Prediction of Geometrical Characteristics of an Inclined Negatively Buoyant Jet Using Group Method of Data Handling (GMDH) Neural Network

Abstract

A new approach to predicting the geometrical characteristics of the mixing behavior of an inclined dense jet for angles ranging from 15° to 85° is proposed in this study. This approach is called the group method of data handling (GMDH) and is based on the artificial neural network (ANN) technique. The proposed model was trained and tested using existing experimental data reported in the literature. The model was then evaluated using statistical indices, as well as being compared with analytical models from previous studies. The results of the coefficient of determination (R²) indicate the high accuracy of the proposed model, with values of 0.9719 and 0.9513 for training and testing for the dimensionless distance from the nozzle to the return point $x_r/D$ and 0.9454 and 0.9565 for training and testing for the dimensionless terminal rise height $y_t/D$. Moreover, four previous analytical models were used to evaluate the GMDH model. The results showed the superiority of the proposed model in predicting the geometrical characteristics of the inclined dense jet for all tested angles. Finally, the standard error of the estimate (SEE) was applied to demonstrate which model performed the best in terms of approaching the actual data. The results illustrate that all fitting lines of the GMDH model performed very well for all geometrical parameter predictions and it was the best model, with an approximately 10% error, which was the lowest error value among the models. Therefore, this study confirms that the GMDH model can be used to predict the geometrical properties of the inclined negatively buoyant jet with high performance and accuracy.

1. Introduction

The unwanted wastewater produced from seawater desalination plants is normally discharged back into the coastal water body adjacent to the plant. This is performed with a submerged pipe, creating a buoyant effluent jet. This effluent (brine) has a high level of salinity, which has negative impacts on the marine ecosystem, particularly in the near-field zone of the discharge point [1,2]. Currently, most new desalination plants use reverse osmosis (RO) systems due to their significantly lower energy consumption [3]. This method (RO) still produces large quantities of brine effluent with a high level of salinity, meaning that it is denser than the receiving water (forming a negatively buoyant jet). Consequently, the submerged pipe can enhance the dilution of this effluent, thereby minimizing the adverse impacts on the marine environment.

A large body of previous research has focused on laboratory studies of the mixing of the negatively buoyant jet, either alone or in combination with modeling efforts [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. However, less effort has been dedicated to numerical modeling and developing an analytical solution to predict the behaviors and geometric characteristics of inclined dense jets [27,28,29,30], which still require further investigation. Therefore, studies using numerical modeling as an efficient tool could be useful for the prediction the behaviors of brine discharge in seawater.

As is known, there are two hypotheses for the development of models, either based on entrainment or jet spreading, which have been developed and studied previously. In 2003, Lee and Chu [27] developed a model called JETLAG/VISJET to predict the mixing of the near-field zone of an inclined round buoyant jet. This model is a combination of the two approaches, i.e., entrainment and jet spreading, and presents empirical correlations for the dilution of the geometrical characteristics. In 2004, Jirka [28] developed an integral model (CORJET) that could predict negatively buoyant jets for a wide range of angles (0° to 90°) discharged into ambient water with a slope or flat bottom. A few years later, Kikkert et al. [29] presented an analytical method to predict the behavior of an inclined dense jet, using experimental data reported in previous studies to validate their model. Their model has limitations as it is only valid for angles ranging from 0° to 75° and for densimetric Froude numbers ($F{r}_d$), i.e., the ratio of inertia to buoyancy, ranging from 14 to 99. In the same manner as Kikkert et al. [29], Oliver et al. [30] developed a modified integral model called the reduced buoyancy flux (RBF) model to predict the behavior of negatively buoyant jets in near-fields. One of the advantages of this modified model is its ability to predict the effects of the additional mixing noted in previous studies. However, more data are required to further improve the understanding of the behaviors and geometric characteristics of inclined dense jets. These data would also be useful for the calibration, validation, and comparison of diverse numerical models in future research studies. In addition, there has been limited research conducted on the prediction of inclined dense jets using different analytical methods, indicating a need for further investigation in this area.

In recent years, efforts in different areas of the engineering science field have used artificial neural network (ANN) methods to address a wide range of applications, such as multigene genetic programming (MGGP), genetic algorithm artificial neural networks (GAA), genetic programming (GP), gene expression programming (GEP), and the GMDH. Therefore, regarding the outcomes obtained from previous studies, it is worth implementing some of these methods to predict the geometrical characteristics of inclined buoyant jets.

