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The static aeroelastic behaviours of a flat-plate forward-swept wing model in the vicinity of static divergence are investigated by numerical simulations and wind tunnel tests. A medium fidelity model based on the vortex lattice method (VLM) and nonlinear structural analysis is proposed to calculate the displacements of the wing structure with large deformation. Follower forces effect and geometric nonlinearity are considered to calculate the deformation of the wing by finite element method (FEM). In the wind tunnel tests, the divergence dynamic pressure is predicted by the Southwell method, and the static aeroelastic displacement is measured by a photogrammetric method. The results obtained by the medium fidelity model calculations show reasonable agreement with wind tunnel test results. A high fidelity model based on coupled computational fluid dynamics (CFD) and computational structural dynamics (CSD) predicts better results of the wing tip displacement when the freestream dynamic pressure is approaching the divergence dynamic pressure.

In static aeroelasticity analysis, the interaction between aerodynamics and structural deflections determines the wing bending and twist at every flight condition. The static aeroelastic deformation in the steady flight condition is of great importance because it governs the aerodynamic performance and flight control characteristics [

It is well known that the forward-swept wing and straight wing with high aspect ratio are susceptible to large deformation. When the aerodynamic load is heavy, the deformation of the wing structure will become very large. According to the aerodynamic and structural models used in the nonlinear aeroelastic analysis, numerical models can be categorized into three levels, namely, low, medium, and high fidelity models [

Though the low fidelity model provides essential insight and knowledge about the aeroelastic characteristics, it has limitation in evaluating the 3D effect of real flow. The strip theory without tip effects correction may overestimate the outboard wing lift and results in greater vertical wing displacement and bending rotation [

The high fidelity simulation model is based on CFD/CSD coupled method and has been developed rapidly and applied widely in the study of computational aeroelasticity in last decades. Even at low subsonic speeds, much attention should be paid to the effects of structural nonlinearity on the aeroelastic behaviour. The geometric nonlinearity changes the aerodynamic loads, thus leading to considerable errors in the static aeroelastic predictions [

The static divergence of wing structure must be predicted accurately by wind tunnel test, because the divergence speed directly reflects the general stiffness of the wing structure and must be considered in the certification process (CS-25 and FAR-25) [

Before the occurrence of divergence, we are usually concerned about the maximal wing tip displacements. Photogrammetry is a nonintrusive measurement technique commonly used to determine the geometrical information of object by analyzing images recorded by camera. This technique is useful when the object to be measured is inaccessible and noncontact measurement is required, and it is especially suitable for static aeroelastic wind tunnel test. A detailed description of the related theory can be found in [

The study aims to obtain an improved understanding of the static aeroelastic behaviours of a wing model. According to the performance of the available wind tunnel, a forward-swept layout model is used. This layout has a decreased divergence speed compared to the unswept wing due to increased effective AOA. When the free stream dynamic pressure is in the vicinity of the static divergence boundary, the forward-swept wing model will encounter large deformation. The static aeroelastic deformation is calculated by medium and high fidelity models. Wind tunnel tests are performed to validate the simulation results. Finally, we make a comparison between the results of medium and high fidelity models in the aspects of accuracy and efficiency.

An aluminum flat-plate wing model is used in the current study, as shown in Figure

Forward-swept wing structural and aerodynamic models. (a) Finite element model. (b) Aerodynamic model.

The equation of motion used for general static aeroelastic calculation can be expressed as follows [

The present wing model has neither control surface nor rigid body motions; equation (

The eigenvalues

In static aeroelastic analysis, the downwash can be calculated as [

To improve the accuracy of aerodynamic calculations, two experimental corrections may be introduced to adjust each theoretical aerodynamic panel lift and moment [

As the downwash of the wing cannot be measured directly in experiments, we use the deformation of the wing model to modify the downwash vector; and an iterative method is used to calculate the static aeroelastic deformation of the wing model.

In static aeroelastic analysis, two types of data transformations are required: the structural equivalent forces from aerodynamic panels to structural grids and the interpolation from the structural deflections to the aerodynamic deflections. The spline methods lead to an interpolation that relates the components of structural grid displacements to the aerodynamic grid displacements. When the deformed structural grids and aerodynamic panels are not coplanar, the local AOA of aerodynamic panel needs to be modified according to structural displacements.

