# Zonation of Positively Buoyant Jets Interacting with the Water-Free Surface Quantified by Physical and Numerical Modelling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Physycal Modelling

#### 2.1.1. Description of the PIV and LIF Techniques

#### 2.1.2. Setup of the Physical Modelling

^{2}with physical dimensions of 15.15 × 15.15 mm

^{2}. Each CCD camera was mounted with a Nikon AF Nikkor 50 mm 1:1.8D objective with an aperture varying between 1.8 and 22. The CCD cameras were placed parallel to each other and approximately perpendicular to the laser sheet and the jet centerline. Both cameras had the same horizontal and vertical field-of-view (FOV) of 71 cm with a total overlap of both FOV. One camera took the image at time t and the other camera at t + ∆t. The velocity composition was carried out by an inter-correlation algorithm [42], which divides the two successive images of the recorded particles at time t and t + ∆t into small interrogation areas. The position of the correlation peak between the two images corresponds to the most probable displacement of the particles for each of these interrogation areas [42]. The effects of geometric distortion, resulting from small angles in camera alignment, were automatically corrected during image processing and post-processing using LaVision DaVis software. The cameras were moved forward or backward depending on the test configurations in order to zoom in or zoom out on the flow area. Firstly, the cameras were relocated to define their results in the plane that marks the central axis of the jet advance. Secondly, the cameras were placed in such a way that the plane was wide enough to contain from the start of the jet until it was in the so-called lateral dispersion dominant zone, taking into account the number of pixels available for the camera to define accurately the jet evolution.

^{2}and an acquisition rate of 5 Hz. For the PIV measurements, an interrogation window size of 32 × 32 pixel was used. The separation time (∆t) was 20 µs for the jet centerline path around the nozzle and 7 ms for the spreading layer where the jet presents a marked upward trajectory, according to [43].

^{3}were used as seeding to follow the flow velocity variations. The standard cyclic Fourier Transform (FT) correlation function was applied to eliminate the PIV-generated noise and detect the peak that shows the velocity for each time step. For the LIF measurements, the exposure time was 20 ms. The absence of surfactants in the water-free surface was checked by liquid chromatography, since small amounts of them may alter the turbulence phenomena [44]. As passive tracer, Rhodamine WT (C

_{29}H

_{29}ClN

_{2}Na

_{2}O

_{5}) with a concentration of 100 µg/L was used. This concentration was selected to assure the right measurement in the highest dilution zones and to avoid any significant laser attenuation caused by self-absorption of the Rhodamine [39,45]. The data acquisition system presents uncertainties of around 10

^{−4}m/s for velocities and 1 µg/L for concentrations.

_{a}, T

_{a}, and ρ

_{a}are the ambient salinity, temperature and density, respectively. S

_{o}, T

_{o}, ρ

_{o}, and Q

_{o}are the effluent salinity, temperature, density and flow, respectively. Re, Fr, and H are the Reynolds number (see in Equation (3)), the Froude number (see in Equation (4)), and the distance from the jet nozzle axis to the surface of every test, respectively.

_{o}is the jet velocity, D the nozzle diameter, g the gravity acceleration, and ∆ρ = ρ

_{a}− ρ

_{o}the difference between the ambient and effluent density.

_{MAX}) and the TKE at the nozzle (TKE

_{MAX}), respectively. LIF measurements were processed calculating the time-averaged concentration (C) and the time-averaged fluctuation of the concentration (C’), nondimensionalized with the maximum concentration, i.e., the initial concentration (C

_{MAX}) discharged through the nozzle.

#### 2.2. Numerical Modelling

#### 2.2.1. Description of Numerical Models

#### 2.2.2. Setup of the Computational Fluid Dynamics Model

^{5}elements, (2) intermediate grid with 6 × 10

^{5}elements, and (3) fine grid with 2 × 10

^{6}elements. Independently of the simulation case and/or the mesh grid, the numerical discretization was a Crank–Nicholson second order scheme in time and a Van Leer second order scheme for the advection terms. In order to ensure the numerical model stability, a variable time step depending on the turbulent flow was used. The time step started with 10

^{−5}s at the beginning of any simulation and finished with 5 × 10

^{−4}s at the end of any simulation. The simulation time was 200 s, recording model results every 10

^{−2}s for every numerical test.