One of the mentioned methods is the GMDH, which is used to solve nonlinear engineering problems while providing an explicit equation to predict and model each phenomenon. The use of this method has led to accurate results compared to those of other methods [31,32,33,34,35,36,37,38,39,40]. For example, Alitaleshi and Daghbandan [32] conducted a study to evaluate the quality of treated water in a water treatment plant using a GMDH-type neural network. The results showed the success of the used (GMDH) model for prediction as the results of the determination coefficient were higher than 0.97. Naeini et al. [33] developed a model using the GMDH to estimate the elasticity modulus ($E_s$) of clayey deposits, which was used in predicting the elastic settlement of foundations. The results obtained were compared with previous empirical equations, and it was found that the GMDH model showed higher performance (32 to 42% improvement) with respect to the other correlations. Another study was conducted by Gholami et al. [35] to examine the ability of the GMDH model in predicting the geometric variables of stable channels (width, depth, and slope). The results showed that the GMDH model performed well. Moreover, they compared the results obtained from the GMDH with those from previous theoretical equations (based on regression analysis), and this comparison showed the ability of the GMDH model to predict the geometric variables of channels better than other theoretical equations. Najafzadeh et al. [39] used the GMDH to predict the scour depth around a vertical pile exposed to regular waves and compared their results with those of different previous methods. They concluded that the use of the GMDH model provided the most accurate prediction of the scour depth compared to the other models.

Recently, Alfaifi et al. [41] have used a GMDH model to predict the geometrical characteristics of inclined dense jets for angles of 15° and 52°. Their study presents experimental findings on inclined buoyant dense jets at angles of 15° and 52°, focusing on key geometric characteristics like $x_m$,$y_m$, $x_r$, and $y_t$ It includes 42 experiments with varying densities and densimetric Froude numbers $F{r}_d$, investigating how these factors influence the jet trajectories. Equations for dimensionless geometric characteristics are derived using a GMDH neural network, showing that the jet behavior is primarily affected by the $F{r}_d$ values. The results obtained were compared with previous data, showing good agreement for the 15° angle and slight variations for 52°, attributed to data limitations. They found that the GMDH model can be highly useful in providing good predictions for the mixing of inclined negatively buoyant jets. Their study was pioneering in its use of the GMDH to predict the mixing behavior of inclined buoyant jets, marking a significant step towards the comparison of this method with other commonly used technical tools in the field.

It is believed that the above study (Alfaifi et al. [41]) represents the first time that this new approach has been used in the field of mixing jet behavior with a wide range of angles. These researchers have established empirical equations for the estimation of the geometrical characteristics of inclined buoyant jets. The GMDH was selected over other AI techniques due to its unique capability to self-organize models and its robustness in handling complex, non-linear relationships. Additionally, the GMDH results provided accurate predictions for the parameters tested compared to other analytical models, demonstrating its effectiveness and reliability in this application. While Alfaifi et al. [41] utilized a GMDH model to predict the geometrical characteristics of inclined dense jets specifically for angles of 15° and 52°, our study represents a novel application of the GMDH in the domain of mixing jet behavior. Unlike the previous work, which focused on dense jets, we have directed our attention towards inclined negatively buoyant jets spanning a wider range of angles from 15° to 85°. This expansion in angle coverage is significant as it enables a more comprehensive understanding of the behavior of buoyant jets across varying inclinations, which is crucial for environmental impact assessments and engineering design considerations. Furthermore, our study introduces empirical equations tailored to estimating the geometrical characteristics of inclined buoyant jets, addressing a specific gap in the literature. The decision to employ the GMDH over other AI techniques was based on its unique ability to self-organize models and effectively handle the complex, non-linear relationships inherent in jet behavior phenomena. The results obtained from our application of the GMDH model not only demonstrate its accuracy in predicting the desired parameters but also highlight its effectiveness and reliability compared to alternative analytical models, as highlighted in our findings. Therefore, while Alfaifi et al. [41] laid the groundwork by employing the GMDH model for inclined jets at specific angles, our study significantly advances this research area by extending the angle range, introducing tailored empirical equations, and displaying the superior performance of the GMDH in predicting the behavior of inclined negatively buoyant jets.

Therefore, the goal of the present study is to examine the superiority of the GMDH model in predicting the main geometrical characteristics of inclined negatively buoyant jets by performing a comprehensive study for a wide range of angles ranging from 15° to 85°. Moreover, the performance of the GMDH model is assessed by comparing its results to those of other analytical and numerical methods. To the best of the authors’ knowledge, the GMDH has not been used before for the prediction of inclined negatively buoyant jets.

2. Methodology

2.1. Analysis of Inclined Negatively Buoyant Jet

In Figure 1, an inclined turbulent round dense jet being discharged into a body of stagnant water body is illustrated (the ambient velocity is $U_a=0$). The port (jet) for the discharge has a diameter $D$ and is positioned at an initial angle (${\theta }_0$) compared to the horizontal plane. The ambient water into which the jet is being discharged is unstratified and has a constant density ${\rho }_a$. This water is deep enough to allow the jet to fully develop below the water surface, without any effects from the water surface. The initial discharge velocity of the jet is $U_0$ and it has a density ${\rho }_0$ (${\rho }_a<{\rho }_0$).

