# Numerical Investigation of Influence of Entropy Wave on the Acoustic and Wall Heat Transfer Characteristics of a High-Pressure Turbine Guide Vane

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## Abstract

**:**

_{θ}transition model is coupled with both models. The baseline cases are first simulated with the two different turbulence models without any incoming perturbation. Then one forced case with an entropy wave train set at the turbine inlet at a given frequency and amplitude is simulated. Results show that the downstream maximum Mach number is rising from 0.98 to 1.16, because the entropy waves increase the local temperature of the flow field; also, the torque of the vane varies as the entropy waves go through, the magnitude of the oscillation is 7% of the unforced case. For the wall (both suction and pressure side of the vane) heat transfer, the entropy waves make the maximum heat transfer coefficient nearly twice as the large at the leading edge, while the minimum heat transfer coefficient stays at a low level. As for the averaged normalized heat transfer coefficient, a maximum difference of 30% appears between the baseline case and the forced case. Besides, during the transmission process of entropy waves, the local pressure fluctuates with the wake vortex shedding. The oscillation magnitude of the pressure wave at the throat is found to be enhanced due to the inlet entropy wave by applying the dynamic mode decomposition (DMD) method. Moreover, the transmission coefficient of the entropy waves, and the reflection and transmission coefficients of acoustic waves are calculated.

## 1. Introduction

## 2. Numerical Setup

_{wall}). In order to investigate the effect of an incoming flow entropy wave on the heat transfer of the vane surface and the downstream pressure field, an entropy wave train E(t) is given via a user defined function (UDF) at the inlet, while the total pressure stays the same as the experiments. Besides, two different turbulence models (k-ð and SAS model) are compared under different incoming flow conditions before setting the entropy wave.

## 3. Results and Discussions

#### 3.1. Case without Entropy Wave

_{wall}stands for the constant vane wall temperature and q

_{wall}is the wall heat flux. As Figure 6 indicates, for the test case MUR129, the results of both turbulence models match the experimental data on the pressure side, while on the suction side, a laminar to turbulent transition shows up and leads to a huge rise of heat flux. In the experiments, the transition position is estimated at S = 75 mm. Gourdain’s work (RANS) [18] predicted a transition position at S = 62 mm. Meanwhile, the transition position given by MUR129_SAS is about S = 67 mm, which is closer to the experimental measurement; however, the k-ω model misses this phenomenon. For the MUR241 test cases, the profiles of heat transfer coefficients on the pressure side are matching the experiments well. Yet, on the suction side, with the rising inlet turbulence intensity, the transition position could not be perfectly estimated by all RANS models. Clearly, an early or late prediction of transition would cause some big differences in heat transfer, as shown in Figure 6b. Overall, the heat transfer coefficients along the two vane surfaces simulated by the two models here in CFX seem to be better predicted than in the previous pioneering RANS simulations by Gourdain, et al. [17]. Even though LES can capture more flow physics and provide more accurate results, like in Morata’s work [16] on the same cases, the preliminary RANS results, notably those obtained with the SAS model, provide improved and reasonable agreement with the experiment, especially in the MUR129 case. As LES requires much higher computing resources and is much more costly to get convergence of the simulation for sufficient periods of inlet entropy waves, the SAS model is chosen to yield a numerical parametric investigation of the influence of entropy wave on the pressure field and wall heat transfer characteristics.

#### 3.2. Case with Entropy Wave

_{i,0}is set to a constant value (see Table 1), whereas the inlet total temperature is varied with the time. Figure 7 shows the distribution of instantaneous temperature field. The frequency of the inlet plane entropy wave is 1000 Hz, and ${t}_{0}=2\times {10}^{-3}\mathrm{s}$ is defined as the moment when the entropy wave is just about to impinge on the vane leading edge. After the leading edge splits the entropy wave train into two parts, one each passing along the pressure and suction sides, the entropy wave interacts with the shed vortices, then goes out at the outlet.

