# Large Eddy Simulation of Film Cooling with Bulk Flow Pulsation: Comparative Study of LES and RANS

^{*}

## Abstract

**:**

_{rms}profiles with the corresponding RANS results. The adiabatic film cooling effectiveness predicted using LES agreed well with the experimental data, whereas RANS highly overpredicted the centerline effectiveness. RANS could not properly predict the injectant topology change in the streamwise normal plane, but LES reproduced it properly. In the case of the injectant trajectory, which greatly influences film cooling effectiveness, RANS could not properly predict the changes in the streamwise velocity peak due to flow pulsation, but they were predicted well with LES. RANS greatly underpredicted the streamwise velocity fluctuations, which determine the mixing of main flow and injectant, whereas LES prediction was close to the experimental data.

## 1. Introduction

## 2. Geometry and Boundary Conditions

^{+}units in the x-streamwise direction. The y

^{+}value of the first cell above the test plate was less than 1 to ensure that the gradient of the wall-normal velocity in the viscous sublayer could be captured accurately, and there were 25 cell layers up to y

^{+}= 30. The mesh spacing value in the z-spanwise direction was 20 throughout the domain in z

^{+}units. If h was assumed as the hole wall-normal coordinate, the h

^{+}value of the first cell was set to less than 2 to accurately capture the gradient of the wall-normal velocity in the viscous sublayer. There were 15 cell layers up to h

^{+}= 40. Figure 1c illustrates a close-up of the mesh around the jet injection region. The black regions in the figure show that the fine cells were concentrated around the boundary layer and in the region where complex flow structures were induced by the interactions between the mainstream and the jet. Table 1 summarizes the boundary conditions of the domain.

_{main}= A sin(2πft) + 10 m/s.

_{plenum}= B sin(2πft) + 0.164 m/s

_{2}) and the static pressure around the tube exit (P

_{1}) oscillates due to oscillations in the main flow. The measurement locations of P

_{1}and P

_{2}are illustrated in Figure 1d. Therefore, in the present study, sinusoidal waveforms of the same frequency were applied to both the main inlet and the plenum inlet simultaneously. The B values for each frequency were not reported, but the P

_{2}–P

_{1}variation plots for each frequency were given in Seo, Lee, and Ligrani [12]. The B values were obtained by matching the variation plots with the trial and error method. The B values, that is, the amplitudes of the plenum inlet velocity oscillation in terms of frequency and Strouhal number, are listed in Table 3.

## 3. Numerical Methods

^{−6}s, which corresponded to the time required for convection of the mainstream through the length of the hole diameter in 400 time steps [19,20]. The time step for unsteady RANS was set to 3.125 × 10

^{−4}s, which corresponded to the period of the 32 Hz oscillation divided by 100. The solution statistics were collected for multiples of the period after the steady state was statistically achieved. For each time step, approximately 10 subiterations were performed to ensure that the data were well resolved and to reduce factorization and linearization errors [21]. Twenty cores of an Intel Xeon Gold 6148 processor were used for the computations, and the LES and RANS calculation times were approximately 2 months and less than 8 h, respectively.

#### 3.1. Governing Equations and Turbulence Models

#### 3.1.1. Unsteady RANS approach

#### 3.1.2. LES Approach

## 4. Results and Discussion

#### 4.1. Mesh Sensitivity Test

#### 4.2. Film Cooling Effectiveness

_{1}and P

_{2}increased because of large phase shifts between them [12]. Then, the amplitude of the coolant velocity oscillation increased, and more jet lift-off was generated periodically, resulting in less effective film cooling. At 32 Hz, the computed cooling effectiveness of the centerline film was more accurate than that at 0 Hz because the jet lift-off effect was considerably stronger than the other vortex effects, such as horseshoe vortex or CRVP, which are detrimental to the film cooling effectiveness. Moreover, LES yielded a good prediction of jet lift-off. As shown in Figure 3, as the main flow passed downstream, the centerline effectiveness decreased, regardless of the frequency, because of the generation of the turbulence and mixing between the main flow and the jet.

#### 4.3. Dimensionless Temperature Contours at x/D = 5

#### 4.4. Dimensionless Temperature Contours at y/D = 0 and z/D = 0

#### 4.5. Q-criterion Contours at z/D = 5 and y/D = 0

_{z}for the 32 Hz pulsation, and obtained with LES and URANS, respectively. The Q-criterion contours are colored with y-vorticity in each phase. Compared to the Q-criterion contours predicted using URANS, those predicted by LES indicate that LES can predict more complex vortex structures, including CRVP, hairpin, jet shear layer, and horseshoe vortices. URANS predicted only simple vortex structures, such as the horseshoe vortex and the hairpin vortex, as shown in Figure 9b. In the Q contour obtained using LES, complex vortical structures continuously exist under the injectant, but in the URANS result, they appear partially under the hairpin vortices.

