# Numerical Modeling of Multiple Inclined Dense Jets Discharged from Moderately Spaced Ports

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dimensional Analysis

_{j}from multiple ports with a spacing of s. The jets leave the nozzle at an angle of θ, reach the highest location, and fall back to the horizontal bed. The highest height is hereafter referred to as the terminal right height (y

_{t}), the length between the diffuser nozzle and the return point is referred to as the impact distance (x

_{i}), and the dilution at the return point is referred to as the impact dilution (S

_{i}).

_{j}is the initial jet velocity; g’ is the reduced gravitational acceleration, which is equal to gravitational acceleration times initial density difference divided by the ambient water density. These variables can also be utilized to determine the jet’s Reynolds number, R = U

_{j}d/ν, where ν is the kinematic viscosity.

_{t}can be expressed as

_{M}and L

_{Q}are two length scales, which can be expressed as

_{i}and S

_{i}can be expressed as

_{t}, x

_{i}, and S

_{i}could be expressed as

#### 2.2. Governing Equations

**U**is velocity, ρ is density, p

_{rgh}is static pressure minus hydraulic pressure, h is height of fluid column, g is gravitational acceleration, α is volume fraction, μ is dynamic viscosity, μ

_{t}is turbulent viscosity, and subscript i denotes either fluid 1 or 2. In the current model, Fluids 1 and 2 represent the jet and ambient water, respectively.

_{ab}is the molecular diffusivity, ν

_{t}is the turbulent eddy viscosity, and S

_{C}is the turbulent Schmidt number.

#### 2.3. Turbulence Modeling

_{ε}, c

_{1ε}, c

_{2ε}, c

_{μ}are model constants equal to 1.3, 1.44, 1.92, and 0.09, respectively.

_{k}, σ

_{ε}, c

_{1ε}, c

_{2ε}, c

_{μ}, η

_{0}, and β are model constants equal to 0.71942, 0.71942, 1.42, 1.68, 0.0845, and 0.012, respectively.

#### 2.4. Model Setup

^{−9}, 1 × 10

^{−7}, 1 × 10

^{−7}, respectively.

## 3. Results

#### 3.1. General Observations

#### 3.2. Terminal Rise Height

_{t}/(d·F) for the eight cases considered in this paper obtained from the experiments and numerical simulations. In the figure, the results of y

_{t}/(d·F) are plotted against the values of s/(d·F) for the convenience of showing the effects of the port spacing. The results for single discharges calculated from the empirical equations proposed by Roberts et al. [35] were also plotted in the same figure. As can be seen, the values of y

_{t}/(d·F) generally increased with increasing values of s/(d·F), which is in agreement with the experimental observations for multiple inclined dense jets discharged from moderately spaced ports [2]. The phenomenon that the values of y

_{t}/(d·F) increased with increasing values of s/(d·F) can be explained by the Coanda effect; namely, the entrainment of ambient water into the jets is less restricted for the multiple jets with greater port spacing, and thus the dilution is higher, resulting in greater values of y

_{t}/(d·F).

_{t}/(d·F) very well, except for case C5, in which the normalized port spacing is quite large. Theoretically, the mixing behaviors of multiple jets with infinitely large port spacing should be identical to those of single jets, and the values of y

_{t}/(d·F) generally increase with increasing port spacing, so the solutions of y

_{t}/(d·F) for single jets should be the largest possible values. The markers corresponding to the numerical results for case C5 were lower than the line for the empirical solutions for the single jet case, implying that the present numerical results are consistent with the experiments reported by Roberts et al. [35]. Overall, the markers corresponding to the RNG k-ε model are quite close to those corresponding to the experimental data, showing the good prediction ability of the RNG k-ε closure. The symbols for the standard k-ε model were roughly close to those for the RNG k-ε model, because most of the terms in the two closures were the same. However, the results obtained by the standard k-ε model departed farther from the data.

^{2}), were calculated (Table 2). It should be noted that the size of each data sample was relatively small, so the relative measure of fit, R

^{2}, may not be able to provide very meaningful conclusions. Both Figure 4 and Table 2 show that the RNG k-ε model predicted better results than the standard k-ε model. Based on the value of MAPE, the errors for the numerical results were well below 15%, so the model accuracy was acceptable for practical purposes.

#### 3.3. Impact Distance

_{i}(d·F) for the cases considered in this study obtained from the experiments and numerical simulations are shown in Figure 5, and the solutions for point discharges are also plotted in the same figure. As can be seen, the values of x

_{i}/(d·F) generally increased with increasing values of s/(d·F), which can also be explained by the Coanda effect mentioned earlier.