The four crucial characteristics of the mixing of the jet with regard to geometry are its terminal rise height ($y_t$); the horizontal distance from the point of maximum height of the centerline to the nozzle ($x_m$); its centerline height ($y_m$); and its return point ($x_r$), i.e., the distance horizontally at which the jet’s centerline returns to the same elevation as the tip of the nozzle (Figure 1). The distribution pattern of a jet being discharged into a homogeneous and stagnant ambient fluid can be affected by various parameters, including the source angle ${\theta }_0$, $D$, $U_0$ and the initial density difference ($\Delta \rho ={\rho }_0-{\rho }_a$) between the ambient fluid and the discharge fluid. In addition, inclined dense jets can be characterized by their discharge volume flux $Q_0{=U}_0{\pi D}^2/4$, momentum flux $M_0=U_0^2{D}^2\pi /4$, and buoyancy flux $B_0={\mathrm{g}}_0^{\prime } U_0\pi D^2/4$, where $g^{\prime }=g\left({\rho }_0-{\rho }_a\right)/{\rho }_a$, which is defined as the effective gravitational acceleration, and $\mathrm{g}$ is the gravitational acceleration. Two length scales can be formed from these fluxes: the momentum length scale $L_M=M_0^{3/4}/B_0^{1/2}$, which determines the distance over which the buoyancy of the jet is less important than its momentum, and the source length scale $L_Q=Q_0/M_0^{1/2}$, which determines the distance over which the source discharge is important. These two length scales are employed to determine the mixing and geometric characteristics of a turbulent buoyant jet. The equation defining the jet’s densimetric Froude number ($F{r}_d$) is as follows, according to Roberts et al. [19]:

$$F{r}_d={\displaystyle \frac{U_0}{\sqrt{g^{\prime }D}}}.$$

(1)

The terminal rise height of an inclined dense jet can be written in terms of the length scales as

$${\displaystyle \frac{y_t}{L_M}}=f\left({\displaystyle \frac{L_M}{L_Q}},{\theta }_0\right).$$

(2)

Besides the length scales, $y_t$ can also be alternatively expressed by using the jet’s densimetric Froude number and nozzle diameter as

$${\displaystyle \frac{y_t}{F{r}_{d}D}}=C_{y_t}\left({\theta }_0\right).$$

(3)

Similarly, the other geometric parameters of the jet (i.e.,$x_r$, $x_m$, and $y_m$) can be derived by using the coefficient values ($C_{x_r}$, $C_{x_m}$, and $C_{y_m}$), which can be established through experiments.

2.2. GMDH Method

An artificial neural network (ANN) method is an artificial intelligence technique that operates based on the functions of the human brain’s neural network. A well-established and effective type of artificial neural network (ANN) utilized for prediction and modeling is a self-organizing network (SON), which includes the polynomial network. Neural networks and linear regression methods are integrated into these network types, with one training algorithm for self-organizing networks (SONs) involving the group method of data handling (GMDH). This algorithm was first proposed by Ivakhnenko [42], and it is a computer program-based algorithm that is often used to model problems that involve data series for multi-input–single-output systems. Second-order polynomials serve as network functions in the GMDH, and its primary advantage over traditional neural networks lies in its ability to create mathematical models for a given procedure. The concept of the GMDH in terms of selection and hybridization corresponds to the genetic algorithm. Like other types of neural networks, the GMDH can have one or more hidden layers. Typically, when establishing a model for a multi-input–single-output system, Volterra–Kolmogorov–Gabor (VKG) polynomials are useful. An example of this utility is demonstrated in Equation (4) according to Ebtehaj et al. [38] and Gholami et al. [35], which states

$$y=a_0+{\displaystyle \sum }_{i=1}^n{a}_i{x}_i+{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^n{a}_{ij}{x}_i{x}_j+{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^n{a}_{ijk}{x}_i{x}_j{x}_k+\dots$$

(4)

where $x=\left(x_1, x_2, \dots , x_n\right)$ represents the vector for the input, while $y$ is the output. The coefficients for the polynomial, or weight vectors, are denoted as $a_i$, $a_{ij}$, and $a_{ijk}$. VKG polynomials are approached using quadratic polynomials, with the second-order polynomials determined by various input pairs. Hence, the GMDH can be employed to “train” a system. A neural network based on the GMDH will form a multilayer, feed-forward network that combines input pair “neurons” through a quadratic polynomial. If the number of input variables is $m$, the number of neurons in the first layer is

$$L_1=\left(\begin{array}{c}2\\ m\end{array}\right)=m\left(m-1\right)/2$$

(5)

Each network layer comprises one or more processing units, each having one output port and two input ports, assumed to be second-order polynomials. The undetermined parameters in the GMDH are often the polynomial coefficients. Figure 2 provides a schematic of the GMDH approach, including the structure of each neuron (N-Adaline).

