# Hybrid RANS/LES Turbulence Model Applied to a Transitional Unsteady Boundary Layer on Wind Turbine Airfoil

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation and CFD-based Experimental Design

## 3. Turbulence Model and Simulation Setups

#### 3.1. Turbulence Model

#### 3.1.1. Delayed-Detached-Eddy Simulation

_{DDES}is directly multiplied with the dissipation term in TKE equation, i.e., for ${F}_{DDES}=1$, the original RANS formulation is recovered; for ${F}_{DDES}>1$, the turbulence model switches to LES mode.

#### 3.1.2. Transition Model Formulation

#### 3.2. Simulation Setups

## 4. Circular Cylinder Flow

#### 4.1. Instantaneous Flow Field

#### 4.2. Mean Flow Field and Flow Statistics

## 5. Unsteady Boundary Layer on Airfoil

#### 5.1. Instantaneous Flow Field

#### 5.2. Mean Flow Field and Flow Statistics

#### 5.2.1. Mean Pressure and Velocity Field

#### 5.2.2. Reynolds Stress

#### 5.2.3. Mean Turbulence Kinetic Energy

#### 5.2.4. Phase-Averaged Boundary Layer Profiles

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

D | =circular cylinder diameter |

c | =airfoil chord length |

${C}_{d}$ | =airfoil section drag coefficient |

${C}_{D}$ | =drag coefficient |

CFD | =Computational Fluid Dynamics |

${C}_{l}$ | =airfoil section lift coefficient |

${C}_{L}$ | =lift coefficient |

${C}_{p}$ | =pressure coefficient |

EFD | =Experimental Fluids Dynamics |

F_{DDES} | =DDES blending function |

k | =airfoil reduced frequency |

L | =distance from cylinder center to airfoil leading edge |

LM | =Langtry-Menter (turbulence model) |

PIV | =particle image velocimetry |

$R{e}_{c}$ | =airfoil Reynolds number |

$R{e}_{D}$ | =cylinder Reynolds number |

RMS | =root-mean square |

$St$ | =Strouhal number |

Ti | =turbulence intensity |

${U}_{\infty}$ | =freestream velocity |

${U}_{\tau}$ | =friction velocity |

$\left|V\right|$ | =velocity vector magnitude |

${y}^{+}$ | =dimensionless wall distance |

$\gamma $ | =LM turbulence intermittency |

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**Figure 2.**Overview of experimental and numerical procedure used in the research. Experimental efforts in blue, numerical efforts in red and joint contributions in black.

**Figure 5.**Particle image velocimetry (PIV) near-surface planes on airfoil suction side. Colors are visualizations of mean velocity magnitude.

**Figure 7.**Lift (blue) and drag (red) coefficients of circular cylinder, $\mathrm{T}=t/(D/{U}_{\infty})$.

**Figure 8.**Langtry-Menter (LM) turbulence intermittency field ($\gamma $) on circular cylinder over one shedding cycle.

**Figure 9.**Normalized instantaneous spanwise vorticity ${\tilde{\mathsf{\omega}}}_{\mathrm{z}}={\mathsf{\omega}}_{\mathrm{z}}/\left({\mathrm{U}}_{\infty}\mathrm{D}\right)\text{}$(

**a**) Simulation, (

**b**) Experiment.

**Figure 10.**Iso-surface of Q criterion of the circular cylinder flow (flow field is mirrored in spanwise direction).

**Figure 12.**Mean velocity components in cylinder wake. (

**a**) contours of streamwise velocity; (

**b**) contours of transverse velocity; (

**c**) EFD/CFD line plot comparisons at $x/D$ = 1.

**Figure 16.**Instantaneous snapshot of blending function ${F}_{DDES}$ around airfoil in cylinder wake.

**Figure 17.**Instantaneous snapshot of normalized eddy viscosity ${\nu}_{t}/\nu $ around airfoil in cylinder wake.

**Figure 18.**Instantaneous snapshot of normalized spanwise vorticity magnitude ${\tilde{\mathsf{\omega}}}_{\mathrm{z}}={\mathsf{\omega}}_{\mathrm{z}}/\left({\mathrm{U}}_{\infty}\mathrm{D}\right)$ from cylinder to airfoil.

