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

^{1}

^{2}

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## Abstract

**:**

^{2}) 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

## 2. Methodology

#### 2.1. Analysis of Inclined Negatively Buoyant Jet

#### 2.2. GMDH Method

#### GMDH Modeling Setup

## 3. Results and Discussion

#### 3.1. Statistical Assessment of Model Performance

#### 3.2. Comparing GMDH Results with Previous Models

#### 3.3. Uncertainty Analysis

## 4. Conclusions

- 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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the classic GMDH (Alfaifi et al. [41]).

**Figure 3.**A scatter plot with the fitting lines of the actual and GMDH-predicted results for angles from 15° to 85°: (

**a**) ${x}_{r}/D$ and (

**b**)${y}_{t}/D$.

**Figure 4.**The performance of the GMDH model in the training and testing stages: (

**a**) ${x}_{r}/D$ and (

**b**)${y}_{t}/D$.

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° |

Geometrical | GMDH Proposed Equations | |
---|---|---|

${x}_{m}/D$ | $=-31.538+1.94873{\theta}_{0}-0.0234242{\theta}_{0}^{2}+1.63203F{r}_{d}$ | (13) |

${x}_{r}/D$ | $=-0.35311+2.9030F{r}_{d}+0.065091F{r}_{d}N3-0.13909{F{r}_{d}}^{2}-0.005735N{3}^{2}$ $N3=-47.2107+2.77506{\theta}_{0}-0.0340387{\theta}_{0}^{2}+2.82939F{r}_{d}$ | (14) |

${y}_{m}/D$ | $=2.37956-0.139279{\theta}_{0}+0.0263993{\theta}_{0}F{r}_{d}+0.0026574{\theta}_{0}^{2}+0.113531F{r}_{d}$ | (15) |

${y}_{t}/D$ | $=1.00174+0.0336764{\theta}_{0}F{r}_{d}-0.000250209{\theta}_{0}^{2}+0.0418149F{r}_{d}$ | (16) |

Geometrical Parameter | R^{2} | MAE | RMSE | R^{2} | 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 |

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 |

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 |

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**MDPI and ACS Style**

Alfaifi, H.; Bonakdari, H.
Prediction of Geometrical Characteristics of an Inclined Negatively Buoyant Jet Using Group Method of Data Handling (GMDH) Neural Network. *Fluids* **2024**, *9*, 198.
https://doi.org/10.3390/fluids9090198

**AMA Style**

Alfaifi H, Bonakdari H.
Prediction of Geometrical Characteristics of an Inclined Negatively Buoyant Jet Using Group Method of Data Handling (GMDH) Neural Network. *Fluids*. 2024; 9(9):198.
https://doi.org/10.3390/fluids9090198

**Chicago/Turabian Style**

Alfaifi, Hassan, and Hossein Bonakdari.
2024. "Prediction of Geometrical Characteristics of an Inclined Negatively Buoyant Jet Using Group Method of Data Handling (GMDH) Neural Network" *Fluids* 9, no. 9: 198.
https://doi.org/10.3390/fluids9090198