This paper presents a study of transonic wings whose planform shape is curved. Using fluid structure interaction analyses, the dynamic instability conditions were investigated by including the effects of the transonic flow field around oscillating wings. To compare the dynamic aeroelastic characteristics of the curved wing configuration, numerical analyses were carried out on a conventional swept wing and on a curved planform wing. The results confirm that, for a curved planform wing, the dynamic instability condition occurs at higher flight speed if compared to a traditional swept wing with similar profiles, aspect ratio, angle of sweep at root, similar structural layout, and similar mass. A curved wing lifting system could thus improve the performances of future aircrafts.

Modern technologies are aimed at increasing efficiency, in order to reduce operative costs and pollution and/or to increase the performances of aircrafts. For several years the Aerospace Engineering Unit of the Department of Civil and Industrial Engineering of the University of Pisa has been studying a novel geometry for wings with high aspect ratio. The wing has a curved planform: both the leading and trailing edges of the wing are described by curved lines. The in-plane curvature of the wing considerably reduces the aerodynamic drag especially in the transonic regime where the nonuniform distribution of the sweep angle of the curved wing leads to a reduction in the wave drag effects.

In the literature, various works focus on wing configurations with a curved leading edge or curved planform. The main topics discussed are a reduction in the induced drag or classical application of low aspect ratio wings for high supersonic configurations [

At the same time patents concerning the curved planform concept have been deposited, but only for a tip extension of wings of transport aircrafts [

To the best of our knowledge, there are no studies on a fully curved planform shape with high aspect ratio wings (with both the leading edge and the trailing edge curved). From an engineering point of view, the research interest in such a wing configuration particularly concerns the strong reduction in drag and the important reduction in structural weight. Both of these synergic effects could lead to significant reduction in fuel consumption and pollution.

In transonic flight conditions, the flow field on the wings is strongly nonlinear, which is why the theoretical modelling of realistic aircraft configurations represents a challenge for researchers. Often, the technical literature regards the validation procedures of numerical techniques, adopted to describe the aerodynamic performances of wings or aircrafts operating in the transonic regime. Several works try to represent pressure and lift distributions. In these cases often the numerical results agree very well with the experimental data [

The present work compares the dynamic aeroelastic behaviour of wings, with different planform shapes, in a three-dimensional fully transonic flow field.

Today by means of a fluid structure interaction (FSI) technique, it is possible to represent the physics of transonic phenomena which develop around deformable lifting surfaces. There are several works that study the dynamic behaviour in the transonic regime of profiles mounted on elastic supports. However in these cases there are no three-dimensional effects of the flow around a real wing. On the other hand, the dynamic oscillations of a three-dimensional lifting surface make the problem very complex. The analysis becomes more complicated if dynamic interactions arise between the transonic flow field and the structural response of the wings. Thus numerical results obtained by comparative analyses carried out with a well-structured procedure for the construction of the aerodynamic grids with a similar topology, a similar number of cells, and a similar layout of cell dimensions for the problems analysed can guarantee a reliable technical comparison of the physical behaviours of the wing configurations under observation without having to use prohibitive computational resources. This is also true if the absolute values of the computed technical coefficients are affected by modelling errors. In fact, in the case of structured and similar fluid dynamic grids, these errors will have the same quantitative effects. Thus several numerical activities have been carried out at the University of Pisa to compare the aerodynamic behaviour of a curved wing with that of a conventional swept wing.

The results of the studies on the drag reduction obtained with a curved wing configuration can be found in [

We applied the FSI technique by means of ANSYS Workbench® Rel. 15 commercial platform. The dynamic responses of a swept wing and a curved wing were studied and compared. The models of the two wings, as assumed in previous researches, were constructed with similar aerodynamic profiles, aspect ratios, sweep angles at the root section, and structural layouts. The geometry of the curved wing was obtained by shearing the swept wing in the longitudinal direction. Figures

Geometrical data of the two half-wing models.