**Figure 21.**Instantaneous velocity profiles of ${U}_{x}$ parallel to airfoil surface. (

**a**) $\text{}x/c=0.3$; (

**b**) $x/c=0.5$; (

**c**) $x/c=0.8$.

**Figure 23.**Normalized mean velocity magnitude in clean flow, (

**a**) EFD, (

**b**) CFD, and cylinder wake, (

**c**) EFD, (

**d**) CFD. Line plot comparisons for both cases in (

**e**).

**Figure 25.**Experimental (dots) and numerical (dashed lines) profiles of the streamwise and transverse Reynolds stress components at $x/c=0.27$. Note HG is theory from Hunt and Graham [34].

Time Scale (s) | Length Scale (m) | Blade Re | Reduced Frequency | Modeling Technique |
---|---|---|---|---|

$\sim \mathcal{O}\left({10}^{-3}-{10}^{1}\right)$ | $\sim \mathcal{O}\left({10}^{-6}-{10}^{2}\right)$ | $\sim \mathcal{O}\left({10}^{7}\right)$ | $\sim \mathcal{O}\left({10}^{-2}-{10}^{0}\right)$ | CFD, actuator methods, BEM |

Freestream Conditions | Test Section Dimensions | Circular Cylinder | NACA63215b Airfoil |
---|---|---|---|

${U}_{\infty}=26\mathrm{m}/\mathrm{s}$ $Ti=1\%$ | Streamwise (x): $121.9\text{}\mathrm{cm}$ | Diameter $\text{}D=3.81\text{}\mathrm{cm}$ | Chord $\text{}c=10.16\mathrm{cm}$ |

Transverse (y): $71.1\mathrm{cm}$ | $R{e}_{D}=6.4\times {10}^{4}$ | $R{e}_{c}=1.7\times {10}^{5}$ | |

Spanwise(z): $71.1\mathrm{cm}$ | Spacing $L=10.7D$ | Reduced frequency $k=1.5$ |

Grid | Average ${\mathit{y}}^{+}$ on Surface | Cells around Cylinder | Total Cell Count | $\overline{{\mathit{C}}_{\mathit{D}}}$ | ${\mathit{C}}_{\mathit{L}}^{\prime}$ | $\mathit{S}\mathit{t}$ |
---|---|---|---|---|---|---|

Coarse | $\approx 2\left(y=2\times {10}^{-5}\mathrm{m}\right)$ | $120$ | 328,680 | 0.64 | 0.16 | 0.256 |

Medium | $\approx 1\left(y=1\times {10}^{-5}\mathrm{m}\right)$ | 240 | 2,768,566 | 1.31 | 0.67 | 0.193 |

Fine | $\approx 0.5\left(y=5\times {10}^{-6}\mathrm{m}\right)$ | 480 | 21,790,080 | 1.35 | 0.71 | 0.190 |

${\mathit{C}}_{\mathit{l}}$ Mean | ${\mathit{C}}_{\mathit{l}}$ RMS | ${\mathit{C}}_{\mathit{d}}$ Mean | ${\mathit{C}}_{\mathit{d}}$ RMS | |
---|---|---|---|---|

Airfoil in Clean flow | 0.16 | $0.16$ | 0.016 | 0.016 |

XFOIL | 0.17 | / | 0.015 | / |

Airfoil in Cylinder wake | 0.10 | 0.43 | −0.026 | 0.042 |

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**MDPI and ACS Style**

Zhang, D.; Cadel, D.R.; Paterson, E.G.; Lowe, K.T.
Hybrid RANS/LES Turbulence Model Applied to a Transitional Unsteady Boundary Layer on Wind Turbine Airfoil. *Fluids* **2019**, *4*, 128.
https://doi.org/10.3390/fluids4030128

**AMA Style**

Zhang D, Cadel DR, Paterson EG, Lowe KT.
Hybrid RANS/LES Turbulence Model Applied to a Transitional Unsteady Boundary Layer on Wind Turbine Airfoil. *Fluids*. 2019; 4(3):128.
https://doi.org/10.3390/fluids4030128

**Chicago/Turabian Style**

Zhang, Di, Daniel R. Cadel, Eric G. Paterson, and K. Todd Lowe.
2019. "Hybrid RANS/LES Turbulence Model Applied to a Transitional Unsteady Boundary Layer on Wind Turbine Airfoil" *Fluids* 4, no. 3: 128.
https://doi.org/10.3390/fluids4030128