#
Simulation of Indexing and Clocking with Harmonic Balance^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. The Harmonic Balance Method

#### 2.1. Example 1: Up- and Downstream Disturbances within a Component

#### 2.2. Example 2: Up- and Downstream Disturbances with Different Relative Rotational Speeds

## 3. Zero-Frequency Harmonics

#### 3.1. Mode Coupling and Solution Method

#### 3.2. Numerical Example

## 4. Fan Simulation with Inlet Distortion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

HB | Harmonic Balance |

NLH | Nonlinear Harmonic |

BDF2 | (Second order) Backward Differencing Scheme |

MUSCL | Monotonic Upstream-Centered Scheme for Conservation Laws |

## Nomenclature

f | frequency |

$\mathbf{i}$ | complex unit |

k | harmonic index |

l | passage index |

m | circumferential mode order |

${\dot{m}}_{red.}$ | mass flow reduced to ISA conditions |

q | vector of conservative flow variables |

${\widehat{q}}_{\omega}$ | Fourier coefficient of q |

w.r.t. the angular frequency $\omega $ | |

s | specific entropy |

${t}_{j}$ | sampling point |

$(x,r,\vartheta )$ | cylindrical coordinates |

$\mathcal{F}$ | Fourier transform |

K | number of higher harmonics |

${N}_{i}$ | number of sampling points |

${N}_{\mathrm{R}}$$\left({N}_{\mathrm{S}}\right)$ | number of blades (vanes) |

R | flow residual |

${\mathcal{S}}_{i},\mathcal{S}$ | harmonic sets |

$\sigma $ | interblade phase angle |

${\eta}_{is}$ | isentropic efficiency |

$\Delta \vartheta $ | pitch |

$\omega $ | angular frequency |

${\Pi}_{t}$ | total pressure ratio |

$\Omega $ | shaft speed |

## References

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**Figure 1.**Red and violet harmonic sets represent the unsteady disturbances caused by two neighbouring blade rows with equal rotational speeds, e.g., wakes and potential effects in a rotor system due to neighbouring stators whose vane count ratio is 2:3.

**Figure 2.**Green and blue harmonic sets represent the unsteady disturbances caused by two neighbouring blade rows with different rotational speeds whose blade count ratio is 2:3.

**Figure 3.**Two harmonic sets representing the disturbances caused by a rotor and a stator. The blade count ratio of the first two rows is 2:3.

**Figure 4.**Density distribution over circumferential coordinate at inlet, outlet, and third interface. analytic solution (

**a**); numerical solutions with HB configuration 1 (

**b**); and HB configuration 2 (

**c**).

**Figure 5.**Full annulus setup of the fan stage (

**a**) and distorted relative total pressure distribution on fan stage inlet plane (

**a**).

**Figure 6.**Compressor map for design speed (

**a**) and impact of the inlet distortion on the operating point. Zoom of total pressure ratio (

**b**) and isentropic efficiency (

**c**) characteristics.

**Figure 7.**Entropy distribution at 90% relative radial height for time domain simulation (

**a**); harmonic balance simulation with full annulus stator (

**b**); and harmonic balance simulation with single stator and zero-frequency harmonic set (

**c**).

**Table 1.**Harmonic set configurations for the numerical experiment of jet propagation through duct segments.

Stator1 | Rotor | Stator 2 | |
---|---|---|---|

# Segments | 400 | 400 | 720 |

Rotational Speed | 0 | ${\Omega}_{R}$ | 0 |

Base frequency, $\omega $ | 0 | $720\phantom{\rule{0.166667em}{0ex}}{\Omega}_{\mathrm{R}}$ | 0 |

Base interblade phase angle, $\sigma $ | 0 | 0 | ${160}^{\circ}$ |

Harmonic Set (Conf. 1) | $(0,0)$ | $(0,0),(\omega ,0),(2\omega ,0)$ | $(0,0),(0,\sigma )$ |

Harmonic Set (Conf. 2) | $(0,0)$ | $(0,0),(\omega ,0),\dots ,(7\omega ,0)$ | $(0,0),(0,\sigma ),\dots ,(0,4\sigma )$ |

Time Domain | HB Full Annulus Stator | HB Single Passage Stator | |
---|---|---|---|

mesh size (cells) | $8.7\times {10}^{6}$ | $4.5\times {10}^{6}$ | $1.2\times {10}^{6}$ |

relative computational effort | $16.1$ | $11.7$ | 1 |

inlet duct | full annulus | full annulus | full annulus |

rotor row | full annulus | single passage | single passage |

stator row | full annulus | full annulus | single passage |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

## Share and Cite

**MDPI and ACS Style**

Frey, C.; Ashcroft, G.; Kersken, H.-P.; Schönweitz, D.; Mennicken, M. Simulation of Indexing and Clocking with Harmonic Balance. *Int. J. Turbomach. Propuls. Power* **2018**, *3*, 1.
https://doi.org/10.3390/ijtpp3010001

**AMA Style**

Frey C, Ashcroft G, Kersken H-P, Schönweitz D, Mennicken M. Simulation of Indexing and Clocking with Harmonic Balance. *International Journal of Turbomachinery, Propulsion and Power*. 2018; 3(1):1.
https://doi.org/10.3390/ijtpp3010001

**Chicago/Turabian Style**

Frey, Christian, Graham Ashcroft, Hans-Peter Kersken, Dirk Schönweitz, and Maximilian Mennicken. 2018. "Simulation of Indexing and Clocking with Harmonic Balance" *International Journal of Turbomachinery, Propulsion and Power* 3, no. 1: 1.
https://doi.org/10.3390/ijtpp3010001