Influencing Factors and Wavelet Coherence of Waves Generated by Submerged Jet

Abstract

This paper investigates the significance of various physical factors affecting the wave generated by submerged jet and the synchronization relationship between the wave surface process and different fluid dynamic parameters, based on three-dimensional numerical simulations using a large eddy simulation (LES) model. An orthogonal experimental design was employed, and range analysis and variance analysis revealed that the orifice contraction ratio has the most significant effect on wave height, followed by upstream water depth and orifice elevation. Through wavelet coherence and spectral correlation analysis, the wave surface process was examined in relation to fluid kinetic energy, Reynolds stress, and vortex structure parameters along the jet axis. The results indicate that regions of strong wavelet coherence are concentrated between 0.01 and 1.0 Hz. In the low-frequency range (0.01~1.0 Hz), there are narrow yet continuous coherence bands, while in the slightly higher frequency range (1.0~5.0 Hz), intermittent coherence relationships with wider bands are observed. Additionally, there is a certain degree of correlation between the power spectral density of the wave surface process and these physical quantities, with a maximum spectral correlation coefficient reaching 0.91. This study contributes to a deeper understanding of the factors affecting waves generated by submerged jets, enabling better prediction and control of their effects.

1. Introduction

Submerged jets are a complex flow phenomenon that occurs when fluid is discharged from an orifice into another fluid space, particularly in marine environments with a free surface. In ocean and coastal engineering, submerged jets are prevalent in applications such as forced stormwater discharge [1,2], wastewater discharge from desalination plants [3,4,5], and cooling water discharge from power plants [6,7]. The interaction between jets and free surfaces generates significant surface waves and turbulence, which can have adverse effects on marine ecosystems, the safe operation of vessels, and the stability of coastal and offshore structures. Therefore, a deeper understanding of the dynamics of wave generation by submerged jets is essential for developing effective solutions, optimizing fluid control strategies, and mitigating adverse impacts in ocean and coastal environments.

Submerged jets with a free surface experience significant changes in their flow structure due to the influence of the free surface boundary. Some studies have investigated the effects of different Reynolds numbers and Froude numbers on the flow structure, kinetic energy, turbulence intensity, and Reynolds stress at the free surface of submerged jets through experimental and numerical simulations [8,9,10,11,12,13,14], as well as the influence of nozzle geometry on the flow structure of the jets [15,16,17]. Additionally, other research has focused on the interactions between various jets and surface waves [7,18,19,20]. These studies focus on the impact of the free surface and waves on jet structure, but there is limited research on waves generated by submerged jets [21,22]. In particular, the effect and significance levels of various physical factors and their interactions on these waves are not yet clearly understood, and the correlation between wave processes and various turbulence parameters requires further study, necessitating advanced computational techniques for a better understanding.

Due to the limitations of measurement equipment, it is challenging for experiments to simultaneously obtain detailed flow field information and wave height time series without disturbing the wave surface, which is essential for analyzing their correlation [21]. However, numerical modeling is not subject to these limitations, providing an important tool for studying wave propagation processes and the interactions between waves, currents, and structures [23]. The LES method, by solving the motions of all turbulent scales above a certain threshold, can capture the large-scale effects and coherent structures in unsteady, non-equilibrium processes that the RANS method cannot handle, while avoiding the enormous computational load required by the DNS method, which must solve all turbulent scales. It has become a powerful tool for capturing turbulent structures and detailed flow patterns [24,25]. The waves generated by submerged jets are transient phenomena under the influence of large-scale turbulent structures, and the wave surface elevation-time series requires high-frequency data sampling. Therefore, the LES method is more suitable for this study.

The correlation between the wave surface process and fluid dynamic parameters such as kinetic energy, Reynolds stress, and vortex structures is significant for predicting and controlling submerged jets [26]. The wavelet coherence method and cross-wavelet transform methods offer a promising approach for this purpose, as they can detect the synchronicity between two time series in the time-frequency space, identify coherent regions across different frequency bands, and have been widely applied in natural sciences and engineering fields. Jiang et al. [27] analyzed the coherent relationship between water quality and water quantity in the Potomac River, USA, using wavelet coherence, and found significant differences in the coherent fluctuation behavior of turbidity between high-frequency and low-frequency regions. Guyennon et al. [28] studied the internal wave field of Lake Como, North Italy, using wavelet transform and wavelet coherence analysis. Lian et al. [29] studied the synchronization of vibrational energy between different discharge structures under pulsating water flow loads using cross-wavelet transform, analyzing the impact of various pulsating water flow loads on ground vibration. Chi et al. [30] analyzed the wavelet coherence relationship between coastal winds and surface and bottom currents using observation data and satellite remote sensing data, discussing the impact of typhoons on coastal currents in the East China Sea during different seasons.

Given these challenges and opportunities, this study aims to systematically and comprehensively investigate the factors influencing downstream waves generated by submerged jets, as well as the relationship between the wave surface process and fluid dynamics parameters. First, we establish a numerical model for submerged jets based on large eddy simulation (LES) and perform a grid independence test to determine a grid division method that ensures both computational accuracy and efficiency. Subsequently, the reliability of the numerical simulation results and the reasonableness of the parameters were validated. Using an orthogonal experimental design method, the significance levels of the factors affecting wave height were analyzed. Finally, we explore the fluctuation synchronization between the wave surface process and parameters, such as specific kinetic energy, Reynolds stress, and vortex structures based on the wavelet coherence method.

