# Investigation of the Trailing Edge Modification Effect on Compressor Blade Aerodynamics Using SST k-ω Turbulence Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Geometry and Mesh Details

^{−3}mm for four cases were obtained. The reasonable mesh model provided results that did not change, despite increasing the number of boundary layers and decreasing the first cell size. Table 1 shows the mesh details for the simulation.

#### 2.2. Mathematical Model

#### 2.3. Turbulence Flow of Compressible Fluid

- Wilcox STD k-ω$$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial}{\partial {x}_{j}}\left(\rho {U}_{j}k\right)=\frac{\partial}{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu}_{t}}{{\sigma}_{k1}}\right)\frac{\partial k}{\partial {x}_{j}}\right]+{P}_{k}-{\beta}^{\prime}\rho k\omega $$$$\frac{\partial \left(\rho \omega \right)}{\partial t}+\frac{\partial}{\partial {x}_{j}}\left(\rho {U}_{j}\omega \right)=\frac{\partial}{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu}_{t}}{{\sigma}_{\omega 1}}\right)\frac{\partial \omega}{\partial {x}_{j}}\right]+{\alpha}_{1}\frac{\omega}{k}{P}_{k}-{\beta}_{1}\rho {\omega}^{2}$$
- Transformed k-ε,$$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial}{\partial {x}_{j}}\left(\rho {U}_{j}k\right)=\frac{\partial}{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu}_{t}}{{\sigma}_{k3}}\right)\frac{\partial k}{\partial {x}_{j}}\right]+{P}_{k}-{\beta}^{\prime}\rho k\omega $$$$\begin{array}{cc}\hfill \frac{\partial \left(\rho \omega \right)}{\partial t}+\frac{\partial}{\partial {x}_{j}}\left(\rho {U}_{j}\omega \right)=& \frac{\partial}{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu}_{t}}{{\sigma}_{\omega 2}}\right)\frac{\partial \omega}{\partial {x}_{j}}\right]+2\rho \frac{1}{{\sigma}_{\omega 2}\omega}\frac{\partial k}{\partial {x}_{j}}\frac{\partial \omega}{\partial {x}_{j}}\hfill \\ & +{\alpha}_{2}\frac{\omega}{k}{P}_{k}-{\beta}_{3}\rho {\omega}^{2}\hfill \end{array}$$

_{1}, the transformed k-ω equations by function (1 − F

_{1}), and the corresponding continuity equation, momentum equation, k and ω equations were added to give the BSL k-ω model, as follows:

_{r1}to f

_{r1}P

_{k}in Equation (7) and α

_{2}(ωk

^{−1}) f

_{r1}P

_{k}in Equation (8). The f

_{r1}formulation was given by:

#### 2.4. Initial and Boundary Conditions

#### 2.5. Numerical Investigation

^{+}less than 5, the accuracy of base line case was verified by the experimental data from EGAT. The computational resource is 16 cores of Intel Xeon 2.20 GHz with 64 GB of RAM, which required 24 h per case.

## 3. Results and Discussion

#### 3.1. Validation Study

#### 3.2. Fluid Flow in Rotational Domain

^{−4}at 918 iterations, so the calculation was stopped. We obtained accurate results. Readers should read the Basic Solver Capability Theory for further details on setting other parameters [22].

_{a}) and velocity distribution were used to describe the flow field. The drag (C

_{d}) and lift (C

_{l}) coefficients represented the aerodynamic parameters in this study. Focusing on pressure ratio of the modification of R6, R7, and R8, Figure 8 shows the pressure ratio between the middle plane of R6 and S6 for the blade (a) before modification, after modifications of (b) 1 mm, (c) 5 mm, and (d) 10 mm. As shown in Figure 8, the result of R6–S6 interface obtained from turbulent flow model, high pressure ratio appeared to have a 5% increase at the hub corner in the 10 mm trailing trimming, as well as at the top of trailing edge, indicating that the maximum pressure ratio appears in lower positions than other cases. It means that the modification effect pressured field distribution in the radial direction. The same behavior is also observed in the results of R7–S7 and R8–S8, as shown in Figure 9 and Figure 10, respectively. This phenomenon tends to lead to non-symmetry pressure forces impacting the turbine, which may cause force fluctuation that affects vibration. However, we still lack the characteristics to validate the vortex dynamics, since this is a steady-state study. Comparing the pressure ratio in Figure 8, Figure 9 and Figure 10 could confirm that increasing the number of stages increases the pressure ratio, as expected. This agrees with the gas turbine’s working principle. Fresh atmospheric air flows through the compressor causing higher pressure. The greater number of stages, the higher the pressure flowing outside.

