# Identify the Rotating Stall in Centrifugal Compressors by Fractal Dimension in Reconstructed Phase Space

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulations of the Rotating Stall in the Centrifugal Impeller

#### 2.1. The Model and Numerical Method

#### 2.2. The Model and Numerical Method

**Figure 2.**(

**a**) Pressure distribution as the flow rates is 30 kg/s at 42.8% blade height; (

**b**) pressure distribution as the flow rates is 23.6 kg/s at 42.8% blade height; (

**c**) pressure distribution as the flow rates is 12.6 kg/s at 42.8% blade height.

## 3. Phase Space Reconstruction of Pressure Time Series

#### 3.1. Phase Space Reconstruction

#### 3.2. C–C Method

#### 3.3. Pressure Time Series of the Sampling Points

#### 3.4. Phase Space Reconstruction in Pre- and Post- Rotating stalls

**Figure 6.**The relationship of correlation coefficients and time delay applying the C–C method to time series at P1.

#### 3.4.1. Dynamics Analysis of the Pre- and Post- Rotating stall by Phase Space Reconstruction

**Figure 8.**Phase space reconstruction for the pressure time series at P1 under the rotating stall. (k is the time delay.)

#### 3.4.2. The Fractal Dimension in the Reconstructed Phase Space in the Post-Rotating Stall

**Figure 9.**Fractal dimension in the reconstructed phase space for pressure time series at P1 under the rotating stall.

## 4. Phase Space Reconstruction of the Pressure Time Series at Different Locations

#### 4.1. Parameters for Phase Space Reconstruction at Different Locations

P2 | P3 | P4 | P5 | |
---|---|---|---|---|

$\tau $ | 12 | 12 | 11 | 10 |

${\tau}_{w}$ | 15 | 12 | 14 | 10 |

$m$ | 2.25 | 2.00 | 2.27 | 2.00 |

#### 4.2. Phase Space Reconstruction for Pressure Time Series at Different Locations

**Figure 10.**(

**a**) Reconstructed phase space for pressure time series at P2 under the rotating stall; (

**b**) Reconstructed phase space for pressure time series at P3 under the rotating stall; (

**c**) Reconstructed phase space for the pressure time series at P4 under the rotating stall; (

**d**) Reconstructed phase space for the pressure time series at P5 under the rotating stall.

#### 4.3. Fractal Dimension in the Reconstructed Phase Space for Pressure Time Series at Different Locations

#### 4.3.1. Fractal Dimension in the Reconstructed Phase Space at Circumference Locations

**Figure 11.**(

**a**) Fractal dimension in reconstructed phase space for pressure time series at P2 under the rotating stall; (

**b**) Fractal dimension in reconstructed phase space for pressure time series at P3 under the rotating stall.

#### 4.3.2. Fractal Dimensions in Reconstructed Phase Space at Different Radial Locations

**Figure 12.**(

**a**) Fractal dimension in reconstructed phase space for pressure time series at P4 under the rotating stall; (

**b**) Fractal dimension in reconstructed phase space for pressure time series at P5 under the rotating stall.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, L.; Zhang, J.; Zhang, W.
Identify the Rotating Stall in Centrifugal Compressors by Fractal Dimension in Reconstructed Phase Space. *Entropy* **2015**, *17*, 7888-7899.
https://doi.org/10.3390/e17127848

**AMA Style**

Wang L, Zhang J, Zhang W.
Identify the Rotating Stall in Centrifugal Compressors by Fractal Dimension in Reconstructed Phase Space. *Entropy*. 2015; 17(12):7888-7899.
https://doi.org/10.3390/e17127848

**Chicago/Turabian Style**

Wang, Le, Jiazhong Zhang, and Wenfan Zhang.
2015. "Identify the Rotating Stall in Centrifugal Compressors by Fractal Dimension in Reconstructed Phase Space" *Entropy* 17, no. 12: 7888-7899.
https://doi.org/10.3390/e17127848