The iteration procedure starts with assigning appropriate initial conditions, as shown in Figure

Flowchart of nonlinear static aeroelastic analysis.

To take the case at AOA

Calculation process by medium fidelity model.

In the detailed design phase of a real wing, a more accurate prediction of static aeroelastic behaviour of the wing structure is necessary. In particular, it is important to predict the static aeroelastic behaviour in the vicinity of divergence with sufficient accuracy. Besides the efficient medium fidelity model, a steady-state CFD/CSD coupling simulation of the wing model is also performed to obtain more accurate results. The simulation is carried out using the commercial software package ANSYS for both the structural analysis and aerodynamic analysis.

The flow field consisting of hexahedral cells around the wing is generated using the ICEM CFD as shown in Figure

Fluid mesh of the CFD/CSD coupling simulation.

Total lift on the rigid wing at

A diagram of the photogrammetric measurement with single camera is shown in Figure

Schema of single-camera measurement system.

During the deformation of the wing, the whole process is recorded by the camera. Later, frames are grabbed from video and used to track the targets. For this purpose, a Sony digital camera with

Target tracking in image plane for the wind-off experiment.

Before conducting the wind tunnel test, it is necessary to perform GVT for model validation and updating. As shown in Figure

Ground vibration test of the forward-swept wing.

The first five natural frequencies and modal shapes of the wing model.

Order (mode) | GVT (Hz) | Calculated by FEM (Hz) | Relative error (%) |
---|---|---|---|

1st (1B) | 2.39 | 2.41 | 0.84 |

2nd (2B) | 15.45 | 15.16 | 1.88 |

3rd (1T) | 24.15 | 24.07 | 0.33 |

4th (3B) | 43.93 | 42.87 | 2.41 |

5th (2T) | 72.46 | 72.61 | 0.21 |

After the GVT, a model updating procedure is applied to match the dynamic characteristics of the numerical and the experimental models. The updating procedure performed by solving the optimization problem is defined as follows:

The forward-swept wing model is tested in the NWPU NF-2 acoustic wind tunnel, which is an opening circuit tunnel with a test section of

Forward-swept wing model in the test section of wind tunnel (upstream view).

The spanwise bending strain is used in the prediction of the static divergence. For this purpose, two strain gages are used. The lower gage was located at the root of the wing model and the upper one at the fifth semispan station (

The divergence dynamic pressure results predicted by experiment and simulation are listed in Table

Divergence dynamic pressure predicted by wind tunnel test and simulation.

By upper strain gage | 371.6 | 355.1 | 338.1 | 382.5 |

By lower strain gage | 383.0 | 359.5 | 337.7 | |

Average | 377.3 | 357.3 | 337.9 |

A typical deformation process of wing model in the wind tunnel tests is shown in Figure

Wing deformation process at

Static aeroelastic deformations of the wing model are computed at two angles of attack (

For

Wing tip displacements versus dynamic pressure for

Figure

Wing tip displacements versus dynamic pressure for

The data of the curves in Figures

Wing tip displacements when

LE | TE | |||||||
---|---|---|---|---|---|---|---|---|

VLM + linear | VLM + nonlinear | Experiment | CFD/CSD | VLM + linear | VLM + nonlinear | Experiment | CFD/CSD | |

