# Large Eddy Simulation of Pulsed Film Cooling with a Dielectric Barrier Discharge Plasma Actuator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Domain and Boundary Conditions (BCs)

_{2}and N

_{2}at a temperature of 188 K. The velocity-inlet boundary was set at the inlet of the crossflow channel (crossflow velocity u

_{∞}= 20 m/s), temperature (T

_{∞}= 298 K), and boundary layer thickness (δ = 1.0 d). We attempted to apply the pressure-outlet boundary at the outlet of the crossflow channel (static pressure, 101,325 Pa), Reynolds number (Re = u

_{∞}d/v = 15,625), and kinematic viscosity (v = 1.6 × 10

^{−5}m

^{2}/s). The periodic conditions were set in the lateral direction of the crossflow channel (at y/d = ±1.5) for superimposing the influences of adjacent cooling jets. All walls were set to be adiabatic and non-slip.

## 3. Governing Equations and Phenomenological Plasma Model

_{i}is the EHD force vector generated by the DBDPA. We describe the phenomenological plasma model in the following sections.

_{0}= V

_{pp}/l), where l is the space between the two electrodes in the x-axis. A linear decrease in the electric field intensity is achievable with the movement from Source A, and the electric field intensity fluctuation can be formulated as

_{b}= 30 kV/cm was employed to assess k

_{1}= (E

_{0}− E

_{b})/b and k

_{2}= (E

_{0}− E

_{b})/a at the plasma boundary. Hence, the electric field intensity components in diverse directions are formulated as

_{p}is about 67 μs, during which the plasma forms. Thus, the EHD force components in the x and z directions can be formulated as

_{e}= 10

^{17}/m

^{3}is the electron number density, e is the elementary charge, ϑ = 6.0 kHz is the frequency of the applied voltage, and ζ is the factor explaining the collision efficiency. A detailed explanation of the above values can be found in [37]. Additionally, it is crucial to note that the force is linearly attenuated along the internal normal direction of the triangular side, reaching its maximum value at the minimum gap between the plates. The high frequency of the plasma discharge (6.0 kHz) is correlative with the EHD force acting on the airflow as a constant; therefore, the DBDPA-generated EHD force was assumed to be a steady-state body force acting on the film cooling flow.

^{−5}s; maximum number of iterations, 40/time step). We considered a time-averaging approach, taking the main flow through the computational domain over a period ∆T = 0.017 s as one statistical cycle. Throughout this process, a total of 17,000 iterations were carried out, statistical averaging was performed over a duration of 5 × ∆T, and, in the current investigation, the CFL number was approximately 0.946. When a quasi-steady state could be reached by the film cooling flow, six cycles were allocated to the flow field to obtain time-averaged statistical information, in which the time of the crossflow through the crossflow channel was taken to be indicative of one cycle.

_{aw}is the temperature of the adiabatic wall. We attempted to average the adiabatic film cooling efficiency along the full lateral length in order to attain the lateral-averaged film cooling efficiency, which can be formulated as

_{c}/ρ

_{∞}), the velocity-inlet boundary was applied at the inlet of the cooling hole, and the adjustment of the cooling jet inlet velocity was performed on the basis of the BR (M = ρ

_{c}u

_{c}/ρ

_{∞}u

_{∞}) and the momentum ratio I = ρ

_{c}u

_{c}

^{2}/ρ

_{∞}u

_{∞}

^{2}. For the un-pulsed film cooling, the BR was 1.0; thus, 12.5 m/s was allocated to the cooling jet velocity (u

_{c}). For the pulsed film cooling, the total amount of the coolant was the same as for the steady film cooling in an operating cycle; thus, the time-averaged BR of the pulsed film cooling was 1.0 and the BR was in the range of 0.5–1.5 in a cycle (Figure 4). Therefore, the cooling jet velocity was specified as u

_{c}(1.0 + 0.5sin(2πft)), where f is the pulsation frequency of the cooling jet. The Strouhal number (St) of the coolant pulsation was defined as St = df/u

_{∞}, and St was set to 0.25.

