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Flutter Analysis of a Transonic Steam Turbine Blade with Frequency and Time-Domain Solvers^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

## 3. Test Case

## 4. Results

#### 4.1. Linear Solver Set-Up and Results

#### 4.2. Harmonic Balance and Time-Domain Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HB | Harmonic Balance |

NRBC | Nonreflecting Boundary Conditions |

## Nomenclature

b | rotor blade height |

c_{m} | chord at midspan |

${d}_{\mathrm{max}}$ | maximal displacement amplitude |

v_{out} | velocity magnitude at outlet |

i | square root of $-1$ |

${p}_{\mathrm{dyn}}$ | dynamic pressure at rotor inlet |

q | vector of conservative flow variables |

${\widehat{q}}_{\omega}$ | Fourier coefficient of q with respect to the angular frequency $\omega $ |

$\mathcal{F}$ | Fourier transform |

R | flow residual |

V | cell volume |

$x,\dot{x}$ | grid coordinates and velocities |

$\alpha $ | non-dimensionalised amplitude |

$\sigma $ | interblade phase angle |

$\omega $ | angular frequency |

## References

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**Figure 5.**Relative error of work coefficient and ${L}^{2}$ residual over GMRES iteration for $\sigma ={0}^{\circ}$.

**Figure 6.**Aerodynamic work coefficient predicted by linearTRACE in comparison with results published by Sun et al. [8].

**Figure 7.**Aerodynamic work coefficient predicted by different harmonic balance set-ups $\sigma =-{72}^{\circ}$.

**Figure 8.**Local work coefficient for $\sigma =-{72}^{\circ}$ on the upper half of the suction side, predicted by time-domain, harmonic balance, and time-linearised solvers. Black lines mark the boundaries between stable and unstable areas.

**Figure 9.**Local work coefficient for $\sigma =-{72}^{\circ}$ on the upper half of the pressure side, predicted by time-domain, harmonic balance, and time-linearised solvers. Black lines mark the boundaries between stable and unstable areas.

**Figure 10.**Local work coefficient for $\sigma =-{72}^{\circ}$ predicted by time-domain, harmonic balance, and time-linearised solvers at $50\%$ (

**left**) and $90\%$ (

**right**) channel height.

**Figure 11.**Relative error of aerodynamic work coefficient over time step (time-domain) and pseudo time step (HB) for $\sigma =-{72}^{\circ}$.

R | $461.52$ | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}$ |

$\gamma $ | $1.12$ | - |

$\mu $ | $1.032\times {10}^{-5}$ | $\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$ |

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## Share and Cite

**MDPI and ACS Style**

Frey, C.; Ashcroft, G.; Kersken, H.-P.; Schlüß, D.
Flutter Analysis of a Transonic Steam Turbine Blade with Frequency and Time-Domain Solvers. *Int. J. Turbomach. Propuls. Power* **2019**, *4*, 15.
https://doi.org/10.3390/ijtpp4020015

**AMA Style**

Frey C, Ashcroft G, Kersken H-P, Schlüß D.
Flutter Analysis of a Transonic Steam Turbine Blade with Frequency and Time-Domain Solvers. *International Journal of Turbomachinery, Propulsion and Power*. 2019; 4(2):15.
https://doi.org/10.3390/ijtpp4020015

**Chicago/Turabian Style**

Frey, Christian, Graham Ashcroft, Hans-Peter Kersken, and Daniel Schlüß.
2019. "Flutter Analysis of a Transonic Steam Turbine Blade with Frequency and Time-Domain Solvers" *International Journal of Turbomachinery, Propulsion and Power* 4, no. 2: 15.
https://doi.org/10.3390/ijtpp4020015