# Effects of Bulk Flow Pulsation on Film Cooling Involving Compound Angle

^{*}

## Abstract

**:**

_{rms}, the root mean squared fluctuating velocity in the streamwise direction around the coolant core increased due to intensive mixing, and v

_{rms}, the root mean squared fluctuating velocity in the wall-normal direction, increased along the trajectory of the injected coolant. Moreover, w

_{rms}, the root mean squared fluctuating velocity in the spanwise direction, increased around the wall compared to those at a steady state. The dimensionless temperature fluctuations increased in the region of the core of the coolant compared with those at a steady state. When the orientation angle was 30°, the distribution of the results moved in the z-direction; however, the overall trend was similar to that of a simple angle.

## 1. Introduction

_{c}and underpredicted the coolant spreading in the spanwise direction. Tyagi and Acharya (2003) conducted an LES for the film cooling on a flat plate [5]. They demonstrated that not only could the LES produce better predictions of η compared with RANS but also accurately predict coherent structures generated by the film-cooling injection. Rozati and Tafti (2007) investigated the effect of M on η and the heat transfer coefficient with an LES [6] and found that if M was increased, a more turbulent shear layer was generated and η decreased while the heat transfer coefficient increased. Na et al. (2007) investigated the effects of using a ramp through a RANS realizable k-ε model, demonstrating that a ramp located upstream of the film-cooling holes increased η because the ramp caused the interaction between the mainstream and the coolant to occur further away from the wall, rendering the formation of a horseshoe vortex weak [7]. Johnson et al. (2011) studied the effects of the ratio of the hole length (L) to the hole diameter (D) for a cylindrical hole, momentum ratio of the coolant injection, and grid resolution on η using a realizable k-ε model [8] and showed that when the coolant jet had a high momentum and small L/D ratio, η decreased due to the increase in coolant jet lift-off. Bianchini, Andrei, Andreini and Facchini (2013) modified RANS turbulence models and compared the results with conventional two equation RANS models and experimental data [9]. They showed that standard RANS models failed to predict η accurately due to the assumption of isotropic turbulence. Yu and Yavuzkurt (2020) modified the DES model by carrying out the correlation for anisotropic eddy viscosity to obtain better film cooling simulations [10]. They stated that their modified model improved the spreading of the coolant in the spanwise direction compared to the original DES model. Zamiri, You, and Chung (2020) investigated optimum geometry of the hole of the laidback fan-shape in order to improve the cooling performance by using LES [11]. They investigated the effects of three parameters (the metering length, forward expansion angle, and lateral expansion angle) and η from the hole of the optimum geometry was raised by around 50%. According to numerous numerical studies, when an orientation angle was implemented and the coolant was injected in the spanwise direction, η improved. Lee et al. (1997) experimentally investigated the effects of the orientation angle β ranging from 15° to 90° on film-cooling performance [12]. They showed that the counter-rotating vortex pair became asymmetry when the orientation angle was 15° and the vortex pair became a single vortex when the orientation angle was 30°. Jung et al. (1999) also investigated the variations in η with the orientation angle by the experiment [13]. When a compound angle was adopted, η increased from 20% to 80% and it depended on the blowing ratio and the orientation angle.

_{rms}, v

_{rms}, w

_{rms}, Reynolds stress such as uv, and uw, and temperature fluctuations are investigated. The film-cooling effectiveness and temperature contours under 36-Hz pulsations are demonstrated in a previous study [20].

## 2. Numerical Method

^{−}

^{6}s and this was the time for the main flow to convect the length of D with 360 time steps [24,25]. The time step for the URANS with 36-Hz pulsations was set as 5.55 × 10

^{−4}s and this was the time of the 36-Hz pulsation period divided by 50. For each time step, 10 sub-iterations were added to resolve the data well [21]. The CFD simulation was conducted on a 20-core Intel Xeon Gold 6148 processor and the computation times for the LES and RANS calculations were approximately 2 months and 10 h, respectively.

