# Cooling Characteristic of a Wall Jet for Suppressing Crossflow Effect under Conjugate Heat Transfer Condition

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## Abstract

**:**

## 1. Introduction

## 2. Computational Model & Numerical Method

#### 2.1. Computational Model

^{3}turbine was simulated to conduct the study. The fluid domain contains the mainstream path (marked in red) and the internal cooling chambers at the leading edge of the blade (marked in blue), and the two regions are connected at the coolant outlet. In the passage with pin-fin rows in Figure 3, the solid domain is only the leading-edge part of the blade (marked in dark gray) to avoid the additional heat transfer of double-wall cooling. In other words, the region marked in light gray is not simulated. The periodic boundary condition was utilized in the study to reduce computational cost. Figure 3 illustrates the internal cooling structures investigated in the study, including the impingement cooling with a single row of jet (IC), swirl cooling (SC), and wall jet cooling (WJ). For the blades with impingement cooling and swirl cooling, the coolant and cooling chambers were designed based on the original design of GE-E

^{3}turbine [33]. The film holes were ignored and the coolant outlet was at the tip clearance according to engineering practice, the height of which is 0.4 mm and 1% of the blade height.

_{inner}), channel width (W

_{ch}), and channel height (H

_{ch}) equal to the outer wall thickness (t). Hence, t was set as the characteristic length in the study, which is 1.2 mm according to Ref. [33]. The orifice width (W

_{in}) and turning internal radius (R) were set as 0.5t. The impingement distance is 0.63t.

_{j}) of the impingement cooling. The pitches of the jetting holes and orifices (p) are all 2t. The geometrical parameters mentioned above are summarized in Table 1.

#### 2.2. Boundary Conditions

#### 2.3. Grid Independence

_{w}is the wall heat flux, D

_{hy}is the hydraulic diameter of the jetting hole/orifice, T

_{c}is the total temperature at the coolant plenum inlet, T

_{w}is the target wall temperature, and $\lambda $ is the cooling air thermal conductivity.

#### 2.4. Turbulence Model Validation

_{ew}is the external surface temperature, T

_{rec}is the temperature of the blade wall without cooling, and T

_{c}is the total temperature at the coolant inlet. The curve of simulation agrees well with that of the experimental data [39], which means that the SST k-ω model has enough prediction accuracy for the simulation of cascade model under conjugate heat transfer condition. According to the validation results above, the SST k-ω model was selected to conduct the simulations of all cases in this study.

## 3. Results and Discussion

#### 3.1. Overall Performance Evaluations

_{j}= 20,000, the coolant mass flow of IC is lower than that of SC and WJ, and the cooling performance of IC is also the worst.

_{chm}among all the cooling methods. However, the ‘ocs’ area’s huge preponderance ensures that WJ has the largest total heat flux compared with IC and SC, no matter under which coolant inlet boundary condition. Besides, WJ’s total heat flux in Figure 12c only increased 12% relative to that in Figure 12b, with a 154% gain in coolant mass flow rate. It means that the jetting Reynolds number of 20,000 might be close to an optimal value for WJ’s current design, and the Re

_{j}over 20,000 would lead to unnecessary flow resistance. If more coolant is needed to dissipate the suction side wall’s heat load in a real WJ design, the jetting orifice width should be increased to reduce the pressure loss.

#### 3.2. Flow Characteristics

_{j}would increase dramatically with coolant mass flow.

_{cou}is the area-averaged static pressure at the outlet of the cooling chamber.

#### 3.3. Heat Transfer Characteristics

_{sp}on the suction side is low, on the pressure side is high, and there is a peak on each side. The WJ has significant advantages near the leading edge stagnation line and on the suction side. With the same jetting Reynolds number, the Φ

_{msp}value of WJ is 37.3% and 18.8% higher than that of IC and SC at the leading edge stagnation line, respectively, and 37–59% and 19–54% higher on the suction side, respectively. With the same total pressure at the coolant inlet, the Φ

_{msp}value of WJ is 54.5% and 41.4% higher than IC and SC at the stagnation line, respectively, and 30–80% and 20–90% higher, respectively, at the whole leading edge.

## 4. Conclusions

_{msp}(main region laterally-averaged overall cooling effectiveness) value of WJ is 37.3% and 18.8% higher than that of IC and SC at the leading edge stagnation line respectively, and 37–59% and 19–54% higher on the suction side. With the same total pressure at the coolant inlet, the Φ

_{msp}value of WJ is 54.5% and 41.4% higher than IC and SC at the stagnation line respectively, and 30–80% and 20–90% higher respectively at the whole leading edge.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbols | |

