# Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Heavy-Gas Dispersal

#### 1.2. Two-Layer Shallow Water Approach

- Flows of the same liquid but at different temperatures, resulting in density differences, such situations being typical of oceanic flows.
- Flows of two liquids of different densities.
- Flows of two gases evolving at low Mach number.

- The vertical velocity component is neglected.
- The velocity is assumed uniform in cross sections of each layer.

- Consider turbulent effects in the phases, as they result in the appearance of a “turbulent sound speed” (Forestier et al., 1997 [20], Saurel et al., 2003 [21], Lhuillier et al., 2013 [22]). In the frame of shallow water flows, these effects have been studied in Richard and Gavrilyuk, 2012 [10] and Gavrilyuk et al., 2016 [14].

## 2. Hyperbolic Two-Layer Shallow Water Model

- Resolution of the hyperbolic step, i.e., the resolution of System (1) in the absence of source terms. An HLL-type Riemann solver has been developed in Chiapolino and Saurel, 2018 [8] and is summarized in the following.
- Stiff pressure relaxation towards the hydrostatic and atmospheric pressures and reset of the heights. This step makes the solution tend to the one of the conventional models [28].

## 3. HLL-Type Riemann Solver and Godunov-Type Method

## 4. Experimental Apparatus

#### 4.1. Description of the Experimental Setup

#### 4.2. Course of Operations

#### 4.3. Processing of the Shadowgraphs

#### 4.4. Experimental Configurations and Data

^{−1}is the molar mass of krypton and $\widehat{\mathrm{R}}=8.314$ J/mol/K is the ideal gas constant. Pressure and temperature equilibria between krypton and atmospheric air are assumed. It could be objected that, since krypton is expanded from a high-pressure bottle, it could be colder than the atmosphere. However, krypton flows slowly in a several-meter long pipe before arriving in the container and this pipe acts as a heat exchanger. Equation (16) is only used to provide the order of magnitude of the cloud Froude and Reynolds numbers for the various container heights as will be seen in Section 4.7. Consequently, assuming pressure and temperature equilibria between krypton and air seems reasonable.

^{−1}, which corresponds to the precision of the flowmeter. For every case, the actual initial height of krypton is close to the target value and the difference between the two is always below the calculated uncertainty, which confirms repeatability of the initial conditions.

#### 4.5. Scaling Law

#### 4.6. Expected Evolution of the Cloud

#### 4.7. Results and Discussion

^{2}. A slight inflection can for example be seen at 0.23 s on the graphs in Figure 7 corresponding to the initial height of 15 cm. Such inflection appears because part of the cloud is then located over the two beams that support the floor of the vessel, which hinders the determination of the circle.

^{3}. The evolution of the cloud radius and area was estimated thanks to aerial photographs taken from a helicopter located 300 m above the test site [35]. The analysis of the collected data allowed to determine that the cloud Froude number was 1.05 ± 0.12 [1]. This value is higher than the ones found during the current trials, which tends to corroborate the previous observation made about the Froude number.

^{−1}. For the current trials, the value of the constant is $\mathrm{K}\simeq 0.06$ m

^{2}/s for ${\mathrm{h}}_{\mathrm{K}\mathrm{r}}^{}=15$ cm, $\mathrm{K}\simeq 0.05$ m

^{2}/s for ${\mathrm{h}}_{\mathrm{K}\mathrm{r}}^{}=10$ cm and $\mathrm{K}\simeq 0.03$ m

^{2}/s for ${\mathrm{h}}_{\mathrm{K}\mathrm{r}}^{}=5$ cm. The cloud Reynolds number is thus smaller for small heights of container and viscous effects are indeed more prominent in that case. In the case of the Thorney Island trials [35], $\mathrm{K}\simeq 68$ m

^{2}/s, which tends to confirm the previous conclusion.

## 5. Numerical Results with Two-Layer Model and Laboratory Experiments Comparison

#### 5.1. Two-Layer Flow Model with Cylindrical Symmetry

#### 5.2. Velocity Relaxation

#### 5.3. Results and Comparisons

^{3}. However, for symmetry reasons, the following 1D computations only consider half of the domain. These initial data correspond to the ones of the experiments of Section 4.

^{3}for the krypton and ${\mathsf{\rho}}_{2}^{}=1.29$ kg/m

^{3}for the air. The gravity constant is $\mathrm{g}=9.81$ m/s

^{2}. Fluids are initially at rest ${\mathrm{u}}_{1}^{}={\mathrm{u}}_{2}^{}=0$ m/s. Reflective boundary conditions are used (symmetry condition on the left and wall on the right). Acoustic impedances ${\mathrm{Z}}_{\mathrm{k}}^{}={\mathsf{\rho}}_{\mathrm{k}}^{}{\mathrm{c}}_{\mathrm{k}}^{}={\left({\mathsf{\rho}}_{\mathrm{k}}^{}{\mathrm{c}}_{\mathrm{k}}^{}\right)}_{\mathrm{a}\mathrm{t}\mathrm{m}}^{}$ are considered constant and are computed with ${\mathsf{\rho}}_{1}^{}=3.506$ kg/m

^{3}and ${\mathrm{c}}_{1}^{}=218$ m/s for the krypton and ${\mathsf{\rho}}_{2}^{}=1.29$ kg/m

^{3}and ${\mathrm{c}}_{2}^{}=340$ m/s for the air. The atmospheric pressure is ${\mathrm{p}}_{0}^{}=101325$ Pa. The “sound speed” numerical parameters are ${\mathsf{\theta}}_{1}^{}={\mathsf{\theta}}_{2}^{}=2$ for both fluids during the hyperbolic step.

