# On the Effect of Regular Waves on Inclined Negatively Buoyant Jets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{D}the discharged fluid density and ρ

_{R}the receiving fluid density. The Reynolds number is defined here as Re = UD/υ, where υ is the kinematic viscosity of the discharged fluid. Two sets of experiments were performed, with Fr equal to 18.0 and 28.0, and a Re of 10

^{3}(higher than the critical value of around 500 for the present set-up, as shown by Ferrari and Querzoli, 2010 [1]) and five monochromatic waves, with wave periods T of 0.50 s, 1.00 s and 1.50 s (corresponding to wave lengths L of 0.39 m, 1.56 m and 3.51 m) and wave amplitudes A of 5.00 mm and 12.50 mm (see Table 1). A reference run with a NBJ released in a stagnant environment was performed for each Fr. The ratio of the water depth d to the wave length L was in the deep-water regime (d/L = 1.02) or in the intermediate-depth water regime (d/L = 0.26 and 0.11); the ratio between the wave height H = 2A and L was between 0.003 and 0.026, typical values of a long wave. The experimental set-up dimensions and wave parameters were chosen in order to simulate a typical submerged discharge in the Mediterranean Sea, with respect to the geometrical similarity (scale model K

_{L}= L

_{M}/L

_{P}= 1/40; the subscript M stands for Model and the subscript P for Prototype), to the kinematic similarity and to the dynamic (Froude) similarity, achieved through the respect of the following equations (see e.g., Von Ellenrieder and Dhanak, 2016 [36]):

_{0}. The subtraction of the background from the images with the jet allows us to remove sources of light not linked to the discharged fluid concentration. The subtraction of the background explains why the cylindrical pipe with the outlet on its wall is visible in Figure 2 and Figure 3 (before the background removal) and not visible in Figure 4, Figure 5 and Figure 6, Figure 10 and Figure 11 (after the background removal). Under the assumption of ergodicity, the non-dimensional fields of the mean concentration C/C

_{0}were obtained by time averaging the measured values of C/C

_{0}on each pixel. The value of C/C

_{0}is reported in false colors in Figure 4, Figure 5 and Figure 6, Figures 10 and 11, according to the color bar shown on the left of each Figure (dark red is linked to the highest concentrations, dark blue to the lowest ones). As a consequence, C/C

_{0}is a measure of the reduction of the mean concentration compared to the outlet concentration C

_{0}: For instance, C/C

_{0}= 1 (dark red) means that the concentration C in that point is the same as C

_{0}(no dilution), C/C

_{0}= 0.5 (green) means that the concentration C in that point is one half of C

_{0}, C/C

_{0}= 0 (dark blue) means that the concentration C in that point is zero (external fluid not reached by the jet). The x-axis and y-axis are non-dimensionalised by the outlet diameter D, with the origin on the outlet.

_{h}/D, Y/D = Y

_{h}/D) and an impact point (defined as the point where the jet axis reaches again the outlet height, with coordinates X/D = X

_{d}/D, Y/D = 0). The point of maximum height and the impact point for the stagnant case are drawn, respectively, as a white circle and a white asterisk in Figure 4, Figure 5 and Figure 6, Figures 10 and 11. The vertical distance between the jet origin and the point of maximum height is defined as the maximum height (vertical orange line in Figure 4) and the horizontal distance between the jet origin and the impact point is defined as the impact distance (horizontal orange line in Figure 4).

_{0}field for phase 1/8 is shown. For the NBjs released into a wave environment, the impact point was measured as the point of maximum height of C/C

_{0}on the horizontal line Y/D = 0 for X/D > X

_{h}/D (i.e., beyond the horizontal coordinate of the point of maximum height; yellow asterisk in Figure 5 and Figure 6), while the point of maximum height for the jet undergoing a bifurcation (see Section 3.1.1 for the phenomenological discussion) was measured in the upper branch of the two branches caused by the bifurcation. In Figure 5 and Figure 6, the point of maximum height is drawn as a magenta circle. In Figure 5, the jet axis of the upper branch as a magenta line, the jet axis of the lower branch as a black line and the jet axis of the descent branch as a yellow line.

