## 1 Introduction
[25] used the short-term Fourier transform (STFT) to study the time coefficient, and obtained a power spectral density diagram of the time coefficient corresponding to the typical modal fluid drag force. [PERSON] et al. [26] studied the power spectral density (PSD) of the far-field noise of an unheated axisymmetric jet at subsonic speeds by using the published noise spectrum and the obtained spectra, and obtained the Strouhal number scaling of the PSD at \(\theta=30^{\circ}\) or higher. [PERSON] [27] studied the spectral region using the principal oscillation patterns (POPs), and provided a synchronous optimization of the spatial and frequency-dependent representation of the second moments of the field. [PERSON] et al. [28] studied the traditional main oscillation mode, and pointed out that, in many practical situations, the time series is not stationary; that is to say, the first and second moments depend on the external cycle. The energy dissipation of hydraulic structures is crucial to the overall safety and stability of hydraulic engineering projects [29]. The energy dissipators in existing hydraulic engineering projects are often arranged on the surfaces of hydraulic structures within the project [30]. However, they typically dissipate energy by intensifying the aeration of the flow to dissipate energy, which can sometimes lead to problems such as cavitation and erosion in drainage structures, and even induce vibration damage to discharge structures [31]. [PERSON] et al. [32; 33] conducted a study on stepped spillways using model experiments, analyzing the variation patterns of conventional hydraulic parameters such as flow velocity, Froude number, and energy dissipation rate, as well as three relative hydraulic parameters along the spillway. Jet energy dissipation refers to the process in which, within a certain range, the water flow mixes with the medium, undergoes sudden expansions and contractions, and generates strong turbulence, effectively dissipating energy [34; 35]. [PERSON] et al. [36] used the Flow3D model to study the air entrainment and energy dissipation process on stepped spillway surfaces, discussing the initial air entrainment point, the division of air-entrainment zones, and scale effects related to air entrainment. [PERSON] et al. [37] conducted a comprehensive study of the Xiangjiaba Hydroelectric Plant using the 1:100 hydraulic model tests. They demonstrated that using multiple layers of horizontal submerged jets for energy dissipation in the flood discharge process is feasible, and the relevant parameters can meet the design requirements. [PERSON] et al. [38] presented a kind of jet-swirl cascade internal energy dissipator to solve the energy dissipation problem in the reconstruction of discharge tunnels with diversion tunnels and perform numerical simulation research on its feasibility and effectiveness, and the results showed that this energy dissipator has a good energy dissipation effect. [PERSON] et al. [39] studied two propagation models of pulsating pressure, namely, the transient flow model and the seepage model, to investigate the pressure propagation mechanism in the gap. The research results indicate that energy-dissipating water flow can impact downstream hydraulic structures and induce strong vibrations. It is urgent to explore energy dissipation patterns that can isolate energy dissipation structures from hydraulic engineering projects. Existing engineering studies such as underflow, pick flow, new stepped overflow, flexible overflow, etc. are often fixed on the surface of the discharge building and rigidly connected to the discharge building. The energy dissipation method can sometimes bring cavitation and erosion to the discharge building and even have the potential to induce vibration damage to the discharge building. Isolating the energy dissipator from the discharge structure is a feasible way to solve the above hazards, especially to solve the vibration damage of the discharge structure. The vertical jet energy dissipator is a kind of jet energy dissipator.