## 1 Introduction
### Numerical Approaches to Modeling High-Velocity Jets  
Numerical simulations of free-water jets in the atmosphere [15] and within plunge pools have provided insights into jet behavior, including pressure fields at the stagnation point. However, challenges remain in accurately modeling the complex interactions, particularly in high-velocity, turbulent flows, and air-water interactions [16; 17; 18]. These include accurately modeling the turbulence and shear forces in the flow, assessing the turbulent flow components, and capturing air-water interactions. The complexity of accurately simulating plunge pool flows is exacerbated by mesh sensitivity and near-wall boundary layer dynamics, mainly using methods like two-phase volume-of-fluid and Euler-Euler. Addressing these challenges requires a careful selection of flow simulation and turbulence models, as well as discretization schemes, along with rigorous validation against reliable experimental data.  
[PERSON] et al. [19] analyzed the velocity profile and pressures at the stagnation point of free rectangular jets, providing valuable data for understanding jet behavior in plunge pools. Their numerical simulations agreed with experimental data regarding the velocity profile and pressure estimates near the stagnation point. [PERSON] et al. [20] conducted a numerical simulation of a free water jet in the atmosphere, which yielded reasonable agreement with experimental data for velocity, pressure, and volume fraction. [PERSON] et al. [21] further analyzed mean velocity and turbulent kinetic energy within a plunge pool, achieving good correspondence between experimental data and theoretical models. [PERSON] et al. [22] simulated a physical model of a spillway using the Volume of Fluid (VoF) method, concluding that it adequately simulated important variables such as water depths in the plunge pool and pressures at the jet impact area. [PERSON] et al. [23] conducted numerical simulations of a jet impacting a plunge pool, comparing their findings with experimental data obtained using k-Epsilon and Detached Eddy Simulation (DES) models for turbulence. They considered a jet velocity of 6.0 m/s, and good agreement was found for pressure readings at the plunge pool bottom. These authors successfully replicated pressure fluctuations that aligned with experimental results using the DES model, although their simulations were limited to a single-phase (water) flow, excluding air-water interactions. Notably, both experimental and CFD pressure fluctuation data were acquired at a frequency of 100 Hz, which, given the presence of significant energy content at higher frequencies, may introduce uncertainty in the measured pressure fluctuations [10; 11; 12; 13; 14; 24].  
[PERSON] and [PERSON] [16] compared the various two-phase volume-of-fluid and Euler-Eulers, concluding that the Euler-Euler approach is more suitable for modeling two-phase water-air flows. Similarly, [PERSON] et al. [18] evaluated the efficiency of twoPhaseEulerFoam relative to the InterFoam-VoF solver in modeling a spillway aerator. They found that the Euler-Euler solver provided better cost-efficiency and significant advantages for hydraulic engineering applications involving aerated spillway flows.