#### 3.2.1. Heat Transfer on the Vane Surface

#### 3.2.2. Downstream Acoustic Field

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Nesbitt, E. Towards a quieter low pressure turbine: Design characteristics and prediction needs. Int. J. Aeroacoustics
**2010**, 10, 1–15. [Google Scholar] [CrossRef] - Candel, S. Analytical Studies of Some Acoustic Problems of Jet Engines. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, USA, 1972. [Google Scholar]
- Marble, F.; Candel, S. Acoustic disturbance from gas non-uniformities convected through a nozzle. J. Sound Vib.
**1977**, 55, 225–243. [Google Scholar] [CrossRef] - Cumpsty, N.; Marble, F. The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise. Proc. R. Soc. Lond. A Math. Phys. Sci.
**1977**, 357, 323–344. [Google Scholar] - Yoon, M. The entropy wave generation in a heated one-dimensional duct. J. Fluid Mech.
**2019**, 883, A44. [Google Scholar] [CrossRef] - Morgans, A.S.; Duran, I. Entropy noise: A review of theory, progress and challenges. Int. J. Spray Combust. Dyn.
**2016**, 8, 285–298. [Google Scholar] [CrossRef][Green Version] - Moreau, S.; Duran, I. (Eds.) Analytical and Numerical Study of the Entropy Wave Generator Experiment on Indirect Combustion Noise. In Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference (32nd AIAA Aeroacoustics Conference), Portland, OR, USA, 5–8 June 2011. [Google Scholar]
- Bake, F.; Kings, N.; Fischer, A. Experimental investigation of the entropy noise mechanism in aero-engines. Int. J. Aeroacoustics
**2009**, 8, 125–141. [Google Scholar] [CrossRef] - Leyko, M.; Nicoud, F.; Moreau, S. Numerical and analytical investigation of the indirect combustion noise in a nozzl. Comptes Rendus Mécanique
**2009**, 337, 415–425. [Google Scholar] [CrossRef][Green Version] - Duran, I.; Moreau, S. Study of the Attenuation of Waves Propagating through Fixed and Rotating Turbine Blades. In Proceedings of the 18th AIAA/CEAS Aeroacoustics Conference (33rd AIAA Aeroacoustics Conference), Colorado Springs, CO, USA, 4–6 June 2012. [Google Scholar]
- Wang, G.; Sanjose, M.; Moreau, S.; Papadogiannis, D.; Duchaine, F.; Gicquel, L. Noise mechanisms in a transonic high-pressure turbine stage. Int. J. Aeroacoustics
**2016**, 15, 144–161. [Google Scholar] [CrossRef] - Papadogiannis, D.; Wang, G.; Moreau, S.; Duchaine, F.; Gicquel, L.; Nicoud, F. Assessment of the indirect combustion noise generated in a transonic high-pressure turbine stage. J. Eng. Gas Turbines Power
**2016**, 138, 041503. [Google Scholar] [CrossRef] - Ceci, A.; Gojon, R.; Mihaescu, M. Large Eddy Simulations for Indirect Combustion Noise Assessment in a Nozzle Guide Vane Passage. Flow, Turbul. Combust.
**2019**, 102, 299–311. [Google Scholar] [CrossRef][Green Version] - Becerril, C.; Moreau, S.; Gicquel, L. (Eds.) Study of Combustion Noise Generation in a Realistic Turbine Stage Configuration. In Proceedings of the ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, Oslo, Norway, 11–15 June 2018. [Google Scholar]
- Wheeler, A.P.; Sandberg, R.D.; Sandham, N.D.; Pichler, R.; Michelassi, V.; Laskowski, G. Direct numerical simulations of a high-pressure turbine vane. J. Turbomach.
**2016**, 138, 071003. [Google Scholar] [CrossRef] - Morata, E.C.; Gourdain, N.; Duchaine, F.; Gicquel, L. Effects of free-stream turbulence on high pressure turbine blade heat transfer predicted by structured and unstructured LES. Int. J. Heat Mass Transf.
**2012**, 55, 5754–5768. [Google Scholar] [CrossRef] - Gourdain, N.; Gicquel, L.Y.; Collado, E. Comparison of RANS and LES for prediction of wall heat transfer in a highly loaded turbine guide vane. J. Propuls. Power
**2012**, 28, 423–433. [Google Scholar] [CrossRef] - Phan, H.M.; Duan, P.H.; Dinh, C.T. Numerical aero-thermal study of high-pressure turbine nozzle guide vane: Effects of inflow conditions. Phys. Fluids
**2020**, 32, 034111. [Google Scholar] [CrossRef] - Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef][Green Version] - Menter, F.R.; Kuntz, M.; Bender, R. A Scale-Adaptive Simulation Model for Turbulent Flow Predictions. In Proceedings of the 41st Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 6–9 January 2003. [Google Scholar]
- Schmid, P.J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech.
**2010**, 656, 5–28. [Google Scholar] [CrossRef][Green Version] - Antares Development Team, Antares Documentation Release 1.14.0. Available online: https://cerfacs.fr/antares/ (accessed on 3 June 2020).
- Consigny, H.; Richards, B. Short duration measurements of heat-transfer rate to a gas turbine rotor blade. J. Eng. Power
**1982**, 104, 542–550. [Google Scholar] [CrossRef] - Arts, T.; Lambertderouvroit, M.; Rutherford, A. Aero-Thermal Investigation of a Highly Loaded Transonic Linear Turbine Guide Vane Cascade; Technical Note 174. Von Karman Institute for Fluid Dynamics: Sint-Genesius-Rode, Belgium, 1990. [Google Scholar]
- Pouangu, A.F.; Sanjos, M.; Moreau, S. Subsonic Jet Noise Simulations Using Both Structured and Unstructured Grids. AIAA J.
**2015**, 53, 55–69. [Google Scholar] [CrossRef][Green Version] - Hiller, S.; Seitz, P. Interaction Between a Fluidic Actuator and Main Flow Using SAS Turbulence Modelling. In Proceedings of the 3rd AIAA Flow Control Conference, San Francisco, CA, USA, 5–8 June 2006. [Google Scholar]
- Winkler, C.; Dorgan, A.; Mani, M. Scale Adaptive Simulations of Turbulent Flows on Unstructured Grids. In Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, Honolulu, GA, USA, 27–30 June 2011. [Google Scholar]
- Wang, Y.; Liu, K.; Song, W.-P. Scale-Adaptive Simulations of Unsteady Flow around NACA0021 Airfoil at 60° angle of attack. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019. [Google Scholar]
- Menter, F.R. A Correlation-Based Transition Model Using Local Variables—Part I: Model Formulation. ASME J. Turbomach.
**2006**, 128, 413–42230. [Google Scholar] [CrossRef] - Ansys CFX. CFX User Manual. Ansys Inc.: Canonsburg, PA, USA, 2017. [Google Scholar]
- Leyko, M.; Duran, I.; Moreau, S. Simulation and modelling of the waves transmission and generation in a stator blade row in a combustion-noise framework. J. Sound Vib.
**2014**, 333, 6090–6106. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Sketch of coarse mesh (