#### 4.6. Streamwise Velocity and Fluctuation Profiles at x/D = 5 and z/D = 0

_{rms}profiles at x/D = 5 and z/D = 0 for 0 and 32 Hz. The almost-zero U

_{rms}values obtained using the RANS realizable k-ε model are considerably much smaller than those obtained experimentally, as illustrated in the figure. The film cooling efficiency predicted using the realizable k-ε model was the closest to the experimental value, but the model yielded the weakest prediction of velocity fluctuation. As shown in Figure 11a, the time-averaged U

_{rms}values at 0 Hz obtained by LES matched well with the experimental result. A peak of U

_{rms}values at around y/D = 0.9 was predicted well by LES. The Reynolds stress model or the k-ω model predicted the velocity fluctuation better than the realizable k-ε, but the predicted value was still less than half the experimental value.

_{rms}values at x/D = 5 and z/D = 0 under 32 Hz pulsation obtained by LES were slightly smaller than the experimental values, even though the shape of the plot is similar to the experimental plot. Figure 11 shows that the LES approach is superior to the URANS approach in predicting the U

_{rms}values of pulsating flows. Compared with the case of no pulsation, the turbulence intensity increased below y/D = 0.5 near the wall, and LES predicted this increase appropriately. Figure 12 illustrates the phase-averaged dimensionless U

_{rms}profiles at x/D = 5 and z/D = 0 for 32 Hz pulsation in three phases during a period as obtained by LES and URANS.

_{rms}profiles at x/D = 5 and z/D = 0 as obtained by the LES Smagorinsky–Lilly model and the URANS realizable k-ε model under 32 Hz pulsation are also shown in the figure. As shown in Figure 12a, as t/τ decreases from 0 to 0.4, the y/D values corresponding to the maximum values of U

_{rms}decrease because the heights of the large vortex at x/D = 5 decrease, as illustrated in Figure 8a. When t/τ increases from 0.4 to 0.8, the y/D values corresponding to the maximum U

_{rms}values increase because the height of the large vortex at x/D = 5 increases. Therefore, the y/D values corresponding to the maximum U

_{rms}values are periodic because of the outline of the large coolant vortices. This outline is attributed to the periodic blowing ratios expressed by Equation (2), even though the averaged blowing ratio is 0.5.

## 5. Conclusions

_{rms}profiles. The experimental U

_{rms}values under 32 Hz pulsation between y/d = 0 and 0.5 were higher than those between y/D = 0.5 and 1, and the U

_{rms}profile obtained with LES was closer to the experimental data comparing to that obtained using URANS. The phase-averaged dimensionless U

_{rms}profiles obtained using LES showed that, in each phase, the shape of the U

_{rms}profile was attributable to the outline of the large coolant vortices.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

C_{s} | Smagorinsky constant | |

D | diameter of single hole (mm) | |

D′ | diameter of primary hole for triple holes (mm) | |

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

L | delivery tube length (mm) | |

L_{s} | mixing length of subgrid scales = $\mathrm{min}\left(\kappa d,{C}_{s}\u2206\right)$ | |

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

P | pitch between holes (mm) | |

Sr | Strouhal number = $\frac{2\pi fD}{{U}_{G}}$ | |

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} | 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)) | |

