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Modelling Turbine Acoustic Impedance^{ †}

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^{†}

## Abstract

**:**

## 1. Introduction

- improvement of the accuracy of analytical models for turbine acoustic impedance;
- demonstration of an efficient CFD approach for predicting acoustic impedance; and
- quantification of the sensitivity of acoustic impedance to a complete set of aerodynamicparameters across the turbine design space.

## 2. Cambered Semi-Actuator Disk Analytical Model

#### 2.1. Mean Flow

#### 2.2. Characteristic Waves

- high hub-to-tip radius ratio, so spanwise variations are negligible;
- small amplitude perturbations, so products of perturbation quantities are negligible;
- high Reynolds number, so viscous and heat conduction effects are negligible.

#### 2.3. Blade Row Modelling

#### 2.4. Method of Solution

## 3. Non-Linear Time-Domain Computations

#### 3.1. Domain and Boundary Conditions

#### 3.2. Post-Processing

- Area average. Area-averaging at each time step makes the unsteady flow at a given location spatially uniform, restricting the method to plane waves. Alternatively, applying constant-area mixed-out averaging (conserving mass, momentum, and energy) gives identical results;
- Fourier transform. Subtracting time-mean pressure and entropy from the instantaneous values at each axial location forms perturbations, which are transformed to the frequency domain using a fast Fourier transform algorithm;
- Wave separation. Applying the least-squares multi-microphone technique of Poinsot et al. [5] separates upstream- and downstream-running wave amplitudes from static pressure perturbations in the inlet and exit ducts;
- Reflectivity correction. We account for reflectivity of the computational boundary conditions using a ‘black-box’ technique after Emmanuelli et al. [6], first proposed by Cremer [7]. Expressing the wave amplitudes as a superposition of reflections, results for all three forcing cases form a system of six simultaneous linear equations. A matrix inversion yields the solution for the complex reflection and transmission coefficients.

#### 3.3. Computational Details

## 4. Analytical Model Validation

#### 4.1. Uncooled Linear Vane Cascade

#### 4.2. Cooled Linear Vane Cascade

#### 4.3. Linear Stage Cascade

## 5. Design Space Study

#### 5.1. Velocity Triangles

#### 5.2. Other Parameters

#### 5.3. Parameter Sensitivity Rankings

## 6. Conclusions

- Predictions of turbine acoustic impedance using a cambered semi-actuator disk model agree with two-dimensional CFD simulations to within ±7% and ±16% for incident pressure and entropy waves, an improvement over simpler flat-plate models;
- For representative coolant flow rates, turbine cooling has a limited effect on acoustic impedance of ±3%. Increasing coolant flow rate reduces entropy–acoustic reflectivity;
- A parametric study of the turbine design space shows that acoustic impedance is most sensitive to flow coefficient, then axial velocity ratio, Mach number, and stage loading coefficient. Acoustic impedance is relatively insensitive to polytropic efficiency and degree of reaction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Roman | Greek | ||

a | Acoustic wave speed [${\mathrm{ms}}^{-1}$] | $\alpha $ | Flow angle [${}^{\circ}$] |

c | Blade chord [$\mathrm{m}$] | $\gamma $ | Specific heat ratio [–] |

f | Frequency [$\mathrm{Hz}$] | $\eta $ | Polytropic efficiency [–] |

h | Specific enthalpy [${\mathrm{Jkg}}^{-1}$] | $\Theta $ | Coolant flow fraction [–] |

$M\phantom{\rule{0.0pt}{0ex}}a$ | Mach number [–] | $\kappa $ | Reduced frequency [–] |

p | Pressure [$\mathrm{Pa}$] | $\Lambda $ | Degree of reaction [–] |

$\mathcal{R}$ | Reflection coefficient [–] | $\rho $ | Density [${\mathrm{kgm}}^{-3}$] |

s | Specific entropy [${\mathrm{JK}}^{-1}{\mathrm{kg}}^{-1}$] | $\upsilon $ | Vorticity [${\mathrm{s}}^{-1}$] |

T | Temperature [$\mathrm{K}$] | $\varphi $ | Flow coefficient [–] |

$\mathcal{T}$ | Transmission coefficient [–] | $\psi $ | Stage loading coefficient [–] |

U | Rotor blade speed [${\mathrm{ms}}^{-1}$] | $\zeta $ | Axial velocity ratio [–] |

V | Velocity [${\mathrm{ms}}^{-1}$] | ||

Subscripts | Accents | ||

0 | Stagnation conditions | $\widehat{\square}$ | Characteristic wave |

i | Control volume index | ${\square}^{\prime}$ | Perturbation relative to mean |