^{2}is recorded, the krypton front position is considered at $\frac{{\mathrm{x}}_{\mathrm{i}}^{}+{\mathrm{x}}_{\mathrm{i}+1}^{}}{2}$ where ${\mathrm{x}}_{\mathrm{i}}^{}$ and ${\mathrm{x}}_{\mathrm{i}+1}^{}$ represent locations of the cell centers $\mathrm{i}$ and $\mathrm{i}+1$.

#### 5.4. Concluding Remarks

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Two-Layer Flow Model with Cylindrical Symmetry

#### Appendix A.2. Velocity Relaxation

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**Figure 1.**Global view of the experimental setup. The retro-reflective back panel was removed when the picture was taken.

**Figure 2.**Detailed view of the retractable, metallic, cylindrical container used to contain the dense gas at the initial state.

**Figure 3.**Detailed view of the lens of one of the cameras showing the small plane mirror used to direct the light on the same axis as the camera optical axis.

**Figure 6.**Shadowgraphs (front view and top view) of trial 15E showing the evolution of the krypton cloud. For each row, the front and top view pictures are taken at the same time. The origin of time corresponds to the beginning of the descent of the container.

**Figure 8.**Temporal evolution of the cloud area (defined by Equation (21)) measured on the shadowgraphs in scaled coordinates.

**Figure 9.**Temporal evolution of the krypton front position. Comparison of the one-dimensional solution of the present two-layer shallow water model versus experimental data. The initial krypton column is 5-cm high. Computed results are in good agreement with experimental measurements.

**Figure 10.**Temporal evolution of the krypton front position. Comparison of the one-dimensional solution of the present two-layer shallow water model versus experimental data. The initial krypton column is 10-cm high. Computed results are in good agreement with experimental measurements.

**Figure 11.**Temporal evolution of the krypton front position. Comparison of the one-dimensional solution of the present two-layer shallow water model versus experimental data. The initial krypton column is 15-cm high. Computed results are in good agreement with experimental measurements.

Case | ${\mathbf{t}}_{\mathbf{K}\mathbf{r}}^{}$ | ${\mathbf{Q}}_{\mathbf{K}\mathbf{r}}^{}$ | ${\mathbf{T}}_{\mathbf{a}}^{}$ | ${\mathbf{p}}_{\mathbf{a}}^{}$ | ${\mathbf{\rho}}_{\mathbf{K}\mathbf{r}}^{}$ | ${\mathbf{V}}_{\mathbf{K}\mathbf{r}}^{}$ | ${\mathbf{h}}_{\mathbf{K}\mathbf{r}}^{}$ | $\mathbf{\Delta}{\mathbf{h}}_{\mathbf{K}\mathbf{r}}^{}$ |
---|---|---|---|---|---|---|---|---|

[L·min^{−1}] | [°C] | [mbar] | [kg·m^{−3}] | [L] | [cm] | [cm] | ||

5A | 1 min 34 s | 0.25 ± 0.01 | 16.8 | 976 | 3.39 | 0.39 ± 0.02 | 5.0 ± 0.3 | 0.0 ± 0.3 |

5B | 1 min 35 s | 0.24 ± 0.01 | 16.9 | 976 | 3.39 | 0.38 ± 0.02 | 4.9 ± 0.3 | −0.1 ± 0.3 |

5C | 1 min 34 s | 0.25 ± 0.01 | 17.8 | 981 | 3.40 | 0.39 ± 0.02 | 5.0 ± 0.3 | 0.0 ± 0.3 |

5D | 1 min 35 s | 0.24 ± 0.01 | 17.7 | 981 | 3.40 | 0.38 ± 0.02 | 4.9 ± 0.3 | −0.1 ± 0.3 |

5E | 1 min 35 s | 0.24 ± 0.01 | 17.6 | 981 | 3.40 | 0.38 ± 0.02 | 4.9 ± 0.3 | −0.1 ± 0.3 |

5F | 1 min 36 s | 0.25 ± 0.01 | 18.1 | 981 | 3.39 | 0.40 ± 0.02 | 5.1 ± 0.3 | 0.1 ± 0.3 |

5G | 1 min 34 s | 0.24 ± 0.01 | 18.0 | 981 | 3.40 | 0.38 ± 0.02 | 4.9 ± 0.3 | −0.1 ± 0.3 |

5H | 1 min 35 s | 0.24 ± 0.01 | 18.3 | 980 | 3.39 | 0.38 ± 0.02 | 4.9 ± 0.3 | −0.1 ± 0.3 |