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 | |

rms | root mean squared | |

$\mu $ | local dynamic viscosity (m^{2}/s) |

## References

- Moran, M.; Shapiro, H.; Boettner, D.; Bailey, M. Fundamentals of Engineering Thermodynamics, 8th ed.; Wiley: New York, NY, USA, 2014. [Google Scholar]
- Leedom, D.H.; Acharya, S. Large eddy simulations of film cooling flow fields from cylindrical and shaped holes. In ASME Turbo Expo; ASME: Berlin, Germany, 2008; pp. 865–877. [Google Scholar] [CrossRef]
- Bogard, D.G. Airfoil film cooling. In The Gas Turbine Handbook; National Energy Technology Laboratory: Pittsburgh, PA, USA, 2006; Section 4.2.2.1. [Google Scholar]
- Bogard, D.; Thole, K. Gas turbine film cooling. J. Propul. Power
**2006**, 22, 249–270. [Google Scholar] [CrossRef] [Green Version] - Fluent Incorporated. RAMPANT User’s Guide; Fluent Incorporated: New York, NY, USA, 1993. [Google Scholar]
- Walters, D.K.; Leylek, J.H. Impact of film-cooling jets on turbine aerodynamic losses. ASME J. Turbomach.
**2000**, 122, 537–545. [Google Scholar] [CrossRef] - Tyagi, M.; Acharya, S. Large eddy simulation of film cooling flow from an inclined cylindrical jet. ASME J. Turbomach.
**2003**, 125, 734–742. [Google Scholar] [CrossRef] - Rozati, A.; Tafti, D. Large eddy simulation of leading edge film cooling: Part II—Heat transfer and effect of blowing ratio. In ASME Turbo Expo 2007; ASME: Montreal, QC, Canada, 2007. [Google Scholar] [CrossRef]
- Na, S.; Shih, T. Increasing adiabatic film cooling effectiveness by using an upstream ramp. J. Heat Transfer
**2007**, 129, 464–471. [Google Scholar] [CrossRef] - Johnson, P.; Shyam, V.; Hah, C. Reynolds-Averaged Navier-Stokes Solutions to Flat Plate Film Cooling Scenarios. NASA/TM-2011-217025, 1 May 2011. [Google Scholar]
- Wojtas, K.; Makowski, Ł.; Orciuch, W. Barium sulfate precipitation in jet reactors: Large eddy simulations, kinetics study and design considerations. Chem. Eng. Res. Des.
**2020**, 158, 64–76. [Google Scholar] [CrossRef] - Seo, H.J.; Lee, J.S.; Ligrani, P.M. The effect of injection hole length on film cooling with bulk flow pulsations. Int. J. Heat Mass Transfer
**1998**, 41, 3515–3528. [Google Scholar] [CrossRef] - Coulthard, S.; Volino, R.; Flack, K. Effect of jet pulsing on film cooling—Part I: Effectiveness and flow-field temperature results. ASME J. Turbomach.
**2007**, 129, 232–246. [Google Scholar] [CrossRef] - Nikitopoulos, D.; Acharya, S.; Oertling, J.; Muldoon, F. On active control of film-cooling flows. In ASME Turbo Expo 2006; ASME: Barcelona, Spain, 2006. [Google Scholar] [CrossRef]
- Jung, I.S.; Ligrani, P.M.; Lee, J.S. Effects of bulk flow pulsations on phase-averaged and time-averaged film-cooled boundary layer flow structure. J. Fluids Eng.
**2001**, 123, 559–566. [Google Scholar] [CrossRef] - Han, J.; Dutta, S.; Ekkad, S. Gas Turbine Heat Transfer and Cooling Technology, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Ansys Fluent Theory Guide, v.14. Available online: http://ansys.com/products/fluids/ansys-fluent (accessed on 7 November 2020).
- Pointwise version 16.04. Available online: http://pointwise.com (accessed on 7 November 2020).
- Renze, P.; Schroder, W.; Meinke, M. Large-eddy simulation of film cooling flows with variable density jets. Flow Turbul. Combust.
**2008**, 80, 119–132. [Google Scholar] [CrossRef] - Iourokina, I.; Lele, S. Towards large eddy simulation of film cooling flows on a model turbine blade leading edge. In Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 2005. No. 2005-0670. [Google Scholar]
- Acharya, S.; Leedom, D. Large eddy simulations of discrete hole film cooling with plenum inflow orientation effects. J. Heat Transfer
**2013**, 135, 011010. [Google Scholar] [CrossRef] - White, F. Fluid Mechanics, 8th ed.; McGraw-Hill: NewYork, NY, USA, 2015. [Google Scholar]
- Cengel, Y.; Cimbala, J. Fluid Mechanics, 3rd ed.; McGraw-Hill: New York, NY, USA, 2014. [Google Scholar]
- Tannehill, J.; Anderson, D.; Pletcher, R. Computational Fluid Mechanics and Heat Transfer, 2nd ed.; Taylor & Francis: Abington-on-Thames, UK, 1997. [Google Scholar]
- Blazek, J. Computational Fluid Dynamics: Principles and Applications, 3rd ed.; Butterworth-Heinemann: Oxford, UK, 2015. [Google Scholar]
- Jung, I.S. Effects of Bulk Flow Pulsations on Film Cooling with Compound Angle Injection Holes. Ph.D. Thesis, Seoul National University, Seoul, Korea, August 1998. [Google Scholar]
- Hou, R.; Wen, F.; Luo, Y.; Tang, X.; Wang, S. Large eddy simulation of film cooling flow from round and trenched holes. Int. J. Heat Mass Transfer
**2019**, 144, 118631. [Google Scholar] [CrossRef] - Kolar, V. Vortex identification: New requirements and limitations. Int. J. Heat Fluid Flow
**2007**, 28, 638–652. [Google Scholar] [CrossRef] - Schroder, A.; Willert, C. (Eds.) Particle Image Velocimetry: New Developments and Recent Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2008; p. 382. [Google Scholar]