Swept wing | Curved wing | |
---|---|---|

AR (aspect ratio) | 9.5 | 9.5 |

Angle of sweep at root | 32° | 32° |

Angle of sweep at tip | 32° | 53° |

Half-wing span ( |
30 m | 30 m |

Reference surface area | 379 m^{2} |
379 m^{2} |

Root chord | 13.18 m | 13.18 m |

Tip chord | 1.7 m | 1.7 m |

Kink section position ( |
9.3 m | 9.3 m |

Kink chord | 7.373 m | 7.373 m |

Dihedral angle | 5° | 5° |

Swept wing model.

(a) Curved wing model. (b) Definition of leading edge geometry of the curved wing model.

In (

The

The combined fluid dynamic and structural analyses were carried out by taking into account the gravity effects and setting the proper geometrical angle of attack in order to get the same lift coefficient for both wings. Following previous experiences [

The elasticity and damping effects obviously influence the final response of the wings providing, for example, suitable displacement histories for fixed control points. The nodes of the finite element models positioned at the leading edge and at the trailing edge of the tip section of the two wings were assumed as control points. Thus, for each flight condition, the overall damping coefficient (which involves both structural and aerodynamic effects) was extracted by processing the displacement time history of the two wings.

The two wing models were constructed by adopting similar structural layouts (aluminium alloy material), similar thickness distribution for skins, and similar geometry for spars and stringers. The structural mass of the models is the same, also including the effect of fuel mass distributed along the span.

During a preliminary numerical campaign, carried out at sea level, neither of the wing models suffered from instability. In fact, the overall damping was always negative in the range values of the Mach number examined. This result depended on the scheme used to define the structural models. In fact in the present models, not only does the wing box affect the structural response but also the front and rear portions of the wing cross sections outside the wing box do. Thus in particular the estimated torsional natural frequencies of the wings were found to be unrealistic: the first torsional frequency was too high with respect to the first bending frequency. To overcome this problem, fictitious rotational inertia was added to the last three ribs at the tip region of the wings, thus keeping the total mass of the two wings unchanged.

This adaptation of the models provided the desired results: the reduction of the torsional frequency, the interaction of first bending and torsional modes, and the onset of dynamic instability for both wings.

Comparing the two wings, with similar aerodynamic profiles and structural layouts, reveals that for a curved geometry the dynamic instability conditions are reached with higher flight velocity values. This occurs at sea level for low subsonic flight conditions and at cruise altitude for high subsonic flight conditions (transonic flight).

The results obtained highlight the need for further research because, as demonstrated in previous studies [

To carry out the fluid dynamic analyses the FLUENT® code was adopted. For the two wings structured meshes were constructed, maintaining a similar topology for both models. Thus numerical effects and/or numerical errors can be assumed as similar for the two models (this approach has also been adopted in previous works).

Firstly, a blocking procedure was used to define the control volumes around the wings models (e.g., Figures

(a) Block layout around the curved wing model (partial view). (b) Block layout near the curved wing model (partial view).

The section profile geometry that we adopted is similar to that used in previous research campaigns [

The whole aerodynamic field analysed has the following dimensions: height 131 m, width 90 m, and length 278 m. To minimize the time needed for the analyses a whole grid of only 389 766 hexahedral cells and 400 544 nodes was used. Figure

Surface grid of the curved wing model.

The boundary conditions fixed for the lateral surfaces of the overall mesh volume are summarized in Table

CFD boundary conditions.

Swept wing model | Curved wing model | |
---|---|---|

Altitudes analysed | 0 m–10000 m | 0 m–10000 m |

Pressure far field | Front side up down | Front side up down |

Pressure outlet | Rear | Rear |

Symmetry | Center-line plane | Center-line plane |

Wall/no slip | Wing surface | Wing surface |

The models of the swept and curved wings were constructed by assigning the properties to the structural components (skin, stringers, ribs, and spars) in ANSYS R15.0. All the components of both structural wing models have the same dimensions. The structures were modelled with a metallic material (aluminium alloy). A three-spar configuration was assumed for the wing box layout. Upper and lower skin, ribs, and spar webs were modelled with shell elements, whereas the stringers and spar flanges were modelled with beam elements. Both the models consist of 8436 nodes and 4157 elements. Figures

(a) The finite element model of the curved wing (upper and lower skins partially removed). (b) The finite element model of the curved wing (wing structure layout).

