# Numerical Simulation of Kelvin–Helmholtz Instability and Boundary Layer Stripping for an Interpretation of Melt Jet Breakup Mechanisms

^{*}

## Abstract

**:**

## 1. Introduction

**Table 1.**Physical property of corium and Woods metal [13].

Property | Corium [4] | Woods Metal [12] |
---|---|---|

Composition | UO_{2} 80% + ZrO_{2} 20% | Bi 50% + Pb 27% + Sn 13% + Cd 10% |

Melting temperature, °C | 2609 | 72 |

Density, kg/m^{3} | 7300 | 9383 |

Specific heat, J/kgK | 510 | 168 |

Thermal conductivity, W/mK | 3.0 | 18.8 |

Dynamic viscosity, Pa/s | 0.005 | 0.002 |

Surface tension, N/m | 0.573 | 0.43 |

Latent heat of fusion, J/kg | 280,000 | 33,500 |

## 2. Mathematical Model

_{p}and α

_{q}, represents the position and the interface in computational cells. If α

_{q}= 0, phase p only exists in the cell and phase q only exists where α

_{q}= 1. Naturally, the interface locates in the cell where 0 < α

_{q}< 1. The sum of volume fractions of all phases is unity,

^{5}of the Reynolds number. Therefore, the flow was assumed laminar.

## 3. Results and Discussions

^{−6}s fulfilled the Courant number requirement, and the time step of 1 × 10

^{−5}s was good for the mesh size of 0.2 mm in BLS simulations.

^{−3}. Most of the boundaries, except the no-slip walls, are shear-free walls because the simulation domain is a part of fluid volume.

#### 3.1. Two-Dimensional Simulation of COLDJET Experiment

#### 3.2. Two-Dimensional Simulation of Kelvin–Helmholtz Instability

_{D}at the earliest time when it can be identified was 0.45 mm for relative velocity of 4.0 m/s and 5.64 mm for 1.0 m/s. The linear analysis of KHI (Equation (2)) gives 0.28 mm for relative velocity of 4.0 m/s and 4.8 mm for 1.0 m/s. Comparison of these numbers implies that λ

_{D}at the beginning of wave growth obtained by CFD simulation is close to the estimate by Equation (2). However, as the wave grows further, λ

_{D}increases gradually.

#### 3.3. Three-Dimensional Simulation of Boundary Layer Stripping

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

D | diameter [J/kg] |

E | specific internal energy [J/kg] |

F | volumetric force [N/m^{3}] |

g | gravitational constant [m/s^{2}] |

k | thermal conductivity [W/mK] |

$\dot{m}$ | volumetric mass transfer rate [kg/m^{3}s] |

p | pressure [Pa] |

S | source term |

S_{h} | heat source [J/m^{3}] |

t | time [s] |

T | temperature [K] |

V | velocity [m/s] |

Greek letters | |

α | volume fraction |

λ | wavelength [m] |

ρ | density [kg/m^{3}] |

σ | surface tension [N/m] |

μ | viscosity [Pa·s] |

Subscripts | |

c | critical |

d | fastest growing |

j | jet |

l | liquid |

p | phase p |

q | phase q |

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**Figure 1.**Woods metal jet breakup in water: T

_{j}= 85 °C, T

_{l}= 40 °C, D

_{j}= 50 mm, V

_{j,in}= ~1.0 m/s [13].

**Figure 8.**Fastest growing wavelength v.s. time (

**left**) and corresponding contour of volume fraction (

**right**) in Kelvin–Helmholtz instability (ΔV = 4 m/s); the contour size is 3.0 mm vertically and 6.2 mm horizontally. These correspond to 150 vertical and 310 horizontal mesh elements.

**Figure 9.**Wavelength analysis by Fast Fourier Transform (FFT) technique on melt volume fraction: melt volume fraction near interface (

**left**), FFT spectrum (

**right**).

**Figure 10.**Particles produced by Kelvin–Helmholtz instability: volume fraction contour (

**left**), melt-water interface outline (

**right**).

**Figure 11.**Time variation of particle size distribution in jet breakup by Kelvin–Helmholtz instability.

**Figure 17.**Identification of breakup modes on the particle size distribution of COLDJET experiment [13]: red box indicates particle sizes obtained from KHI simulation, and blue box indicates particle sizes obtained from BLS simulation.

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**MDPI and ACS Style**

Kim, M.-S.; Bang, K.-H.
Numerical Simulation of Kelvin–Helmholtz Instability and Boundary Layer Stripping for an Interpretation of Melt Jet Breakup Mechanisms. *Energies* **2022**, *15*, 7517.
https://doi.org/10.3390/en15207517

**AMA Style**

Kim M-S, Bang K-H.
Numerical Simulation of Kelvin–Helmholtz Instability and Boundary Layer Stripping for an Interpretation of Melt Jet Breakup Mechanisms. *Energies*. 2022; 15(20):7517.
https://doi.org/10.3390/en15207517

**Chicago/Turabian Style**

Kim, Min-Soo, and Kwang-Hyun Bang.
2022. "Numerical Simulation of Kelvin–Helmholtz Instability and Boundary Layer Stripping for an Interpretation of Melt Jet Breakup Mechanisms" *Energies* 15, no. 20: 7517.
https://doi.org/10.3390/en15207517