To establish the network structure and determine the output value of the input vectors $x=\left(x_{i1}, x_{i2}, \dots , x_{in}\right)$, the target function is minimized. The objective function aims to reduce the errors in estimating the modeling coefficients of the GMDH algorithm (Ebtehaj et al. [43]), i.e.,

$$AIC=n log\left(MSE\right)+2\left(N+1\right)$$

(6)

where the number of neurons in the model is represented as $N$, while $n$ is the sample number, and the mean square error is denoted as $\mathit{MSE}$. Consequently, there are six unknown coefficients in each equation, as shown below in Equation (7):

$$\{\begin{array}{c}{y}_1=a_0+a_1{x}_{1p}+a_2{x}_{1q}+a_3{x}_{1p}{x}_{1q}+a_4{x_{1p}}^2+a_5{x_{1q}}^2\\ y_2=a_0+a_1{x}_{2p}+a_2{x}_{2q}+a_3{x}_{2p}{x}_{2q}+a_4{x_{2p}}^2+a_5{x_{2q}}^2\\ ------------------\\ ------------------\\ y_N=a_0+a_1{x}_{Np}+a_2{x}_{Nq}+a_3{x}_{Np}{x}_{Nq}+a_4{x_{Np}}^2+a_5{x_{Nq}}^2\end{array}$$

(7)

The matrix form of Equation (7) is represented as follows:

$$Aa=Y$$

(8)

where

$$A=\left[\begin{array}{cccccc}1& x_{1p}& x_{1q}& x_{1p}{x}_{1q}& {x_{1p}}^2& {x_{1q}}^2\\ 1& x_{2p}& x_{2q}& x_{2p}{x}_{2q}& {x_{2p}}^2& {x_{2q}}^2\\ \hspace{1em}& ·& ·& ·\hspace{1em}··& · ·& \hspace{1em}\\ \hspace{1em}& ·& ·& ·\hspace{1em}··& · ·& \hspace{1em}\\ 1& x_{Np}& x_{Nq}& x_{Np}{x}_{Nq}& {x_{Np}}^2& {x_{Nq}}^2\end{array}\right]$$

(9)

$$a={\left\{a_0, a_1, a_2, a_3, a_4, a_5 \right\}}^T$$

(10)

$$Y=\left\{y_1, y_2, y_3, \dots \dots , y_N \right\}$$

(11)

Using multiple regression analysis, the least squares method is applied to determine the values of each coefficient as follows:

$$a={\left(A^{T}A\right)}^{-1} A^{T}Y$$

(12)

However, there are significant limitations in the GMDH approach, including (i) the requirement for second-order polynomials, (ii) the restriction that each neuron’s inputs come only from adjacent layers, and (iii) the limitation that each neuron’s inputs consist of only two parameters.

GMDH Modeling Setup

To use the GMDH for the prediction of the geometrical characteristics of a negatively buoyant jet, the output variables were set to be dimensionless jet characteristics (i.e., ${\mathrm{x}}_{\mathrm{m}}/D$,${ \mathrm{y}}_{\mathrm{m}}/D$, ${\mathrm{x}}_{\mathrm{r}}/D$, and ${\mathrm{y}}_{\mathrm{t}}/D$) by using one output variable for each run. For all cases, the ${\mathrm{Fr}}_{\mathrm{d}}$ and ${\theta }_0$ parameters were used as the input variables. The explicit equations for the GMDH were developed using the GMDH Shell 4.2 software [44]. The dataset used in this study covered all discharge angles presented in previous studies, ranging from 15° to 85°. The data collected in earlier investigations are illustrated in

Table 1. The collected data reported in earlier investigations.

No.	Investigators	Angles
1	Roberts et al. [19]	60°
2	Cipollina et al. [4]	30°, 45°, and 60°
3	Kikkert [5]	15°, 30°, 45°, and 60°
4	Lai [21]	15°, 30°, 38°, 45°, 52°, and 60°
5	Shao and Law [17]	30° and 45°
6	Papakonstantis et al. [13]	45°, 60°, 75°, 80°, and 85°
7	Oliver [24]	15°, 30°, 45°, 60°, 70°, and 75°
8	Bashitialshaaer et al. [10]	30°, 45°, and 60°
9	Abessi and Roberts [23]	60°
10	Roberts and Abessi [45]	15°
11	Jiang et al. [20]	30° and 45°
12	Abessi and Roberts [9]	15°, 20°, 30°, 40°, 45°, 50°, 55°, 60°, 65°, 70°, 75°, 80°, and 85°
13	Abessi and Roberts [7]	30°, 45°, and 60°
14	Crowe [15]	15°, 30°, 45°, 60°, 65°, 70°, and 75°
15	Papakonstantis and Tsatsara [25]	15°, 30°, 35°, 50°, and 70°
16	Papakonstantis and Tsatsara [18]	35°, 50°, and 70°
17	Alfaifi et al. [41]	15° and 52°