^{3}, which is compatible with the previously achieved value [18,19], and the power consumption of the plasma actuator was around 0.75 watts. However, it is worth noting that DBD plasma actuators are characterized by a low efficiency of electrical-to-fluid energy conversion, with a maximum efficiency below 0.1%. This efficiency is significantly lower than that of the other control devices we studied. Nevertheless, DBD plasma actuators offer such advantages as simplicity, a lack of moving parts, and a wide frequency band, making them suitable for practical flow control applications.

## 4. Model Validation

## 5. Results and Discussion

#### 5.1. Time-Averaged Flow Fields

#### 5.2. Instantaneous Flow Fields

_{c})/(T

_{∞}− T

_{c}) (Figure 17, Figure 18 and Figure 19). In the UBL case, the classical coherent structures can be clearly observed. It can be seen that horseshoe vortices form upstream of the cooling hole exit due to the pressure gradient. The fluid temperature in the horseshoe vortices is higher, indicating the aggravation of the mixing of the coolant and the crossflow via the horseshoe vortices. Notably, a series of structurally complete hairpin vortices can be clearly identified, and these hairpin vortices actually dominate the entrainment and mixing of the crossflow. There is a close association of heads, normal legs, and horizontal legs of the hairpin vortices with a shear layer vortex, an upright wake vortex, and the CRVP, respectively [41]. The heads of the hairpin vortices are caused by a shearing effect that leads to a strong entrainment of the coolant with the crossflow. As a result, an increase in the mixed fluid temperature can be found around the heads of the hairpin vortices. An enlarged size of the hairpin vortices and their gradual shifting away from the wall can be found. Small-scale turbulent vortices eventually result, highlighting their noticeable role in the far-field region.

_{0}to t = t

_{7}. The coherent structure groups persist downstream over a short distance and happen to break up earlier, leading to a low film cooling efficiency in the wake region.

_{0}, the coherent structure groups happen to break up at x/d = 6, followed by turbulent incorporation of the coolant into the crossflow, which is remarkably enhanced, and a rapid increase in the CT. The last coherent structure group is regularly and gradually extended downstream, the locations of the minimal CT are accordingly altered downstream, and there is an increment in the minimum CT due to the turbulent incorporation effect. In the far-field region, the CT is also regularly altered with time as the small-scale turbulent vortices still propagate downstream in an orderly manner.

**ur**comprehension of the jet–crossflow interactions, the three-dimensional streamlines are drawn in Figure 24. As can be seen from the purple streamlines, a horseshoe vortex forms upstream of the cooling hole exit. In the UBL case, the purple streamlines wrap around the cooling jet, flow into the recirculation region, exacerbate the turbulent mixing, and then eventually intertwine with the red streamlines and contribute to the development of the CRVP. While approaching downstream, the purple streamlines and the red streamlines gradually shift away from the wall together, accompanied by the cooling jet flow. In the PBL-off case, the purple streamlines have larger heights upstream of the cooling hole exit due to the higher instantaneous BR, meaning that the horseshoe vortex has higher strength. The majority of the purple streamlines flow directly downstream, and some of the purple streamlines intertwine with the red streamlines. Moreover, just downstream of the cooling hole exit, the red streamlines stay closer to the wall surface since the BR is low at this moment. Then, the red streamlines rapidly move away from the wall surface due to the coherent structure group. In the PBL-on case, the purple streamlines shift downstream, indicating that a horseshoe vortex formed slightly further downstream. This may have resulted from the PAA-generated downward force that may push the cooling jet closer to the wall surface, thereby attenuating the pressure gradient upstream of the cooling hole. Downstream of the cooling hole, the red streamlines and the purple streamlines are closer to the wall surface, indicating that the PAA weakens the lift-off effects of the CRVP.

