# Investigation of Reynolds Number Effects on Aerodynamic Characteristics of a Transport Aircraft

^{*}

## Abstract

**:**

^{6}to 35 × 10

^{6}. In addition, an in-house developed CFD tool that has been validated by extensive experimental data was used to correct the wing deformation effect of the test model and achieve detailed flow structures. The results show that the Reynolds number has a significant impact on the boundary layer displacement thickness, surface pressure distribution, shock wave position, and overall aerodynamic force coefficients of the transport aircraft in the presence of shock wave and the induced boundary layer separation. The wind tunnel data combined with flow fields achieved from CFD show that the essence of the Reynolds number effect on the aerodynamic characteristics of transport aircraft is the difference of boundary layer development, shock wave/boundary layer interaction, and induced flow separation at different Reynolds numbers.

## 1. Introduction

^{6}to 35 × 10

^{6}. These results are beneficial to understanding the mechanism of Reynolds number effects of transport aircraft, which are briefly analyzed in this paper and also are conducive to developing Reynolds number effect extrapolation techniques, which is the priority of our next research.

## 2. Wind Tunnel and Experimental Setup

#### 2.1. Wind Tunnel

^{6}at cruise conditions, and up to 90 × 10

^{6}with vertically mounted semi-span models.

#### 2.2. Model Configuration and Test Campaigns

^{6}. The supercritical wing was tested without the roughness band when the Reynolds number is higher than 15 × 10

^{6}. During the whole wind tunnel test campaign, Ballotini grit strips with diameters from 0.1 to 0.125 mm were attached to the model fuselage 25 mm away from the fuselage nose for transition fixing. The grit size and distribution were determined according to the ETW standard wind tunnel procedure and criterion. Typically, grit strip height and grit density are 2 mm and 4%, respectively. The grit strip is stuck at the position of 7% local section chord along the wingspan and at the position of 10% model body length from the nose.

_{0}, z

_{0}and x

_{d}, z

_{d}represent the coordinate values of the feature points on the section before and after the wing deformation. The accuracy of ETW SPT system was 0.1° and 1 mm.

## 3. Computational Setup

#### 3.1. Computing Platform and Simulation Methods

#### 3.2. Grid Generation

^{+}≈ 1. To verify the mesh dependency, 4 sets of grids were generated with the same y

^{+}and O-H topology. Table 1 presents the parameters of the 4 sets of grids. By comparing the aerodynamic coefficient convergence curves of the 4 sets of grids, only the coarsest mesh (2 M) does not run to monotonic convergence and the densest mesh (20 M) has the best gird convergence. However, considering the results of medium grid (10 M) were close to the convergence solution and can meet the requirements for engineer application with much less calculation amount, the medium mesh (10 M) was used in the current study.

## 4. Results and Discussion

#### 4.1. Wing Deformation Effect

#### 4.2. Influence of Transition Strips on Reynolds Number Effects

^{6}). The envelope area of the pressure distribution curve with fixed transition is smaller than that of free boundary layer transition. When transition grit strips are attached, the boundary layer becomes thicker, which is similar to the boundary development state at a lower Reynolds number. The thicker boundary layer can induce boundary layer separation early and the shock wave moves to a new position ahead of the flow separation accordingly. As the test Reynolds number increases, the boundary transition will occur ahead of the grit strips and the thickness of the boundary layer will mainly be impacted by Reynolds numbers, which makes the difference of the pressure distribution between the fixed transition mode and free transition mode smaller. However, the difference can be neglected when the Reynolds number is above 15 × 10

^{6}. Based on the above analysis, it can be concluded that the transition strip can simulate the flight transition location but cannot simulate the boundary layer thickness and development on the supercritical wing surface. The shock wave location and pressure distribution with transition fixing at a low Reynolds number are significantly different to those in flight conditions. The transition band is mainly used in low Reynolds number wind tunnel tests to make the experimental data steadier. However, it can be removed when the test Reynolds number is high enough. For the supercritical wing employed in this research, the transition band is removed when the Reynolds number is higher than 15 × 10

^{6}.

#### 4.3. Reynolds Number Effect on the Pressure Distribution of Supercritical Wing

^{6}to the flight Reynolds number 35 × 10

^{6}. However, the variation of pressure coefficient distribution is not generally apparent.

^{6}to the flight Reynolds number 35 × 10

^{6}.

^{6}and 35 × 10

^{6}in the case of M = 0.76, α = 4°. As illustrated in Figure 13, the shock wave and separation bubble appear near the outer part of the upper supercritical wing surface. As the Reynolds number increases from 3.3 × 10

^{6}to 35 × 10

^{6}, the shock wave and the separation are pushed downward and the size of the separation bubble is reduced significantly, resulting in the stronger span flow behind the separation bubble. Figure 14 illustrates the numerical flow structures at Re = 3.3 × 10

^{6}and 35 × 10

^{6}in the case of M = 0.76, α = 6°. As the angle of attack increases from 4° to 6° at M = 0.76, the flow separation becomes much more severe and crossflow appears in the most domain of the upper wing. When the Reynolds number increases to 35 × 10

^{6}, the size of backflow behind the shock wave is significantly reduced and the crossflow near the wing root becomes much weaker.