2. Materials and Methods

2.1. Governing Equations

The governing equations of the LES are the grid-filtered, unsteady incompressible Navier–Stokes equations, including the following continuity and momentum equations [31]:

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_i}}=0$$

(1)

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial t}}+{\displaystyle \frac{\partial {\overline{u}}_i{\overline{u}}_j}{\partial x_j}}=\overline{f_i}-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{u}{\rho }}{\displaystyle \frac{{\partial }^2{\overline{u}}_i}{\partial x_j\partial x_j}}+{\displaystyle \frac{\partial {\overline{\tau }}_{ij}}{\partial x_j}}$$

(2)

where $x_i$ and ${\overline{u}}_i$ denote the coordinate and filtered velocity in the i direction (i = 1, 2, 3), respectively; $\overline{p}$ represents filtered pressure; µ stands for the kinetic viscosity; ρ stands for fluid density; and $f_i$ is the external force on the fluid per unit mass. The subgrid-scale stress ${\overline{\tau }}_{ij}$, which represents the effect of smaller-scale motions, is defined as follows:

$${\overline{\tau }}_{ij}={\overline{u}}_i{\overline{u}}_j-\overline{u_i{u}_j}$$

(3)

According to the Smagorink–Lilly subgrid model [32,33]:

$${\overline{\tau }}_{ij}=2v_t{\overline{S}}_{ij}+{\displaystyle \frac{{\delta }_{ij}{\overline{\tau }}_{kk}}{3}}$$

(4)

$$v_t={\displaystyle \frac{{\mu }_t}{\rho }}={\left(C_S\Delta \right)}^2\sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}}$$

(5)

where ${\nu }_t$ denotes the subgrid-scale eddy viscosity coefficient, and $\Delta ={\left(\delta x_1\delta x_2\delta x_3\right)}^{1/3}$ represents the grid filter scale. ${\overline{S}}_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_i}}\right)$ stands for the rate of strain tensor for the resolved scale. The quantity $C_S$ is the Smagorinsky coefficient, which can be derived from the Kolmogorov coefficient and is taken as 0.18. δ is the Kronecker delta.

The VOF method is employed to capture the free surface and obtain the wave surface process, with a volume of fluid function F to define the water region. The physical meaning of the F function is the fractional volume of a cell occupied by water, with values ranging between 0 and 1. F is governed by the following equation [34]:

$${\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{\partial \left(u_{i}F\right)}{\partial x_i}}=0$$

(6)

where t represents time, and u_i and x_i are velocity components and coordinate components, respectively.

2.2. Numerical Method

The simulation of submerged jets in this study was conducted using the commercial software FLOW-3D (version V11.2) in conjunction with the LES model. The governing equations are discretized by the finite difference method. The pressure equation is solved implicitly, and the viscous stress term, free surface pressure, and convective property are solved explicitly. The convection term of the momentum equation is solved by the second-order monotonicity preserving difference scheme [35], with higher accuracy. The pressure solver selects the generalized minimum residual algorithm (GMRES) [36,37].

The finite difference solver is based on the structured Cartesian grid system. A smaller uniform grid was employed in key regions, such as the jet development zone, while non-uniform grid stretching was implemented near the boundaries of the computational domain to optimize efficiency without compromising solution accuracy. The fluid–solid boundaries are handled using the FAVOR™ (Fractional Area/Volume Obstacle Representation) method, which accurately represents solid boundaries within the structured grid by assigning fractional volumes and areas to cells intersected by solid surfaces.

2.3. Geometric Configuration and Boundary Conditions

The geometric configuration for the numerical simulation consists of an upstream pool, jet orifices, and a downstream pool, as shown in Figure 1a. There are two symmetrical rectangular orifices with a width of b = 0.10 m, a height of h = 2b = 0.20 m, and a length of 50 cm between the upstream and downstream pools, with a distance of 3.4b = 0.34 m between the orifices, as shown in Figure 1b. The structure of the rectangular orifice is shown in Figure 2, where Shape 1 has no contraction, Shape 2 is a rectangular orifice with a top contraction ratio of 5:10 at the outlet, and Shape 3 is a rectangular orifice with a top contraction ratio of 10:10 at the outlet. The downstream pool has a calculated length of 6 m, a width of 13b = 1.3 m, and a slope ratio of i = 1:0.3. Three blocks are used to grid the upstream reservoir, jet orifice, and downstream pool. All grids are structured orthogonal grids. In order to describe the propagation characteristics of water waves, a Cartesian coordinate system is established based on the projection of the right orifice center point on the floor as the coordinate origin O. The center line of the right orifice is along the flow direction as the x-axis, perpendicular to this centerline towards the left bank is the y-axis, and vertically positioned is the z-axis.

