# Large Eddy Simulation of Film Cooling with Forward Expansion Hole: Comparative Study with LES and RANS Simulations

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## Abstract

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## 1. Introduction

_{c}); the results showed less lateral spreading of the coolant on the wall. McGovern and Leylek (2000) compared the effects of holes with 45°, 60°, and 90° compound angles with those of an ordinary cylindrical hole with the standard k–ε turbulence model [8]. They stated that the compound angle improved the lateral spreading of the coolant and increased the cooling effectiveness of the laterally averaged film compared to that created via the inclined inline injection from a hole. However, they discovered that the heat transfer coefficient increased owing to the effects of the compound angle injection. Moreover, Tyagi, and Acharya (2003) conducted LES to study the film cooling effect of a cylindrical hole [9]. The film cooling process resulted in complex flow fields and coherent structures; the LES results showed better predictions of η than the RANS simulations. Rozati and Tafti (2007) studied the effects of freestream turbulence on film cooling with a cylindrical hole using a leading-edge model with LES [10]. The fully turbulent jet increased the mixing with the main flow, which decreased the adiabatic film cooling effectiveness. They also reported that the turbulent injectant did not greatly affect the heat transfer coefficients, which were significantly affected by the near-wall turbulence. Johnson et al. (2011) investigated the effects of the length-to-diameter ratio, momentum ratio, and grid resolution on the film cooling process with a cylindrical hole with the realizable k–ε turbulence model [11]. They stated that the injectant with high momentum and short L/D ratio decreased η via the generation of a jet lift-off. In addition, refining the grid in the region near the hole’s trailing edge was important for obtaining good numerical results at high momentum ratios.

## 2. Geometry and Boundary Conditions

## 3. Validation of Numerical Methods

^{−6}s time step means that the mainstream convects the hole diameter after 400-time steps [23,24]. The maximum Courant–Friedrichs–Lewy (CFL) number was less than one (U∆t/∆x = 0.3125) [25]. After the solution had reached a statistical steady state, the statistics of the solution were collected for about 20 flow-through times for the main flow (about 0.5 s). The LES runtime with an Intel Xeon Gold 6148 processor 16 cores was approximately 6–7 weeks in each case. In addition, the fluid was assumed to be Newtonian and incompressible with temperature-dependent properties. The compressibility was negligible because the main flow velocity was 10 m/s (Mach number of 0.029), and the jet injection velocities were 5 and 10 m/s at blowing ratios of M = 0.5 and 1.0, respectively [26]. The governing equations are the incompressible Navier–Stokes and energy equations. Moreover, the Semi Implicit Method for Pressure Linked Equations (SIMPLE) scheme was used; the convection term was solved with the second-order upwind scheme, and the diffusion term was solved via central differencing. In the LES, the governing equations were transformed into a filtered equation [20,27,28]. The governing equations [29,30] were not shown here because a commercial CFD code was used. Figure 3 illustrates the meshes of the forward expansion hole system, including the mesh configuration in the x–y plane, the mesh configuration in the y–z plane, close-up of the mesh near the cooling hole, and mesh configuration at the hole exit. The LES and RANS simulations were conducted in the same mesh in the figure.

## 4. Results and Discussion

#### 4.1. Contours of Time-Averaged η

#### 4.2. Time-Averaged Film Cooling Effectiveness

#### 4.3. Contours of Instantaneous Film Cooling Effectiveness on Wall

#### 4.4. Contours of Velocity Fluctuation on Streamwise-Normal Plane

_{rms}/U

**values of the forward expansion hole were greatly decreased compared to those of the cylindrical hole as the injection velocity in the normal direction was decreased by increased cross-sectional area at the forward expansion hole exit. When the compound angle was 30°, the contours of the velocity fluctuations showed that the distribution of the turbulence intensity had shifted along the −z-direction because the coolant was injected at a spanwise velocity, as shown in Figure 6 and Figure 7.**

_{∞}#### 4.5. Contours of Temperature Fluctuations on Streamwise-Normal Plane

_{rms}at x/D = 2.5. Like the contours of the velocity fluctuations in Section 4.4, the prediction results of the RANS model showed extremely small temperature fluctuations compared to those of the LES contour range, and only the temperature fluctuations obtained with the LES are shown in the figure. High θ

_{rms}is connected to the large gradients in the mean temperature contour illustrated in Figure 8 and Figure 9, indicating that the mixing intensity between the hot mainstream flow and injectant around the area with high θ

_{rms}is high and the area with high-temperature fluctuation was larger than the area with large velocity fluctuations in Figure 16 and Figure 17. As indicated by the contours, for the shaped hole with β = 0°, the maximum of θ

_{rms}was higher than that for β = 30°; thus, the mixing intensity for β = 0° was higher than that for β = 30°, which resulted in a higher η on the test plate for β = 30°. The mixing of the hot main flow and coolant was supported by numerous vortices such as hairpin, horseshoe, and roller vortices; these vortices were weakened in the forward expansion hole with β = 30° compared to that in the case with β = 0°. In addition, the maximal θ

_{rms}of the shaped hole was much lower relative to the contours of the cylindrical hole; this resulted in a higher film cooling performance in the shaped hole configuration compared to that in the cylindrical hole system.

