#
Modeling Aerodynamics, Including Dynamic Stall, for Comprehensive Analysis of Helicopter Rotors^{ †}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Aerodynamics

#### 2.1. Modeling Dynamic Stall Phenomenon

#### 2.2. 3D Effects

#### 2.3. Aerodynamic Correction for the High-Speed Flight

- Correction 1: ${c}_{ind}^{0}=0.010,\phantom{\rule{0.277778em}{0ex}}{\varphi}_{0}=38.0\times \pi /180.0$
- Correction 2: ${c}_{ind}^{0}=0.015,\phantom{\rule{0.277778em}{0ex}}{\varphi}_{0}=55.0\times \pi /180.0$
- Correction 3: ${c}_{ind}^{0}=0.010,\phantom{\rule{0.277778em}{0ex}}{\varphi}_{0}=38.0\times \pi /180.0$

#### 2.4. Wind Tunnel Effects

## 3. Implementation of the Stall Model

## 4. Application to the Rotor 7A

#### 4.1. High-Speed Test Point 312

#### 4.2. High-Thrust Test Point 293

#### 4.3. High-Thrust Test Point 596

**Remark**

**1.**

## 5. Summary and Conclusions

- The action of modeling aerodynamics has been made at the level of the global flow, outside of the stall model, for the accounting of the extreme aerodynamic condition of the high-speed flight. The proposed correction can be checked independently by high-fidelity CFD techniques. The methodology for the extraction of the induced velocity and the angle of attack has been established in wind turbines.
- The Hopf bifurcation stall model has been revised to emphasize the boundary-layer effects accompanying the formation of the dynamic stall vortex. This phenomenon is responsible for about half of the important variation of the pitching moment at stall onset. This stall model is the only one to treat the vortex-shedding phenomenon as a non-linear effect governed by ordinary differential equations. The number of parameters of the model is limited to those contained in Equations (3)–(6), for characterizing the vortex-shedding phenomena and the boundary-layer effects. The determination of the parameters of the stall model requires some expense in CFD simulation or in experimentation, but once their values are established, they could be used for a large variety of test configurations.
- Comprehensive analysis uses look-up tables for the values of aerodynamic coefficients and, thus, is a time-efficient means for the predictions of the aeromechanical behavior of the rotor. These tables are up to now 2D values of aerodynamic coefficients. If appropriate 3D corrections are made on these tables, based on experiments or CFD computation, the predictions of CA will be improved. Such a procedure was made for rotation effects in wind turbines. The main issue is therefore to replace the 2D polars by appropriate 3D polars.
- The procedure for the implementation of the “Hopf bifurcation model” to the comprehensive analysis code of ONERA has been presented. The numerical code established is robust and allows convergence in the deep-stall test point of the Rotor 7A.
- The application of the comprehensive analysis code equipped with the stall model has been illustrated on three test points of the Rotor 7A carried out in the Wind Tunnel S1 of ONERA, involving high-speed and high-thrust test points. For wind tunnel experiments, it was necessary to make corrections for the aerodynamic tunnel environment, rotor stand and wind tunnel walls, as well as corrections for Reynolds number effects. For all test points, the predictions of air loads are in reasonable agreement with experiments and require about 30 min on a PC for stalled test points and less for the high-speed test point. Such test cases request weeks of computation for coupled analysis involving CFD techniques, on multiprocessor machines. Improvement of air load predictions leads to improved predictions of structural loads.

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${C}_{i}\phantom{\rule{0.277778em}{0ex}}(i=1,2,3)$ | aerodynamic coefficients (i = 1: lift; i = 2: pitching moment, i = 3: drag) |

${C}_{1,i}\phantom{\rule{0.277778em}{0ex}}(i=1,2,3)$ | in Regime 1 of attached flow |

${C}_{2,i}\phantom{\rule{0.277778em}{0ex}}(i=1,2,3)$ | in Regime 2 of separated flow |