The upstream pool inlet and downstream pool outlet are defined as pressure inlet and pressure outlet, respectively, with water depths h₁ and h₂ measured from the bottom of the downstream pool. The inlet pressure and outlet pressure are calculated hydrostatically as p(z) = ρg(h − z). The top boundary is defined as a pressure boundary with a pressure value of 0 Pa, simulating atmospheric pressure. The orifices are arranged at the altitude h₃ from the bottom of the downstream pool. The dimensionless water depths of the upstream and downstream pools are given by l = (h₁ − h₃)/h and l’ = (h₂ − h₃)/h. Considering that the downstream water depth typically varies only slightly in practical engineering, the downstream dimensionless water depth l’ is kept constant at 1.8. In order to avoid errors in wave reflection at the outlet boundary, a 50 cm thick wave-absorbing layer that only acts on wave motion is installed at the outlet. This layer introduces additional damping in a designated region before the open boundary, effectively dissipating wave energy and reducing reflection. All solid boundaries are treated as non-slip walls with u = 0 and the standard wall function is used to resolve the near-wall turbulence. At time t = 0, the entire computational domain is initialized with still water, characterized by a hydrostatic pressure distribution and zero velocity where u(x,y,z,0) = 0, and p(x,y,z,0) = ρg(h(x,y) − z). Two flow monitoring baffles are arranged in the downstream pool to verify whether the calculation has reached stability. A set of measuring points is arranged along the central axis of the two orifices every 20 cm to obtain data such as flow velocity, pressure, and water surface elevation, as shown in Figure 1c. The sampling frequency is 100 Hz. The simulation time of the model is 1000 s.

2.4. Wavelet Coherence

Wavelet transform can simultaneously reflect the characteristics of time series in both time and frequency domains. The wavelet coherence coefficient $R_{xy}^2$ of two time series x(t) and y(t) [38,39] is defined as follows:

$$W{T}_{xy}\left(\alpha ,\tau \right)=W{T}_x\left(\alpha ,\tau \right)W{T}_y^{*}\left(\alpha ,\tau \right)$$

(7)

$$R_{xy}^2\left(\alpha ,\tau \right)={\displaystyle \frac{{\left|S\left({\tau }^{-1}W{T}_{xy}\left(\alpha ,\tau \right)\right)\right|}^2}{S\left({\tau }^{-1}{\left|W{T}_x\left(\alpha ,\tau \right)\right|}^2\right)S\left({\tau }^{-1}{\left|W{T}_y\left(\alpha ,\tau \right)\right|}^2\right)}}$$

(8)

where $W{T}_{xy}\left(\alpha ,\tau \right)$ represents the wavelet cross spectrum of x(t) and y(t); $W{T}_x\left(\alpha ,\tau \right)$ and $W{T}_y\left(\alpha ,\tau \right)$, respectively, represent the continuous wavelet transform of time series x(t) and y(t); and * denotes complex conjugation. $R_{xy}^2\left(\alpha ,\tau \right)$ is the wavelet coherence coefficient, ranging from 0 to 1, with values closer to 1 indicating stronger synchronization of fluctuations between the two sequences at that time and frequency. $\alpha$ is the scaling factor that corresponds to frequency; $\tau$ is the time factor that corresponds to time; and $S\left( \right)$ represents the smoothing operator on the scale factor and time factor.

The complex argument $\arg \left[W{T}_{xy}(\alpha ,\tau )\right]$ can be interpreted as the relative phase or phase difference between x(t) and y(t) in the time-frequency domain, which is given by

$$\arg \left[W{T}_{xy}(\alpha ,\tau )\right]=\arctan \left[\mathrm{Im}\left(W{T}_{xy}\left(\alpha ,\tau \right)\right)/\mathrm{Re}\left(W{T}_{xy}\left(\alpha ,\tau \right)\right)\right]$$

(9)

A phase angle of 0 indicates in-phase, while a phase angle of π indicates anti-phase. A positive phase angle means x(t) leads y(t), and a negative phase angle means x(t) lags behind y(t).

3. Numerical Model Validation

3.1. Grid Independence Test

To achieve a grid-independent numerical solution, taking Shape 1 under the conditions of l = 5.0 and l′ = 1.8 as an example, four grid division schemes have been considered. Block 1 in the upstream pool adopts a 0.04 × 0.04 × 0.04 m grid, while Block 2 in the jet orifices and Block 3 in the downstream pool are as shown in

Table 1. Grid size and flow relative error.

Schemes	Block 1 Upstream Pool	Block 2 Jet Orifices	Block 3 Downstream Pool	Grid Quantity (Million)	Inlet Flow (L/s)	Outlet Flow (L/s)	Relative Error (%)
Mesh 1	0.04 × 0.04 × 0.04 m	0.04 × 0.04 × 0.04 m	0.02 ×0.02 ×0.02 m	1.548	130.4	145.1	11.27
Mesh 2	0.04 ×0.04 × 0.04 m	0.02 × 0.02 × 0.02 m	0.02 ×0.02 ×0.02 m	1.768	130.4	121.5	−6.82
Mesh 3	0.04 × 0.04 × 0.04 m	0.02 × 0.02 × 0.02 m	x- and y-directions 0.02 m, z-direction 0.01–0.02 m	2.314	130.4	130.0	−0.31
Mesh 4	0.04 × 0.04 × 0.04 m	0.02 × 0.02 × 0.02 m	x- and y-directions 0.015 m, z-direction 0.005–0.015 m	5.246	130.4	130.6	0.15

. In Mesh 3 and Mesh 4, the grid size in the z-direction gradually increases from 0.01 m and 0.005 m to 0.02 m and 0.015 m, respectively, as the distance from the free surface increases. As the number of grids increases, the relative error gradually decreases. However, when the grid number increased from 2.314 million to 5.246 million, the relative error did not significantly decrease despite the doubling of the grid number, as shown in Table 1. Therefore, considering computational accuracy and time, Mesh 3 was selected for the numerical simulations in this study.