The engine nacelle was modelled with beams with a very high stiffness and three-point masses describe the inertial effects of the engine. The structural mass and the fuel mass distributed along the wing are the same for both wings (swept and curved).

Fictitious inertia values were added on the tip region of the two wings to facilitate the dynamic instability and to overcome the effects of the boundary conditions at the root section of the wing models (the root section was assumed to be clamped for the wings). Two distinct moment inertia distributions were analysed: in the first case (Case

Structural analyses data.

Swept wing | Curved wing | |
---|---|---|

Material | Aluminium alloy | Aluminium alloy |

Skin thickness | 7 mm to 2.5 mm | 7 mm to 2.5 mm |

Rib thickness | 7 mm to 1 mm | 7 mm to 1 mm |

Front spar thickness | 12 mm to 5 mm | 12 mm to 5 mm |

Central spar thickness | 12 mm to 8 mm | 12 mm to 8 mm |

Rear spar thickness | 12 mm to 7.5 mm | 12 mm to 7.5 mm |

Total structural mass | 15 372.2 kg | 15 372.2 kg |

Total fuel mass | 20 000 kg | 20 000 kg |

Engine masses | 4000 kg front, 4000 kg centre, and 2000 kg rear | 4000 kg front, 4000 kg centre, and 2000 kg rear |

Total model mass | 45 394 kg | 45 394 kg |

Fictitious inertia (Case |
1300 |
1300 |

Fictitious inertia (Case |
750 |
1050 |

^{2}.

The structural damping factor

The modal analyses enabled the values of

Before activating the fluid structure interaction, modal analyses of the wing models were carried out to study the distribution of the natural frequencies and the shape of the associated normal modes. For the two models, Table

Results of modal analysis: Case

Swept wing | Curved wing | |
---|---|---|

First bending mode: Case |
1.058 Hz |
0.916 Hz |

First torsion mode: Case |
12.633 Hz |
12.733 Hz |

Table

Results of modal analysis: Case

Swept wing | Curved wing | |
---|---|---|

First bending mode (N. 1) |
1.057 Hz | 0.913 Hz |

First torsion mode (N. 4) |
3.402 Hz | 3.089 Hz |

First bending mode (N. 1) |
1.057 Hz | 0.914 Hz |

First torsion mode (N. 5) |
4.417 Hz | 3.729 Hz |

Finally, as expected the first bending frequencies remain unchanged in the three cases examined (Case

On the basis of the preliminary results of the modal analyses, the final fluid structure interaction analyses were carried out for two altitude values: sea level (0 m) and cruise condition (10000 m).

The time step for the fluid dynamic analyses was fixed at 0.01 s. The time step for the structural analyses was fixed at 0.0025 s. To minimize errors during the data exchange between the structural and the fluid dynamic modules five coupling iterations per time step were set. To obtain the overall damping factor

Assuming that, close to the flutter condition, a damped harmonic oscillation occurs,

Time history of LE vertical displacement of the curved wing (Case

From a practical point of view a dynamic instability condition exists if the parameter

For both wings, Case

For Case

Results of the FSI analyses of the swept wing (Case

Results of the FSI analyses of the curved wing (Case

At the sea level (

The time histories for the cruise flight conditions (

Results of the FSI analyses of the swept wing (Case

Results of the FSI analyses of the curved wing (Case

On the basis of this first set of analyses, related to the distribution of fictitious moments of inertia corresponding to Case

To confirm this last result also for a transonic flight condition, typical of modern transport aircrafts of a medium and/or long operative range, a second distribution of fictitious moments of inertia was adopted. In Case

In this second case, for both wings, the instability condition corresponds to higher Mach numbers and the supersonic zones around the two wings occupy large zones of the aerodynamic field near the surfaces of wings (see Figures

(a) Front view of the supersonic zone around the curved wing (Case

(a) Front view of the supersonic zone around the swept wing (Case

The Reynolds numbers related to the represented supersonic zones are Re = 5.835 × 10^{7} for ^{7} for

The aerodynamic fields are now fully transonic and shock waves develop on both the wings during the dynamic oscillations. As it is well known, this physical phenomenon represents a source of a strong nonlinearity from a mathematical point of view.