_{l}and C

_{d}of the blade before modification. The results precisely confirmed that the maximum values appeared in the modification of 10 mm. The drag coefficients (C

_{d}) change for +2.30% in R6, +4.94% in R7, and −0.76% in R8. The lift coefficients (C

_{l}) change for, +5.73% in R6, +12.97% in R7, and −1.07% in R8, respectively. For all cases, the modification had less effect on the Cd in R6, R7 and R8 regions, but it had significant effect on the C

_{l}. Table 3 demonstrated C

_{l}and C

_{d}, as plotted in Figure 11, including the related parameters employed in the calculation. Results in the bracket are the percentage changes of C

_{l}and C

_{d}as compared to parameters of the blade before the modification. From Figure 8, Figure 9, Figure 10 and Figure 11, it can be interpreted that modifications of trailing edges with more than 5 mm significantly affect the aerodynamic parameters and change the forces of F

_{z}and F

_{y}. The changes of these forces affected C

_{l}and C

_{d}. It is possible to claim that these modifications can cause the change in power consumption, as well as the compressor performance.

## 4. Conclusions and Outlook

_{l}and C

_{d}, and 6.55–13.58% of blade loading ratio, respectively. These indications suggested that modifying less than 5 mm was suitable to avoid the change in aerodynamics behavior, but we still cannot confirm to validate the axial compressor performance as data from the power plant was confidential. We still lacked the characteristics to validate flow-induced vibrations according to pressure fluctuation on flow field since this was a steady state study. Unsteady flow simulation for a vibrating compressor rotor blade can be conducted by transient blade row flutter simulations using both time integration and harmonic balance transient methods, in combination with the Fourier Transformation pitch change model. The method allowed the capabilities of multi-stage passages for all vibration modes. The blades were forced to vibrate at one of the natural frequency modes, then, the unsteady force characteristics were calculated. The aerodynamic damping parameter was calculated, allowing for the evaluation of blade flutter in later aeroelastic analysis. McMullan et al. [28] suggested that the Large Eddy Simulation (LES) results may offer advantages over traditional RANS methods when off-design conditions are considered, as well as Duchaine et al. [29], who confirmed that, if we need to conjugate heat transfer around the turbine blade, the LES gives a comprehensive view of the main flow features that are responsible for heat transfer and the separation bubble. In the future, more studies should be carried out regarding an unsteady turbulence model, such as the LES, to validate and analyze the instant of the flow characteristic effects from blade modification.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

f_{r1} | production multiplier term |

k | turbulent kinetic energy (m^{2}/s^{2}) |

P_{k} | production of turbulent kinetic energy (kg/m·s^{3}) |

$\tilde{r}$ | argument in the determination of production multiplier |

r * | ratio of the strain rate and rotation rate tensor magnitudes |

S | strain rate magnitude (s^{−1}) |

S_{ij} | strain rate tensor (s^{−1}) |

u_{i} | fluctuation velocity component in the ith direction (m/s) |

F_{i} | force in the ith direction (m/s) |

U_{i} | mean velocity component in the ith direction (m/s) |

x_{i} | Cartesian coordinate in the ith direction (m) |

y | minimum distance to a no-slip wall (m) |

y^{+} | dimensionless wall distance |

ε | turbulence dissipation rate (m^{2}/s^{3}) |

δ_{ijk} | permutation tensor |

μ | molecular dynamics viscosity (kg/m·s) |

μ_{t} | eddy viscosity (kg/m·s) |

μ_{eff} | effective viscosity accounting for turbulence (kg/m·s) |

p_{a} | atmospheric pressure (Pa) |

p | pressure (Pa) |

Ω | rotation rate magnitude (s^{−1}) |

Ω_{ij} | rotation rate tensor (s^{−1}) |

Ω_{ij}^{rot} | rotation rate of the system (rad/s) |

ω | specific dissipation rate (s^{−1}) |

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**Figure 2.**Computational solid model of axial compressor blade: (