3.2. Velocity Validation

To validate the simulation accuracy of the jet flow field, a numerical simulation model was established that matches the square submerged jet experiment conducted by Sankar et al. [40] using the methods described earlier. Figure 3a–d shows the comparison between the numerical simulation and experimental velocities at the centerline of the jet orifice for cross-sections at x/h = 0.5, 2, 5, and 10, with a downstream submergence depth of 1.5h. The velocity U and the vertical distance z were non-dimensionalized by the jet exit velocity U_j and the orifice dimension h, respectively. The results agree well, with a maximum error of 10%, indicating the reliability of the numerical simulation method.

3.3. Wave Height Validation

To validate the simulation accuracy of the wave surface processes, a mathematical model with the same dimensions as the experiments conducted by Zhang et al. [22] was established, and simulations for Case 2~Case 5 were performed. As shown in Figure 4, the root mean square wave height amplitude ${\eta }_{RMS}$ obtained from numerical simulations shows a consistent trend with the experimental measurements, with a maximum error not exceeding 30%. As the relative distance x/h increases, the ${\eta }_{RMS}$ gradually decreases. There is some systematic bias because the numerical model fails to capture certain smaller waves.

4. Results and Discussion

4.1. Influencing Factors of the Wave Height

The orifice altitude h₃, orifice shape, and upstream pool water depth h₁ were selected as factors influencing wave height, with each factor having three levels, as shown in

Table 2. Three factors and their corresponding levels in the study.

Level	Factors
	A Orifice Altitude (h3/b)	B Upstream Water Depth (l)	C Orifice Shape
1	2.4	4.5	Shape 1
2	1.4	5.0	Shape 3
3	0.4	5.5	Shape 2

. Since the orthogonal experiment conditions are evenly distributed among all experiment conditions, the optimal conditions found exhibit a consistent trend with comprehensive experiment. The orthogonal experiment design table, based on the L₉(3⁴) orthogonal array, is shown in

Table 3. The orthogonal experiment schemes with three factors and three levels.

Experiment Sequence Number	Factors
	A Orifice Altitude (h3/b)	B Upstream Water Depth (l)	C Orifice Shape
1	A1 (2.4)	B1 (4.5)	C1 (Shape 1)
2	A1 (2.4)	B2 (5.0)	C3 (Shape 3)
3	A1 (2.4)	B3 (5.5)	C2 (Shape 2)
4	A2 (1.4)	B1 (4.5)	C3 (Shape 3)
5	A2 (1.4)	B2 (5.0)	C2 (Shape 2)
6	A2 (1.4)	B3 (5.5)	C1 (Shape 1)
7	A3 (0.4)	B1 (4.5)	C2 (Shape 2)
8	A3 (0.4)	B2 (5.0)	C1 (Shape 1)
9	A3 (0.4)	B3 (5.5)	C3 (Shape 3)

, and LES was used to simulate these experiments listed in Table 3. An example of the instantaneous velocity contour of the jet profile for orifice Shape 1 at h₃/b = 1.4 and l = 5.5 is shown in Figure 5. Since the position where the submerged jet diffuses to the free surface varies with different orifice shapes and upstream water depths, the dimensionless value of the root mean square wave height amplitude ${\eta }_{RMS}/h$ at two measurement points x/h = 10 and 15 was selected as an indicator of the influence on wave height, as shown in Figure 6. These two positions correspond to characteristic zones in the jet’s development, representing the transition from the potential core region to the fully developed turbulent region. The larger the value of ${\eta }_{RMS}/h$, the greater the influence. ${\eta }_{RMS}/h$ is always greater than x/h =15 at x/h =10, which is consistent with Zhang et al. [22].

4.1.1. Range Analysis

Range analysis was used to assess the influence of various factors and their levels on the orthogonal experiment results. T_ij represents the average value of experimental indicators for the jth factor at the ith level. R_j denotes the range of T_ij for the jth factor. The larger the R_j value, the greater the impact of the jth factor on the experimental indicator, indicating higher sensitivity of the indicator to the jth factor. The results of the range analysis are presented in

Table 4. Range analysis of factors affecting wave height.

Factors	${\mathit{\eta }}_{\mathit{R}\mathit{M}\mathit{S}}/\mathit{h}$ at x/h = 10			${\mathit{\eta }}_{\mathit{R}\mathit{M}\mathit{S}}/\mathit{h}$ at x/h = 15
	A	B	C	A	B	C
$T_{1j}\left(\times {10}^{-2}\right)$	8.14	6.44	12.56	5.41	4.83	7.90
$T_{2j}\left(\times {10}^{-2}\right)$	10.59	7.85	9.11	5.90	6.92	6.66
$T_{3j}\left(\times {10}^{-2}\right)$	6.72	11.15	3.78	6.32	5.87	3.06
Dominant level	2	3	1	3	2	1
$R_j\left(\times {10}^{-2}\right)$	3.87	4.71	8.78	0.91	2.09	4.84
The influence order of factors	C > B > A			C > B > A

.