A_{chm} | area of the coolant chamber surface, m^{2} |

A_{ocs} | area of the other cooling surface, m^{2} |

A_{tar} | area of the target surface, m^{2} |

BR | blowing ratio |

D_{hy} | hydraulic diameter, mm |

D_{im} | diameter of impingement cooling hole, mm |

H_{ch} | height of bended channel, mm |

h | heat transfer coefficient, W/(m^{2}·K) |

I | impingement distance, mm |

Ma_{is} | isentropic Mach number |

MR | ratio of jet mass to total coolant mass |

${\dot{m}}_{\mathrm{cin}}$ | mainstream mass flow rate, kg/s |

m_{ideal} | calculational mass flow rate in film hole with incompressible assumption, kg/s |

m_{m} | coolant air mass flow rate, kg/s |

Nu | Nusselt number |

Nu_{a} | area-averaged Nusselt number |

Nu_{c} | area-averaged Nusselt number on the whole internal cooling surfaces |

Nu_{ci} | circumferentially-averaged Nusselt number |

Nu_{sp} | span-wise averaged Nusselt number |

Nu_{tar} | area-averaged Nusselt number on the target surface |

q_{w} | wall heat flux, W/m^{2} |

P | static pressure, Pa |

p | jetting orifice/hole pitch, mm |

P_{t,cin} | total pressure at the coolant inlet, MPa |

P_{cou} | total pressure at the coolant outlet, MPa |

PR | pressure ratio |

R | turning internal radius of the cooling channel, mm |

Re | Reynolds number |

Re_{j} | Reynolds number based on the hydraulic diameter of the jetting orifice/hole |

r | radius of swirl pipe, mm |

s | stream-wise surface coordinate, mm |

s’ | stream-wise surface coordinate on the target surface, mm |

t | outer wall thickness of the blade, mm |

t_{inner} | inner cooling wall thickness of the blade, mm |

T_{ew} | external wall temperature, K |

T_{c} | total temperature at coolant plenum inlet, K |

T_{rec} | uncooled blade wall temperature, K |

T_{w} | target wall temperature, K |

U | mean velocity, m/s |

W_{ch} | width of cooling channel, mm |

W_{in} | jetting orifice width, mm |

x | stream-wise distance, mm |

y | height direction coordinate, mm |

Greek Symbols | |

μ | fluid dynamic viscosity, kg/(m·s) |

ρ | fluid density, kg/m^{3} |

λ | fluid thermal conductivity, W/(m·K) |

η_{j} | mass flow nonuniformity coefficient |

Φ | overall cooling effectiveness |

Φ_{msp} | main region span-wise averaged overall cooling effectiveness |

Φ_{sp} | span-wise averaged overall cooling effectiveness |

Abbreviation | |

CHT | Conjugate Heat Transfer |

IC | Impingement Cooling |

RANS | Reynolds-Averaged Navier-Stokes |

SC | Swirl Cooling |

WJ | Wall Jet cooling |

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**Figure 6.**Effect of fluid grids size on Nusselt number at the stagnation line of the target surface.

**Figure 8.**Comparison of circumferentially averaged Nusselt number results at Re = 10,000 and temperature ratio = 0.83.

**Figure 10.**Cooling surfaces among different internal cooling structures; (

**a**) Impingement cooling; (

**b**) Swirl cooling; (

**c**) Wall jet cooling.

**Figure 12.**Total heat flux among different internal cooling structures; (

**a**) Cases with the same ${\dot{m}}_{\mathrm{cin}}$; (

**b**) Cases with the same Re

_{j}; (

**c**) Cases with the same P

_{t,cin}.

**Figure 13.**Mass flow rate ratio and jetting Reynolds number comparison among different internal cooling structures; (

**a**) Cases with the same ${\dot{m}}_{\mathrm{cin}}$; (

**b**) Cases with the same Re

_{j}; (

**c**) Cases with the same P

_{t,cin}.

**Figure 14.**Pressure ratio distribution of different internal cooling structures (Re

_{j}= 20,000); (

**a**) Impingement cooling; (

**b**) Swirl cooling; (

**c**) Wall jet cooling.

**Figure 15.**Streamline comparison between different internal cooling structures (Re

_{j}= 20,000); (

**a**) Impingement cooling; (

**b**) Swirl cooling; (

**c**) Wall jet impingement cooling.

**Figure 16.**Streamline and velocity contours in mid-span section with different internal cooling structures (Re

_{j}= 20,000).

**Figure 17.**Span-wise averaged overall cooling effectiveness around blade leading edge with three cooling structures; (

**a**) Cases with the same ${\dot{m}}_{\mathrm{cin}}$; (

**b**) Cases with the same Re

_{j}; (

**c**) Cases with the same P

_{t,cin}.

**Figure 18.**Overall cooling effectiveness distribution of different internal cooling structures (Re

_{j}= 20,000).

**Figure 19.**Temperature distributions in sections with different heights of the studied internal cooling structures (Re

_{j}= 20,000).

**Figure 20.**Heat transfer coefficient distribution on target surfaces of different internal cooling structures (Re

_{j}= 20,000).

**Figure 21.**Heat transfer coefficient distribution at the stagnation line of the target surfaces (Re

_{j}= 20,000).