Nevertheless, in the present work, the well-structured aerodynamic grids were able to describe the complex aeroelastic behaviour of the two wing models very well, even though the adopted level of grid refinement was not very high due to the available computational resources. In a previous work it was also demonstrated that adopting well-structured grids (similar to fluid dynamic grids used in the present work), also with a low level of refinement, the numerical distributions of the pressure coefficient agree very well with the available experimental data [

During the oscillating motion of the wings (flexural and torsional) the shock waves move in a chordwise direction due to the continuous change in the angle of attack along the wing span. The motion of the shock wave increases the complexity of the aeroelastic phenomena and the computational difficulties.

For Case

Results of the FSI analyses of the swept wing model (Case

Results of the FSI analyses of the curved wing model (Case

Damping ratio versus Mach (the damping data are represented with the opposite sign).

This phenomenon related to the swept wing is clearly evident in Figure

Likewise, for the curved wing, as shown in Figure

As the present analyses are aeroelastic, both conditions considered, for which the lift and the deformed shape fall down, represent unstable motions of the wings for Case

As said above, the instability condition of the swept wing model examined in the present work seems to be related to a flutter-buffet interaction. In Figure

Figures

Figures

Damping ratio versus True Air Speed (the damping data are represented with the opposite sign).

Case

An increase in the flutter speed enables aircrafts to be operated with higher commercial velocity values, thus increasing the productivity of a fleet. On the contrary, the better dynamic response of the curved planform wing enables lighter aircrafts to be designed with a similar flutter boundary of traditional swept wing configurations. In this second case, the saving in weight means a reduction in operative costs. The consequent reduction in fuel consumption also reduces the level of pollution.

A campaign of dynamic fluid structure interaction analyses was carried out to investigate the effects of the planform on the stability properties of high aspect ratio wings. As in previous studies, a comparison between a traditional swept wing and a curved planform wing was performed. The numerical models of the wings were constructed maintaining the same aerodynamic profiles, the same span, aspect ratio, value of the sweep angle at root section, structural layout, and total weight. To carry out the analyses commercial software was used (ANSYS Workbench Rel. 15). Using a blocking procedure (Figures

Previous studies have demonstrated that a curved planform wing reduces the drag coefficient. Present aeroelastic analyses show that this type of wing improves the dynamic performances of transport aircrafts. Adopting a fictitious distribution of moments of inertia applied at the tip of the two finite element models, aimed at reducing the first torsion natural frequency of clamped half wings, the fluid structure interaction analyses provided unstable conditions for both wing models. In addition from the numerical time histories of structural displacements, an estimation of the global damping ratio was obtained.

Our results highlight that (a) in the first case analysed both the swept wing and the curved wing reached the flutter condition (at the sea level and cruise altitude) for a subsonic flow field, and the bending-torsion flutter speeds of the curved wing were greater than swept wing; (b) in the second case (with lower values of the fictitious moments of inertia) at the cruise altitude, the flow fields analysed were fully transonic (as shown in Figures

The results of the present study demonstrate that from a dynamic point of view (i) a curved planform wing shows an excellent performance also in fully subsonic flight conditions and (ii) in the transonic regime, for a curved planform wing, shock phenomena are less critical compared to those occurring on a conventional swept wing.

Figure

These results agree with the preliminary aeroelastic results obtained with the use of the NASTRAN code discussed in [

The authors declare that there are no competing interests regarding the publication of this paper.