## 3. Results

#### 3.1. Temporal Evolution

#### 3.1.1. Mean Concentration Fields

#### 3.1.2. Geometrical Features

#### 3.1.3. Concentration Profiles

_{0}measured on the point of maximum height (a and c, vertical profiles) and on the impact point (b and d, horizontal profiles), in eight different phases of the wave cycle, are shown, for two NBJs with Fr = 28.0 (a and b) and Fr = 18.0 (c and d) released into a receiving body affected by a wave motion with T = 1.00 s and A = 12.5 mm. The different colors highlight the different phases the profiles are measured in. The bimodal distribution of the concentration maxima in Figure 7c and Figure 9a confirms what was previously stated about the bifurcation of NBJs released in a wave environment.

#### 3.2. Overall Time Averaged Concentration Fields

#### 3.3. Dilution

_{DIL}: if R

_{DIL}is higher than one, the waves increase the dilution and vice versa. In Figure 12, R

_{DIL}has been plotted versus Fr for all the experiments. The horizontal dotted black line at R

_{DIL}= 1 highlights the stagnant case, while, as in the previous Figures, colored lines highlight the wave cases (blue circles for T = 0.5 s and A = 5.0 mm, green asterisks for T = 1.0 s and A = 5.0 mm, magenta stars for T = 1.0 s and A = 12.5 mm, cyan xs for T = 1.5 s and A = 5.0 mm and red +s for T = 1.5 s and A = 12.5 mm). The dilution is measured, in all the cases, as the inverse of the maximum non-dimensional concentration at the impact point on the overall time-averaged fields, in order to avoid phase variations (see Figure 9b,d).

_{DIL}= 1.81 for A = 5.0 mm, T = 0.5 s, Fr = 18.0; R

_{DIL}= 1.08 for A = 5.0 mm, T = 1.0 s, Fr = 18.0; R

_{DIL}= 1.26 for A = 5.0 mm, T = 0.5 s, Fr = 28.0; R

_{DIL}= 1.08 for A = 5.0 mm, T = 1.0 s, Fr = 28.0), whilst the remaining waves (with higher values of A and T) reduce the dilution. The lowest values for R

_{DIL}have been measured for the strongest wave (A = 12.5 mm, T = 1.0 s) for both the Fr (R

_{DIL}= 0.36 for Fr = 28.0; R

_{DIL}= 0.45 for Fr = 18.0).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Wave period T versus wave height H recorded by the Alghero wave buoy in the period 2002–2014; data from the Rete Ondametrica Nazionale (Italian National wavemetric System) [37].

**Figure 2.**An instantaneous visualization of the concentration field for an inclined negatively buoyant jet with Fr = 28.0 and θ = 67° released into a stagnant receiving body.

**Figure 3.**An instantaneous visualization of the concentration field for the same inclined NBJ of Figure 2 released into a receiving body affected by regular waves with A = 12.5 mm and T = 1.00 s.

**Figure 4.**Non-dimensional mean concentration C/C

_{0}fields for the same inclined NBJ of Figure 2 released into a stagnant receiving body, with the jet geometrical features; the maximum uncertainty in the mean concentration measurement is ±4.41%.

**Figure 5.**Non-dimensional mean concentration fields for the same NBJ of Figure 3, in phase 1/8, with the jet geometrical features; the maximum uncertainty in the mean concentration measurement is ±4.38%.

**Figure 6.**Non-dimensional mean concentration fields for the same NBJ of Figure 3, divided into 8 phases; the white line is the stagnant case jet axis; the maximum uncertainty in the mean concentration measurement is ±4.38%.

**Figure 7.**Trajectories of the maximum height points for the NBJs with Fr = 28.0 (

**left**) and Fr = 18.0 (

**right**) released into a receiving body affected by regular waves (wave parameters in the legend).

**Figure 8.**Non-dimensional impact distance x

_{d}/D for the NBJs with Fr = 28.0 (

**left**) and Fr = 18.0 (

**right**) released into a receiving body affected by regular waves (wave parameters in the legend).

**Figure 9.**Cross-sectional non-dimensional mean concentration profiles for a NBJ with Fr = 28.0 (

**a**,

**b**) and for a NBJ with Fr = 18.0 (

**c**,

**d**) released in a receiving body affected by regular waves with T = 1.00 s and A = 12.5 mm; (a,c) vertical profiles on the point of maximum height; (b,d) horizontal profiles on the impact point; R/D is the non-dimensional span-wise abscissa.