#### 2.1. Computational Domain and CFD Mesh

#### 2.2. Governing Equations and Boundary Conditions

#### 2.2.1. Unsteady RANS Approach

#### 2.2.2. LES Approach

#### 2.2.3. Boundary Conditions

_{main flow}= A sin(2πft) + 10 m/s

_{plenum inlet}= B sin(2πft) + C m/s

#### 2.3. Validation of Numerical Methods

_{c}, the centerline (z = 0, y = 0) film cooling effectiveness when the orientation angle β was 0° at steady state with M = 0.5 based on the LES Smagorinsky-Lilly model are illustrated in Figure 5. Five meshes were tested and the specifications for each mesh are indicated in Table 4. The results for the 2.04 million cells are almost the same values of η

_{c}as those for the finer meshes. Therefore, the 2.04 million hexahedron cells were selected for the CFD simulations. The y

^{+}value of the first cell above the wall was less than 2 as illustrated in Figure 6 and there were layers of 25 cells up to y

^{+}= 30.

## 3. Results and Discussion

#### 3.1. Contours of the Time-Averaged Effectiveness on the Wall

#### 3.2. Time-Averaged Temperature Contours on the Streamwise-Normal Planes

#### 3.3. Time-Averaged Streamwise Velocity Contour

#### 3.4. Turbulence Statistics

_{rms}, v

_{rms}, and w

_{rms}do not exceed 10% of the values in the LES results, and the contours of the RANS could not be shown within the contour range in LES and they are not included. As shown in the figure, in the experimental data, the turbulence intensity below y/D = 0.2 could not be measured; however, the LES contours show the contours of u

_{rms}, v

_{rms}, and w

_{rms}below y/D = 0.2, which are highest around the core area of the coolant.

_{rms}and w

_{rms}do not converge to zero at the wall but show high values along the wall, whereas v

_{rms}converges to zero at the wall. When the orientation angle β was 30°, the CRVP changed into a single vortex and the contours of the normalized u

_{rms}, v

_{rms}, and w

_{rms}became asymmetric and the contours of the coolant core moved in the +z-direction. However, the overall trends of the distribution are similar to those of the simple angle β = 0° and the maximum values of the normalized u

_{rms}, v

_{rms}, and w

_{rms}are also similar to those of the simple angle. At M = 0.5, the overall distributions of the components of the turbulence intensities are downsized compared with those at M = 1.0. When the pulsation of 36 Hz was applied to the flow, the values of u

_{rms}around the coolant core increased, and the change in the contour is attributed to the periodic change in the blowing ratio, as shown in Equation (10), resulting in periodic variations in the injection velocity of the coolant. The values of w

_{rms}around the wall increase in Figure 10c. However, the contour of v

_{rms}does not show any significant differences from that at steady state due to the existence of the wall. When the orientation angle β is 30° and the pulsation of 36 Hz is applied, the overall trends of the contours are similar to those of the simple angle (β = 0°); however, the values of w

_{rms}near the wall are higher than those in the case with the simple angle.

_{rms}and v

_{rms}of the turbulence intensity and the Reynolds stress uv for the plane at z = 0 obtained by the LES at steady state and with flow pulsations of 36 Hz at M = 0.5 and 1.0 for the orientation angle β of 0° in comparison with the experimental data from a study by Coletti et al. [35] at M = 1.0. In Figure 12a–c, the LES-obtained results at M = 1.0 are similar to those in the experiment. The LES results illustrate the u

_{rms}, v

_{rms,}and uv below y/D = 0.2, the area of which these values could not be measured experimentally. In Figure 12a, the values of u

_{rms}obtained by the LES increase around the hole center (x/D = 0) and along the shear layer under the core of the coolant in the region between 1 ≤ x/D ≤ 3 and are not strong in the far field. The LES results match the experimental data with an acceptable accuracy even though the LES results are slightly overpredicted. As shown in Figure 12b, the values of v

_{rms}are strong along the injectant and the LES results are slightly overpredicted compared with the experimental results. In the contour of Reynolds stress uv in Figure 12c, the Reynolds stress is strongly generated around the shear layer under the core of the injected coolant downstream of the hole and expands in the streamwise direction with the trajectory of the coolant, while the Reynolds stress generated upstream of the hole does not expand in the streamwise direction.