10A | 3 min 9 s | 0.25 ± 0.01 | 18.0 | 980 | 3.39 | 0.79 ± 0.03 | 10.1 ± 0.4 | 0.1 ± 0.4 |

10B | 3 min 9 s | 0.24 ± 0.01 | 17.9 | 980 | 3.39 | 0.76 ± 0.03 | 9.8 ± 0.4 | −0.2 ± 0.4 |

10C | 3 min 8 s | 0.24 ± 0.01 | 15.9 | 978 | 3.41 | 0.75 ± 0.03 | 9.6 ± 0.4 | −0.4 ± 0.4 |

10D | 3 min 8 s | 0.25 ± 0.01 | 16.5 | 978 | 3.40 | 0.78 ± 0.03 | 10.0 ± 0.4 | 0.0 ± 0.4 |

10E | 3 min 9 s | 0.24 ± 0.01 | 16.3 | 978 | 3.41 | 0.76 ± 0.03 | 9.8 ± 0.4 | −0.2 ± 0.4 |

10F | 3 min 8 s | 0.25 ± 0.01 | 16.7 | 978 | 3.40 | 0.78 ± 0.03 | 10.0 ± 0.4 | 0.0 ± 0.4 |

15A | 4 min 46 s | 0.24 ± 0.01 | 15.0 | 975 | 3.41 | 1.14 ± 0.05 | 14.6 ± 0.6 | −0.4 ± 0.6 |

15B | 4 min 45 s | 0.24 ± 0.01 | 15.1 | 975 | 3.41 | 1.14 ± 0.05 | 14.6 ± 0.6 | −0.4 ± 0.6 |

15C | 4 min 44 s | 0.25 ± 0.01 | 15.4 | 975 | 3.41 | 1.18 ± 0.05 | 15.1 ± 0.6 | 0.1 ± 0.6 |

15D | 4 min 44 s | 0.25 ± 0.01 | 15.5 | 975 | 3.41 | 1.18 ± 0.05 | 15.1 ± 0.6 | 0.1 ± 0.6 |

15E | 4 min 44 s | 0.25 ± 0.01 | 16.3 | 975 | 3.40 | 1.18 ± 0.05 | 15.1 ± 0.6 | 0.1 ± 0.6 |

15F | 4 min 44 s | 0.25 ± 0.01 | 16.2 | 975 | 3.40 | 1.18 ± 0.05 | 15.1 ± 0.6 | 0.1 ± 0.6 |

Case | Froude Number [–] | |||
---|---|---|---|---|

Top View | Front View–Left | Front View–Right | Average | |

5A | 0.61 | 0.80 | 0.70 | 0.70 |

5B | - | 0.79 | 0.65 | 0.72 |

5C | 0.63 | 0.72 | 0.78 | 0.71 |

5D | 0.58 | 0.80 | 0.63 | 0.67 |

5E | 0.61 | 0.82 | 0.75 | 0.73 |

5F | 0.61 | 0.71 | 0.58 | 0.63 |

5G | 0.58 | 0.71 | 0.69 | 0.66 |

5H | 0.61 | 0.76 | 0.73 | 0.70 |

10A | 0.63 | 0.78 | 0.81 | 0.74 |

10C | 0.57 | 0.79 | 0.73 | 0.70 |

10D | 0.62 | 0.77 | 0.81 | 0.73 |

10E | 0.61 | 0.86 | 0.93 | 0.80 |

10F | 0.63 | 0.86 | 0.71 | 0.73 |

15A | 0.80 | 0.80 | 1.01 | 0.87 |

15B | 0.81 | 1.00 | 0.84 | 0.88 |

15C | 0.77 | 0.81 | 0.93 | 0.84 |

15D | 0.81 | 0.80 | 0.85 | 0.82 |

15E | 0.72 | 0.90 | 0.82 | 0.81 |

15F | 0.75 | 0.87 | 0.96 | 0.86 |

${\mathbf{h}}_{\mathbf{K}\mathbf{r}}\text{}\left[\mathbf{cm}\right]$ | $\mathbf{Average}\text{}\mathbf{F}\mathbf{r}$ [–] | Standard Deviation [–] |
---|---|---|

5 | 0.69 | 0.03 |

10 | 0.74 | 0.03 |

15 | 0.85 | 0.03 |

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**MDPI and ACS Style**

Chiapolino, A.; Courtiaud, S.; Lapébie, E.; Saurel, R.
Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations. *Fluids* **2021**, *6*, 2.
https://doi.org/10.3390/fluids6010002

**AMA Style**

Chiapolino A, Courtiaud S, Lapébie E, Saurel R.
Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations. *Fluids*. 2021; 6(1):2.
https://doi.org/10.3390/fluids6010002

**Chicago/Turabian Style**

Chiapolino, Alexandre, Sébastien Courtiaud, Emmanuel Lapébie, and Richard Saurel.
2021. "Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations" *Fluids* 6, no. 1: 2.
https://doi.org/10.3390/fluids6010002