# Assessment of Machine-Learned Turbulence Models Trained for Improved Wake-Mixing in Low-Pressure Turbine Flows

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## Abstract

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## 1. Introduction

_{2}emissions from air transport in half by 2050, compared to the base year of 2005, in order to keep global warming below 1.5 ${}^{\xb0}$C as recommended by the Intergovernmental Panel on Climate Change (IPCC) [1]. In 2018, commercial airliners consumed approximately 360 billion liters of aviation fuel globally (source: www.statista.com (accessed on July 2019)). The reduction of specific fuel consumption directly impacts the carbon footprint of aeroengines and thus remains one of the primary objectives for the aviation industry. In addition to that, aircraft noise is the most significant cause of adverse community reaction related to the operation and expansion of airports. Noise reduction at the source has proven to be effective, and it remains a priority for aircraft engine manufacturers.

## 2. Test Case and Boundary Conditions

## 3. Domain Discretization

## 4. Computational Framework

## 5. Discussion of Results

#### 5.1. Blade Boundary-Layers and Effect of Transition Modelling

#### 5.2. Wake Diffusion and Effect of the EARSMs

#### 5.3. Secondary Flows

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DNS | Direct Numerical Simulation |

EARSM | Explicit Algebraic Reynolds Stress Model |

FSTI | Free Stream Turbulence Intensity |

GEP | Gene Expression Programming |

LES | Large Eddy Simulation |

LKE | Laminar Kinetic Energy |

LPT | Low Pressure Turbine |

TVD | Total Variation Diminishing |

## Nomenclature

${a}_{ij}$ | Anisotropy tensor |

$AR$ | Blade aspect ratio, $AR=h/{C}_{x}$ |

C | blade chord |

${C}_{p}$ | Pressure coefficient |

${C}_{pt}$ | Total Pressure Loss Coefficient |

${C}_{f}$ | Skin Friction Coefficient |

$DR$ | Diffusion rate, $DR=\frac{{v}_{peak}-{v}_{2}}{{v}_{2}}\frac{{s}_{tot}}{{s}_{tot}-{s}_{peak}}$ |

g | Tangential gap |

h | Blade height |

I | Invariant |

k | Turbulent Kinetic Energy |

p | Pressure |

$R{e}_{2,is}$ | Isentropic exit Reynolds number, $R{e}_{2,is}=\frac{\rho {v}_{2,is}{C}_{x,ms}}{\mu}$ |

s | Curvilinear abscissa along airfoil surface |

${S}_{ij}$ | Strain rate tensor |

v | Flow velocity |

${V}_{ij}$ | Tensor basis element |

x | Axial, axial coordinate |

y | Pitchwise coordinate |

z | Spanwise coordinate |

$Zw$ | Zweifel coefficient, $Zw=2\left(\frac{g}{{C}_{x}}\right){\mathrm{cos}}^{2}{\alpha}_{1}\left(\mathrm{tan}{\alpha}_{1}+\mathrm{tan}{\alpha}_{2}\right)$ |

${\delta}_{ij}$ | Kronecker tensor |

$\mu $ | Dinamic viscosity |

$\rho $ | Fluid density |

$\tau $ | Turbulence time scale |

${\tau}_{ij}$ | Reynolds stress tensor |

$\omega $ | Specific dissipation rate |

${\mathrm{\Omega}}_{ij}$ | Rotation tensor |

## Subscripts

0 | Total quantity |

1 | Inlet |

2 | Outlet |

$is$ | Isentropic |

$ms$ | Midspan |

$peak$ | Relative to the suction peak velocity |

$REF$ | Reference |

$tot$ | Total |

w | Wall |

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**Figure 3.**Blade surface distributions of (

**a**) pressure-coefficient (

**b**) skin friction ($R{e}_{2,is}=3.0\times {10}^{5}$).

**Figure 4.**Blade surface distributions (

**a**) pressure-coefficient (

**b**) skin friction ($R{e}_{2,is}=0.7\times {10}^{5}$).

**Figure 5.**Wake loss profiles at $R{e}_{2,is}=0.7\times {10}^{5}$, (

**a**) obtained with different values of ${x}_{mask}$ (GEP-EARSM, $x/{C}_{x,ms}=1.46$), (

**b**) in different locations downstream of blade TE (GEP-EARSM H).

**Figure 6.**Wake loss profiles w/ and w/o GEP-EARSM ($R{e}_{2,is}=0.7\times {10}^{5}$, $x/{C}_{x,ms}=1.46$).

**Figure 7.**Nondimensional turbulent kinetic energy at midspan (

**a**) without GEP-EARSM (

**b**) with GEP-EARSM ($R{e}_{2,is}=0.7\times {10}^{5}$).

**Figure 10.**Spanwise distribution of (

**a**) total pressure loss coefficient (

**b**) blade-to-blade flow angle ($R{e}_{2,is}=1.5\times {10}^{5}$, $x/{C}_{x,ms}=1.46$).

**Figure 11.**Contours of total pressure loss coefficient (

**a**) Experimental (

**b**) without GEP-EARSM (

**c**) with GEP-EARSM.

Zw | DR | AR |
---|---|---|

1.1 | 0.5 | 3.0 |

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**MDPI and ACS Style**

Pacciani, R.; Marconcini, M.; Bertini, F.; Rosa Taddei, S.; Spano, E.; Zhao, Y.; Akolekar, H.D.; Sandberg, R.D.; Arnone, A.
Assessment of Machine-Learned Turbulence Models Trained for Improved Wake-Mixing in Low-Pressure Turbine Flows. *Energies* **2021**, *14*, 8327.
https://doi.org/10.3390/en14248327

**AMA Style**

Pacciani R, Marconcini M, Bertini F, Rosa Taddei S, Spano E, Zhao Y, Akolekar HD, Sandberg RD, Arnone A.
Assessment of Machine-Learned Turbulence Models Trained for Improved Wake-Mixing in Low-Pressure Turbine Flows. *Energies*. 2021; 14(24):8327.
https://doi.org/10.3390/en14248327

**Chicago/Turabian Style**

Pacciani, Roberto, Michele Marconcini, Francesco Bertini, Simone Rosa Taddei, Ennio Spano, Yaomin Zhao, Harshal D. Akolekar, Richard D. Sandberg, and Andrea Arnone.
2021. "Assessment of Machine-Learned Turbulence Models Trained for Improved Wake-Mixing in Low-Pressure Turbine Flows" *Energies* 14, no. 24: 8327.
https://doi.org/10.3390/en14248327