# Large-Eddy Simulations with an Immersed Boundary Method: Pollutant Dispersion over Urban Terrain

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## Abstract

**:**

## 1. Introduction

## 2. The Immersed Boundary Method in Meso-NH

## 3. The AZF Case

## 4. Presentation of the AZF Case Simulation

## 5. Simulations of the AZF Case

#### 5.1. Sensitivity to Different Numerical Schemes

#### 5.2. Nitrogen Dioxide Dispersion

## 6. Discussion on Population Exposure

#### 6.1. The Initial Plume Structure

- Barthélémy et al. [21] mentioned an explosion of $\mathcal{O}($10${}^{5}-$10${}^{6})$ kg of ammonium nitrate. The maximum produced NO${}_{2}$ mass can be evaluated assuming that oxidation was complete and that all the nitrogen atoms formed the nitrogen dioxide. Assuming [4:8] dozen of tons of detonated ammonium nitrate [20], a plume with a volume of ${\mathcal{V}}_{0}=$ 5.10${}^{7}$ m${}^{3}$ and an uniform NO${}_{2}$ distribution results in a mean concentration approaching [1:2] g m${}^{-3}$.
- Considering the ORAMIP measurement at the Jacquier Station, the passive sampler used for the measurement provides a NO${}_{2}$ value mean over 15 min which gives a dose estimate. Knowing the experimental value of ${\widehat{C}}_{jaq}^{exp}=$ 350 $\mathsf{\mu}$g m${}^{-3}$ (the difference of the observed concentrations at Jacquier between 0830 and 0815 UTC), considering the plume dilution to be well–modelled by MNH-IBM and following the numerical results for ${\widehat{\tilde{C}}}_{jaq}$ (Figure 7b), the range of the presumed initial NO${}_{2}$ concentration is estimated to [10:30] mg m${}^{-3}$ with a more likely value of ${C}_{ini}\sim $ 20 mg m${}^{-3}\sim $ 10 ppm (ratio defined by the mole fraction).
- Deedi [22] reported several concentration values from a study of the plume opacity. The few pictures of the initial plume lead to a $\left[\mathcal{O}\right($10${}^{1}):\mathcal{O}($10${}^{2}\left)\right]$mg m${}^{-3}$ NO${}_{2}$ concentration range.

#### 6.2. The Population Exposure

## 7. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AZF | Azote de France |

BL | Boundary Layer |

CCT | Cut-Cell Technique |

DFT | Discrete Fourier Transform |

ERK | Explicit Runge-Kutta scheme |

GCT | Ghost-Cell Technique |

IBM | Immersed Boundary Method |

LES | Large Eddy Simulation |

MNH-IBM | IBM adapted to the Meso-NH model |

MUST | Mock Urban Setting Test |

ORAMIP | Observatoire Régional de l’Air en Midi-Pyrénées |

PPM | Piecewise Parabolic Method |

SSF | Surface State Function |

SSI | Surface State Index |

WENO | Weight-Essential-Non-Oscillatory |

## Appendix A. Surface State and LevelSet Function

- P1: If $|SS{I}_{l}|=3$, no process is expected (Figure A1a).
- P3: If $|SS{I}_{l}|=1$ and $|SS{I}_{s}|<2$, the ‘filtering’ process is activated. The obstacle shown in Figure A1c disappears.
- P4: The proximity of the obstacles (or spacings) can induce a ‘merging’ process testing $SS{I}_{l}$ = ±1 and $SS{I}_{s}\mp 1$ (opposite sign). This is performed once in all grid directions and a second time only in the diagonal directions to limit the ‘stair-step shaped edges’ effect (Figure A1d–f). Figure A1d shows a small obstacle in the domain centre and another obstacle in the top-left corner. The ’increasing’ process (P2) is activated and the obstacle in the centre is extended. The first step of the ’merging’ process (P4) affects the median height of the two obstacles for cells detected at the blue points (Figure A1e). The second ’merging pass’ softens the border of the merged blue region (Figure A1f).