_{rms}, v

_{rms}, and uv appear to be similar to those at M = 1.0, but the values are lower. If the pulsation of 36 Hz is applied to the flow, the area with high values of u

_{rms}, v

_{rms}, and uv obtained by the LES becomes much larger than that at steady state with M = 0.5. The definition of the Strouhal number is Sr = $\frac{2\pi fL}{{U}_{C}}=\frac{L/{U}_{c}}{1/2\pi f}$ and this value is the ratio of the time needed for the injectant to pass through the hole to the time of a pulsation cycle. Thus, if the Strouhal number is greater than 1, it implies that the coolant is greatly affected by the pulsation while the coolant passes through the hole [14]. Therefore, under the pulsation of 36 Hz, the trajectory of the injectant differs considerably from that at steady state, and the mixing intensity is increased, leading to higher values of u

_{rms}, v

_{rms}, and uv than the values at steady state with M = 1.0. Specifically, u

_{rms}increases around the center of the hole and the trailing edge of the hole. v

_{rms}increases along the trajectory of the injected coolant. Moreover, the height of high value of v

_{rms}in the region between −1 ≤ x/D ≤ 0 increases and the shape of high value of v

_{rms}becomes blunt. Additionally, the values of v

_{rms}are the lowest around the wall at x/D > 2, the values of uv around the injected coolant increase, and the signs of the values at the main flow inlet and the outlet are opposite.

#### 3.5. Temperature Fluctuations

_{rms}for the plane at x/D = 2.5 obtained by the LES. In the RANS results, θ

_{rms}is too small to be fitted within the same contour range, and only the LES results are shown in Figure 14. As can be observed in Figure 14a,b, at steady state, θ

_{rms}is high around the region of the shear layer for the mean temperatures in Figure 8, while the components of the turbulence intensities are high in the region of the core area of the coolant, as observed in Figure 10.

_{rms}in the region of the core of the coolant is also increased compared with that at steady state due to the intensive mixing, which is indicated by the increase in the red area. In the case of the orientation angle β of 30°, the distribution of the temperature fluctuations moves in the z-direction; however, the overall trend is similar to that of the simple angle (β = 0°).

## 4. Conclusions

- The contours of the time-averaged effectiveness and the dimensionless temperatures of the coolant on the streamwise-normal plane obtained by the LES show a better agreement with the experimental data than with the contours at steady state.
- The streamwise velocity gradients in the shear layer predicted by the LES and URANS are smaller than those at steady state because of the intensive mixing between the coolant and the main flow.
- The URANS results predict a weaker streamwise velocity of the coolant jet that blocks the main flow compared with the LES.
- The values of u
_{rms}around the coolant core, the center of the hole and the trailing edge of the hole increase as well as the values of w_{rms}around the wall, while the contour of v_{rms}increased along the trajectory of the injected coolant. Additionally, in the contour of uv, the secondary peaks became stronger while the main peaks weakened. - The dimensionless temperature fluctuations increase in the region of the core of the coolant compared with those at steady state.
- For the orientation angle of 30°, the secondary peaks became stronger, similar to those for the simple angle, although the main blue streak weakened.
- This paper only covered the cylindrical hole, however, the effects of the main flow pulsations on film cooling for the shaped hole such as the forward expansion hole will be covered in future studies. Moreover, the effects for the two sister holes positioned more downstream of the primary hole and the optimized length between the sister hole and the primary hole will be investigated to obtain the best film cooling performance. For the parametric study, the CFD results of this study could be used as the baseline.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

D = hole diameter | |

L = hole length | |

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

P = pitch between holes [mm] | |

Sr = Strouhal number = $\frac{2\pi fL}{{U}_{C}}$ | |

T = temperature [K] | |

t = time [s] | |

U = flow velocity [m/s] | |

u = fluctuating velocity [m/s] | |

x = streamwise coordinate | |

y = wall-normal coordinate | |

z = spanwise coordinate | |

Greek symbols | |

β | = orientation angle |

= angle between the streamwise direction and projected injection vector on the x–z plane | |

κ = von Karman’s universal constant = 0.41 | |

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

${\eta}_{m}$ = 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)] | |

μ = dynamic viscosity | |

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

Subscripts | |

C = coolant | |

G = mainstream gas | |

m = spanwise-averaged | |

rms = root mean squared |

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**Figure 2.**Details of the cylindrical hole configuration used in the current study with an orientation angle β of 0° and 30°. (

**a**) β = 0°; upper: top view, lower: side view; (

**b**) β = 30°; upper: top view, lower: side view.

**Figure 3.**Computational domain configuration: (

**a**) β = 0°; (

**b**) β = 30°. (

**a**) Orientation angle β = 0°; (

**b**) Orientation angle β = 30°.

**Figure 4.**CFD meshes [20]. (

**a**) View of the mesh at z = 0; (

**b**) Close-up of mesh around the hole; (

**c**) Close-up of mesh at the hole exit.

**Figure 5.**Comparison of η

_{c}values for the grid sensitivity test based on the LES Smagorinsky-Lilly model [20].