#### 2.2.3. Setup of the Semiempirical Models

#### 2.2.4. Performance Metrics of Numerical Modelling

^{2}) was calculated as the difference between modelled results and observed values, as expressed in Equation (5). Secondly, the error between both series was assessed using the root-mean square error (RMSE) and the normalized RMSE (NRMSE), displayed in Equations (6) and (7), respectively. Finally, the error between both series was also calculated using the model efficiency (CE), developed by [51] and showed in Equation (8).

_{i}is the i-data of the physical tests, S

_{i}is the i-data of the numerical tests, $\overline{\mathrm{R}}$ is the average data of the physical tests, and i is the ith value from 1 to N data of the physical tests.

#### 2.3. Zonation of Positively Buoyant Jets

## 3. Results

#### 3.1. Physycal Modelling

#### 3.1.1. PIV Measurements

_{MAX}) in a longitudinal view of the positively buoyant jet advance at the nozzle XY plane for the tests V1 (a), V2 (b), V3 (c) and V4 (d).

#### 3.1.2. LIF Measurements

_{MAX}) and the dimensionless fluctuations of the concentration (C’/C

_{MAX}) in a longitudinal view of the positively buoyant jet advance at the nozzle XY plane for the tests V1 (a), V2 (b), V3 (c) and V4 (d), respectively.

_{o}) at Limit 1. Between X/D = 0 and X/D = 20, the test results presented outliers since the rhodamine concentrations used showed a saturation in their emission values and could not provide reliable results in this area. Next, the concentration was reduced to values of the order or less than 5% of C

_{o}at the Limit 2. However, in the physical test V2, the development of the buoyant jet generates a higher concentration due to the relationship between the jet momentum and the water depth until it hits the water-free surface. As can be seen in Figure 6, the fluctuation values are less than 1% of C

_{MAX}in all the zones of the buoyant jet. Nonetheless, these fluctuations can present values corresponding to 2% of C

_{MAX}in the central area of the buoyant jet before reaching the Zone 3, independently of the physical test.

#### 3.2. Numerical Modelling

#### 3.2.1. Setup of the Computational Fluid Dynamics Model

_{MAX}along the centerline (Figure 7b) for the fine grid versus the intermediate grid (circle markers), and the fine grid versus the coarse grid (star markers), respectively.

_{MAX}along the centerline, the coarse and intermediate grids displayed a RMS of 0.2 and 0.03, respectively. Furthermore, the simulation time was 72 h, 9.5 h and 3 h for the fine, intermediate and coarse grids, respectively. These results support the use of the intermediate grid due to the trade-off between computational time and model accuracy.

#### 3.2.2. Performance of Numerical Models

_{MAX}along the centerline (right panels) for the tests V1 (a, b), V2 (c, d), V3 (e, f) and V4 (g, h).

^{2}, RMSE, NRMSE, and CE were calculated (Table 2). Figure 9 displays a scatter plot of the centerline trajectory (Figure 9a) in dimensionless positions (X/D and Y/D), and a scatter plot of the C/C

_{MAX}along the centerline (Figure 9b) for the physical results versus the numerical results obtained with CORJET (green markers), VISJET (blue markers) and OpenFOAM (red markers) for the momentum dominant zone (square markers), the momentum to buoyancy transition zone (circle markers), the buoyancy dominant zone (triangle markers), and the lateral dispersion dominant zone (diamond markers), respectively.

_{MAX}, R

^{2}values were above 0.8 for OpenFOAM. Moreover, CE values for OpenFOAM were considered as excellent (CE > 0.8) and good (CE > 0.6) for centerline position and dilution along the centerline, respectively. On the other hand, this metric shows semiempirical models provided good results (CE > 0.6) for the centerline position but their results were poor (CE < 0.4) in defining the dilution of a positively buoyant jet.

^{2}, the three models obtained a very high value (>0.87), being especially high for OpenFOAM (0.98).

^{2}greater than 0.8. However, in the case of semiempirical models, the CORJET obtained a CE of 0.45 (acceptable result) and the VISJET of 0.37 (poor result), a NRMSE of 16% and 20%, respectively, and a R

^{2}less than 0.8 for both semiempirical models.

#### 3.3. Zonation of Positively Buoyant Jets

## 4. Discussion

#### 4.1. Physycal Modelling

_{Maxi}and velocities U

_{Maxi}in a selected profile (see Equation (9)). According to [59], the value of λ follows a non-linear pattern in the centerline advance along the longitudinal axis.