152 | 2.12 | 2.13 | 2.11 | 2.59 | 1.78 | 1.78 | 1.85 | 2.17 |

192 | 3.25 | 3.25 | 3.56 | 4.08 | 2.72 | 2.72 | 3.04 | 3.42 |

238 | 5.31 | 5.29 | 5.64 | 7.00 | 4.45 | 4.43 | 4.85 | 5.86 |

262 | 7.01 | 6.92 | 8.08 | 9.57 | 5.87 | 5.80 | 6.67 | 8.02 |

275 | 8.26 | 8.07 | 9.43 | 11.48 | 6.91 | 6.76 | 8.09 | 9.63 |

287 | 9.71 | 9.34 | 12.40 | 13.78 | 8.13 | 7.83 | 10.42 | 11.56 |

310 | 13.83 | 12.58 | 17.09 | 21.65 | 11.57 | 10.53 | 14.57 | 18.22 |

324 | 17.93 | 15.16 | 28.55 | 15.01 | 12.71 | 24.07 | ||

338 | 24.65 | 18.32 | 33.09 | 20.63 | 15.36 | 27.92 | ||

352 | 37.62 | 22.02 | 36.40 | 31.49 | 18.46 | 30.73 | ||

367 | 78.19 | 26.47 | 40.00 | 65.44 | 22.21 | 33.78 |

Wing tip displacements when

LE | TE | |||||||
---|---|---|---|---|---|---|---|---|

VLM + linear | VLM + nonlinear | Experiment | CFD/CSD | VLM + linear | VLM + nonlinear | Experiment | CFD/CSD | |

117 | 2.13 | 2.13 | 2.57 | 2.51 | 1.78 | 1.78 | 1.90 | 2.11 |

153 | 3.22 | 3.22 | 3.42 | 3.88 | 2.70 | 2.70 | 2.92 | 3.25 |

194 | 4.97 | 4.96 | 5.44 | 6.16 | 4.16 | 4.16 | 4.50 | 5.17 |

216 | 6.27 | 6.23 | 6.77 | 7.90 | 5.25 | 5.22 | 5.69 | 6.62 |

239 | 8.06 | 7.92 | 9.62 | 10.33 | 6.75 | 6.64 | 8.25 | 8.66 |

251 | 9.24 | 9.01 | 11.57 | 12.01 | 7.73 | 7.54 | 9.76 | 10.07 |

264 | 10.78 | 10.37 | 15.05 | 15.72 | 9.03 | 8.69 | 12.71 | 13.19 |

276 | 12.55 | 11.85 | 18.69 | 16.75 | 10.51 | 9.92 | 15.81 | 14.07 |

289 | 14.98 | 13.71 | 21.75 | 21.25 | 12.54 | 11.49 | 18.62 | 17.89 |

310 | 20.74 | 17.42 | 28.81 | 17.37 | 14.60 | 24.30 | ||

324 | 26.91 | 20.42 | 32.50 | 22.52 | 17.12 | 27.42 | ||

338 | 36.98 | 23.84 | 35.47 | 30.96 | 19.99 | 29.93 | ||

352 | 56.44 | 27.62 | 38.29 | 47.24 | 23.17 | 32.34 | ||

367 | 117.30 | 31.89 | 41.51 | 98.18 | 26.77 | 35.07 |

In order to find the primary factor for the difference among the three numerical approaches, we compare the status of deformed structure at the beginning of the iterations. Here, we only focus on the case of

The wing tip displacements when applying the aerodynamics of rigid wing (mm).

Medium fidelity model | High fidelity model (CFD/CSD) | ||
---|---|---|---|

Linear | Nonlinear | ||

LE point | 34.74 | 34.65 | 34.59 |

TE point | 29.10 | 29.03 | 29.10 |

When the iteration process is finished, the wing structure is in a static state under the balance of the aerodynamic load and structure elastic restoring forces. It is well reflected from Figure

Streamlines of deformed wing at different dynamic pressure,

Further insights can be gained by checking the pressure difference distribution when the iteration is finished. A comparison between the pressure difference distributions provided by the medium and high fidelity models is shown in Figure

Pressure difference distribution at

It is worth mentioning here that, with the same computation resource (CPU: Intel Xeon E5-

This study has been one of the first attempts to find out the application condition for the medium and high fidelity models in nonlinear static aeroelastic analysis. The capability of an iterative method in calculating the deformation of forward-swept wing is investigated by comparing experimental data and simulation results. In the given experimental arrangements, the simulation results have an acceptable accuracy compared with the wind tunnel test. The results show that the proposed method is suitable for the static aeroelastic analysis of the flexible wing undergoing large deformation with high computation efficiency.

The conclusions are summarized as follows:

Although it may underestimate the displacements when large deformation occurs, it is advisable to use the proposed method in the framework of preliminary design and optimization when the computation time is concerned.

For large wing deformation, the high fidelity model generates more accurate results compared to the medium fidelity model; however, its capability is limited by being time-consuming. The high fidelity model is recommended to be used in the detailed design phase, in which the accuracy of the result is more important than the computation time.

The numerical and experimental data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grants nos. 11472216 and 11672240).