## 6. Conclusions

- (1)
- The coolant pulsation might cause a slight reduction in the film cooling efficiency as the averaged pulsation BR was 1.0, while the PAA could effectively improve the pulsed film cooling efficiency and it would be superior to steady-state film cooling;
- (2)
- The pulsed cooling jet can penetrate more deeply than steady-state film cooling in the near-hole region. Thus, the jet–crossflow interactions produced a large-scale CRVP, promoting the turbulent integration. Because of the downward force generated by the PAA, the penetration depth of the pulsed cooling jet was greatly reduced, which could be attributable to the downward force. The detrimental lift-off effect and entrainment of the CRVP were weakened.
- (3)
- Rather than hairpin vortices, intermittent coherent structure groups formed in the pulsed film cooling, and these groups also had upcast behavior and moved away from the wall surface while evolving downstream, thereby aggravating the turbulent integration of the coolant with the crossflow. The coherent structure groups were reduced in size and strength owing to the PAA, and their upcast behavior was attenuated; thus, the turbulent integration was suppressed and the film cooling efficiency was enhanced.
- (4)
- The three-dimensional streamlines also confirmed that the PAA could effectively control the unsteady dynamic behavior of the LSCSs. The height of the three-dimensional streamlines was significantly reduced, indicating that the pulsed cooling jet flow was positioned close to the wall surface owing to the PAA.
- (5)
- Operating with an AC voltage of V(t) = V
_{max}× sin(2ft), where V_{max}= 8 kV and f = 6 kHz, the PAA demonstrated compelling results. Importantly, the power supply characteristics (a high frequency of 6.0 kHz and a high voltage of 8.0 kVpp) translated into an estimated EHD force of about 2 MN/m3. Remarkably, the power consumption of the plasma actuator remained minimal, at approximately 0.75 watts.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

E | electric field intensity (V/m) | Greek symbols | |

E_{b} | breakdown electric field intensity (V/m) | ρ | gas density (kg/m^{3}) |

E_{0} | electric field intensity at the tip (V/m) | λ | wavelength of plasma actuator (m) |

E_{s} | electric field intensity at serrated edge (V/m) | α | intersection angle of actuator (°) |

V_{0} | peak AC voltage (V) | β | angle of plasma force at the tip (°) |

l | space between the two electrodes (m) | η | film cooling efficiency (-) |

l_{a} | distance from the root (m) | ζ | collision efficiency (-) |

f | pulsation frequency (Hz) | δ | thickness of boundary layer (m) |

e | elementary charge (C) | ν | kinematic viscosity (m^{2}/s) |

∆t | space of time (s) | ν_{sgs} | eddy viscosity (m^{2}/s) |

a | height of plasma region (m) | ϑ | applied voltage frequency (Hz) |

b | length of plasma region (m) | Subscripts | |

k | constants in plasma model (-) | aw | adiabatic wall |

d | diameter of film-cooling hole (m) | ∞ | crossflow |

x, y, z | cartesian coordinates (m) | c | jet flow |

u, v, w | velocity component index (m/s) | lat | lateral-averaged cooling efficiency |

T | local fluid temperature (K) | p | plasma actuation in a cycle |

t | time (s) | ||

k_{1}, k_{2} | constants in plasma model (-) |

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**Figure 12.**The contours of the time-averaged dimensionless temperature in the cross-sections downstream of the cooling hole.

**Figure 17.**The instantaneous Q iso-surface colored by temperature in the UBL case (Q = 5.0 × 10

^{5}).

**Figure 18.**The instantaneous Q iso-surface colored by temperature in the PBL-off case (Q = 5.0 × 10

^{5}).

**Figure 19.**The instantaneous Q iso-surface colored by temperature in the PBL-on case (Q = 5.0 × 10

^{5}).

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## Share and Cite

**MDPI and ACS Style**

Shen, Z.; Hu, B.; Li, G.; Zhang, H.
Large Eddy Simulation of Pulsed Film Cooling with a Dielectric Barrier Discharge Plasma Actuator. *Aerospace* **2024**, *11*, 28.
https://doi.org/10.3390/aerospace11010028

**AMA Style**

Shen Z, Hu B, Li G, Zhang H.
Large Eddy Simulation of Pulsed Film Cooling with a Dielectric Barrier Discharge Plasma Actuator. *Aerospace*. 2024; 11(1):28.
https://doi.org/10.3390/aerospace11010028

**Chicago/Turabian Style**

Shen, Zhou, Beimeng Hu, Guozhan Li, and Hongjun Zhang.
2024. "Large Eddy Simulation of Pulsed Film Cooling with a Dielectric Barrier Discharge Plasma Actuator" *Aerospace* 11, no. 1: 28.
https://doi.org/10.3390/aerospace11010028