M | Mach number |

${R}_{e}$ | Reynolds number |

$\alpha $ | aerodynamic incidence angle or angle of attack |

${\alpha}_{cr}$ | critical stall angle |

${r}_{n}$ | normalized radial distance |

$rev$ | revolution |

R | blade radius |

${\mathrm{\Lambda}}_{deg}$ | sweep angle, expressed in degrees |

Contr. Sett. | control settings |

BM | bending moment |

w/o | without |

w | with |

## Appendix A. 3D Pitching Moment Coefficient

## Appendix B. Values of the Parameters Used for the Test Points

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**Figure 2.**Various corrections of the induced velocity (Ind. Vel.) and the corrected values for a typical high-speed flight at radius $90\%R$: Correction 1: ${c}_{ind}^{0}=0.010,\phantom{\rule{3.33333pt}{0ex}}{\varphi}_{0}=38.0\times \pi /180.0$, Correction 2: ${c}_{ind}^{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.015,\phantom{\rule{3.33333pt}{0ex}}{\varphi}_{0}=\phantom{\rule{3.33333pt}{0ex}}55.0\times \pi /180.0$, Correction 3: ${c}_{ind}^{0}=0.010,\phantom{\rule{3.33333pt}{0ex}}{\varphi}_{0}=38.0\times \pi /180.0$.

**Figure 3.**Correction of the induced velocity due to the presence of the rotor stand and wind tunnel walls.

**Figure 4.**Variation of the critical stall angle of various NACA airfoils with respect to the Reynolds numbers.

**Figure 9.**Structural loads for the high-speed test point of the Rotor 7A: (

**a**) flap bending moment; (

**b**) chord bending moment; (

**c**) torsion moment.

**Figure 13.**Structural loads for the high-thrust test point 293 of the Rotor 7A: (

**a**) flap bending moment; (

**b**) chord bending moment; (

**c**) torsion moment.

**Figure 14.**Air loads for the high-thrust test point 596 of the Rotor 7A under conditions of fixed and trimmed control settings (Contr. Sett.): (

**a**) normal force; (

**b**) pitching moment; (

**c**) vibratory loads.

${\mathit{\theta}}_{\mathbf{0}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{c}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{q}}$ | P (kW) | |
---|---|---|---|---|---|

Experiments | 10.41 | 3.43 | $-3.70$ | $-13.75$ | 88.0 |

QS aero. 1 | 11.28 | 0.99 | $-1.33$ | $-13.48$ | 98.17 |

QS aero. 2 | 9.87 | 1.58 | $-3.26$ | $-14.38$ | 92.51 |

Stall model | 10.97 | 1.04 | $-1.38$ | $-13.33$ | 84.70 |

**Table 2.**High-thrust Flight 293: Control angles and power given by experiments and computations without Stall Model ( w/o St. Mod.) and with Stall Model ( w Stall Model).

${\mathit{\theta}}_{\mathbf{0}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{c}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{q}}$ | P (kW) | |
---|---|---|---|---|---|

Experiments | 8.40 | 3.16 | $-3.51$ | $-6.70$ | 74.80 |

w/o St. Mod. | 10.16 | 1.90 | $-1.98$ | $-7.90$ | 78.13 |

w Stall Model | 6.31 | 1.32 | $-1.47$ | $-7.36$ | 84.09 |

**Table 3.**High-thrust Flight 596: control angles and power under fixed and trimmed control settings (C.T.).

${\mathit{\theta}}_{\mathbf{0}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{c}}$ | ${\mathit{\theta}}_{\mathbf{1}\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{q}}$ | P (kW) | |
---|---|---|---|---|---|

Experiments | 13.47 | 5.25 | $-7.06$ | $-5.40$ | 78.30 |

Fixed C.T. | 13.47 | 5.25 | $-7.06$ | $-5.40$ | 114.44 |

Trimmed C.T. | 14.43 | 4.02 | $-4.87$ | $-8.01$ | 110.54 |

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**MDPI and ACS Style**

Truong, K.V.
Modeling Aerodynamics, Including Dynamic Stall, for Comprehensive Analysis of Helicopter Rotors. *Aerospace* **2017**, *4*, 21.
https://doi.org/10.3390/aerospace4020021

**AMA Style**

Truong KV.
Modeling Aerodynamics, Including Dynamic Stall, for Comprehensive Analysis of Helicopter Rotors. *Aerospace*. 2017; 4(2):21.
https://doi.org/10.3390/aerospace4020021

**Chicago/Turabian Style**

Truong, Khiem Van.
2017. "Modeling Aerodynamics, Including Dynamic Stall, for Comprehensive Analysis of Helicopter Rotors" *Aerospace* 4, no. 2: 21.
https://doi.org/10.3390/aerospace4020021