# Large Eddy Simulation of Film Cooling with Triple Holes: Injectant Behavior and Adiabatic Film-Cooling Effectiveness

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometry and Boundary Conditions

## 3. Validation of Numerical Methods

^{−6}s. Assuming the mainstream convected the hole diameter after 400 time steps, the simulations were carried out with the computational time step $\Delta t=0.0025D/{U}_{\infty}$ [18,19]. Once the system had reached a statistical steady state, the statistics of the solution was collected in multiples of the period. In each time step, approximately 10 subiterations were executed to obtain well-resolved data [20]. The LES runtime using 18 cores of an Intel Xeon Gold 6148 processor was 1–2 months in each case.

_{ij}, the subgrid-scale turbulent stress, needs modeling using the Boussinesq hypothesis like RANS models as

_{t}is the turbulent viscosity on the subgrid scale. In the Smagorinsky–Lilly model, the turbulent viscosity is modeled as [18]

## 4. Results and Discussion

#### 4.1. Centerline and Spanwise-Averaged Film Cooling Effectiveness

#### 4.2. Contours of Film Cooling Effectiveness and Dimensionless Temperature

#### 4.3. Contours of x-Vorticity and Mean Velocity Vectors

#### 4.4. Root-Mean-Square Temperature Contours

_{rms}in the x/D = 2 plane reached maximum values of 6.25, 6.24, 6.08, and 5.19, as shown in Figure 18a–d, respectively, which indicates a similar but slightly decreased mixing intensity in the triple-hole system relative to the single-hole system. This reduced mixing intensity is beneficial to film cooling because increased mixing between the coolant and mainstream flow consequently decreases film-cooling effectiveness [22]. Nevertheless, it could be stated that in the triple-hole system, the effect of the sister holes on the jet trajectory contributes greatly to improve film-cooling effectiveness as compared to the effect of less mixing between the main flow and coolant.

#### 4.5. Q-Criterion Iso-Surfaces

#### 4.6. Cross-Sectional Contours of the Mean Velocity Magnitude in the Hole

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

C_{s} | Smagorinsky constant |

D | hole diameter for a single hole [mm] |

D’ | diameter of primary hole for triple holes [mm] |

${E}_{ij}$ | rate of deformation |

L | delivery tube length [mm] |

L_{s} | mixing length of sub-grid scales = $\mathrm{min}\left(\kappa d,{C}_{s}\Delta \right)$ |

M | blowing ratio = $\left({\rho}_{C}{U}_{C}\right)/\left({\rho}_{G}{U}_{G}\right)$ |

P | pitch between holes [mm] |

T | temperature [K] |

t | time [s] |

U | flow velocity [m/s] |

V_{mi} | main flow velocity at the main inlet [m/s] |

V_{pi} | coolant velocity at the plenum inlet [m/s] |

x | streamwise coordinate |

y | wall-normal coordinate |

z | spanwise coordinate |

Greek symbols | |

κ | von Karman’s universal constant = 0.41 |

$\eta $ | adiabatic film cooling effectiveness $=\frac{\left({T}_{G}-{T}_{aw}\right)}{{T}_{G}-{T}_{C}}$ |

${\eta}_{C}$ | centerline film cooling effectiveness |

${\eta}_{m}$ | spanwise-averaged film cooling effectiveness |

$\rho $ | density [kg/m^{3}] |

τ_{ij} | sub-grid scale turbulent stress $=\rho \overline{{u}_{i}{u}_{j}}-\rho \overline{{u}_{i}}\overline{{u}_{j}}$ |

μ_{t} | sub-grid scale turbulent viscosity [kg/(m·s)] |

ν | local kinematic viscosity [m^{2}/s] |

∆ | local grid scale |

Θ | dimensionless temperature $=\frac{\left({T}_{G}-T\right)}{{T}_{G}-{T}_{C}}$ |

Subscripts | |

aw | adiabatic wall |

c | centerline |

C | coolant |

G | mainstream gas |

m | spanwise averaged |

$\mu $ | local dynamic viscosity [m^{2}/s] |

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**Figure 1.**Computational domains of the single- and triple-hole configurations (orange dashed lines).

**Figure 3.**Cross-sectional schematic of a counter rotating vortex pair generated in the single- and triple-hole configurations.

**Figure 7.**Grid sensitivity test of the large eddy simulation (LES) calculation in the single-hole system.

**Figure 10.**Ratios of spanwise-averaged effectiveness in the triple hole system to those in the single-hole system.

**Figure 13.**Central cross-sectional contours of the dimensionless mean temperatures in the z = 0 plane.

Surface | Boundary Condition |
---|---|

Main inlet | Velocity inlet |

Plenum inlet | Velocity inlet |

Top | Symmetry |

Test plate | Adiabatic wall |

Outflow | Pressure outlet |

Main sides | Periodic |

Sides of plenum | Wall |

Tube wall | Wall |

**Table 2.**Specifications of mesh arrangements in the grid sensitivity test of the single-hole system.

Grid | Number of Cells in the x Direction | Number of Cells in the y Direction | Number of Cells in the z Direction | Number of Cells in Cross Flow Block (million) | Total Cells Number (million) |
---|---|---|---|---|---|

First | 320 | 50 | 32 | 0.52 | 1.14 |

Second | 334 | 60 | 48 | 0.98 | 1.60 |

Third | 352 | 80 | 50 | 1.42 | 2.04 |

Fourth | 364 | 94 | 56 | 1.94 | 2.56 |

Fifth | 390 | 110 | 64 | 2.78 | 3.4 |

**Table 3.**Specifications of mesh arrangements in the grid sensitivity test of the triple-hole system.

Grid | Number of Cells in the x Direction | Number of Cells in the y Direction | Number of Cells in the z Direction | Number of Cells in Crossflow Block (million) | Total Cells Number (million) |
---|---|---|---|---|---|

First | 300 | 46 | 65 | 1.01 | 1.78 |

Second | 330 | 60 | 75 | 1.62 | 2.39 |

Third | 350 | 70 | 80 | 1.92 | 2.69 |

Fourth | 370 | 80 | 85 | 2.41 | 3.19 |

Fifth | 390 | 90 | 90 | 2.94 | 3.71 |

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**MDPI and ACS Style**

Baek, S.I.; Ahn, J.
Large Eddy Simulation of Film Cooling with Triple Holes: Injectant Behavior and Adiabatic Film-Cooling Effectiveness. *Processes* **2020**, *8*, 1443.
https://doi.org/10.3390/pr8111443

**AMA Style**

Baek SI, Ahn J.
Large Eddy Simulation of Film Cooling with Triple Holes: Injectant Behavior and Adiabatic Film-Cooling Effectiveness. *Processes*. 2020; 8(11):1443.
https://doi.org/10.3390/pr8111443

**Chicago/Turabian Style**

Baek, Seung Il, and Joon Ahn.
2020. "Large Eddy Simulation of Film Cooling with Triple Holes: Injectant Behavior and Adiabatic Film-Cooling Effectiveness" *Processes* 8, no. 11: 1443.
https://doi.org/10.3390/pr8111443