# Large Eddy Simulation of Film Cooling Involving Compound Angle Holes: Comparative Study of LES and RANS

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods and Validation

#### 2.1. Computational Domain and Boundary Conditions

^{+}= 30 on the wall and the value of y

^{+}of the first cell above the wall was set as about 1. Figure 1e illustrates the close-up of the mesh near the injection region of the hole.

#### 2.2. Governing Equations and Turbulence Models

_{ij}, the subgrid-scale stress was modeled using the Boussinesq hypothesis:

_{t}is the subgrid-scale viscosity. The Smagorinsky–Lilly model was adopted based on a validation test [29,30]:

_{j}refers to the subgrid-scale heat flux and can be expressed as follows by introducing the subgrid-scale diffusivity α

_{t}.

_{t}was considered as α

_{t}, assuming that the subgrid-scale turbulent Prandtl number is unity [31].

^{−6}s, and the number of steps was 400 considering that the mainstream flowed as much as the hole diameter in this time interval. The film cooling flow is inherently unsteady and the current study performed the statistically steady state analysis by post-processing. A total of 20,000 time steps were required from the initial condition to reach steady state and 30,000 integrated time steps were considered to obtain the average flow field and statistics. The complete process required approximately one month of computation time on a 20-core-cluster computer. The RANS analyzed the time-averaged continuity, Navier–Stokes, and energy equations, as shown in Equations (7)–(9), respectively.

^{−6}level for all the equations in approximately 10 h on a 20-core-cluster computer.

#### 2.3. Code Validation

## 3. Results and Discussion

#### 3.1. Time-Averaged Flow and Thermal Fields

#### 3.2. Adiabatic Film Cooling Effectiveness

#### 3.3. Turbulence Statistics and Instantaneous Flow Fields

_{rms}and w

_{rms}components not blocked by the wall appeared high along the wall, whereas the v

_{rms}component converged to zero at the wall. In the case of the compound angle (β = 30°), the overall trend was similar to that of the simple angle. However, the CRVP changed to a single vortex and consequently, the contours became asymmetric, and the distribution moved in the +z direction.

_{rms}component was larger than that when the blowing ratio was 1.0 near the wall.

_{rms}was large in the region surrounding the area with large velocity fluctuation, as shown in Figure 10. In the same range of the contours, when β = 0°, the value was slightly higher than that when β = 30°. This finding supports the result of the high film cooling effectiveness under a compound angle as the injectant flowed downstream.

## 4. Conclusions

- In the time-averaged flow field, the RANS data exhibited a difference from the experiment and LES in terms of the rising point of the CRVP as the vortices collided with each other on the wall. When the injection ratio was 0.5 and the orientation angle is 30°, the LES predicted that the counter-rotating vortex remained weak and the RANS predicted that the vortex completely changed to a single vortex;
- The RANS did not accurately predict the lift-off of the injectant or mixing with the main flow, and thus, it could not accurately predict the film cooling performance. The corresponding predictions obtained using the LES were better. The reattachment of the injectant at the blowing ratio of 1.0 was better predicted by the RANS in the compound angle case than that in the case of the simple angle;
- The turbulence intensity was large in the region in which the upward flow of the vortex in the injectant was generated and the temperature fluctuation was large at the boundary of the turbulent intensity peak. The temperature fluctuation slightly decreased when the injectant was supplied at a compound angle;
- In the compound angle case, the insulation film was eliminated near the leeward rim of the film cooling hole due to the influence of the single vortex however, at the injection ratio of 1.0, the injectant flowed along the wall more smoothly than that in the simple injection angle, thereby enhancing the downstream film cooling performance.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C_{s} | Smagorinsky constant |

C_{p} | Specific heat [J/(kgK)] |

D | diameter of a single hole [mm] |

d | wall distance [mm] |

L | delivery tube length [mm] |

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

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

P | pitch of the holes [mm] |

q_{j} | heat flux [W/mm^{2}] |

T | emperature [K] |

t | time [s] |

t* | non-dimensional time = U_{∞} t/D |

U | time-averaged flow velocity [m/s] |

U_{∞} | freestream velocity [m/s] |

u | streamwise velocity [m/s] |

v | wall-normal velocity [m/s] |

w | panwise velocity [m/s] |

x | streamwise coordinate |

y | wall-normal coordinate |

z | spanwise coordinate |

Greek symbols | |

α | thermal diffusivity [m^{2}/s] |

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

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

$\kappa $ | thermal conductivity [W/(mK)] |

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

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

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

Δ | 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 |

rms | root mean square value |

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**Figure 1.**Film cooling geometry and computational domain: (