#### 4.4. Reynolds Number Effect on Shock Wave Position

^{6}to 35 × 10

^{6}at a location of 72.36% of the wingspan. It seems that shock wave positions at the flight Reynolds number might be extrapolated from experiment results obtained in conventional wind tunnels with a relatively low Reynolds number range.

#### 4.5. Reynolds Number Effect on Trailing Edge Pressure Recovery

^{6}to 35 × 10

^{6}at a location of 43.72% of the wingspan. Additionally, an approximate linear growth of the trailing edge pressure with the logarithm of the Reynolds number can be found, which is consistent with the trend of shock wave position variation.

#### 4.6. Reynolds Number Effect on Boundary Layer Thickness

^{6}to 35 × 10

^{6}. To be specific, the boundary layer displacement thickness is reduced by 0.07 mm at the location of 20% of the local chord and 0.25 mm at the location of 80% of the local chord when the Reynolds number is increased from 3.3 × 10

^{6}to 35 × 10

^{6}. By contrast, the shock wave appears on the upper surface in the case of M = 0.76, α = 4°. In this condition, the displacement thickness of the boundary layer increases rapidly around the shock wave foot, but it still decreases with the growth in Reynolds number. In addition, the starting location of the boundary layer displacement thickness has a dramatic increase that corresponds to the shock wave position moving toward the trailing edge along the local chord as the Reynolds number increases, which is consistent with the trend obtained from the experimental surface pressure distribution discussed in Section 4.4.

#### 4.7. Pure Reynolds Number Effect on Aerodynamic Characteristics of the Transport Aircraft

^{6}. The curves of lift coefficient increment (ΔCL) curves versus the logarithm of the Reynolds number are given in Figure 19. It can be seen that the curve of the lift coefficient increment versus the logarithm of the Reynolds number is approximately linear, but the change in ΔCL starts to be less steep when the test Reynolds number is higher than 25 × 10

^{6}. The effect of the Reynolds number on the pitch moment coefficient (Cm) and the polar curve (Cm-CL) is shown in Figure 20. As the Reynolds number increases, extra nose-down pitch moment is produced and the critical angle of attack for the pitching instability is postponed, resulting in an extended linear segment of the pitch moment curve. On the other hand, the wind tunnel results show that the aerodynamic center keeps approximately unchanged when the Reynolds number varies from 3.3 × 10

^{6}to 35 × 10

^{6}and the Mach number is less than 0.79. However, while the aerodynamic center moves toward the trailing edge of the local airfoil to some extent when the Mach number is larger than 0.79. For example, the aerodynamic center position change is about 3.88% of the mean aerodynamic chord in the case of M = 0.82.

#### 4.8. Analysis of the Mechanism of Reynolds Number Effect on Flow over the Supercritical Wing

^{6}. When the Reynolds number increases to 35 × 10

^{6}, the flow separation is significantly weakened, the strong vortex is replaced by a separation bubble and the full separation flow behind the shock wave is replaced by the reattached flow behind the separation bubble.

## 5. Conclusions

^{6}. Combining the ETW wind tunnel test data and numerical results from CFD, it can be concluded that the Reynolds number has a weak impact on the aerodynamic characteristics of the supercritical wing when the shock wave does not appear, but dramatic Reynolds number effects can be found in the presence of the shock wave. Reynolds number effect on aerodynamic characteristics over the supercritical wing are essentially a kind of mutual interference among boundary layer, shock wave, and induced flow separation. The lift coefficient and the slope of lift curve become larger as the test Reynolds number increases; however, the impact of Reynolds number on the lift coefficient becomes smaller when the Reynolds number exceeds 25 × 10

^{6}. Moreover, as the Reynolds number increases, extra nose-down pitch moment is produced when the Reynolds number varies from 3.3 × 10

^{6}to 35 × 10

^{6}and the Mach number is less than 0.79. The results obtained in this study can be used to develop Reynolds number effect correction and extrapolation methods for conventional wind tunnel test data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

C_{a} | axial force coefficient |

C_{L} | lift force coefficient |

C_{m} | pitching moment coefficient taking the center point of the fuselage at the location of ¼ of the mean aerodynamic chord as the reference point |

C_{n} | normal force coefficient |

Cp_{i} | surface pressure coefficient of the orifice tap i |

Cp_{te} | tailing edge pressure coefficient |

M | Mach number |

p_{i} | surface pressure of the orifice tap i |

p_{∞:} | static pressure of the free stream |

Q, q_{∞} | dynamic pressure of the free stream |

Re | Reynolds number |

U | flow velocity |

X_{sh} | shock wave location (non-dimensional) along the span |

α | angles of attack |

η | relative location of local airfoil along the span |

δ | boundary thickness |

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**Figure 4.**Longitudinal aerodynamic characteristics under different dynamic pressures (M = 0.76, Re = 15 × 10