**Table A1.**Modifications of the horizontal obstacle size depending on the Surface State Index ($SSI$).

Process | Class | Direction | Local $\mathbf{SSI}$ | Surrounding $\mathbf{SSI}$ |
---|---|---|---|---|

P1 | Nothing | All | $SS{I}_{l}=\pm 3$ | . |

P2 | Increasing | All | $SS{I}_{l}=\pm 2$ | $\nexists SS{I}_{s}=\pm 3$ |

P3 | Filtering | All | $SS{I}_{l}=\pm 1$ | $\nexists SS{I}_{s}=\pm 2$ |

P4 | Merging | All | $SS{I}_{l}=\pm 1$ | $\exists SS{I}_{s}=\mp 1$ |

**Figure A1.**(

**a**–

**c**): Situations depending on the Surface State Index ($SSI$) inducing processes on the horizontal obstacle size: (

**a**) Nothing ($SSI=3$); (

**b**) Increasing ($SSI=2$); (

**c**) Filtering ($SSI=1$). (

**d**–

**f**): Process combinations on the horizontal obstacle size function of $SSI$: (

**d**) Reference state is two bodies; (

**e**) Increasing one; (

**f**) Merging the two bodies. The red contours indicate the presence of the obstacle. The green contours indicate the non-presence of the obstaclee. The blue points indicate the region to be merged.

## Appendix B. Smoothing Technique and LevelSet Function

**Figure A2.**(

**a**) Errors introduced by the discrete nature of the interface during the LSF computation; (

**b**) Dependence of the LSF-relative uncertainty (${\varphi}_{theo}=r$) with the $\Delta L/\Delta X$ resolution ratio (cylinder case).

## Appendix C. Geometric Properties and LevelSet Function

**Figure A3.**(

**a**) Illustration of the $\frac{\Delta L}{\Delta X}=\pi $ resolution ratio where $\Delta X$ is the mesh (resp. interface) space step in green symbols, where $\Delta L$ is the interface space step in red symbols. The $\varphi $-contours are shown without smoothing for $\frac{\Delta L}{\Delta X}$ = [(

**b**) $\pi $; (

**c**) $\frac{\pi}{2}$; (

**d**) $\frac{\pi}{4}$] (cylinder case).

**Figure A4.**Interface curvature $\kappa $ for the cylinder (

**a**,

**b**) and cube (

**c**,

**d**) cases: (

**a**) and (

**c**) impact of the $\frac{\Delta L}{\Delta X}$ resolution ratio with $N=0$; (

**b**) and (

**d**) Impact of the N smoothing level with (

**b**) $\frac{\Delta L}{\Delta X}=\frac{\pi}{8}$ and (

**d**) $\frac{\Delta L}{\Delta X}=\frac{1}{2}$. For the cylinder case, X is the projection of the interface location on an arbitrary axis of the R radius. For the cube case, R is the radial distance of the interface to the cube centre with a length L.

**Figure A5.**LSF generation: (

**a**) Illustration of the discrete interface (in yellow) not correlated with the mesh. The $\varphi =0$-contour is visualized with $\frac{\Delta L}{\Delta X}=\frac{1}{2}$ for four N smoothing levels: (

**b**) $N=0$; (

**c**) $N=16$; (

**d**) $N=32$; (

**e**) $N=64$ (cube case).

**Figure A6.**Geometric properties: (

**a**) $\varphi =0$ altitude; (

**b**) Horizontal component of the $\overrightarrow{n}$ normal vector to the interface; (

**c**) Vertical component of the $\overrightarrow{n}$ normal vector to the interface; (

**d**) Local curvature $\kappa $. $(x,y)$ indicates the horizontal interface location of the interface. z corresponds to the studied variable. The top panels show the analytical solution in black. In red, the numerical solutions are plotted without the smoothing technique in the middle of the panels. In red, the numerical solutions are plotted with the smoothing technique in the bottom of the panels. The poor numerical curvature without smoothing is not presented (bell case).

