# CFD-CNN Modeling of the Concentration Field of Multiport Buoyant Jets

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^{2}

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## Abstract

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^{2}to conduct evaluations of the models. This work also makes comparisons with the machine learning approach based on multi-gene genetic programming, to assess the performance of the proposed approach. The experimental results show that the models constructed based on the proposed approach meet the accuracy requirement and possess better performance compared with the traditional machine learning method, and they can provide reasonable predictions.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CNN Method

- (1)
- Select the training group and randomly set 60 percent of the samples as the training group, 20 percent as the verification group, and 20 percent as the test group from the sample set;
- (2)
- Set each weight and threshold value to a small random value close to 0 and initialize the precision control parameters and learning rate. Set the parameter epochs to 1000, batch size to 32, and verbose to 2;
- (3)
- Select Fr, x, and y as the input vectors, and normalize the concentration target output vector;
- (4)
- Calculate the output vector of the middle layer and calculate the actual output vector of the network;
- (5)
- The output error is calculated by comparing the elements in the output vector with the elements in the target vector. For the hidden elements of the middle layer, the error also needs to be calculated;
- (6)
- The adjustment amount of each weight and the adjustment amount of the threshold are calculated in turn;
- (7)
- Adjust weights and adjust thresholds;
- (8)
- When M is experienced, it is judged whether the indicator meets the accuracy requirements. If not, it returns (3) and continues to iterate. If satisfied, it proceeds to the next step;
- (9)
- At the end of training, the weights and thresholds in a document are saved. At this moment, it can be considered that the various weights have reached stability and the classifier has been formed. Training is performed again, exporting the weights and thresholds directly from the documents for training; no initialization is required.

#### 2.2. The Joint Method of CFD-CNN

^{2}value, etc. If the validation results are good, the prediction is made from the test set. If the verification effect is not good, the parameters are adjusted to continue training.

#### 2.3. Comparison of Models

## 3. Results

#### 3.1. Results of CFD

^{2}/s

^{2}and 0.002061 m

^{2}/s

^{2}, respectively. The open-source platform Salome was used to compute the geometry and meshes, with regional discretization near ports and local refinement using unstructured computational meshes. According to the method proposed by Yan et al. [2], a sensitivity analysis of the grid was carried out, and the grid size iwas finally determined to be between 0.001 and 0.005 m. The “adjusttimestep” in the OpenFOAM software was used to automatically determine the time step.

#### 3.2. Results of CFD and CNN Joint Method

^{2}values were calculated to further evaluate the model performance. It was calculated that the RMSE values of the training set and the test set were 0.087 and 0.088, and the R

^{2}values were 0.903 and 0.902, respectively. The RMSE value was low and the R

^{2}value was high, so the accuracy of the model prediction is acceptable, though large errors still exist.

^{2}value, and so on, including errors. The test set is only utilized for model testing, and it is absolutely impossible to adjust the network parameter configuration and select the trained model according to the results on the test set, otherwise the model will be overfitted on the test set.

^{2}value is becoming larger and larger, the degree of fit is getting higher and higher, and after 300 generations, the result is not very diverse. According to the prediction of the normalized concentration of diverse cases at diverse locations, the normalized concentration field of the central plane is established, as shown in Figure 7. At large Fr values, the simulation prediction is not very good. This is due to the fact that the dataset is relatively small when Fr is larger, which also shows that machine learning requires a large amount of data for training in order to reduce errors. Overall, the prediction of the buoyant jet trajectory at the nozzle of the CNN method is highly consistent with the actual situation, showing an upward vertical jet. As can be observed from the figure, the CNN joint method predicts that the scope of sewage impact is basically consistent with the actual data, which provides a reference basis for determining the location of sewage diffusers and formulating environmental protection policies. In cases 1 to 8, the sewage has a large impact range and is round or elliptical. In cases 9 to 20, the boundary of the sewage impact range is more obvious, and it is greatly affected by buoyancy, which is consistent with the actual data obtained. The validation set is verified by using the trained model, and the resulting RSME value is 0.031 and the R

^{2}value is 0.985, so it can be proved that the model passes the test and can be tested. The test set was evaluated using this model, and the resulting RSME value was 0.056 and R

^{2}value was 0.956. Figure 8 compares the actual and predicted normalized concentrations of the CNN model. It can be observed from the figure that the CNN model prediction is consistent with the actual data trend, and the numerical size is not very diverse, so the prediction accuracy of the model is relatively high.

^{2}value, and the accuracy rate is significantly improved compared with MGGP. Compared with the normalized concentration distribution maps under diverse models, the concentration distribution maps obtained by MGGP have obvious concentration band fractures, and horizontal concentration distribution bands appear above the port, which is quite diverse from the actual concentration distribution. The concentration distribution map obtained by the CNN method is more consistent with the actual value, and the concentration band is basically the same, so it can be said that the simulation effect is better. The buoyancy jet trajectory was predicted by the CNN combined method, which was more consistent with the measured data, and the vertical normalized concentration distribution and diffusion range were basically consistent with the measured data. In addition, in terms of computing time, MGGP needs several hours to build a model, while CNN training time is relatively short and can be completed in a few minutes; therefore, the computational efficiency is greatly improved. This shows that the results of modeling predictions using the CNN method are superior to the MGGP method, which is an encouraging result in the domain of machine learning.