. Obtaining a good prediction from the GMDH model requires the preparation of the dataset, and therefore the observed dataset used in this study was randomly divided into two groups, one for training and one for testing. For each case, the GMDH model was trained using 70% to 80% of the dataset, while the remaining data were used to test the model. In this study, the layers were limited to a maximum of two in order to ensure the simplicity of the model. The developed models were used to calculate the dimensionality of the jet geometrics as a function of ${\mathit{Fr}}_d$ and ${\theta }_0$ for the training dataset. The optimal models were then selected and used to perform additional calculations for the testing dataset. Finally, the performance of these models was evaluated.

For the training dataset, the models developed were used to determine the dimensionless jet geometrics as a function of ${\mathit{Fr}}_{\mathrm{d}}$ and ${\theta }_0$, with the optimal models being selected. After this, the models selected were used to perform additional calculations for the testing dataset, and then their performance was evaluated.

3. Results and Discussion

In this section, the results of the proposed model (GMDH) are presented. The predicted results of the GMDH are compared with the observed data obtained from previous studies for angles ranging from 15° to 85°. For brevity, only the results of $x_r/D$ and $y_t/D$ are shown in this section. The results of the dimensionless geometrical parameters $x_r/D$ and $y_t/D$ for the actual data for training and testing are plotted versus the predicted values of the GMDH model, as shown in Figure 3a,b. In order to evaluate the performance of the GMDH model, a linear fit with the coefficient of determination ($R^2$) is added to each figure to show the accuracy of the proposed model in predicting and fitting the actual data. From these figures, it can be seen that the $R^2$ values show excellent prediction for the proposed model (GMDH), where 0.9719 and 0.9513 were obtained for training and testing for $x_r/D$ and 0.9454 and 0.9565 were obtained for training and testing for $y_t/D$. The testing results for both parameters show the good prediction performance of the model. Although there was a decrease in the value of the $R^2$ for $x_r/D$ for the testing model compared to the training model, the accuracy remained high and acceptable. This indicates that all fitting lines passed through most of the data, and the proposed model is highly accurate and satisfactory.

Figure 4a,b, show a detailed depiction of the performance of the GMDH model for ${\mathrm{x}}_{\mathrm{r}}/\mathrm{D}$ (Figure 4a) and ${\mathrm{y}}_{\mathrm{t}}/\mathrm{D}$ (Figure 4b), as well as illustrating how the model was trained and tested with the data index. To obtain Figure 4a, the model was trained with 75% of the data, while, to obtain Figure 4b, it was trained with 70% of the data. As mentioned previously, the model was trained on between 70% and 80% of the data. Therefore, the key to changing the training ratio is to obtain a simple model with high accuracy.

The equations extracted from the GMDH for all geometrical parameters are presented in

Table 2. Nonlinear equations extracted from GMDH for dimensionless geometrical parameters.

Geometrical	GMDH Proposed Equations
$x_m/D$	$=-31.538+1.94873 {\theta }_0-0.0234242 {\theta }_0^2+1.63203 F{r}_d$	(13)
$x_r/D$	$=-0.35311+2.9030 F{r}_d+0.065091 F{r}_d N3-0.13909 {F{r}_d}^2-0.005735 N{3}^2$  $N3=-47.2107+2.77506 {\theta }_0-0.0340387 {\theta }_0^2+2.82939 F{r}_d$	(14)
$y_m/D$	$=2.37956-0.139279 {\theta }_0+0.0263993 {\theta }_0 F{r}_d+0.0026574 {\theta }_0^2+0.113531 F{r}_d$	(15)
$y_t/D$	$=1.00174+0.0336764 {\theta }_0 F{r}_d-0.000250209 {\theta }_0^2+0.0418149 F{r}_d$	(16)

. These equations were used to predict the geometrical parameters in this study. As can be seen, all equations in Table 2 are functions of $F{r}_d$ and ${\theta }_0$. These equations are valid for $F{r}_d$ ranging from 10 to 100. All equations presented in Table 2 can be used to predict the geometrical parameters of inclined dense jets for any angles located between 15° and 85°. All of these equations are presented in the form of dimensionless geometric parameters, which depend on the same input variables: $F{r}_d$ and $\theta$.