**a**) Gas turbine blade with film cooling holes; (

**b**) injection geometry of film cooling; (

**c**) computational domain; (

**d**) grid system; and (

**e**) close-up of the mesh near the hole.

**Figure 2.**Code validation based on the centerline adiabatic film cooling effectiveness: (

**a**) Turbulence model validation by Harrison and Bogard [15]; (

**b**) subgrid scale model validation for the large eddy simulation (LES); (

**c**) grid sensitivity test; and (

**d**) code validation test compared to the data by Jung and Lee [11].

**Figure 3.**Time-averaged flow field at x/D = 2.5 for the streamwise film cooling when M = 1.0, compared with data from Burd et al. [35]: (

**a**) Cross-stream velocity vectors and (

**b**) streamwise velocity contour.

**Figure 4.**Time-averaged cross-stream velocity vectors with temperature contours at x/D = 2.5: (

**a**) Cross-stream velocity vectors reported by Lee et al. [10]; (

**b**) cross-stream velocity vectors with streamwise velocity contours, reported by Aga and Abhari [13]; (

**c**) M = 0.5, β = 0°; (

**d**) M = 1.0, β = 0°; (

**e**) M = 0.5, β = 30°; and (

**f**) M = 1.0, β = 30°.

**Figure 5.**Boundary layer temperature distributions in the streamwise normal planes at M = 0.5: (

**a**) x/D = 2.5, β = 0°; (

**b**) x/D = 5.0, β = 0°; (

**c**) x/D = 10.0, β = 0°; (

**d**) x/D = 2.5, β = 30°; (

**e**) x/D = 5.0, β = 30°; and (

**f**) x/D = 10.0, β = 30°.

**Figure 6.**Boundary layer temperature distributions in the streamwise normal planes at M = 1.0: (

**a**) x/D = 2.5, β = 0°; (

**b**) x/D = 5.0, β = 0°; (

**c**) x/D = 10.0, β = 0°; (

**d**) x/D = 2.5, β = 30°; (

**e**) x/D = 5.0, β = 30°; and (

**f**) x/D = 10.0, β = 30°.

**Figure 7.**Local adiabatic film cooling effectiveness distributions at M = 0.5: (

**a**) β = 0° and (

**b**) β = 30°.

**Figure 8.**Local adiabatic film cooling effectiveness distributions at M = 1.0: (

**a**) β = 0° and (

**b**) β = 30°.

**Figure 9.**Spanwise variations in the adiabatic film cooling effectiveness at x/D = 2.5: (

**a**) M = 0.5, β = 0°; (

**b**) M = 0.5, β = 30°; (

**c**) M = 1.0, β = 0°; and (

**d**) M = 1.0, β = 30°.

**Figure 10.**Turbulence intensities at x/D = 2.5 compared to those in the experiment of Burd et al. [32]: (

**a**) u

_{rms}, M = 1.0; (

**b**) v

_{rms}, M = 1.0; (

**c**) w

_{rms}, M = 1.0; (

**d**) u

_{rms}, M = 0.5; (

**e**) v

_{rms}, M = 0.5; and (

**f**) w

_{rms}, M = 0.5.

**Figure 11.**Temperature fluctuations obtained from the LES at x/D = 2.5: (

**a**) M = 0.5, β = 0°; (

**b**) M = 0.5, β = 30°; (

**c**) M = 1.0, β = 0°; and (

**d**) M = 1.0, β = 30°.

**Figure 12.**Isosurface of the second invariant of the velocity gradient tensor (Q): (

**a**) M = 0.5, β = 0°; (

**b**) M = 0.5, β = 30°; (

**c**) M = 1.0, β = 0°; and (

**d**) M = 1.0, β = 30°.

**Figure 13.**Instantaneous adiabatic film cooling effectiveness distributions: (

**a**) M = 0.5, β = 0°; (

**b**) M = 0.5, β = 30°; (

**c**) M = 1.0, β = 0°; and (

**d**) M = 1.0, β = 30°.

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) |

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

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**MDPI and ACS Style**

Baek, S.I.; Ahn, J. Large Eddy Simulation of Film Cooling Involving Compound Angle Holes: Comparative Study of LES and RANS. *Processes* **2021**, *9*, 198.
https://doi.org/10.3390/pr9020198

**AMA Style**

Baek SI, Ahn J. Large Eddy Simulation of Film Cooling Involving Compound Angle Holes: Comparative Study of LES and RANS. *Processes*. 2021; 9(2):198.
https://doi.org/10.3390/pr9020198

**Chicago/Turabian Style**

Baek, Seung Il, and Joon Ahn. 2021. "Large Eddy Simulation of Film Cooling Involving Compound Angle Holes: Comparative Study of LES and RANS" *Processes* 9, no. 2: 198.
https://doi.org/10.3390/pr9020198