**Figure 10.**Non-dimensional mean concentration fields for an NBJ with Fr = 18.0 released into a receiving body affected by regular waves with T = 0.0 s and A = 0.0 mm (stagnant case,

**a**), T = 0.5 s and A = 5.0 mm (

**b**), T = 1.0 s and A = 5.0 mm (

**c**), T = 1.0 s and A = 12.5 mm (

**d**), T = 1.5 s and A = 5.0 mm (

**e**), and T = 1.5 s and A = 12.5 mm (

**f**), with the stagnant case jet axis; the maximum uncertainty in the mean concentration measurement is ±3.44% in (

**a**), ±3.93% in (

**b**), ±3.35% in (

**c**), ±3.72% in (

**d**), ±3.45% in (

**e**), ±3.99% in (

**f**).

**Figure 11.**Non-dimensional mean concentration fields for an NBJ with Fr = 28.0 released into a receiving body affected by regular waves with T = 0.0 s and A = 0.0 mm (stagnant case,

**a**), T = 0.5 s and A = 5.0 mm (

**b**), T = 1.0 s and A = 5.0 mm (

**c**), T = 1.0 s and A = 12.5 mm (

**d**), T = 1.5 s and A = 5.0 mm (

**e**) and T = 1.5 s and A = 12.5 mm (

**f**), with the stagnant case jet axis; the maximum uncertainty in the mean concentration measurement is ±4.41% in (

**a**), ±4.52% in (

**b**), ±4.16% in (

**c**), ±4.38% in (

**d**), ±4.28% in (

**e**), ±4.31% in (

**f**).

**Figure 12.**R

_{DIL}versus Fr; R

_{DIL}is the ratio of the dilution of NBJs under waves to the dilution of jets in stagnant environment; (wave parameters in the legend).

**Table 1.**Main parameters for the experiments; A is the wave amplitude, H the wave height, T the wave period, L the wave length, d the water depth (0.40 m), Fr the densimetric Froude number; experiments 1 and 7 were performed without waves as reference cases.

Exp. | A [mm] | H [mm] | T [s] | L [m] | d/L | H/L | Fr |
---|---|---|---|---|---|---|---|

1 | 0 | 0 | - | - | - | - | 18.0 |

2 | 5.00 | 10.00 | 0.50 | 0.39 | 1.02 | 0.026 | 18.0 |

3 | 5.00 | 10.00 | 1.00 | 1.56 | 0.26 | 0.006 | 18.0 |

4 | 12.50 | 25.00 | 1.00 | 1.56 | 0.26 | 0.015 | 18.0 |

5 | 5.00 | 10.00 | 1.50 | 3.51 | 0.11 | 0.003 | 18.0 |

6 | 12.50 | 25.00 | 1.50 | 3.51 | 0.11 | 0.007 | 18.0 |

7 | 0 | 0 | - | - | - | - | 28.0 |

8 | 5.00 | 10.00 | 0.50 | 0.39 | 1.02 | 0.026 | 28.0 |

9 | 5.00 | 10.00 | 1.00 | 1.56 | 0.26 | 0.006 | 28.0 |

10 | 12.50 | 25.00 | 1.00 | 1.56 | 0.26 | 0.015 | 28.0 |

11 | 5.00 | 10.00 | 1.50 | 3.51 | 0.11 | 0.003 | 28.0 |

12 | 12.50 | 25.00 | 1.50 | 3.51 | 0.11 | 0.007 | 28.0 |

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**MDPI and ACS Style**

Ferrari, S.; Badas, M.G.; Querzoli, G.
On the Effect of Regular Waves on Inclined Negatively Buoyant Jets. *Water* **2018**, *10*, 726.
https://doi.org/10.3390/w10060726

**AMA Style**

Ferrari S, Badas MG, Querzoli G.
On the Effect of Regular Waves on Inclined Negatively Buoyant Jets. *Water*. 2018; 10(6):726.
https://doi.org/10.3390/w10060726

**Chicago/Turabian Style**

Ferrari, Simone, Maria Grazia Badas, and Giorgio Querzoli.
2018. "On the Effect of Regular Waves on Inclined Negatively Buoyant Jets" *Water* 10, no. 6: 726.
https://doi.org/10.3390/w10060726