#### 4.2. Numerical Modelling

#### 4.3. Zonation of Positively Buoyant Jets

_{M}) proposed by [11] because both mark the limit of momentum dominant zone-based characteristics (single-port diameter, momentum and buoyancy). The obtained values were very similar to those obtained with the proposed methodology. Regarding the Limit 2, the classical length scales do not include a specific methodology to obtain it. For instance, the calculation of the buoyancy length scale (L

_{B}), proposed by [11], is diffused and constrained to the existence of stratification in the receiving medium. Finally, according to the Roberts experiments [35,36,37], the Limit 3 is obtained when the vertical concentration profile in the layer where the buoyant jet is progressing presents longitudinal variations lower than 5%. This calculation was in line with the lengths obtained by the proposed method for the Limit 3. Table 5 shows the values of these scales for the laboratory tests.

_{M}are closer to the semiempirical models results than the laboratory test data, displaying an error of 44% compared with physical test data and displaying more error than modelling errors obtained by means of OpenFOAM (error value of 6.4% compared with test data). In the case of Limit 2, L

_{B}could not be calculated because the environment did not present stratification. Finally, the point marked by [35,36,37], as Limit 3, shows a variation of 25% compared with the physical tests; this variation is greater than the variation for OpenFOAM (a variation for the Limit 3 location of 11.9%).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram and photography of the physical model (

**a**), profile view of a preliminary test (

**b**), and plane view of a preliminary test (

**c**).

**Figure 2.**Mesh grid scheme for the simulation of a jet in a stagnant medium interacting with the water-free surface by means of the computational fluid dynamics (CFD) model: (

**a**) profile view, (

**b**) plan view, and (

**c**) example of CFD mesh grid.

**Figure 3.**Proposed zonation of positively buoyant jets using the angle (α) shaped by the tangent of the centerline trajectory and the longitudinal axis (X-axis).

**Figure 4.**Dimensionless time-averaged velocities (U/U

_{MAX}) in a longitudinal view of the positively buoyant jet advance at the nozzle XY plane for the physical tests V1 (

**a**), V2 (

**b**), V3 (

**c**), and V4 (

**d**).

**Figure 5.**Dimensionless time-averaged concentrations (C/C

_{MAX}) in a longitudinal view of the positively buoyant jet advance at the nozzle XY plane for the physical tests V1 (

**a**), V2 (

**b**), V3 (

**c**), and V4 (

**d**).

**Figure 6.**Dimensionless time-averaged fluctuations of the concentration (C’/C

_{MAX}) in a longitudinal view of the positively buoyant jet advance at the nozzle XY plane for the physical tests V1 (

**a**), V2 (

**b**), V3 (

**c**), and V4 (

**d**).

**Figure 7.**(

**a**) Scatter plot of the centerline trajectory in dimensionless positions (X/D and Y/D), and (

**b**) scatter plot of the C/C

_{MAX}along the centerline for the fine grid versus the medium grid (circle markers), and the fine grid versus the coarse grid (star markers). <-> indicates the used grids for the RMS calculation.

**Figure 8.**Comparison of the physical results (black dots) versus the numerical results obtained with CORJET (green line), VISJET (blue line) and Open Field Operation And Manipulation model (OpenFOAM) (red line) of the centerline trajectory (left panels) in dimensionless positions (X/D and Y/D) and the C/C

_{MAX}along the centerline (right panels) for the physical tests V1 (

**a**,

**b**), V2 (

**c**,

**d**), V3 (

**e**,

**f**) and V4 (

**g**,

**h**), respectively.

**Figure 9.**(

**a**) Scatter plot of the centerline trajectory in dimensionless positions (X/D and Y/D). (

**b**) Scatter plot of the C/C

_{MAX}along the centerline for the physical results versus the numerical results obtained with CORJET (green markers), VISJET (blue markers) and OpenFOAM (red markers) in the momentum dominant zone (square markers), the momentum to buoyancy transition zone (circle markers), the buoyancy dominant zone (triangle markers), and the lateral dispersion dominant zone (diamond markers), respectively.

