#
Numerical Prediction of the Aerodynamics and Acoustics of a Tip Leakage Flow Using Large-Eddy Simulation^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Set-Up

## 3. Numerical Settings

**AVBP**, an explicit, unstructured, massively parallel solver [16] which solves the compressible Navier–Stokes equations. The package

**pyhip**[17] to handle unstructured computational grids and their associated datasets is used in combination with the

**antares**[18] pre-postprocessing library. In this paper, each LES is performed using the same following set-up. The convective fluxes are computed using the Two-Step Taylor-Galerkin C (TTGC) finite element scheme [19]. This scheme is third-order accurate in time and space. The viscous fluxes are computed using the $2\Delta $ diffusion operator from Colin [20]. Finally, the closure of the LES equations is done using the SIGMA subgrid scale model from Nicoud et al. [21]. Regarding the boundary condition, each simulation shares the wall modelling approach and the outlet boundary modelling: a wall law [22] is applied on each wall, and a characteristic boundary condition (NSCBC) based on static pressure is applied at outlet [23]. The inlet boundary conditions are detailed below.

## 4. Effects of Inflow Conditions

#### 4.1. Instantaneous Flow

#### 4.2. Mean Flow

## 5. Tip Leakage Vortex Trajectory

## 6. Tip Leakage Vortex Convection

**pyhip**[17] tool, 38 × 10${}^{6}$ tetrahedrons are added to the initial mesh, and the minimal edge size is divided by a factor of 1.12. The magnification factor is set to $\alpha $ = 100, and the minimum of the metric field to $\u03f5$ = 0.7. The spatial extension of the adaptation is limited to ${z}_{max}$/c = 0.5 spanwise and to ${x}_{max}$/c = 1.25 streamwise.

## 7. Spectral Signature of the Tip Leakage Flow

**AVBP**exhibits the same shape than the current LES with a shift in frequency.

**antares**[18] is used following the advanced time formulation of Casalino [33].

## 8. Conclusions

## Author Contributions

## Funding

^{o}2018/1057.

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\beta $ | Angle of attack (${}^{\circ}$) |

${\mathsf{\Gamma}}_{1}$ | Vortex identification function |

$\overline{\tilde{\mathsf{\Phi}}}$ | Time-average dissipation |

${C}_{p}$ | Pressure coefficient |

f | Frequency (Hz) |

p | Static pressure (Pa) |

T | Static temperature (K) |

U, V, W | Mean velocity components (m·s${}^{-1}$) |

c | Chord (m) |

Ma | Mach number (-) |

Q | Q criterion (s${}^{-2}$) |

Re | Reynolds number (-) |

s | Gap height (m) |

Acronyms | |

CFL | Courant–Friedrichs–Lewy |

LBM | Lattice Boltzmann Method |

LES | Large Eddy Simulation |

PIV | Particle Image Velocimetry |

PSD | Power Spectral Density |

RANS | Reynolds-Averaged Navier-Stokes |

RMS | Root-Mean-Square |

URANS | Unsteady Reynolds-Averaged Navier-Stokes |

ZLES | Zonal Large-Eddy Simulation |

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**Figure 1.**Sketch of the experimental set-up from Jacob et al. [13]. Dimensions are in millimeters.

**Figure 2.**(

**a**) Sketch of the computational domains and the boundary conditions. (

**b**) Edge size of the mesh around the airfoil at z/c = 0.1 and close-ups at the airfoil leading and trailing edges and in the gap.

**Figure 3.**LES instantaneous iso-surfaces of Q criterion (Q = 3.0 × 10${}^{2}$ (U${}_{0}$/c)${}^{2}$) coloured by the velocity magnitude in the tip leakage flow region.

**Figure 4.**Instantaneous vorticity and dilatation fields with (

**a**) and without (

**b**) the convergent at z/c = 0.1.

**Figure 6.**Mean pressure coefficients on the airfoil at midspan, z/c = 0.45 (

**a**) and at the tip, z/c = 0.005 (

**b**).

**Figure 7.**Streamwise U, horizontal V, and vertical W mean velocity components of the tip leakage vortex at the airfoil trailing edge (x/c = 0.01).

**Figure 11.**Longitudinal velocity component U of the tip leakage vortex at the airfoil trailing edge (x/c = 0.01).

**Figure 13.**PSD of the wall pressure on the airfoil suction side (

**a**) and on the airfoil tip (

**b**) at 77.5% of the chord.

**Figure 14.**PSD of acoustic pressure 2-m away from the airfoil suction side, forming an angle of 90${}^{\circ}$ with the airfoil chord.

LES NO CONV | LES CONV | |
---|---|---|

Mesh size | $229\times {10}^{6}$ | $252\times {10}^{6}$ |

Wall resolution $\Delta {x}^{+}$ = $\Delta {y}^{+}$ = $\Delta {z}^{+}$ | 100 | 100 |

Wall model | yes | yes |

Convective scheme | TTGC | TTGC |

Subgrid scale model | SIGMA | SIGMA |

$\Delta $t c/U${}_{0}$ | 3.5 × 10${}^{-5}$ | 3.5 × 10${}^{-5}$ |

T${}_{sim}$ c/U${}_{0}$ | 14 | 14 |

CPU time | 70 h | 84 h |

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**MDPI and ACS Style**

Lamidel, D.; Daviller, G.; Roger, M.; Posson, H.
Numerical Prediction of the Aerodynamics and Acoustics of a Tip Leakage Flow Using Large-Eddy Simulation. *Int. J. Turbomach. Propuls. Power* **2021**, *6*, 27.
https://doi.org/10.3390/ijtpp6030027

**AMA Style**

Lamidel D, Daviller G, Roger M, Posson H.
Numerical Prediction of the Aerodynamics and Acoustics of a Tip Leakage Flow Using Large-Eddy Simulation. *International Journal of Turbomachinery, Propulsion and Power*. 2021; 6(3):27.
https://doi.org/10.3390/ijtpp6030027

**Chicago/Turabian Style**

Lamidel, David, Guillaume Daviller, Michel Roger, and Hélène Posson.
2021. "Numerical Prediction of the Aerodynamics and Acoustics of a Tip Leakage Flow Using Large-Eddy Simulation" *International Journal of Turbomachinery, Propulsion and Power* 6, no. 3: 27.
https://doi.org/10.3390/ijtpp6030027