet. Fluid simulation helps a lot in the visualization of flow structures and capturing intricate flow details, while the erosion test can be used to study the damage pattern and failure mechanisms of cavitating jet [32]. [PERSON] et al. [33] simulated cavitating flow in a venturi nozzle by several turbulence models and found that the models can reproduce the cavitating process well. [PERSON] et al. [34] provided a comprehensive explanation of the underlying mechanism responsible for the diverse distribution patterns observed on the impingement surface. They employed an inverse finite element method grounded in pitting analysis. The hybrid model was utilized due to its accurate depiction of both cavitation instability and periodic shedding [35; 36; 37; 38; 39]. [PERSON] et al.