_{i}/(d·F) showed a good match with the experimental results [2]. Similar to the comparisons for y

_{t}/(d·F), the match for case C5 was relatively poor. However, the numerical results were smaller than the empirical solutions for the single jet, which were expected to be the upper limit of possible values, so the current numerical results were in accordance with those of Roberts et al. [35]. Generally, the results obtained by the RNG k-ε model were quite close to measurements, demonstrating the prediction capability of the RNG k-ε model for multiple inclined dense jets discharged from moderately spaced ports. The results obtained by the standard k-ε model were similar to those obtained by the RNG k-ε model, but showed larger differences from the experimental measurements. Table 3 lists the results for error analyses. Both Figure 5 and Table 3 show that the RNG k-ε model predicted better results than the standard k-ε model. According to the values of MAPE, the errors for the results obtained by the standard k-ε model exceeded 15%, whereas those by the RNG k-ε model were within 15%.

#### 3.4. Impact Dilution

_{i}/(d·F) obtained from the experiments and numerical simulations for the eight cases were plotted against the normalized port spacing in Figure 6. Similar to the geometrical features (y

_{t}and x

_{i}), the impact dilution also increased when the port spacing increased. The agreements between the experimental and numerical results were satisfactory, demonstrating that the numerical model can satisfactorily predict the impact dilution for multiple inclined dense jets. The various fit measures were calculated and summarized in Table 4. The plots shown in Figure 6 and error indices summarized in Table 4 show that the RNG k-ε model predicted better results than the standard k-ε model, and both models provided predictions with errors smaller than 15%.

## 4. Discussion

_{t}using the gradient diffusion hypothesis. The RNG k-ε turbulence closure outperformed the standard one in modeling multiple inclined dense jets, probably due to three reasons: 1) the RNG k-ε turbulence closure calculates ν

_{t}using the scale elimination procedure, which takes into account the modification of effective turbulent transport due to the Reynolds number; 2) the RNG k-ε turbulence closure calculates the inverse effective Prandtl numbers using a RNG-based formula, which is of a lower level of simplification; and 3) the RNG k-ε turbulence closure improved the ε equation by adding an additional term, which can more accurately calculate ν

_{t}

_{.}

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Concentration fields at various downstream locations; Case C1, F = 62.9, and s/(d·F) = 0.94.

Cases | d (mm) | s (mm) | F (—) | s/(d·F) (—) |
---|---|---|---|---|

C1 | 1.93 | 114 | 62.9 | 0.94 |

C2 | 1.93 | 114 | 56.8 | 1.04 |

C3 | 1.93 | 114 | 77.3 | 0.76 |

C4 | 1.93 | 114 | 81.1 | 0.73 |

C5 | 1.93 | 57 | 17.3 | 1.71 |

C6 | 1.93 | 57 | 34.7 | 0.85 |

C7 | 1.93 | 57 | 63.1 | 0.47 |

C8 | 1.93 | 57 | 56.8 | 0.52 |

Model | MBE | MAE | MAPE (%) | RMSE | NRMSE (%) | R^{2} |
---|---|---|---|---|---|---|

Standard k-ε | −0.24 | 0.24 | 12.10 | 0.29 | 0.16 | 0.88 |

RNG k-ε | −0.22 | 0.22 | 11.81 | 0.26 | 0.14 | 0.91 |

Model | MBE | MAE | MAPE (%) | RMSE | NRMSE (%) | R^{2} |
---|---|---|---|---|---|---|

Standard k-ε | −0.37 | 0.37 | 18.56 | 0.38 | 0.19 | 0.98 |

RNG k-ε | −0.27 | 0.27 | 13.26 | 0.30 | 0.15 | 0.84 |

Model | MBE | MAE | MAPE (%) | RMSE | NRMSE (%) | R^{2} |
---|---|---|---|---|---|---|

Standard k-ε | −0.13 | 0.13 | 13.82 | 0.18 | 0.22 | 0.99 |

RNG k-ε | −0.04 | 0.08 | 10.76 | 0.09 | 0.11 | 0.96 |

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**MDPI and ACS Style**

Yan, X.; Mohammadian, A.
Numerical Modeling of Multiple Inclined Dense Jets Discharged from Moderately Spaced Ports. *Water* **2019**, *11*, 2077.
https://doi.org/10.3390/w11102077

**AMA Style**

Yan X, Mohammadian A.
Numerical Modeling of Multiple Inclined Dense Jets Discharged from Moderately Spaced Ports. *Water*. 2019; 11(10):2077.
https://doi.org/10.3390/w11102077

**Chicago/Turabian Style**

Yan, Xiaohui, and Abdolmajid Mohammadian.
2019. "Numerical Modeling of Multiple Inclined Dense Jets Discharged from Moderately Spaced Ports" *Water* 11, no. 10: 2077.
https://doi.org/10.3390/w11102077