Geometry | T [mm] | D_{j}[mm] | W_{in}/t[-] | P/t [-] | t_{inner}/t[-] | W_{ch}/t[-] | H_{ch}/t[-] | R/t [-] | I/t [-] |
---|---|---|---|---|---|---|---|---|---|

IC | 1.2 | 0.8 | - | 2 | - | - | - | - | 1.68 |

SC | 1.2 | - | 0.5 | 2 | - | - | - | - | - |

WJ | 1.2 | - | 0.5 | 2 | 1 | 1 | 1 | 0.5 | 0.63 |

Boundary Conditions | Value |
---|---|

Fluid (mainstream and coolant) | Ideal air |

Solid | DD6 alloy |

Mainstream inlet total pressure [MPa] | 2.526 |

Mainstream inlet total temperature [K] | 1780 |

Mainstream inlet turbulence intensity [%] | 10 |

Mainstream inlet turbulence length scale [mm] | 7.5 |

Mainstream outlet static pressure [MPa] | 1.123 |

Coolant inlet total temperature [K] | 883 |

Coolant inlet turbulence intensity [%] | 5 |

Coolant inlet turbulence length scale [mm] | 0.1 |

Coolant inlet massflow rate (under constant mass condition) [g/s] | 10.4 |

Coolant inlet total pressure in the relative coordinate system (under constant total pressure condition) [MPa] | 3.0 |

Jetting Reynolds number (under constant Re_{j} condition) | 20,000 |

Geometry | ${\dot{\mathit{m}}}_{\mathbf{cin}}$ [g/s] | Re_{j} | P_{t,cin} [MPa] | P_{cou} [MPa] | Φ_{ps} | Φ_{ss} | Φ_{le} | Nu_{tar} | Nu_{c} |
---|---|---|---|---|---|---|---|---|---|

IC | 10.40 | 28,000 | 3.32 | 2.88 | 0.43 | 0.33 | 0.37 | 61.12 | 52.92 |

SC | 10.40 | 20,100 | 3.07 | 2.63 | 0.46 | 0.31 | 0.36 | 70.88 | 50.37 |

WJ | 10.40 | 19,400 | 1.59 | 1.31 | 0.48 | 0.41 | 0.43 | 63.24 | 62.31 |

Geometry | ${\dot{\mathit{m}}}_{\mathbf{cin}}$ [g/s] | Re_{j} | P_{t,cin} [MPa] | P_{cou} [MPa] | Φ_{ps} | Φ_{ss} | Φ_{le} | Nu_{tar} | Nu_{c} |
---|---|---|---|---|---|---|---|---|---|

IC | 7.51 | 20,000 | 2.54 | 2.26 | 0.39 | 0.3 | 0.33 | 44.67 | 39.12 |

SC | 10.32 | 20,000 | 3.05 | 2.61 | 0.46 | 0.31 | 0.36 | 70.17 | 50.34 |

WJ | 10.70 | 20,000 | 1.61 | 1.31 | 0.48 | 0.41 | 0.44 | 64.92 | 64.24 |

**Table 5.**Flow and heat transfer and parameters of the cases with the same coolant inlet total pressure.

Geometry | ${\dot{\mathit{m}}}_{\mathbf{cin}}$ [g/s] | Re_{j} | P_{t,cin} [MPa] | P_{cou} [MPa] | Φ_{ps} | Φ_{ss} | Φ_{le} | Nu_{tar} | Nu_{c} |
---|---|---|---|---|---|---|---|---|---|

IC | 9.26 | 25,000 | 3.00 | 2.62 | 0.42 | 0.32 | 0.35 | 55.68 | 48.11 |

SC | 10.16 | 19,700 | 3.00 | 2.59 | 0.46 | 0.31 | 0.36 | 69.97 | 48.87 |

WJ | 27.13 | 51,500 | 3.00 | 1.42 | 0.59 | 0.51 | 0.54 | 175.34 | 145.07 |

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## Share and Cite

**MDPI and ACS Style**

Deng, Q.; Wang, H.; He, W.; Feng, Z.
Cooling Characteristic of a Wall Jet for Suppressing Crossflow Effect under Conjugate Heat Transfer Condition. *Aerospace* **2022**, *9*, 29.
https://doi.org/10.3390/aerospace9010029

**AMA Style**

Deng Q, Wang H, He W, Feng Z.
Cooling Characteristic of a Wall Jet for Suppressing Crossflow Effect under Conjugate Heat Transfer Condition. *Aerospace*. 2022; 9(1):29.
https://doi.org/10.3390/aerospace9010029

**Chicago/Turabian Style**

Deng, Qinghua, Huihui Wang, Wei He, and Zhenping Feng.
2022. "Cooling Characteristic of a Wall Jet for Suppressing Crossflow Effect under Conjugate Heat Transfer Condition" *Aerospace* 9, no. 1: 29.
https://doi.org/10.3390/aerospace9010029