#### 3.3. The Performance of the Proposed Approach

^{2}values calculated by the CNN model based on the buoyancy jet space coordinates are 0.056 and 0.956, respectively, which are higher than the 0.088 and 0.902 of the MGGP model, confirming that the CNN is greatly improved compared with other machine learning models. To the best of the authors’ knowledge, this is the first time a CNN model has been developed for jets or plumes. The CNN method has recently been employed in similar applications. For example, Syed Kabir et al. [32] developed a fast-prediction flood model based on the deep convolutional neural network method, which further confirmed the performance of the CNN model. We have shown that the CNN model greatly outperforms the SVR. Hossein Hosseiny et al. [33] proposed a general river flood modeling framework based on coupled hydraulics and ML modeling, and the results showed that machine learning can reduce the computational time, resources, and cost of large-scale real-time simulations with high accuracy. Compared with CNN technology, the R

^{2}value of the CNN model in this study is 0.956; the R

^{2}of Syed Kabir et al. using CNN modeling was 0.94, and the CNN model studied by Hossein Hosseiny et al. had an R

^{2}of 0.88, which confirms that the accuracy achieved in this study is comparable to those achieved in previous similar studies. In the later research, more influencing factors will be added to continuously improve the accuracy of the model.

## 4. Conclusions

^{2}value is 0.956; when the Froude number is small, the buoyancy jet trajectory shows a significant upward parabolic shape, and the concentration distribution is consistent with the measured value. Compared with MGGP, the CNN model occupies less memory, the operation rate is faster, the accuracy of the results is higher, and the overall performance of the model is better. This study demonstrates the ability of this method in simulating the concentration field of a rosette buoyancy jet. When more data is available, the composite model can be further improved or the performance of the composite model can be extended. Furthermore, in machine learning, the training method and hyperparameter settings have a significant impact on the model training results, and in the training of neural networks, how to better train a model is a problem that is worth exploring.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Rosette-type diffuser, computational domain, and mesh: ((

**a**) shows a schematic of the diffuser, (

**b**) is a computational grid, and (

**c**) is a refined mesh near the discharge port).

**Figure 3.**The normalized concentration field of the central plane under different conditions obtained by CFD model.

**Figure 4.**The normalized concentration field of the central plane under different conditions obtained by the MGGP model.

**Figure 7.**The normalized concentration fields on the central plane under different conditions obtained by the CNN model.

D | U | Fr | ρ | g |
---|---|---|---|---|

0.0044 m | 0.185 m/s | variable | 997 km/m^{3} | 9.8 |

y/D | x/D (Exp) | x/D (Num) |
---|---|---|

0.40 | 1.28 | 0.99 |

1.76 | 2.56 | 2.84 |

3.12 | 3.52 | 3.67 |

4.88 | 3.92 | 4.39 |

37.72 | 4.48 | 5.13 |

6.57 | 4.64 | 4.80 |

8.01 | 4.96 | 5.10 |

35.48 | 4.96 | 5.09 |

9.69 | 5.28 | 5.35 |

33.15 | 5.28 | 5.20 |

11.13 | 5.52 | 5.53 |

30.75 | 5.52 | 5.32 |

28.11 | 5.68 | 5.47 |

25.71 | 5.76 | 5.64 |

13.53 | 5.84 | 5.79 |

19.14 | 5.84 | 6.23 |

20.98 | 5.84 | 6.19 |

22.74 | 5.84 | 6.02 |

15.22 | 5.92 | 5.99 |

17.06 | 6 | 6.20 |

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**MDPI and ACS Style**

Yan, X.; Wang, Y.; Mohammadian, A.; Liu, J.; Chen, X.
CFD-CNN Modeling of the Concentration Field of Multiport Buoyant Jets. *J. Mar. Sci. Eng.* **2022**, *10*, 1383.
https://doi.org/10.3390/jmse10101383

**AMA Style**

Yan X, Wang Y, Mohammadian A, Liu J, Chen X.
CFD-CNN Modeling of the Concentration Field of Multiport Buoyant Jets. *Journal of Marine Science and Engineering*. 2022; 10(10):1383.
https://doi.org/10.3390/jmse10101383

**Chicago/Turabian Style**

Yan, Xiaohui, Yan Wang, Abdolmajid Mohammadian, Jianwei Liu, and Xiaoqiang Chen.
2022. "CFD-CNN Modeling of the Concentration Field of Multiport Buoyant Jets" *Journal of Marine Science and Engineering* 10, no. 10: 1383.
https://doi.org/10.3390/jmse10101383