#### 4.6. Time-Averaged Velocity Magnitude Contours in Hole

#### 4.7. Mean Dimensionless Temperature Contours at Hole Exit

## 5. Conclusions

_{rms}was larger than the area with high-velocity fluctuations. Additionally, for the forward expansion hole with β = 0°, the maximal θ

_{rms}in the LES contour exceeded that for β = 30°; thus, the mixing intensity for β = 0° exceeded that for β = 30°; this resulted in a higher η on the wall for β = 30°. The flow visualization contours of the experiment and LES results clearly show the flow reversal around its leading edge in the forward expansion hole, whereas the RANS results did not show this phenomenon because the velocity of the injectant near the trailing edge of the hole was lower in the forward expansion hole than that in the cylindrical hole. Considering the reversed main flow into the hole can damage the hole and affect the film cooling performance downstream of the hole, it is important to know the area of hot main flow reversal accurately.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

D | diameter of lower hole |

L | hole length |

M | Blowing ratio = $\frac{{\rho}_{c}{U}_{c}}{{\rho}_{G}{U}_{G}}$ |

P | pitch between holes |

t | time |

${T}_{aw}$ | Adiabatic wall temperature |

${T}_{G}$ | Temperature of mainstream gas |

U | Mean velocity |

${U}_{C}$ | injectant velocity |

${U}_{G}$ | mainstream velocity |

Greek Symbols | |

α | injection angle |

β = | compound angle (spanwise injection angle) |

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

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

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

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

ρ | density |

${\rho}_{C}$ | injectant density |

${\rho}_{G}$ | mainstream density |

Subscripts | |

C | coolant |

aw | adiabatic wall |

m | lateral-averaged |

Abbreviations | |

LES | Large Eddy Simulation |

RANS | Reynolds-Averaged Navier–Stokes |

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**Figure 3.**Computational fluid dynamic (CFD) meshes of forward expansion hole with β = 0°: (

**a**) mesh in x–y plane (

**b**) mesh in y–z plane (

**c**,

**e**) close-up of mesh near the cooling hole, and (

**d**) mesh at hole exit.

**Figure 4.**Spanwise-averaged effectiveness of forward expansion hole with β = 0° in grid sensitivity test with large eddy simulations (LES).

**Figure 5.**Spanwise-averaged effectiveness of forward expansion hole with β = 30° in grid sensitivity test with LES.

**Figure 10.**Contours of time-averaged dimensionless temperature and streamlines at x/D = 2.5 and M = 0.5.

**Figure 11.**Contours of time-averaged dimensionless temperature and streamlines at x/D = 2.5 and M = 1.0.

**Figure 15.**Contours of instantaneous film cooling effectiveness on wall obtained with LES: (

**a**) β = 0°, M = 0.5; (

**b**) β = 30°, M = 0.5; (

**c**) β = 0°, M = 1.0; (

**d**) β = 30°, M = 1.0.

**Figure 17.**Comparison of contours of turbulence intensities obtained with LES for x/D = 2.5 and M = 1.0 with experimental results presented in [34].

**Figure 20.**Mean dimensionless temperature compared with experimental flow visualization [19] at hole exit at M = 0.5.

**Figure 21.**Mean dimensionless temperature compared with experimental flow visualization [19] at hole exit at M = 1.0.

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

Main inlet | Velocity inlet (u = constant) |

Plenum inlet | Velocity inlet (u = constant) |

Top | Symmetry ($\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=0,\text{}\frac{\partial \mathrm{w}}{\partial \mathrm{y}}=0,\text{}\mathrm{v}=0$) |

Test plate | Adiabatic wall (u = v = w = 0) |

Outflow | Pressure outlet |

Main sides | Periodic (u(x, y, z, t) = u(x, y, z + P, t), ΔP = 0) |

Sides of plenum | Wall (u = v = w = 0) |

Hole wall | Wall (u = v = w = 0) |

**Table 2.**Specifications of the cell number for grid sensitivity test of forward expansion hole with β = 0°.

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

First | 320 | 70 | 70 | 1.89 | 4.49 |

Second | 502 | 75 | 76 | 3.18 | 5.78 |

Third | 552 | 80 | 82 | 3.94 | 6.53 |

Fourth | 602 | 85 | 86 | 4.72 | 7.31 |

Fifth | 682 | 90 | 90 | 5.85 | 8.45 |

**Table 3.**Specifications of the cell number for grid sensitivity test of forward expansion hole with β = 30°.

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

First | 426 | 70 | 92 | 3.06 | 5.65 |

Second | 480 | 75 | 102 | 3.99 | 6.59 |

Third | 530 | 80 | 114 | 5.16 | 7.75 |

Fourth | 560 | 85 | 124 | 6.22 | 8.82 |

Fifth | 594 | 88 | 134 | 7.32 | 9.91 |

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**MDPI and ACS Style**

Baek, S.I.; Ryu, J.; Ahn, J.
Large Eddy Simulation of Film Cooling with Forward Expansion Hole: Comparative Study with LES and RANS Simulations. *Energies* **2021**, *14*, 2063.
https://doi.org/10.3390/en14082063

**AMA Style**

Baek SI, Ryu J, Ahn J.
Large Eddy Simulation of Film Cooling with Forward Expansion Hole: Comparative Study with LES and RANS Simulations. *Energies*. 2021; 14(8):2063.
https://doi.org/10.3390/en14082063

**Chicago/Turabian Style**

Baek, Seung Il, Jaiyoung Ryu, and Joon Ahn.
2021. "Large Eddy Simulation of Film Cooling with Forward Expansion Hole: Comparative Study with LES and RANS Simulations" *Energies* 14, no. 8: 2063.
https://doi.org/10.3390/en14082063