3.1. Statistical Assessment of Model Performance

This study presents some of the most commonly used statistical indices to evaluate the performance of the GMDH model. These indices can clearly show the accuracy of the proposed model as they are commonly used in the literature. In the present study, the three statistical indices used were the coefficient of determination ($R^2$), the root mean squared error (RMSE), and the mean absolute error (MAE), which are defined as follows:

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle \sum }{\left(\hat{\mathrm{y}}-\overline{\mathrm{y}}\right)}^2}{{\displaystyle \sum }{\left(\mathrm{y}-\overline{\mathrm{y}}\right)}^2}}$$

(17)

$$\mathrm{MAE}={\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\left|\hat{\mathrm{y}}-\mathrm{y}\right|}{\mathrm{n}}}$$

(18)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left(\hat{\mathrm{y}}-\mathrm{y}\right)}^2}{\mathrm{n}}}}$$

(19)

where $\widehat{y}$ is the predicted value of the GMDH model, $\overline{y}$ is the mean of the actual data, $\mathrm{y}$ denotes the actual data, and $n$ is the number of actual data. As is known, low values of these parameters (except ${\mathrm{R}}^2$) indicate the high accuracy of the model. The statistical results obtained for the GMDH model from these Equations (i.e., (17)–(19)) are presented in

Table 3. Results of statistical indices for the training and testing of the GMDH model.

Geometrical Parameter	R2	MAE	RMSE	R2	MAE	RMSE
Training				Testing
$x_m/D$	0.948	5.911	8.239	0.936	6.809	9.192
$x_r/D$	0.971	6.052	8.556	0.951	6.499	10.143
$y_m/D$	0.962	4.009	5.711	0.947	4.458	6.861
$y_t/D$	0.945	5.471	8.298	0.956	5.236	7.804

. The results of the GMDH model used to predict the geometrical characteristics of the inclined jet show the high accuracy of the prediction, as shown by the ${\mathrm{R}}^2$ results. All geometrical results are higher than 0.94, which means that the proposed model produced excellent prediction results. The highest value of the ${\mathrm{R}}^2$ was 0.967 for $x_r/D$, while the lowest value for the same index was for $x_m/D$, which was 0.948. The most interesting findings in Table 3 are the testing results, which reflect the accuracy of the proposed model. For the MAE, the lowest value was noted for $y_m/D$ (4.458), while the lowest value for the RMSE was found for $x_m/D$ (6.861). On the other hand, the highest value for the MAE was shown for $x_m/D$ (6.809), while, for the RMSE, the highest value was found for $x_r/D$ (10.143). Generally, all results obtained for the statistical indices shown in Table 3 show excellent agreement between the observed dataset and the proposed model (GMDH), indicating that it can be used to predict the geometrical parameters of inclined dense jets with highly accurate results.

3.2. Comparing GMDH Results with Previous Models

In this study, another evaluation of the performance of the GMDH model was conducted by comparing it with several analytical and numerical models reported in previous studies. Four analytical models described in the literature were selected. These analytical solution models were CORJET, introduced by Jirka [46]; the model introduced by Kikkert et al. [29]; VISJET, introduced by Lee and Chu [27]; and the model introduced by Oliver et al. [47]. The results of the GMDH model for the dimensionless geometrical characteristics of $x_r/D$ and $y_t/D$ versus the $F{r}_d$ are shown in Figure 5 and Figure 6, along with those of the previous analytical models. For brevity, only the results for five angles (i.e., 15°, 30°, 45°, 60°, and 75°) for $x_r/D$ and six angles (i.e., 15°, 30°, 45°, 60°, 75°, and 85°) for $y_t/D$ are presented in this study. The existing dataset indicated that the highest angle in the literature was 75° for $x_r/D$, while 85° was the highest for $y_t/D$. Due to a lack of previous data for some of the geometrical parameters, the results of only two analytical models (CORJET and VESJET) are shown in several figures. In Figure 5, the GMDH model’s predicted results are presented in black lines to more effectively demonstrate the predictions as compared to the observed data obtained from the previous experiments using other analytical models (see Figure 5a,c). It is clear that the GMDH model shows accurate predictions and its results are almost identical to and overlap with the previous analytical models at some angles (i.e., 45° and 60°); see Figure 5b,c. This superior performance at these specific angles can be attributed to the higher accuracy of the analytical model for 45° and 60° compared to other angles. The underlying assumptions and conditions of the analytical model align more closely with the characteristics of these angles, leading to better predictive accuracy for the GMDH model.

Another comparison of the GMDH’s predicted results for $y_t/D$ is presented in Figure 6a–e. A fifth angle was added for the $y_t/D$ comparison (i.e., 85°) to test the predictive capabilities of the proposed model. In these figures, two other models are added to the comparison (i.e., CORJET and VISJET). The proposed model (GMDH) showed good predictions for all angles. It should be noted that it occupied a neutral position regarding the data scatter. In addition, we can observe the convergence of the GMDH and VISJET models presented by Lee and Chu [27] in Figure 6b,c. The predicted results for the other analytical models (i.e., Kikkert et al. [29]; Oliver et al. [47]) were slightly higher than those of the other models as the inclination angle increased, while the predicted results of the CORJET model were the lowest in comparison to the actual data and other models.

