# The Differential Entropy Generation Rate as a Unified Measure for Both the Stability and Efficiency of an Axial Compressor

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## Abstract

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## 1. Introduction

## 2. Grid Independence Verification and Numerical Simulation Method Validation

#### 2.1. Model and Grid Independence Verification

#### 2.2. Validation of Numerical Simulation Method

## 3. Results and Discussions

#### 3.1. Casing Treatment Configuration and Comparison with a Solid Wall

#### 3.2. Correlation between Entropy Generation and Peak Efficiency

#### 3.2.1. Correlation of Efficiency and DEGR and its Flow Mechanism

#### 3.2.2. The Metric of the CTs Efficiency Change by DEGR

#### 3.3. Correlation between the DEGR and the Stability

#### 3.3.1. Connection between DEGR and the MF/LF Interface at the near Stall

#### 3.3.2. The Metric of the CTs Stability Enhancement by DEGR

^{3}). Given that CT1 has a more anterior groove position, it may be inferred that its effect in shifting the MF/LF interface is greater. The upper part of Figure 16 confirms this, as the three intersection points between the DEGR curve and the pink dashed line near the leading edge indicate the average MF/LF interface for SW (black), CT1 (red), and CT2 (blue), respectively. The corresponding streamwise values denote their respective positions. The upper part of the figure indicates that both the red and blue circles are positioned behind the SW point, with CT1 (the red circle) having the highest streamwise value and being located at the most posterior position, followed by CT2 (the blue circle). This outcome is consistent with the expected judgment and also correlates with stall margin results of the CTs. The lower part of Figure 16 displays DEGR distributions on the 0.98 span section for these three cases (SW, CT1, and CT2). The CT1-B groove is positioned just above the original DEGR boundary of SW-B, resulting in a low DEGR region below it that effectively inhibits forward movement of the DEGR boundary. In CT2-B, the groove is located behind the original DEGR boundary of SW-B and improves a small area on the pressure side of the blade to form a low DEGR zone, slightly shifting back the stall boundary line in this region.

#### 3.3.3. Flow Mechanism between the DEGR and Stability Enhancement

## 4. Conclusions

- (1)
- The total entropy generation shows a reversed trend to isentropic efficiency as mass flow varies. In the PE operating condition, high DEGR regions are primarily concentrated near the blade tip. The influence of CTs on DEGR is also focused around the blade tip. CTs can enhance efficiency by suppressing the highest DEGR generated by complex flows around the blade tip.
- (2)
- The DEGR boundary aligns completely with the wall shear boundary at the near stall, which means that the DEGR boundary can represent the MF/LF interface. At the near stall, high DEGR regions are mainly concentrated near the blade tip, and the influence of CTs on DEGR is also focused around the blade tip. CTs can narrow down the range of high DEGR and push the DEGR boundary downstream. The flow mechanism shows that the CTs inhibit the TLV turning toward to the leading edge. In other words, CTs suppress the interface of MF/LF moving upstream, thereby delaying the onset of stall.
- (3)
- The method of how to utilize the DEGR to measure the efficiency and stability enhancement of CTs are proposed. Volumes within the 0.95–1 span were chosen for averaging the DEGR. The cumulative distribution of the DEGR along the axial direction provides a measure for efficiency improvement ability of CTs. The location of the DEGR along the axial direction provides a measure for the stability enhancement ability of CTs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathsf{\alpha}$ | Thermal diffusivity, (m^{2}/s) |

$\mathsf{\alpha}$_{t} | Thermal diffusivity of the fluctuating temperature, (m^{2}/s) |

$\theta $ | Dimensionless temperature, K |

$\lambda $ | Thermal conductivity, J s^{−1} m^{−1} K^{−1} |

$\mu $ | Dynamic viscosity, kg m^{−1} s^{−1} |

${\overline{\Phi}}_{\theta}$ | Entropy production term, (WK/m^{3}) |

$\omega $ | Characteristic frequency, MHz |

$n$ | Rotation speed, rpm |

$p$ | Local pressure, N m^{−2} |

${\dot{S}}_{irr,D\prime}^{\prime \prime \prime}$ | Entropy production rate by turbulent dissipation, (W/(m^{3} K)) |

