# Large Eddy Simulation of Leakage Flow in a Stepped Labyrinth Seal

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method and Verification

#### Computational Domain and Boundary Conditions

^{+}= 30, so that the velocity gradient of the boundary layer could be sufficiently resolved.

^{−5}.

^{−6}s, the calculation was assumed to enter a statistically steady state at about 2e

^{−2}s, the result was obtained at the 4000th timestep, and the time average value was used. The calculation took about seven days using the 3.2 GHz Intel Xeon Gold 32-core processor.

^{+}< 3. The above results indicate that the discharge coefficient is not significantly changed at a grid resolution of up to 2.0 million, but grids with a resolution of around 7.0 million cannot adequately resolve a part of the flow structure in which complex flow occurs. Hence, in view of the complex flow structures that may occur in the stepped labyrinth seal, the subsequent analysis was conducted using at least 7.0 million grid units.

## 3. Results

#### 3.1. Discharge Coefficient and Flow Structure

#### 3.1.1. Straight-Through Seal

#### 3.1.2. Stationary Stepped Seal

#### 3.1.3. Rotating Stepped Seal

#### 3.2. Turbulence Kinetic Energy Estimation

#### 3.3. Flow Structure Effects on Mass Flow Rate Oscillation

#### 3.3.1. Stationary Condition

#### 3.3.2. Rotating Condition

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{c} | Cross sectional area of labyrinth seal [m^{2}] |

C_{d} | Discharge coefficient |

C_{p} | Specific heat |

k | Turbulence kinetic energy [m^{2}/s^{2}] |

L_{s} | Mixing length for sub-grid scales |

T | Temperature [K] |

U | x Velocity component [m/s] |

u | Flow velocity [m/s] |

u’ | RMS of the turbulent velocity fluctuations [m/s] |

P | Pressure [Pa] |

R | Specific gas constant |

t | Time [s] |

ṁ | Mass flow rate [kg/s] |

V | y velocity Component [m/s] |

x | coordinate |

Greek Symbols | |

ρ | Density [kg/m^{3}] |

$\tau $ | Turbulence stress |

ε | Turbulence kinetic energy dissipation [m^{2}/s^{3}] |

ν | Kinematic viscosity [m^{2}/s] |

γ | Specific heat ratio |

μ | Dynamic Viscosity [kg/m∙s] |

μ_{t} | Turbulence viscosity |

Subscripts | |

id | ideal |

in | inlet |

out | outlet |

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**Figure 2.**The computational domain of the stepped seal: (

**a**) the grid system; (

**b**) the geometric parameters.

**Figure 5.**Prediction of the discharge coefficient of the labyrinth seal: (

**a**) code validation study; (

**b**) comparison of the LES and RANS prediction results.

**Figure 6.**The velocity distributions in a straight-trough seal, as predicted by (

**a**) RANS (k-ε), and (

**b**) LES.

**Figure 7.**The velocity distributions in a stepped seal, as predicted by (

**a**) RANS (k-ε), and (

**b**) LES.

**Figure 8.**The velocity distributions in a stepped seal with a rotating shaft: (

**a**,

**b**) the meridional velocity according to (

**a**) the RANS and (

**b**) the LES; (

**c**,

**d**) the tangential velocity according to (

**c**) the RANS and (

**d**) the LES.

**Figure 9.**The effects of shaft rotation in the stepped seal upon (

**a**) the discharge coefficient, and (

**b**) the average tangential velocity.

**Figure 10.**The turbulence kinetic energy in a stepped seal according to (

**a**) RANS (k-ε), (

**b**) RANS (RSM), and (

**c**) LES.

**Figure 12.**The selection of data extraction points for power spectrum analysis: (

**a**) the vorticity magnitude distribution obtained by RANS; (

**b**) the selected data extraction points.

**Figure 14.**Power spectrum analysis of the LES data for a stationary stepped seal: (

**a**) the analyzed points and flow fluctuations; (

**b**) the power spectrum at each point; (

**c**) the flow structure at the points where power peaks are observed.

**Figure 15.**Power spectrum analysis of the LES data for a stepped seal with a rotating shaft: (

**a**) the analyzed points and flow fluctuations; (

**b**) the power spectrum at each point; (

**c**) the flow structure at the points where power peaks are observed.

**Figure 16.**The time averaged turbulence kinetic energy inside the stepped seal with (

**a**) a stationary shaft and (

**b**) a rotating shaft, as obtained by LES.

Geometrical Parameter | Straight-Through Seal | Stepped Seal |
---|---|---|

Clearance (C) | 0.5 mm | 0.5 mm |

Cavity width (CW) | 9 mm | 11.5 mm |

Tooth width (W) | 2.5 mm | 2.5 mm |

Tooth height (H) | 10.5 mm | 9 mm |

Minimum radius (R) | 200 mm | 200 mm |

Number of teeth (N) | 6 | 5 |

Step height (SH) | - | 0.7 mm |

Surface | Boundary Conditions |
---|---|

Inlet | Absolute Total Pressure = 202,650 Pa |

Outlet | Absolute Total Pressure = 101,325 Pa |

Shaft | 0 RPM, 20,000 RPM, Adiabatic wall |

Casing | 0 RPM, Adiabatic wall |

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**MDPI and ACS Style**

Kim, J.-H.; Ahn, J.
Large Eddy Simulation of Leakage Flow in a Stepped Labyrinth Seal. *Processes* **2021**, *9*, 2179.
https://doi.org/10.3390/pr9122179

**AMA Style**

Kim J-H, Ahn J.
Large Eddy Simulation of Leakage Flow in a Stepped Labyrinth Seal. *Processes*. 2021; 9(12):2179.
https://doi.org/10.3390/pr9122179

**Chicago/Turabian Style**

Kim, Ji-Hwan, and Joon Ahn.
2021. "Large Eddy Simulation of Leakage Flow in a Stepped Labyrinth Seal" *Processes* 9, no. 12: 2179.
https://doi.org/10.3390/pr9122179