**Figure 9.**Time-averaged streamwise velocity contour at x/D = 2.5 for the streamwise film cooling compared with the experimental data in [32] at M = 1.0: (

**a**) M = 1.0, 0 Hz, β = 0°; (

**b**) M = 0.5, 0 Hz, β = 0°; (

**c**) M = 0.5, 0 Hz, β = 30°; (

**d**) M = 0.5, 36 Hz, β = 0°; (

**e**) M = 0.5, 36 Hz, β = 30°.

**Figure 10.**Normalized turbulence intensities at x/D = 2.5 compared with the experimental data in [27]: (

**a**) u

_{rms}/U

_{∞}; (

**b**) v

_{rms}/U

_{∞}; (

**c**) w

_{rms}/U

_{∞.}

**Figure 11.**Contours of Reynolds stress for the plane at x/D = 2.5: (

**a**) EXP [32] and LES at M = 1.0, 0 Hz, β = 30°; (

**b**) LES at M = 0.5, β = 0°; (

**c**) LES at M = 0.5, β = 30°.

**Figure 12.**The dimensionless u

_{rms}, v

_{rms}, and uv at z/D = 0 compared with the experimental data of Coletti et al. [35]: (

**a**) u

_{rms}/U

_{∞}at M = 1.0; (

**b**) v

_{rms}/U

_{∞}at M = 1.0; (

**c**) w

_{rms}/U

_{∞}at M = 1.0; (

**d**) u

_{rms}/U

_{∞}at M = 0.5; (

**e**) v

_{rms}/U

_{∞}at M = 0.5; (

**f**) w

_{rms}/U

_{∞}at M = 0.5.

**Figure 13.**Reynolds shear stresses along the plane of y/D = 0.5: (

**a**) EXP [35] and LES, M = 1.0, β = 0°, 0 Hz; (

**b**) LES, M = 0.5, β = 0°; (

**c**) LES, M = 0.5, β = 30°.

**Figure 14.**Temperature fluctuations obtained from the LES at x/D = 2.5 for M = 0.5: (

**a**) β = 0°, 0 Hz; (

**b**) β = 30°, 0 Hz; (

**c**) β = 0°, 36 Hz; and (

**d**) β = 30°, 36 Hz.

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 |

Blowing ratio, M | 1.0 | 0.5 | ||||

Frequency, f (Hz) | 0 | 0 | 36 | |||

Sr | 0 | 0 | 3.62 | |||

β | 0° | 30° | 0° | 30° | 0° | 30° |

A | 0 | 0 | 0.54 |

Blowing ratio, M | 1.0 | 0.5 | ||||

Frequency, f (Hz) | 0 | 0 | 36 | |||

Sr | 0 | 0 | 3.62 | |||

β | 0° | 30° | 0° | 30° | 0° | 30° |

B | 0 | 0 | 0.3 | |||

C | 0.328 | 0.164 | 0.164 |

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 | 242 | 52 | 34 | 0.5 | 1.14 |

Second | 248 | 62 | 52 | 0.96 | 1.60 |

Third | 276 | 80 | 56 | 1.41 | 2.04 |

Fourth | 298 | 94 | 60 | 1.93 | 2.56 |

Fifth | 312 | 110 | 68 | 2.76 | 3.40 |

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**MDPI and ACS Style**

Baek, S.-I.; Ahn, J. Effects of Bulk Flow Pulsation on Film Cooling Involving Compound Angle. *Energies* **2022**, *15*, 2643.
https://doi.org/10.3390/en15072643

**AMA Style**

Baek S-I, Ahn J. Effects of Bulk Flow Pulsation on Film Cooling Involving Compound Angle. *Energies*. 2022; 15(7):2643.
https://doi.org/10.3390/en15072643

**Chicago/Turabian Style**

Baek, Seung-Il, and Joon Ahn. 2022. "Effects of Bulk Flow Pulsation on Film Cooling Involving Compound Angle" *Energies* 15, no. 7: 2643.
https://doi.org/10.3390/en15072643