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**Figure 1.**(

**a**) Aerial photograph of Toulouse and its hilly region (www.geoportail.gouv.fr); (

**b**) Topographic map of the same zone. The MNH-IBM study area is bounded by a grey contour. The black circle indicates the location of the factory and the black cross indicates the Jacquier station. The green color indicates a near-flat terrain with a 150 m altitude above sea level.

**Figure 2.**(

**a**) IGN-BDTOPO database related to the bounded area in Figure 1: Yellow–orange–red colours represent buildings of 5–10 m height, purple–blue 10–20 m, and green-grey 30–60 m. (

**b**) Recovery of the fluid–solid interface in the MNH-IBM domain modelled by the zero value of the LevelSet function. The black circle indicates the location of the factory in the two pictures. Figure 1’s legend informs about the space scale.

**Figure 3.**Large-scale vertical profiles at 0815 UTC at the Azote de France (AZF) fertilizer plant location: (

**a**) Potential temperature; (

**b**) zonal (in red line), and meridional (in green line) components of the mean wind. The dashed lines indicate the height of the large-eddy simulations (LES) domain.

**Figure 4.**Surface map of the 15-min averages of the Gaussian integration of variables 1 m above the ground: (

**a**,

**b**) Wind speed; (

**c**,

**d**) wind fluctuations; (

**e**,

**f**) subgrid turbulence kinetic energy. The left column shows the results of the $C4/C2$ advection scheme, and the right column $W5/W3$.

**Figure 5.**Spectral density (

**a**,

**b**) in the boundary layer up to 200 m; (

**c**,

**d**) in the highest levels 200 m $<z<$ 800 m. (

**a**) and (

**c**) present the discrete Fourier transform (DFT) of the zonal wind component, (

**b**) and (

**d**) the DFT of the vertical wind component. The colour code indicates the advection scheme and the line type of the run depending on the spin-up.

**Figure 6.**Four visualisations of the plume dispersion for the C4/C2-R3 simulation from the initial stage (top) until 6 min after the emission (bottom) with two view angles (left and right). The colour code indicates the ratio of pollutant concentration to the initial value (in %) with dark brown/black corresponding to the initial concentration. The ground and the buildings appear in grey. At $t=6$ min, the plume presents a length of 2 km and a width of 500 m.

**Figure 7.**At the ’Maurice Jacquier’ primary school: (

**a**) Temporal evolution of the scalar concentration (according to the initial value); (

**b**) relationship between the mean and the maximum concentrations numerically observed within 15 min after the emission.

**Figure 8.**Exposure time (s) near the ground as a percentage of the initial pollutant concentration: (

**a**) 20%; (

**b**) 10%; (

**c**) 4%, and (

**d**) 2% from the C4/C2-R3 simulation. The black cross indicates the location of the Jacquier Station.

**Figure 9.**Estimate of the population exposure as a function of the NO${}_{2}$ concentration and the exposure time. The colour code and symbol style indicate the simulation type. Two thresholds are given by Tissot and Lafon [23].

Notation | Definition |
---|---|

f | Local and instantaneous variable |

$\tilde{f}$ | Gaussian distribution near the ground |

$\widehat{f}$ | 15 min integration |

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**MDPI and ACS Style**

Auguste, F.; Lac, C.; Masson, V.; Cariolle, D. Large-Eddy Simulations with an Immersed Boundary Method: Pollutant Dispersion over Urban Terrain. *Atmosphere* **2020**, *11*, 113.
https://doi.org/10.3390/atmos11010113

**AMA Style**

Auguste F, Lac C, Masson V, Cariolle D. Large-Eddy Simulations with an Immersed Boundary Method: Pollutant Dispersion over Urban Terrain. *Atmosphere*. 2020; 11(1):113.
https://doi.org/10.3390/atmos11010113

**Chicago/Turabian Style**

Auguste, Franck, Christine Lac, Valery Masson, and Daniel Cariolle. 2020. "Large-Eddy Simulations with an Immersed Boundary Method: Pollutant Dispersion over Urban Terrain" *Atmosphere* 11, no. 1: 113.
https://doi.org/10.3390/atmos11010113