**a**), details of the refined mesh in simulation (

**b**), (

**c**), (

**d**) and the distribution of the normalized wall distance y

^{+}(estimation based on result of MUR129_SAS) (

**e**).

**Figure 4.**Instantaneous contours of the normalized density gradient |▽ρ|/ρ: (

**a**) MUR129_SAS; (

**b**) MUR241_SAS.

**Figure 7.**Distribution of instantaneous temperature fields at different times within one period $\delta =2\pi /{\omega}_{0}=1ms$.

**Figure 9.**Distribution of instantaneous isentropic Mach number at different times within one period $\delta =2\pi /{\omega}_{0}=1ms$; the black line represents the Ma = 1).

**Figure 10.**Influence of the entropy wave on the wall heat transfer (

**a**) and the variation of air thermal conductivity (

**b**) with temperature.

**Figure 11.**Distribution of instantaneous wall heat flux on the vane suction side (top view of the vane).

**Figure 12.**(

**A**): Distribution of instantaneous static pressure (without entropy wave); (

**B**): Distribution of instantaneous static pressure at different times within one period with entropy ($\delta =2\pi /{\omega}_{0}=1ms$) wave.

Case | T_{wall} (K) | P_{i,0} (Mpa) ^{a} | P_{2} (Mpa) ^{b} | T_{i,0} (K) ^{c} |
---|---|---|---|---|

MUR129_k-ω | 297.75 | 0.1849 | 0.1165 | 409.20 |

MUR241_k-ω | 299.75 | 3.2570 | 1.5470 | 416.40 |

MUR129_SAS | 297.75 | 0.1849 | 0.1165 | 409.20 |

MUR241_SAS | 299.75 | 3.2570 | 1.5470 | 416.40 |

MUR129_SAS_Wave | 297.75 | 0.1849 | 0.1165 | T_{1}(t)_{c} |

^{a}footnote i represents the isentropic quantities.

^{b}footnote 0 and 2 represents the inlet and outlet, respectively.

^{c}$E\left(\mathrm{t}\right)={T}_{0}+C\mathrm{sin}\left({\omega}_{0}\xb7t\right)$, ${T}_{0}=409.20K$, ${\omega}_{0}=2000\mathsf{\pi}$, $C=100\text{}K$.

Case | Calculation | Theory |
---|---|---|

${\beta}_{s}$ | 0.6360 | 1 |

${\beta}_{r}$ | 0.0554 | 0.0617 |

${\beta}_{t}$ | 0.0166 | 0.0064 |

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**MDPI and ACS Style**

Hu, K.; Fang, Y.; Zheng, Y.; Wang, G.; Moreau, S.
Numerical Investigation of Influence of Entropy Wave on the Acoustic and Wall Heat Transfer Characteristics of a High-Pressure Turbine Guide Vane. *Acoustics* **2020**, *2*, 524-538.
https://doi.org/10.3390/acoustics2030028

**AMA Style**

Hu K, Fang Y, Zheng Y, Wang G, Moreau S.
Numerical Investigation of Influence of Entropy Wave on the Acoustic and Wall Heat Transfer Characteristics of a High-Pressure Turbine Guide Vane. *Acoustics*. 2020; 2(3):524-538.
https://doi.org/10.3390/acoustics2030028

**Chicago/Turabian Style**

Hu, Keqi, Yuanqi Fang, Yao Zheng, Gaofeng Wang, and Stéphane Moreau.
2020. "Numerical Investigation of Influence of Entropy Wave on the Acoustic and Wall Heat Transfer Characteristics of a High-Pressure Turbine Guide Vane" *Acoustics* 2, no. 3: 524-538.
https://doi.org/10.3390/acoustics2030028