${\dot{S}}_{irr,\overline{D}}^{\prime \prime \prime}$ | Entropy production rate by viscous dissipation, (W/(m^{3} K)) |

${\dot{S}}_{PRO,C\prime}$ | Entropy production rate by heat transfer with gradients of the fluctuating temperature, (W/(m^{3} K)) |

${\dot{S}}_{PRO,\overline{C}}$ | Entropy production rate by heat transfer with mean temperature gradients, (W/(m^{3} K)) |

$T$ | Bulk temperature, K |

u’ v’ w’ | Local fluctuating velocity component, m s^{−1} |

$\mathrm{u}\xaf\mathrm{v}\xaf\mathrm{w}\xaf$ | Local average velocity component, m s^{−1} |

x y z | Coordinate vector component, m |

## Abbreviations

CT | Casing treatment |

CT-A | Casing treatment’s A operating condition |

CT-B | Casing treatment’s B operating condition |

TLV | Tip leakage vortex |

DEGR | Differential entropy generation rate |

$\mathrm{SW}$ | Solid wall |

EXP | Experimental data |

NS | Near stall |

NS-corrected | Corrected near stall |

PE | Peak efficiency |

PE-corrected | Corrected peak efficiency |

R67 | NASA Rotor 67 |

BL | Boundary leakage |

MF/LF | Main flow/leakage flow |

Subscripts | |

gen | Generation rate |

${irr,D}^{\prime}$ | Turbulent dissipation |

$irr,\overline{D}$ | Viscous dissipation |

${PRO,C}^{\prime}$ | Heat transfer with gradients of the fluctuating temperature |

$PRO,\overline{C}$ | Heat transfer with mean temperature gradients |

$rev$ | Reversible |

t | Relative total |

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**Figure 13.**DEGR distribution above the 0.95 span along with streamwise. (

**a**) Local quantity; (

**b**) cumulant quantity.

**Figure 16.**The upper half: DEGR distribution above the 0.95 span along with the axial direction. The bottom half: DEGR distribution at the 0.98 span.

**Figure 18.**DEGR distribution and three-dimensional streamline with the relative Mach number contour.

a | Entropy Generation (W/K) | b and e | Entropy Generation (W/K) | c and d | Entropy Generation (W/K) | f and g | Entropy Generation (W/K) | h (Height) | Entropy Generation (W/K) |
---|---|---|---|---|---|---|---|---|---|

129 | 17.54 | 17 | 16.21 | 17 | 17.07 | 17 | 17.25 | 113 | 16.21 |

177 | 17.79 | 33 | 17.79 | 33 | 17.79 | 33 | 17.79 | 177 | 17.79 |

225 | 17.78 | 49 | 17.85 | 49 | 17.81 | 49 | 17.67 | 225 | 17.85 |

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**MDPI and ACS Style**

Ma, J.; Lin, F.
The Differential Entropy Generation Rate as a Unified Measure for Both the Stability and Efficiency of an Axial Compressor. *Machines* **2023**, *11*, 815.
https://doi.org/10.3390/machines11080815

**AMA Style**

Ma J, Lin F.
The Differential Entropy Generation Rate as a Unified Measure for Both the Stability and Efficiency of an Axial Compressor. *Machines*. 2023; 11(8):815.
https://doi.org/10.3390/machines11080815

**Chicago/Turabian Style**

Ma, Jingyuan, and Feng Lin.
2023. "The Differential Entropy Generation Rate as a Unified Measure for Both the Stability and Efficiency of an Axial Compressor" *Machines* 11, no. 8: 815.
https://doi.org/10.3390/machines11080815