At x/h = 10, because $T_{3A}<T_{1A}<T_{2A}$, the dominant level for factor A (orifice elevation) is A₂ (h₃/b = 1.4). Similarly, B₃ and C₁ are the dominant levels for factors B and C, respectively. Under the experimental conditions A₂B₃C₁, ${\eta }_{RMS}/h$ is the largest at x/h = 10. At x/h = 15, under the experimental conditions A₃B₂C₁, ${\eta }_{RMS}/h$ is the largest. At both x/h = 10 and x/h = 15, the influence of the factors on ${\eta }_{RMS}/h$ follows the order R_C > R_B > R_A. This indicates that the orifice contraction ratio has the greatest influence on wave height, followed by the upstream water depth, with the orifice altitude having the least influence.

4.1.2. Variance Analysis

Range analysis does not account for data fluctuations within the experiment, nor can it determine whether the influence of each factor is significant [41]. Variance analysis effectively addresses these shortcomings and can evaluate the influence of each factor on the experimental results. The formulas for calculating the F-value used to test significance levels are as follows:

$$SST={{\displaystyle {\sum }_{k=1}^r\left({{\eta }_{RMS}/h}_k-\overline{{\eta }_{RMS}/h}\right)}}^2,S{S}_j={\displaystyle \frac{r}{m}}{{\displaystyle {\sum }_{i=1}^m\left(T_{ij}-\overline{{\eta }_{RMS}/h}\right)}}^2,SSE=SST-{\displaystyle {\sum }_{j=1}^{n}S{S}_j}$$

(10)

$$f_T=r-1,f_j=m-1,f_e=f_T-{\displaystyle {\sum }_{j=1}^n{f}_j}$$

(11)

$$F_j={\displaystyle \frac{M{S}_j}{MSE}}={\displaystyle \frac{S{S}_j/f_j}{SEE/f_e}}$$

(12)

where SST, SS, SSE, and MS represent the total sum of squares, the between-groups sum of squares, the within-groups sum of squares, and the mean square, respectively; the ${\eta }_{RMS}/h_k$ is the dimensionless wave surface amplitude of the kth experiment; r is the number of experiments; n is the number of factors; m is the number of levels; f_T is the total degree of freedom; f_j is the degree of freedom of the jth factor; f_e is the degree of freedom of the deviation. When F_j > F_0.05 and F_j > F_0.01, the experiment results are significant and highly significant, respectively.

At x/h = 10 and x/h = 15, the influence of the orifice shape (factor C) on ${\eta }_{RMS}/h$ is generally significant, especially at x/h = 15, where it is extremely significant, as shown in

Table 5. Analysis of variance for factors affecting wave height at x/h = 10.

Source of Variance	SSj	fj	MSj	Fj	Fcrit	Significant
Factor A	22.90 × 10−4	2	11.49 × 10−4	4.21	F0.05(2,2) = 19 F0.01(2,2) = 99	Generally significant
Factor B	35.04 × 10−4	2	17.52 × 10−4	6.42
Factor C	117.51 × 10−4	2	58.76 × 10−4	21.52
Random error	5.46 × 10−4	2	2.73 × 10−4
Sum	180.99 × 10−4	8

and

Table 6. Analysis of variance for factors affecting wave height at x/h = 15.

Source of Variance	SSj	fj	MSj	Fj	Fcrit	Significant
Factor A	1.25 × 10−4	2	0.62 × 10−4	0.75	F0.05(2,4) = 6.94 F0.01(2,4) = 18	Extremely significant
Factor B	6.57 × 10−4	2	3.28 × 10−4	3.96
Factor C	37.87 × 10−4	2	18.94 × 10−4	22.81
Random error	3.31 × 10−4	4	0.83 × 10−4
Sum	47.75 × 10−4	8

. At both positions, the order of influence of the three factors is consistent: C (orifice shape) > B (upstream pool water depth) > A (orifice altitude), which aligns with the results of the range analysis.

4.2. Wavelet Coherence of the Wave Surface Process and Kinetic Energy

Based on the results of LES, the coherence between the kinetic energy per unit weight of the fluid along the centerline of the orifice jet and the wave surface process was analyzed using the wavelet coherence method. The kinetic energy can be characterized by the dimensionless velocity head $P_u$, which is defined as follows:

$$P_u={\displaystyle \frac{u_{x/h}^2/2g}{u_j^2/2g}}={\left({\displaystyle \frac{u_{x/h}}{u_j}}\right)}^2$$

(13)

where $u_j$ denotes the flow velocity at the outlet of the jet orifice; $u_{x/h}$ denotes the flow velocity along the center axis of the orifice at a location x/h from the orifice; and g is the acceleration of gravity.

The following contents of this paper were based on Shape 1 with orifice elevation h₃ = 2.4b. The distributions of the wavelet coherence coefficients $R_{xy}^2$, and the phase angle between the wave surface process and $P_u$, are shown in Figure 7 and Figure 8 for x/h = 10 at upstream water depths l = 3.0~5.5, and x/h = 10~15 at upstream water depths l = 4.0, respectively. The horizontal coordinate in the figure indicates the time t and the vertical coordinate indicates the frequency f. Different colors identify the $R_{xy}^2$ value at the corresponding coherence time and coherence frequency.

