# Numerical Study of Spray-Induced Turbulence Using Industrial Fire-Mitigation Nozzles

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## Abstract

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## 1. Introduction

## 2. Numerical Modeling

#### 2.1. Modeling Assumptions

#### 2.2. Governing Equations

## 3. Study of a Single Spray Nozzle

#### 3.1. Spray Nozzle

#### 3.2. Geometry, Mesh, and Spray Characteristics

#### 3.3. Simulation Convergence

#### 3.3.1. Mesh Size Effects

#### 3.3.2. Time Convergence

#### 3.4. Code Validation

#### 3.4.1. Monodisperse Spray

#### 3.4.2. Polydisperse Spray

#### 3.5. Turbulent Properties of the Monodisperse Spray

#### 3.6. Turbulent Properties of the Polydisperse Spray

#### 3.6.1. Droplet Volume Fraction

#### 3.6.2. Droplet Size Distribution

#### 3.6.3. Surface-Averaged Volume Fraction

## 4. Study of Two Interacting Nozzles

## 5. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${d}_{p}$ | Droplet diameter |

${f}_{g}$ | Gravity acceleration |

h | Vertical distance from the spray nozzle |

${h}_{\kappa}$ | Phase-averaged specific enthalpy for phase $\kappa $ |

${k}_{max}$ | Maximal turbulent kinetic energy |

${L}_{x}$, ${L}_{y}$, ${L}_{z}$ | Geometry sizes |

${L}_{t}$ | Turbulence integral length scale |

M | Interphase momentum transfer term |

${N}_{{d}_{p}}$ | Number of droplets of diameter ${d}_{p}$ |

p | Gas pressure |

$Pa$ | Droplet momentum number |

q | Heat transfer flux |

R | Radius from spray cone center |

${R}_{i,j}$ | Reynolds stress |

${u}_{g}$ | Gas velocity |

${v}_{p}$ | Particle velocity |

${\alpha}_{tot}$ | Total volume fraction of droplets |

$\overline{\alpha}$ | Surface-averaged droplet volume fraction |

$\rho $ | Mass density |

$\epsilon $ | Turbulent dissipation rate |

$\mathrm{\Gamma}$ | Interphase mass transfer rate |

${\Delta}_{t}$ | Time step |

${\Delta}_{x}$ | Grid size |

DNS | Direct Numerical Simulation |

RANS | Reynolds-Averaged Navier–Stokes |

LES | Large Eddy Simulation |

TKE | Turbulent Kinetic Energy |

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**Figure 2.**(

**a**) Ramp bottom nozzle and its section view [41]; (

**b**) hollow cone spray with fine droplets. h, the vertical distance from the nozzle; R, the radial distance from the spray central line; O, the numerical injection plane.

**Figure 3.**Geometry and mesh grid of a single spray nozzle: (

**a**) a sketch of the geometry and spray nozzle position; (

**b**) a grid of mean mesh size $\Delta x=1.5$ cm in the injection zone ${L}_{i}=1.5$ m; ${L}_{x}=3.5$ m, ${L}_{y}=3$ m, and ${L}_{z}=3.5$ m; with O the numerical injection plane; symmetric and wall boundary conditions highlighted.

**Figure 4.**Mesh convergence of different flow parameters as a function of the radial distance from the center R at plane $h=60$ cm; results of the mean mesh size $\Delta x=2.5$ cm ( ), $\Delta x=1.67$ cm ( ), $\Delta x=1.25$ cm ( ), $\Delta x=1.11$ cm ( ), and $\Delta x=0.67$ cm ( ); (

**a**) turbulence kinetic energy k, (

**b**) axial gas velocity ${u}_{g,z}$, and (

**c**) the mesh convergence tendency for ${k}_{max}$ as a function of mean mesh size $\Delta x$ at plane $h=60$ cm ( ).

**Figure 5.**Time convergence of the axial gas velocity ${u}_{g,z}$ and the turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$), with $\Delta x=1.5$ cm, $\Delta t=1$ ms: Position 1 ($h=0.5$ m) ( ), Position 2 ($h=1.5$ m) ( ), and Position 3 ($h=2.3$ m) ( ); (

**a**) vertical gas velocity and (

**b**) turbulent kinetic energy as a function of time.

**Figure 6.**Validation of the droplet velocity at different vertical positions as a function of the radial distance from the center R for a monodisperse spray; (

**a**) axial velocity ${v}_{d,z}$ and (

**b**) radial velocity ${v}_{d,r}$. Experimental measurements at $h=20$ cm, ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ); numerical simulation of [21] at $h=20$ cm ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ); current simulation at $h=20$ cm ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ).

**Figure 7.**Validation of the droplet velocity at different vertical positions as a function of the radial distance from the center R for a polydisperse spray; (

**a**) mean axial velocity ${\overline{v}}_{d,z}$ and (

**b**) mean radial velocity ${\overline{v}}_{d,r}$. Experimental measurements at $h=20$ cm, ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ); Numerical simulation of [21] at $h=20$ cm ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ); current simulation at $h=20$ cm ( ), $h=40$ cm ( ), $h=60$ cm ( ), $h=95$ cm ( ).

