# Interaction Mechanism and Loss Analysis of Mixing between Film Cooling Jet and Passage Vortex

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Setup and Validation

## 3. Flow Field in the Main-Flow Passage

## 4. Entropy Generation Rate Based Loss Analysis Method

#### 4.1. Entropy Generation Rate

#### 4.2. Reference Flat Plate Film Cooling without Passage Vortex

#### 4.3. Scale Analysis on the Magnitude of Thermal and Viscous Entropy Generation Rates

#### 4.4. Verification of the RANS-Solved Entropy Generation Rate

## 5. Effect of Passage Vortex on Film Cooling Performance

#### 5.1. Effect of Hole Location

#### 5.2. Effect of Blowing Ratio

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$Be$ | Bejan number |

$Cp$ | Specific heat |

D | Diameter of the cooling hole |

${D}_{m}$ | Diameter of the half-cylinder |

$DR$ | Density ratio, $\frac{{\rho}_{c}}{{\rho}_{0}}$ |

$EGR$ | Non-dimensionalized entropy generation rate |

h | Wall thickness |

I | Momentum ratio |

k | Thermal diffusion conductivity |

K | Acceleration parameter |

L | Hole length |

M | Blowing ratio, $\frac{{\rho}_{c}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{U}_{c}}{{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{U}_{0}}$ |

$Ma$ | Mach number |

$Pr$ | Prandtl number |

s or S | Entropy |

$\dot{S}$ | Entropy generation rate |

T | Temperature |

U | Velocity |

V | Volume |

x or X | Spanwise direction |

y or Y | Pitchwise direction |

z or Z | Streamwise direction |

Greek | |

$\alpha $ | Injection angle |

${\alpha}_{t}$ | Turbulent thermal diffusion coefficient |

$\beta $ | Expansion angle |

$\eta $ | Adiabatic cooling effectiveness, $\frac{{T}_{aw}-{T}_{0}}{{T}_{c}-{T}_{0}}$ |

$\gamma $ | Deviation angle |

$\lambda $ | Thermal conductivity |

$\nu $ | Viscosity |

${\mathbf{\omega}}^{*}$ | Non-dimensionalized vorticity, $\frac{\mathbf{\omega}}{{U}_{0}/{D}_{m}}$ |

${\Phi}_{\Theta}$ | Thermal irreversible entropy generation |

$\rho $ | Density |

${\tau}_{c}$ | Characteristic time scale |

${\tau}_{i,j}$ | Viscous stress |

$\theta $ | Non-dimensionalized temperature, $\frac{T-{T}_{0}}{{T}_{c}-{T}_{0}}$ |

$\epsilon $ | Tensor of the deformation rate |

Subscript | |

$aw$ | Adiabatic wall |

c | Coolant |

$fwd$ | Forward direction |

$lat$ | Lateral/pitchwise direction |

0 | Main-flow |

t | Turbulent |

$ther$ | Thermal |

$visc$ | Viscous |

Abbreviation | |

BL | Boundary layer |

PV | Passage vortex |

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**Figure 4.**Grid independence analysis of laterally averaged adiabatic cooling effectiveness ${\overline{\eta}}_{lat}$.

**Figure 5.**Validation of the laterally averaged film cooling effectiveness ${\overline{\eta}}_{lat}$ using the experimental data by Schroeder et al. [29].

**Figure 6.**Comparisons of effectiveness distributions between numerical results and experimental data (M = 1.0).

**Figure 8.**Illustration of the endwall limiting streamlines (left: experimental result [33]; right: CFD result in current study).

**Figure 10.**Vortex structures at A-A slice in Figure 9.

**Figure 13.**Contour of thermal entropy generation rate $EG{R}_{ther}$, flat plate film cooling at M = 1.0.

**Figure 14.**Contour of viscous entropy generation rate $EG{R}_{visc}$, flat plate film cooling at M = 1.0.

**Figure 15.**Contour of vorticity with isolines of non-dimensionalized temperatures, flat plate film cooling at M = 1.0.

**Figure 16.**Distributions of the Bejan number $Be$ denoting the proportion of the thermal entropy generation rate, flat plate film cooling at M = 1.0.

**Figure 17.**Demonstration of ${T}_{RMS}$ calculated by LES at z = 0D, M = 3.0 (the isoline of $\theta $ = 0.1).

**Figure 22.**Area-averaged cooling effectiveness under varying blowing ratios and pitchwise locations.

Parameter | Value | Parameter | Value |
---|---|---|---|

${D}_{m}$ | 150 mm | D | 7.75 mm |

$h/D$ | 3.0 | $L/D$ | 2.5 |

${\beta}_{fwd}$ | ${7}^{\circ}$ | ${\beta}_{lat}$ | ${7}^{\circ}$ |

$\alpha $ | ${30}^{\circ}$ | M | 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 |

${U}_{0}$ | 22 m/s | $DR$ | 1.5 |

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**MDPI and ACS Style**

Chen, Z.; Hu, K.; Mao, Y.; Su, X.; Yuan, X.
Interaction Mechanism and Loss Analysis of Mixing between Film Cooling Jet and Passage Vortex. *Entropy* **2022**, *24*, 15.
https://doi.org/10.3390/e24010015

**AMA Style**

Chen Z, Hu K, Mao Y, Su X, Yuan X.
Interaction Mechanism and Loss Analysis of Mixing between Film Cooling Jet and Passage Vortex. *Entropy*. 2022; 24(1):15.
https://doi.org/10.3390/e24010015

**Chicago/Turabian Style**

Chen, Ziyu, Kexin Hu, Yinbo Mao, Xinrong Su, and Xin Yuan.
2022. "Interaction Mechanism and Loss Analysis of Mixing between Film Cooling Jet and Passage Vortex" *Entropy* 24, no. 1: 15.
https://doi.org/10.3390/e24010015