The fluctuation synchronization between the wave surface process and the velocity head at x/h = 10 is stronger in the frequency range of 0 to 1.0 Hz under different upstream water depths. However, this strong coherence is not continuous but appears intermittently. Under upstream water depths l = 4.0 and 4.5, the wave surface process shows stronger intermittent coherence with the velocity head, indicating higher coherence compared to other water depths. In the 0~1.0 Hz, the phase difference varies significantly across different upstream water depths. At upstream water depths l = 3.0 and 3.5, the phase difference is nearly in anti-phase; at l = 4.0 and 4.5, the phase difference is approximately 135°; and at l = 5.0, it is around 100° and −80°. However, it is worth noting that at l = 5.5, the phase difference is about 30° and −30°.

At the same upstream water depth, the distribution of higher $R_{xy}^2$ values and the phase difference between adjacent points remains relatively unchanged during wave propagation, suggesting a consistent coherence pattern along the flow direction and indicating minimal variation in the fluctuation synchronization between the wave surface and the kinetic energy per unit weight of fluid. The phase difference is mostly near 180°, and the wave surface process is 1/2 cycle ahead of the velocity head.

4.3. Wavelet Coherence of the Wave Surface Process and Reynolds Stress

Reynolds stress is the additional stress caused by the intermixing or pulsation of liquid particles, which can reflect the intensity of convective transport in all directions of the turbulent flow field, including normal additional stress and tangential additional stress. In a three-dimensional turbulence field, the Reynolds stress can be expressed as a second-order tensor formed by the correlations of the fluctuating velocity components at a given point in space, defined as ${\tau }_{ij}=-\rho u_i^{\prime }{u}_j^{\prime }$, where ${\tau }_{ij}$ is the ij-component of the Reynolds stress tensor, $u_i^{\prime }$ and $u_j^{\prime }$ denote the fluctuating components of the velocity in the i and j directions, respectively, and $\rho$ is the liquid density.

The wave height time series was collected along the streamwise (x-direction). The wave surface process showed a stronger correlation with the velocity fluctuations $u_z^{\prime }$ in the vertical (z-direction), while the velocity and kinetic energy in the spanwise (y-direction) were relatively smaller. Therefore, the discussion will focus on the shear stresses ${\tau }_{xz}$ and normal stresses ${\tau }_{zz}$ related to the velocity fluctuations $u_x^{\prime }$ in the streamwise direction and $u_z^{\prime }$ in the vertical direction, along the central axis of the orifice jet. Figure 9 and Figure 10 show the distributions of wavelet coherence coefficients $R_{xy}^2$ and the phase angle between the wave surface process and Reynolds shear stress ${\tau }_{xz}$ at x/h = 10 for different upstream water depths (l = 3.0~5.5), and at x/h = 10~15 for the upstream water depth l = 4.0, respectively. The distribution of $R_{xy}^2$ and the phase angle between the wave surface process and Reynolds normal stress ${\tau }_{zz}$ is presented in Figure 11 and Figure 12.

4.3.1. Reynolds Shear Stress

The distribution of $R_{xy}^2$ between the wave surface process and the Reynolds shear stress is similar to that of the kinetic energy. In the frequency domain, there is a strong fluctuation synchrony between the wave surface process and the Reynolds shear stress ${\tau }_{xz}$ between 0 and 1.0 Hz. In the time domain, the time range of the maximum $R_{xy}^2$ at most measurement points is relatively broad but lacks continuity, which indicates that the fluctuation synchrony between the wave surface process and Reynolds shear stress ${\tau }_{xz}$ is intermittent.

At the same measurement point, for different upstream water depths l, the larger $R_{xy}^2$ is concentrated between 0 and 1.0 Hz, while it appears intermittently between 1.0 and 5.0 Hz. The distribution of the larger $R_{xy}^2$ also varies over different time periods. Taking the position at x/h = 10 as an example, as the upstream water depth l increases, there is no obvious pattern in the fluctuation synchrony between the wave surface process and the Reynolds shear stress ${\tau }_{xz}$, which is presumed to be related to the randomness and disorder of turbulence [42]. When l = 3.0, 4.5, and 5.5, the extreme values of $R_{xy}^2$ are larger, and the frequency bands are wider. When l = 3.5, 4.0, and 5.0, the regions of larger $R_{xy}^2$ exhibit stronger temporal continuity. When l = 5.5, there is a stronger fluctuation synchrony at the initial and final time periods of sampling.

At the same upstream water depth, the overall correlation between the distribution regions of $R_{xy}^2$ at various measurement points along the streamwise direction is not strong. However, there is a certain similarity in the distribution of $R_{xy}^2$ maxima and the phase difference between adjacent measurement points, which is different from kinetic energy. Taking l = 4.0 as an example, at x/h = 10, when t = 0~120 s, the $R_{xy}^2$ ranges from 0.50 to 0.80. Additionally, there are several intermittent maxima between 0.1 and 1.0 Hz. At x/h = 11, when t = 0~150 s, the $R_{xy}^2$ increases to between 0.50 and 0.94, indicating enhanced fluctuation synchrony between the wave surface process and the Reynolds shear stress ${\tau }_{xz}$ at this position. At x/h = 12, when t = 0~150 s, $R_{xy}^2$ decreases to between 0.50 and 0.63, and when t = 150~240 s, the $R_{xy}^2$ has an additional extreme region with narrower frequency bands than the previous point. At these points, there are both positive and negative phases, and the phase difference is close to 90°, which means that the wave surface process is 1/4 cycle ahead of the Reynolds shear stress ${\tau }_{xz}$. At x/h = 13 and 14, the distribution regions of $R_{xy}^2$ maxima are quite similar, mainly concentrated at the initial and final sampling periods, and the phase difference is also mainly in-phase. When the wave propagates to x/h = 15, there is no significant change in the distribution of $R_{xy}^2$, but the frequency band becomes narrower.