**Figure 8.**Spatial distribution of the properties on XZ plane at $y=0.05$ m for a monodisperse spray, $t=10$ s, $\Delta x=1.5$ cm, $\Delta t=1$ ms; (

**a**) droplet volume fraction ${\alpha}_{tot}$ and (

**b**) turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$).

**Figure 9.**Spatial distribution of the properties on XZ plane at $y=0.05$ m for a polydisperse spray, $t=10$ s, $\Delta x=1.5$ cm, $\Delta t=1$ ms; (

**a**) droplet volume fraction ${\alpha}_{tot}$ and (

**b**) turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$).

**Figure 10.**(

**a**) Turbulent integral length scale ${L}_{t}$ (m) on the XZ plane at $y=0.05$ m; (

**b**) turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$) on the XY plane at $h=1.5$ m.

**Figure 11.**Evolution of normalized volume fraction ${\alpha}_{{d}_{p}}/{\alpha}_{tot}$ for droplets of different diameters: ${d}_{p}=166$ μm ( ), ${d}_{p}=277$ μm ( ), ${d}_{p}=388$ μm ( ), ${d}_{p}=500$ μm ( ), ${d}_{p}=611$ μm ( ), ${d}_{p}=722$ μm ( ), ${d}_{p}=833$ μm ( ), ${d}_{p}=944$ μm ( ); (

**a**) $h=40$ cm and (

**b**) $h=95$ cm.

**Figure 12.**Histogram for different droplet diameters at vertical distances from the nozzle $h=20$ cm and $h=95$ cm; (

**a**) normalized volume fraction ${\alpha}_{{d}_{p}}/{\alpha}_{tot}$ and (

**b**) normalized number fraction ${N}_{{d}_{p}}/{N}_{tot}$.

**Figure 13.**(

**a**) Evolution of the surface-averaged droplet volume fraction $\overline{\alpha}$ as a function of h; monodisperse spray ( ), polydisperse spray ( ); (

**b**) Evolution of ${k}_{max}$ at different vertical surfaces as a function of the vertical distance; monodisperse spray ( ), polydisperse spray ( ).

**Figure 14.**(

**a**) Geometry and (

**b**) mesh of the two spray nozzles $\Delta x=2$ cm; ${L}_{i}=3$ m, ${L}_{x}=7$ m, ${L}_{y}=3$ m, and ${L}_{z}=3.5$ m with O the numerical injection plane; symmetric and wall boundary conditions highlighted in (

**a**).

**Figure 15.**Properties on the XZ plane at 30 s; (

**a**) droplet volume fraction ${\alpha}_{tot}$, (

**b**) integral length scale ${L}_{t}$ (m), (

**c**) turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$), and (

**d**) turbulent dissipation rate $\epsilon $ (${\mathrm{m}}^{2}/{\mathrm{s}}^{3}$).

**Figure 16.**Evolutions of the droplet volume fraction ${\alpha}_{tot}$ (

**a**) and the turbulent kinetic energy k (${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$) (

**b**) on the XY plane $y=0.05$ m as a function of x, $h=0.2$ m ( ), $h=0.5$ m ( ), $h=1.0$ m ( ), and $h=1.5$ m ( ).

**Figure 17.**(

**a**) Evolution of the surface averaged droplet volume fraction $\overline{\alpha}$ as a function of h: tow nozzles ( ) and single nozzle ( ); (

**b**) evolution of ${k}_{max}$ as a function of the vertical distance h: two nozzles ( ) and single nozzle ( ).

**Table 1.**Particles with different diameters in the polydisperse spray [21].

Particle Class | Diameter (m) |
---|---|

1 | 55 |

2 | 166 |

3 | 277 |

4 | 388 |

5 | 500 |

6 | 611 |

7 | 722 |

8 | 833 |

9 | 944 |

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**MDPI and ACS Style**

Gai, G.; Hadjadj, A.; Kudriakov, S.; Mimouni, S.; Thomine, O.
Numerical Study of Spray-Induced Turbulence Using Industrial Fire-Mitigation Nozzles. *Energies* **2021**, *14*, 1135.
https://doi.org/10.3390/en14041135

**AMA Style**

Gai G, Hadjadj A, Kudriakov S, Mimouni S, Thomine O.
Numerical Study of Spray-Induced Turbulence Using Industrial Fire-Mitigation Nozzles. *Energies*. 2021; 14(4):1135.
https://doi.org/10.3390/en14041135

**Chicago/Turabian Style**

Gai, Guodong, Abdellah Hadjadj, Sergey Kudriakov, Stephane Mimouni, and Olivier Thomine.
2021. "Numerical Study of Spray-Induced Turbulence Using Industrial Fire-Mitigation Nozzles" *Energies* 14, no. 4: 1135.
https://doi.org/10.3390/en14041135