To quantify the variations in the results obtained from these models (the proposed and previous models) and the actual data, the statistical index equation called the standard error of the estimate (SEE) can be used; it can indicate the preferred model and give a more precise prediction of the actual data. The lower the value of the SEE, the more accurate the model. The SEE equation is defined as follows:

$$\mathrm{SEE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }{\left(\hat{\mathrm{y}}-\mathrm{y}\right)}^2}$$

(20)

The results of the models’ predictions of the geometrical parameters at specific angles are summarized in

Table 4. Results of the standard error of the estimate for all models.

Geometrical Parameter	Angle	GMDH Model	CORJET	VISJET	Kikkert et al. [29]	Oliver et al. [30]
$x_r/D$	15°	6.39	-	-	7.94	6.86
	30°	7.26	-	-	12.40	12.58
	45°	10.82	-	-	11.72	10.77
	60°	9.42	-	-	12.29	9.51
	75°	4.35	-	-	20.80	4.99
$y_t/D$	15°	3.35	-	-	3.72	3.66
	30°	3.82	7.86	7.57	7.40	7.57
	45°	11.13	12.16	11.31	15.71	12.44
	60°	9.81	12.17	10.96	18.72	12.31
	75°	3.93	-	-	12.00	12.95
	85°	9.06	-	-	15.53	16.39

. The standard error values show that the GMDH model performed better and was much more accurate than the other models. All of the SEE results for the GMDH model were the lowest for all angles and for both geometrical parameters, except for angle 45° for $x_r/D$, where the result of Oliver et al.’s model was the lowest. Therefore, the SEE results illustrate that all fitting lines of the GMDH model were satisfactory for all geometrical parameter predictions, as the scattered data were located close to the fitting lines, with an approximately 10% error around the lines. As a result, the GMDH model was the most accurate model among those tested.

Accordingly, this provides evidence that the GMDH model is a reliable model for the prediction of the geometrical parameters of the near-field mixing zone of an inclined dense jet.

Figure 7a,b show the results of the constant values for the normalized forms ($x_r/DF{r}_d$) and ($y_t/DF{r}_d$) for the GMDH model plotted against the initial discharge angle, along with the data reported in previous studies. Third-order polynomial fittings were obtained for all models and are shown in both figures. Despite the large dispersion in the actual data, the fitting curve of the proposed model reflects the best prediction compared to the other models in terms of being close to most of the actual data. Moreover, the results obtained from the GMDH model are consistent with what was previously stated by [7], indicating that the horizontal distance of the return point increases with an increasing angle up to approximately 40°, after which the distance begins to decrease. In Figure 7b, the results show that the terminal rise height gradually increases as the angle of the jet increases until reaching an approximate angle of 85°, at which point the fitting curve remains almost constant, without any change. In this figure, the GMDH model demonstrates good agreement with the observed data. Generally, it can be seen that the GMDH model shows better results for the predicted angles regarding the observed data compared to the other models.

3.3. Uncertainty Analysis

Uncertainty analysis is crucial to assess the confidence in the results obtained and evaluate the proposed GMDH model. Therefore, to measure the uncertainty in the prediction of the geometrical characteristics obtained from the GMDH model, several explicit statistics are employed, such as the prediction error ($E_i$), the mean prediction error ($MPE$), the standard deviation ($s_d$), and the width of the uncertainty band ($WUB$). These parameters are calculated for the whole dataset employed in this study. Negative and positive values of the $MPE$ demonstrate that the models underestimate or overestimate the observed values, respectively. These parameters are as follows [48]:

$${\mathrm{E}}_{\mathrm{i}}={\mathrm{Prediction}}_{\mathrm{i}}-{\mathrm{Actual}}_{\mathrm{i}},$$

(21)

$$\overline{\mathrm{E}}={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{E}}_{\mathrm{i}}/\mathrm{n},$$

(22)

$$\mathrm{MPE}={\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{Prediction}-\mathrm{Actual}\right)}{\mathrm{n}}},$$

(23)

$${\mathrm{s}}_{\mathrm{d}}={\displaystyle \frac{\sqrt{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{E}}_{\mathrm{i}}-\overline{\mathrm{E}}\right)}^2}}{\left(\mathrm{n}-1\right)}},$$

(24)

$$\mathrm{WUB}=1.96{\displaystyle \frac{{\mathrm{s}}_{\mathrm{d}}}{\sqrt{\mathrm{n}}}}$$

(25)

where $n$ is the sample size. Moreover, a confidence band was defined around the prediction error values for the $MPE$ and $s_d$ by using the Wilson score method without continuity correction. The difference between the low and high uncertainty bands is defined as the WUB. The model can be considered highly accurate when the value of the WUB is smaller. Therefore, a confidence band of 95% was achieved using ±1.96 $s_d/\sqrt{n}$ values [49].