4.3.2. Reynolds Normal Stress

For the wavelet coherence coefficient $R_{xy}^2$ between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$, the regions of larger $R_{xy}^2$ at each measurement point under different upstream water depths are distributed in the frequency domain between 0 and 1.0 Hz, while more intermittent occurrences are observed between 1.0 and 5.0 Hz. This indicates a strong fluctuation synchrony between the wave surface process and the ${\tau }_{zz}$ between 0 and 1.0 Hz. In the time domain, larger values of $R_{xy}^2$ are distributed over a wider temporal range at most measurement points, but with poor continuity. This suggests that the fluctuations between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$ are intermittently synchronized.

At the same measurement point, with the increase in upstream water depth, the fluctuation synchrony between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$ did not exhibit an obvious pattern. At x/h = 10, it can be observed that when l = 3.0, 3.5, and 4.5, there exist multiple intermittent extreme regions of $R_{xy}^2$. This implies that fluctuations between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$ are predominantly intermittently synchronized at these water depths. When l = 4.0 and 5.0, $R_{xy}^2$ exhibits continuous extreme value regions in the low-frequency range, and the phase difference in the region is consistent. When l = 5.5, the distribution regions of larger $R_{xy}^2$ significantly increase, suggesting an overall better fluctuation synchrony between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$. At the same time, the phase difference changes from near the anti-phase to near the in-phase between 0.01 and 0.1 Hz.

At the same upstream water depth, along the streamwise direction, the distribution of larger wavelet coherence coefficients $R_{xy}^2$ between the wave surface process and the Reynolds normal stress ${\tau }_{zz}$ shows some similarity, indicating that the fluctuations are somewhat synchronized. For instance, when l = 4.0, at x/h = 10, there is a continuous extreme value region around 0.01 Hz. At x/h = 11, 12, 13, 14, and 15, when t = 160~300 s and f = 0.01 ~0.10 Hz, $R_{xy}^2$ shows continuous extreme value regions, with a phase difference of 90~135°. At x/h = 12 and 15, the extreme values of $R_{xy}^2$ are larger. At x/h = 14, between t = 0 s and 70 s, the bandwidth corresponding to the extreme values of $R_{xy}^2$ increases. At x/h = 15, the frequency band broadens.

In summary, at different upstream water depths, the wave surface process and the Reynolds stress exhibit strong fluctuation synchronization between 0 and 1.0 Hz, albeit intermittently. At the same measurement point, with the increase in upstream water depth, the fluctuation synchronization between the wave surface process and Reynolds stress did not show a significant pattern. At the same water depth, the wavelet coherence coefficient and the phase difference distribution of adjacent measurement points show some correlation as the waves propagate, indicating some degree of fluctuation synchronization between them.

4.4. Wavelet Coherence of Wave Surface Process and Eddy Structure Parameter

Eddies play a crucial role in the diffusion and dissipation of energy in turbulence, significantly impacting the generation, maintenance, and development of turbulence. When submerged jets enter the downstream pool, intense turbulent mixing occurs between water bodies, generating a large number of eddies with varying scales, shapes, and rotation directions. This study employs the Q-criterion method proposed by Hunt et al. [43] for eddy structure identification, which is a widely used eddy identification method based on the velocity gradient tensor, especially for regions where the second Galilean invariant of the velocity gradient tensor Q > 0 represent vortex structures. For incompressible fluids, according to the Cauchy–Stokes decomposition, the expression for $\nabla \mathit{V}$ can be written as:

$$\nabla \mathit{V}=\mathit{A}+\mathit{B}$$

(14)

$$\mathit{A}={\displaystyle \frac{1}{2}}\left(\nabla \mathit{V}+\nabla {\mathit{V}}^T\right)$$

(15)

$$\mathit{B}={\displaystyle \frac{1}{2}}\left(\nabla \mathit{V}-\nabla {\mathit{V}}^T\right)$$

(16)

$$Q={\displaystyle \frac{1}{2}}\left({‖\mathit{B}‖}_F^2-{‖\mathit{A}‖}_F^2\right)$$

(17)

where ‖ ‖_F denotes the Frobenius norm of a matrix, and A and B represent the symmetric and antisymmetric tensors resulting from the decomposition of the velocity gradient tensor $\nabla \mathit{V}$, respectively.

Figure 13 and Figure 14 show the distribution of the wavelet coherence coefficient $R_{xy}^2$ and the phase angle between the wave surface process and the eddy structure parameter Q value along the centerline of the Shape 1 jet orifice at x/h = 10 under different upstream water depths (l = 3.0~5.5), and at x/h = 10~15 under an upstream water depth of l = 4.0. The wave surface process and the Q value exhibit strong fluctuation synchronization in the frequency range of 0~1.0 Hz, with no significant variation due to changes in the measurement location or upstream water depth. The distribution of larger $R_{xy}^2$ spans a wide time range at most measurement points but lacks continuity, which suggests that strong fluctuation synchronization appears intermittently rather than continuously. The phase difference of about 0.1 Hz at the same measuring point has the same direction, and most are anti-phase.

