# Transition Modeling for Low Pressure Turbines Using Computational Fluid Dynamics Driven Machine Learning

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## Abstract

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## 1. Introduction

## 2. Transition Modeling

#### 2.1. Laminar Kinetic Energy Model

**$k/(\nu \omega )$**. The transport equation of the specific dissipation rate is given as

#### 2.2. Proposed Model Formulation

#### 2.2.1. Scientific Ideology

#### 2.2.2. Non-Dimensional Pi Groups

## 3. Machine Learning Framework

#### 3.1. Advantages

#### 3.2. Methodology

## 4. Numerical Setup and Configuration

## 5. Results and Discussions

#### 5.1. Single-Objective Optimization

#### 5.1.1. Baseline RANS Calculation

#### 5.1.2. Cost Function

#### 5.1.3. Analysis of the SO-Model

#### 5.2. Multi-Objective Optimization

#### 5.2.1. Cost Functions

#### 5.2.2. Pareto Analysis

#### 5.2.3. Analysis of Solutions

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational fluid dynamics |

DNS | Direct numerical simulation |

EARSM | Explicit algebraic Reynolds stress model |

GEP | Gene expression programming |

LES | Large eddy simulation |

LKE | Laminar kinetic energy |

LPT | Low pressure turbine |

MO | Multi-objective |

RANS | Reynolds averaged Navier–Stokes |

SO | Single-objective |

TE | Trailing edge |

TKE | Turbulent kinetic energy |

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**Figure 1.**‘CFD-driven’ transition model development with the multi-objective and multi-expression approach.

**Figure 3.**(

**a**) Suction-side wall-shear stress from the baseline LKE model and DNS. (

**b**) Zoomed view of the wall-shear stress in the separated flow region, near the blade trailing edge.

**Figure 4.**(

**a**) Suction-side wall-shear stress and (

**b**) pressure and suction-side pressure coefficient for the SO-Model, baseline case and DNS.

**Figure 6.**(

**a**) Pressure coefficient with the various multi-objective optimized models. (

**b**) Zoomed view of the suction-side pressure coefficient in the separated flow region, near the blade trailing edge.

**Figure 7.**(

**a**) Suction-side wall-shear stress with the various multi-objective optimized models. (

**b**) Zoomed view of the suction-side wall-shear stress in the separated flow region, near the blade TE.

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**MDPI and ACS Style**

Akolekar, H.D.; Waschkowski, F.; Zhao, Y.; Pacciani, R.; Sandberg, R.D.
Transition Modeling for Low Pressure Turbines Using Computational Fluid Dynamics Driven Machine Learning. *Energies* **2021**, *14*, 4680.
https://doi.org/10.3390/en14154680

**AMA Style**

Akolekar HD, Waschkowski F, Zhao Y, Pacciani R, Sandberg RD.
Transition Modeling for Low Pressure Turbines Using Computational Fluid Dynamics Driven Machine Learning. *Energies*. 2021; 14(15):4680.
https://doi.org/10.3390/en14154680

**Chicago/Turabian Style**

Akolekar, Harshal D., Fabian Waschkowski, Yaomin Zhao, Roberto Pacciani, and Richard D. Sandberg.
2021. "Transition Modeling for Low Pressure Turbines Using Computational Fluid Dynamics Driven Machine Learning" *Energies* 14, no. 15: 4680.
https://doi.org/10.3390/en14154680