**a**) three-dimensional (3D) model from FARO Edge Arm, and (

**b**) a simplified model.

**Figure 8.**Pressure ratio between the middle plan of R6–S6 of the blade: (

**a**) before modification, after modifications of (

**b**) 1 mm, (

**c**) 5 mm, and (

**d**) 10 mm.

**Figure 9.**Pressure ratio between the middle plan of R7–S7 of the blade: (

**a**) before modification, after modifications of (

**b**) 1 mm, (

**c**) 5 mm, and (

**d**) 10 mm.

**Figure 10.**Pressure ratio between the middle plan of R8–S8 of the blade: (

**a**) before modification, after modifications of (

**b**) 1 mm, (

**c**) 5 mm, and (

**d**) 10 mm.

**Figure 11.**Lift (C

_{l}) and drag (C

_{d}) coefficients of the blade in different stage after modifications of (

**a**) 1 mm, (

**b**) 5 mm, and (

**c**) 10 mm.

Domain | Nodes | Elements | No. of Blades |
---|---|---|---|

S4 | 434,398 | 1,880,465 | 63 |

R5 | 644,654 | 2,689,547 | 58 |

S5 | 413,149 | 1,797,943 | 71 |

R6 | 558,787 | 2,320,838 | 66 |

S6 | 358,677 | 1,520,767 | 73 |

R7 | 468,637 | 1,932,659 | 75 |

S7 | 307,316 | 1,261,484 | 75 |

R8 | 403,899 | 1,663,696 | 77 |

S8 | 285,277 | 1,214,127 | 79 |

R9 | 325,074 | 1,366,770 | 82 |

S9 | 237,098 | 1,002,103 | 81 |

All Domain | 4,436,966 | 18,650,399 | - |

Numerical Parameters | Setting |
---|---|

Solver | Pressure-based |

Special discretization | High resolution scheme for advection term |

High resolution scheme for turbulence quantities | |

Convergence control | Max. Iteration 1000 |

Convergence criteria | 1.0 × 10^{−4} |

Time scale control | Auto Timescale |

Length scale option | Conservative |

Time scale factor | Auto Timescale |

**Table 3.**C

_{l}and C

_{d}for the blade before and after modifications and the parameters using in the calculation.

Stages | F_{z} (N) | F_{y} (N) | Area (m^{2}) | C_{l} | C_{d} |
---|---|---|---|---|---|

Before modification | |||||

R6 | 866.173 | 857.821 | 0.05542 | 0.1362 | 0.1349 |

R7 | 877.440 | 829.332 | 0.04150 | 0.1842 | 0.1741 |

R8 | 725.145 | 755.115 | 0.03564 | 0.1773 | 0.1846 |

After modification of 1 mm | |||||

R6 | 941.792 | 891.838 | 0.05537 | 0.1481 (+8.74%) | 0.1402 (+3.93%) |

R7 | 896.533 | 837.688 | 0.04152 | 0.1882 (+2.17%) | 0.1759 (+1.03%) |

R8 | 725.115 | 755.148 | 0.03566 | 0.1773 (0.00%) | 0.1846 (0.00%) |

After modification of 5 mm | |||||

R6 | 913.889 | 879.026 | 0.05534 | 0.1438 (+5.58%) | 0.1383 (+2.52%) |

R7 | 953.400 | 858.918 | 0.04149 | 0.2003 (+8.74%) | 0.1805 (−3.68%) |

R8 | 714.005 | 748.08 | 0.03563 | 0.1747 (−1.47%) | 0.1830 (−0.87%) |

After modification of 10 mm | |||||

R6 | 915.914 | 877.969 | 0.05530 | 0.1440 (+5.73%) | 0.1380 (+2.30%) |

R7 | 991.220 | 870.310 | 0.04141 | 0.2081 (+12.97%) | 0.1827 (+4.94%) |

R8 | 717.413 | 749.50 | 0.03555 | 0.1754 (−1.07%) | 0.1832 (−0.76%) |

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**MDPI and ACS Style**

Kaewbumrung, M.; Tangsopa, W.; Thongsri, J.
Investigation of the Trailing Edge Modification Effect on Compressor Blade Aerodynamics Using SST *k*-*ω* Turbulence Model. *Aerospace* **2019**, *6*, 48.
https://doi.org/10.3390/aerospace6040048

**AMA Style**

Kaewbumrung M, Tangsopa W, Thongsri J.
Investigation of the Trailing Edge Modification Effect on Compressor Blade Aerodynamics Using SST *k*-*ω* Turbulence Model. *Aerospace*. 2019; 6(4):48.
https://doi.org/10.3390/aerospace6040048

**Chicago/Turabian Style**

Kaewbumrung, Mongkol, Worapol Tangsopa, and Jatuporn Thongsri.
2019. "Investigation of the Trailing Edge Modification Effect on Compressor Blade Aerodynamics Using SST *k*-*ω* Turbulence Model" *Aerospace* 6, no. 4: 48.
https://doi.org/10.3390/aerospace6040048