As the upstream water depth l increases, the fluctuation synchronization between the wave surface process and the Q value at the same measurement point shows little change in the frequency domain but significant change in the time domain. For example, at the measurement point x/h = 10, regions with larger $R_{xy}^2$ are primarily concentrated between 0 and 1.0 Hz, while they appear more intermittently between 1.0 and 5.0 Hz. When l = 3.0, 3.5, 4.5, and 5.0, larger $R_{xy}^2$ intermittently occur around 0.10 Hz, and the phase difference is close to anti-phase. However, when l = 4.0 and 5.5, the $R_{xy}^2$ shows stronger temporal coherence, with longer-lasting synchronized frequency bands and better coherence. Especially when l = 5.5, the phase difference changes to in-phase behavior in significant sections, which is similar to the phase difference change in velocity head and Reynolds normal stress.

Under the same upstream water depth, there is a certain correlation in the distribution regions of larger $R_{xy}^2$ at different measurement points. This indicates that the coherence between the wave surface process and the eddy structure parameter Q shows some similarity along the streamwise direction, which is likely related to the propagation of waves and flow. For l = 4.0, at six measurement points, in the range of t = 100~260 s and f = 0.01~0.1 Hz, there exists a well-defined region of continuity, and the phase difference is between 90~135°. However, at x/h = 13 and 15, the frequency band is narrower.

4.5. Spectral Correlation Analysis

To further investigate the correlation in the frequency domain and the similarity in the power spectral density between the kinetic energy, Reynolds stress, eddy structure parameters, and the wave surface process during wave propagation, the Pearson correlation coefficient $\rho$ of the power spectral density between different measurement points was calculated using the following formula:

$$\rho ={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\left[S_{xx}\left(f\right)-\overline{S_{xx}\left(f\right)}\right]\left[S_{yy}\left(f\right)-\overline{S_{yy}\left(f\right)}\right]}}{\sqrt{{\displaystyle {\sum }_{i=1}^n{\left[S_{xx}\left(f\right)-\overline{S_{xx}\left(f\right)}\right]}^2}}\sqrt{{\displaystyle {\sum }_{i=1}^n{\left[S_{yy}\left(f\right)-\overline{S_{yy}\left(f\right)}\right]}^2}}}}$$

(18)

where $S_{xx}\left(f\right)$ and $S_{yy}\left(f\right)$ represent the power spectral densities of the random time series x(t) and y(t), respectively. To avoid the influence of the direct current component in the power spectral density on the spectral correlation coefficient, it is necessary to remove the mean values of the physical quantities with non-zero means, before estimating the power spectral density [44].

According to the Pearson correlation coefficient $\rho$ shown in Figure 15, there is a correlation between the power spectra. The spectral correlation coefficients at each point are small for upstream depth l = 3.0 and show a tendency to increase as the relative distance x/h increases. The spectral correlation coefficients increase gradually as the upstream water depth l increases. When l is between 4.0 and 5.5, the spectral correlation coefficients at the same measurement point do not differ significantly, and at these upstream depths, the changes in the spectral correlation coefficients are not significant as the relative distance x/h increases.

Except for the upstream water depth l = 3.0, the correlations between the power spectra are strong, and the spectral correlation coefficient reaches up to 0.91, which may be related to the turbulence intensity in the downstream pool. Under l = 3.0, the flow velocity at the jet outlet is relatively low, resulting in weaker turbulence intensity in the downstream pool. The water flow ejected from the orifice cannot easily reach the water surface directly, leading to an insignificant correlation between the wave surface process and the orifice flow velocity. As the upstream water depth increases, the flow velocity of the water ejected from the orifice also increases, allowing it to spread directly to the free surface. Consequently, the correlation between the wave surface process and the flow velocity is enhanced. Since flow velocity head, Reynolds stress, and eddy structure parameters are all physical quantities related to flow velocity and its fluctuations, their spectral correlation coefficients also increase.

5. Conclusions

In this paper, based on LES and wavelet analysis methods, the influencing factors of waves generated by submerged jets and the coherence between the wave surface process and various physical quantities were investigated. The main conclusions are as follows:

(1)

The LES model effectively simulates the flow field of submerged jets and wave propagation, providing reliable data on surface wave behavior and flow characteristics.

(2)

Orthogonal experimental design combined with range analysis and variance analysis revealed that under the conditions of this paper, the orifice contraction ratio has the most significant effect on wave height, followed by upstream water depth, while orifice altitude has the least effect. The most influential combination of orifice altitude and upstream depth varies by location, but Shape 1 always dominates.

(3)

Wavelet analysis revealed that along the jet centerline, strong coherence between the wave surface process and variables such as velocity head, Reynolds stress, and eddy structure parameters mainly occurs in the low-frequency range (0.01–1.0 Hz) with consistent phase. In the higher range (1.0–5.0 Hz), the coherence becomes intermittent. When the upstream water depth l = 5.5, the wave surface process and flow variables are in-phase, while lower upstream depths exhibit anti-phase behavior.

(4)

Spectral correlation analysis indicates that the power spectral density of the wave surface process is correlated with those of the velocity head, Reynolds stress, and eddy structure parameters, with the maximum spectral correlation coefficient reaching 0.91.

This study clarifies the key factors affecting waves generated by submerged jet and reveals their coherence with flow dynamics, contributing to jet flow research. Future work will further investigate how wave surface responses vary with initial conditions at different locations, deepening the understanding of wave behavior induced by submerged jets.