^{6}): (

**a**) lift coefficient; (

**b**) pitching moment coefficient.

**Figure 5.**Increments of the lift and pitching moment coefficients caused by dynamic pressure variation (M = 0.76, Re = 15 × 10

^{6}): (

**a**) increments of the lift coefficient; (

**b**) increments of the pitching moment coefficient.

**Figure 6.**Pressure distributions under different dynamic pressures (α = 0°, M = 0.76, Re = 15 × 10

^{6}): (

**a**) η = 54%; (

**b**) η = 96.61%.

**Figure 7.**Pressure distributions under different dynamic pressures (α = 2°, M = 0.76, Re = 15 × 10

^{6}): (

**a**) η = 54%; (

**b**) η = 96.61%.

**Figure 8.**Numerical flow patterns near the wing tip with and without deformation (M = 0.76, Re = 15 × 10

^{6}, Q = 94 kPa, α = 4°): (

**a**) rigid model; (

**b**) deformed model.

**Figure 9.**Comparison of surface pressure distributions from CFD and ETW wind tunnel tests (M = 0.76, Re = 15 × 10

^{6}, α = 2°): (

**a**) η = 72.3%; (

**b**) η = 86.1%.

**Figure 10.**Pressure distributions with and without transition band under different Reynolds numbers (M = 0.76, α = 4°, η = 72.36%): (

**a**) Re = 3.3 × 10

^{6}; (

**b**) Re = 6.6 × 10

^{6}; (

**c**) Re = 15 × 10

^{6}.

**Figure 11.**Pressure distributions under different Reynolds numbers (M = 0.76, α = 0°): (

**a**) η = 54%; (

**b**) η = 72.36%.

**Figure 12.**Pressure distributions at different angles of attack and Reynolds numbers (M = 0.76, η = 54%): (

**a**) α = 2°; (

**b**) α = 4°.

**Figure 13.**Numerical flow structures at different Reynolds numbers (M = 0.76, α = 4°): (

**a**) Re = 3.3 × 10

^{6}; (

**b**) Re = 35 × 10

^{6}.

**Figure 14.**Numerical flow structures at different Reynolds numbers (M = 0.76, α = 6°): (

**a**) Re = 3.3 × 10

^{6}; (

**b**) Re = 35 × 10

^{6}.

**Figure 15.**Reynolds number effect on shock wave position: (

**a**) M = 0.74, α = 3°, η = 43.72%; (

**b**) M = 0.74, α = 4°, η = 54%; (

**c**) M = 0.76, α = 3°, η = 72.36%.

**Figure 16.**Reynolds number effect on trailing edge pressure recovery (M = 0.74): (

**a**) α = 3°, η = 43.72%; (

**b**) α = 4°, η = 54%.

**Figure 17.**Numerical boundary layer displacement thicknesses of the upper wing surface at different Reynolds numbers (M = 0.76, η = 0.35): (

**a**) α = 0°; (

**b**) α = 4°.

**Figure 18.**Typical lift coefficient curves at different Reynolds numbers: (

**a**) M = 0.76; (

**b**) M = 0.79.

**Figure 19.**Increments of lift coefficient versus Reynolds number (M = 0.76): (

**a**) α = 2°; (

**b**) α = 4°.

**Figure 20.**Pitch moment coefficient and polar curves at different Reynolds number: (

**a**) polar curves; (

**b**) pitch moment coefficient.

**Figure 21.**Numerical flow structures at different Reynolds numbers (M = 0.76, η = 63%, α = 6°): (

**a**) Re = 3.3 × 10

^{6}; (

**b**) Re = 35 × 10

^{6}.

Grid Quality | Flow Direction | Wingspan Direction | Normal Direction | Leading Edge | Grid Quantity (Million, M) |
---|---|---|---|---|---|

Coarsest | 141 | 73 | 69 | 9 | 2 |

Coarser | 191 | 99 | 73 | 13 | 4 |

Medium | 297 | 129 | 105 | 17 | 10 |

Densest | 359 | 175 | 113 | 21 | 20 |

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**MDPI and ACS Style**

Wang, Y.; Liu, D.; Xu, X.; Li, G.
Investigation of Reynolds Number Effects on Aerodynamic Characteristics of a Transport Aircraft. *Aerospace* **2021**, *8*, 177.
https://doi.org/10.3390/aerospace8070177

**AMA Style**

Wang Y, Liu D, Xu X, Li G.
Investigation of Reynolds Number Effects on Aerodynamic Characteristics of a Transport Aircraft. *Aerospace*. 2021; 8(7):177.
https://doi.org/10.3390/aerospace8070177

**Chicago/Turabian Style**

Wang, Yuanjing, Dawei Liu, Xin Xu, and Guoshuai Li.
2021. "Investigation of Reynolds Number Effects on Aerodynamic Characteristics of a Transport Aircraft" *Aerospace* 8, no. 7: 177.
https://doi.org/10.3390/aerospace8070177