The results of the uncertainty analysis obtained for the GMDH model’s predictions of the $x_m/D$, $x_r/D$, $y_m/D$, and $y_t/D$ are presented in

Table 5. Uncertainty analysis for GMDH model’s predictions of the geometrical characteristics.

Predicted Geometrical Characteristic	Sample Size	MPE	${\mathbf{s}}_{\mathbf{d}}$	WUB	95% PEI
$x_m/D$	309	+0.29	8.58	±0.96	−0.66 to +1.25
$x_r/D$	305	−0.11	9.29	±1.04	−1.15 to +0.93
$y_m/D$	341	−0.10	6.11	±0.65	−0.75 to +0.55
$y_t/D$	420	−0.03	8.25	±0.79	−0.82 to +0.76

, along with the mean prediction errors, the width of the uncertainty band, and the 95% prediction error interval ($PEI$).

Based on the results shown in Table 5, the lowest value for the mean prediction error was obtained for the non-dimensional $y_t/D$, while the lowest values of the $s_d$, $WUB$, and $95\% PEI$ were obtained for $y_m/D$, which indicate that this is the most accurate model compared to the other models. On the other hand, the least accurate predictions in this study were found for the non-dimensional geometrical parameters of $x_m/D$ and $x_r/D$ with the higher values of the $s_d$, $WUB$, and $95\% PEI$, respectively. For the $MPE$, it was found that the largest value was obtained for $x_m/D$, i.e., $+0.29$. Although both $x_r/D$ and $y_m/D$ had almost the same $MPE$ value, considering the $WUB$ and $95\% PEI$ values, the prediction model for $y_m/D$ was more accurate. A larger value of the $WUB$ was noted for $x_r/D$, where it ranged from $-1.04$ to $+1.04$, indicating lower accuracy compared to the other prediction models. As a result, the non-dimensional values of the geometrical characteristic $y_m/D$ predicted by the model showed the lowest mean prediction error, the smallest uncertainty band, and a confidence band for the 95% prediction error interval, therefore validating the accuracy of this GMDH model.

4. Conclusions

The discharge of the brine water produced from desalination plants back into the coastal ocean using a submerged pipe is a common method that is widely used in engineering practice. Therefore, studying the mixing behavior of this discharge in the near-field zone has become a research priority in order to reduce the impact of this water on the marine environment. The discharged water can form an inclined negatively buoyant jet when its density is higher than the density of the receiving water. Therefore, predicting the geometrical characteristics of this jet is crucial to improve the understanding of the mixing behavior of this type of discharge and its dilution. In this study, the geometrical characteristics of the inclined negatively buoyant jet for angles ranging from 15° to 85° were modeled and predicted using a type of artificial neural network (ANN) called the group method of data handling (GMDH). The results of this model were evaluated statistically and compared with those of several analytical models that were previously studied. Based on the results and discussion presented, the following conclusions can be drawn.

• The GMDH model demonstrated excellent performance in predicting the dimensionless geometrical parameters ($x_r/D$ and $y_t/D$) of inclined dense jets. The coefficient of determination (${\mathrm{R}}^2$) values indicated high accuracy, with values exceeding 0.94 for all parameters.
• Statistical indices such as the root mean squared error (RMSE) and mean absolute error (MAE) confirmed the high accuracy of the GMDH model. The low values of these indices, along with the high ${\mathrm{R}}^2$ values, validate the reliability of the model in predicting geometrical characteristics.
• A comparative analysis with analytical and numerical models from previous studies showed that the GMDH model outperformed the other models in terms of accuracy and precision. The standard error of the estimate (SEE) results indicated that the GMDH model provided the most accurate predictions compared to the other models tested.
• The GMDH model’s predictive capability was demonstrated across a range of angles and geometrical parameters. It showed consistent and accurate predictions even for higher inclination angles, where other models exhibited limitations.
• The uncertainty analysis further validated the accuracy of the GMDH model, particularly in predicting the dimensionless parameter $y_t/D$, which showed the lowest mean prediction error, the smallest uncertainty bandwidth, and a narrow confidence band for the 95% prediction error interval.
• Considering the comprehensive evaluation, including the statistical indices, the comparison with previous models, and the uncertainty analysis, the GMDH model emerged as a highly reliable and accurate method for the prediction of the geometrical parameters of inclined dense jets.

In conclusion, the GMDH model represents a robust and dependable approach to studying the near-field mixing zones of inclined dense jets, offering accurate predictions of the geometrical characteristics across a wide range of angles compared to other methods.