1 | Vane inlet | ${\square}^{+}$ | Downstream-running |

2 | Vane outlet | ${\square}^{-}$ | Upstream-running |

3 | Blade outlet | $\square \left(t\right)$ | Function of time |

x | Axial component | $\overline{\square}$ | Time average |

## References

- Lamarque, N.; Poinsot, T. Boundary Conditions for Acoustic Eigenmodes Computation in Gas Turbine Combustion Chambers. AIAA J.
**2008**, 46, 2282–2292. [Google Scholar] [CrossRef] - Cumpsty, N.A.; Marble, F.E. The Interaction of Entropy Fluctuations with Turbine Blade Rows: A Mechanism of Turbojet Engine Noise. Proc. R. Soc. Lond.
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**1970**, 11, 339–353. [Google Scholar] [CrossRef] - Bauerheim, M.; Duran, I.; Livebardon, T.; Wang, G.; Moreau, S.; Poinsot, T. Transmission and reflection of acoustic and entropy waves through a stator–rotor stage. J. Sound Vib.
**2016**, 374, 260–278. [Google Scholar] [CrossRef][Green Version] - Poinsot, T.; Le Chatelier, C.; Candel, S.; Esposito, E. Experimental determination of the reflection coefficient of a premixed flame in a duct. J. Sound Vib.
**1986**, 107, 265–278. [Google Scholar] [CrossRef] - Emmanuelli, A.B.; Huet, M.; Le Garrec, T.; Ducruix, S. Study of Entropy Noise through a 2D Stator using CAA. In Proceedings of the 2018 AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 25–29 June 2018; p. 3915. [Google Scholar] [CrossRef]
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**Figure 1.**Annulus control volumes $i=1,2,\dots ,{N}_{i}$ in the semi-actuator disk model. The mean flow is uniform and steady over each of the inlet, outlet, and spaces between blade rows.

**Figure 4.**Reflection and transmission coefficients for an uncooled vane cascade, predicted by analytical model and CFD, for different incident characteristic waves. $\widehat{s}$ Att’n denotes augmentation with Bauerheim et al. [4] streamtube model. The cambered model matches the CFD results to within on average ±2% for downstream-running pressure wave reflection, and ±11% for entropy–acoustic reflection, an improvement over flat plates.

**Figure 5.**Reflection and transmission coefficients for a cooled vane cascade, with varying coolant flow rates $\Theta $, predicted by cambered analytical model and CFD. The change in pressure wave reflection coefficients is small: within ±3% of the mean value over all flow rates.

**Figure 6.**Comparison of reflection and transmission coefficients for a two-dimensional turbine stage, predicted by cambered analytical model and CFD. The predictions agree on average to within ±7% for downstream-running pressure wave reflection, and ±16% for entropy–acoustic reflection.

**Figure 7.**Effect of velocity triangle design on acoustic impedance for incident downstream-running pressure and entropy waves. Pressure wave reflection coefficients are most sensitive to flow coefficient. Entropy–acoustic reflection increases for low-turning designs.

**Figure 8.**Effect of secondary parameter variations on acoustic impedance, for incident downstream-running pressure and entropy waves. Contours of sensitivity calculated by subtracting acoustic impedances at the upper and lower limits of each parameter, with other parameters at datum values.

**Figure 9.**Ranking of turbine design parameters by root-mean-square acoustic impedance sensitivity, for incident pressure and entropy waves. Flow coefficient is the most influential parameter.

Parameter | $\mathit{\varphi}$ | $\mathit{\psi}$ | $\mathbf{\Lambda}$ | ${\mathit{\zeta}}_{1}$, ${\mathit{\zeta}}_{3}$ | $\mathit{M}\phantom{\rule{0.0pt}{0ex}}{\mathit{a}}_{2}$ | $\mathit{\eta}$ | $\mathit{\gamma}$ | $\mathit{\kappa}$ |
---|---|---|---|---|---|---|---|---|

Lower | 0.4 | 0.8 | 0.4 | — | 0.6 | 0.85 | — | — |

Datum | 0.8 | 1.6 | 0.5 | 1.0 | 0.7 | 0.90 | 1.33 | 0.02 |

Upper | 1.2 | 2.4 | 0.6 | parallel annulus | 0.8 | 0.95 | — | — |

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**MDPI and ACS Style**

Brind, J.; Pullan, G. Modelling Turbine Acoustic Impedance. *Int. J. Turbomach. Propuls. Power* **2021**, *6*, 18.
https://doi.org/10.3390/ijtpp6020018

**AMA Style**

Brind J, Pullan G. Modelling Turbine Acoustic Impedance. *International Journal of Turbomachinery, Propulsion and Power*. 2021; 6(2):18.
https://doi.org/10.3390/ijtpp6020018

**Chicago/Turabian Style**

Brind, James, and Graham Pullan. 2021. "Modelling Turbine Acoustic Impedance" *International Journal of Turbomachinery, Propulsion and Power* 6, no. 2: 18.
https://doi.org/10.3390/ijtpp6020018