Test | S_{a}(psu) | T_{a}(°C) | ρ_{a}(kg/m ^{3}) | S_{o}(psu) | T_{o}(°C) | ρ_{o}(kg/m ^{3}) | Q_{o}(m ^{3}/s) | H (m) | Re (-) | Fr (-) |
---|---|---|---|---|---|---|---|---|---|---|

V1 | 0.28 | 21.5 | 998.10 | 0.25 | 50 | 988.25 | 0.81 | 0.210 | 3390 | 31.27 |

V2 | 0.28 | 21.9 | 998.00 | 0.25 | 55 | 985.96 | 0.75 | 0.210 | 3132 | 26.18 |

V3 | 0.28 | 22.1 | 997.96 | 0.20 | 55 | 985.96 | 0.68 | 0.150 | 2840 | 23.78 |

V4 | 0.28 | 22.7 | 997.82 | 0.20 | 50 | 985.96 | 0.66 | 0.158 | 2756 | 23.21 |

**Table 2.**Performance of the numerical models to reproduce the physical tests by the metric errors R

^{2}, RMSE, NRMSE, and CE.

Parameter | Numerical Model | Metrics | |||
---|---|---|---|---|---|

R^{2} | RMSE (y/D or C/C _{max}) | NRMSE (%) | CE | ||

y/D - Centerline | CORJET | 0.88 | 3.94 | 11.93 | 0.78 |

VISJET | 0.88 | 3.87 | 11.54 | 0.79 | |

OpenFOAM | 0.98 | 1.84 | 4.62 | 0.98 | |

C/Cmax - Centerline | CORJET | 0.71 | 0.03 | 16.00 | 0.45 |

VISJET | 0.61 | 0.04 | 20.65 | 0.37 | |

OpenFOAM | 0.81 | 0.02 | 12.03 | 0.64 |

**Table 3.**Summary of the limits of positively buoyant jet zones defined by the physical (Experimental) and the numerical (OpenFOAM, CORJET, VISJET) modelling.

Physical Test | Data Type | x/D | ||
---|---|---|---|---|

Limit 1 | Limit 2 | Limit 3 | ||

V1 | Experimental | 41.5 | 90.0 | 103.9 |

OpenFOAM | 43.0 | 92.0 | 107.9 | |

CORJET | 34.0 | 83.3 | * | |

VISJET | 34.3 | 85.3 | * | |

V2 | Experimental | 29.0 | 65.0 | 91.0 |

OpenFOAM | 28.8 | 66.5 | 88.2 | |

CORJET | 27.0 | 72.9 | * | |

VISJET | 27.8 | 71.6 | * | |

V3 | Experimental | 31.8 | 65.0 | 82.5 |

OpenFOAM | 32.0 | 65.0 | 82.3 | |

CORJET | 28.0 | 63.0 | * | |

VISJET | 25.0 | 64.0 | * | |

V4 | Experimental | 32.5 | 69.0 | 90.3 |

OpenFOAM | 33.0 | 70.0 | 91.8 | |

CORJET | 28.0 | 65.0 | * | |

VISJET | 27.0 | 65.0 | * |

**Table 4.**Performance of the numerical models to define the limits of positively buoyant jet zones by the metric errors: root-mean square error (RMSE) and normalized RMSE (NRMSE).

Numerical Model | RMSE (x/D) | NRMSE (%) | ||||
---|---|---|---|---|---|---|

Limit 1 | Limit 2 | Limit 3 | Limit 1 | Limit 2 | Limit 3 | |

OpenFOAM | 0.8 | 1.3 | 2.6 | 6.4 | 5.4 | 11.9 |

CORJET | 4.9 | 5.7 | * | 39.0 | 22.6 | * |

VISJET | 5.7 | 4.5 | * | 45.6 | 18.2 | * |

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**MDPI and ACS Style**

García-Alba, J.; Bárcena, J.F.; García, A.
Zonation of Positively Buoyant Jets Interacting with the Water-Free Surface Quantified by Physical and Numerical Modelling. *Water* **2020**, *12*, 1324.
https://doi.org/10.3390/w12051324

**AMA Style**

García-Alba J, Bárcena JF, García A.
Zonation of Positively Buoyant Jets Interacting with the Water-Free Surface Quantified by Physical and Numerical Modelling. *Water*. 2020; 12(5):1324.
https://doi.org/10.3390/w12051324

**Chicago/Turabian Style**

García-Alba, Javier, Javier F. Bárcena, and Andrés García.
2020. "Zonation of Positively Buoyant Jets Interacting with the Water-Free Surface Quantified by Physical and Numerical Modelling" *Water* 12, no. 5: 1324.
https://doi.org/10.3390/w12051324