**Figure 1.**Computational domain and grid system. (

**a**) Schematic of the computational domain (orange dashed line). (

**b**) CFD mesh in the z/D = 0 plane. (

**c**) CFD mesh for simulation without bulk flow pulsation. (

**d**) Close-up of the mesh around the jet injection region.

**Figure 2.**Adiabatic film cooling effectiveness at the centerline at 0 Hz (steady state) in (

**a**) the mesh sensitivity test based on large eddy simulation (LES) calculations; (

**b**) centerline effectiveness obtained in various Reynolds-averaged Navier–Stokes simulations (RANSs).

**Figure 4.**Time-averaged dimensionless temperature contours at x/D = 5: (

**a**) experiment [12]; (

**b**) LES (Smagorinsky–Lilly); (

**c**) RANS (realizable k-ε). Numbers in contours, 1: θ = 0.05, 2: θ = 0.1, 3: θ = 0.15, 4: θ = 0.2, 5: θ = 0.25, 6: θ = 0.3, 7: θ = 0.35, 8: θ = 0.4, 9: θ = 0.45, A: θ = 0.5, B: θ = 0.55, C: θ = 0.6, D: θ = 0.65.

**Figure 5.**Phase-averaged dimensionless temperature contours at x/D = 5: (

**a**) experiment [12]; (

**b**) LES (Smagorinsky–Lilly); (

**c**) RANS (realizable k-ε). Numbers in contours, 1: θ = 0.05, 2: θ = 0.1, 3: θ = 0.15, 4: θ = 0.2, 5: θ = 0.25, 6: θ = 0.3, 7: θ = 0.35, 8: θ = 0.4, 9: θ = 0.45, A: θ = 0.5, B: θ = 0.55, C: θ = 0.6, D: θ = 0.65.

**Figure 6.**Time-averaged film cooling effectiveness contours on the test plate: (

**a**) steady (0 Hz) and (

**b**) with flow pulsation (32 Hz).

**Figure 7.**Time-averaged dimensionless temperature contours at z/D = 0: (

**a**) steady (0 Hz) and (

**b**) with flow pulsation (32 Hz).

**Figure 8.**Phase-averaged dimensionless temperature contours at y/D = 0 (upper) and z/D = 0 (lower) for 32 Hz: (

**a**) LES (Smagorinsky–Lilly) and (

**b**) unsteady RANS (URANS) (realizable k-ε).

**Figure 9.**Q-criterion contours colored by y-vorticity in each phase with iso-surfaces of level 1000 for three dimensions under 32 Hz pulsation: (

**a**) LES (Smagorinsky–Lilly) and (

**b**) unsteady RANS (realizable k-ε).

**Figure 10.**Time-averaged streamwise velocity profiles at x/D = 5 and z/D = 0: (

**a**) 0 Hz and (

**b**) 32 Hz.

**Figure 11.**Time-averaged dimensionless U

_{rms}profiles obtained using RANS and LES models at x/D = 5 and z/D = 0: (

**a**) 0 Hz and (

**b**) 32 Hz.

**Figure 12.**Phase-averaged dimensionless U

_{rms}profiles at x/D = 5 and z/D = 0 for 32 Hz: (

**a**) LES (Smagorinsky–Lilly) and (

**b**) URANS (realizable k-ε).

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 |

Frequency (Hz) | 0 | 2 | 16 | 32 |
---|---|---|---|---|

Sr | 0 | 0.0314 | 0.2513 | 0.5027 |

A | 0 | 1.82 | 0.57 | 0.44 |

Frequency (Hz) | 0 | 2 | 16 | 32 |
---|---|---|---|---|

Sr | 0 | 0.0314 | 0.2513 | 0.5027 |

B | 0 | 0.04 | 0.05 | 0.16 |

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 the Main Block (Million) | Total Number of Cells (Million) |
---|---|---|---|---|---|

First | 240 | 50 | 32 | 0.40 | 1.02 |

Second | 248 | 60 | 48 | 0.73 | 1.35 |

Third | 284 | 80 | 50 | 1.15 | 1.77 |

Fourth | 302 | 94 | 56 | 1.60 | 2.22 |

Fifth | 308 | 110 | 64 | 2.18 | 2.80 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baek, S.I.; Ahn, J.
Large Eddy Simulation of Film Cooling with Bulk Flow Pulsation: Comparative Study of LES and RANS. *Appl. Sci.* **2020**, *10*, 8553.
https://doi.org/10.3390/app10238553

**AMA Style**

Baek SI, Ahn J.
Large Eddy Simulation of Film Cooling with Bulk Flow Pulsation: Comparative Study of LES and RANS. *Applied Sciences*. 2020; 10(23):8553.
https://doi.org/10.3390/app10238553

**Chicago/Turabian Style**

Baek, Seung Il, and Joon Ahn.
2020. "Large Eddy Simulation of Film Cooling with Bulk Flow Pulsation: Comparative Study of LES and RANS" *Applied Sciences* 10, no. 23: 8553.
https://doi.org/10.3390/app10238553