Abstract
Wide-speed range flight is a critical design objective and development direction for hypersonic vehicles. However， the complex environmental changes pose design conflicts for fixed-configuration vehicles under different flight conditions. Hypersonic morphing vehicles can adapt to various flight conditions and meet performance requirements by presenting different configurations. This paper introduces numerical simulations to investigate the aerodynamic characteristics of a foldable-wing vehicle， and focuses on the lift-to-drag ratio， longitudinal static stability， and directional static stability of the aerodynamic configuration in different wing folding states at various flight altitudes and Mach numbers. The impact of varying wing folding angles （0°， 45°， 90°） on the aerodynamic performance are compared. The results indicate that across the entire range of speeds studied （Ma=0—5）， the smaller wing folding angles result in the higher lift coefficients， drag coefficients， and lift-to-drag ratios. The wing folding angle of 0° exhibits the highest lift-to-drag ratio. In terms of longitudinal stability， the configuration with a smaller folding angle has better longitudinal stability. As the Mach number increases， the differences in longitudinal stability between different folding angles initially decrease and then increase. The static stability margins change from 1∶0.95∶0.84 to 1∶0.98∶0.88， then to 1∶0.89∶0.79. In addition， configurations with larger wing folding angles exhibit better directional stability. All three wing folding configurations are directionally stable during the low-speed flight phase. As the Mach number increases， the 0° and 45° folding angles gradually become directionally unstable.
The hypersonic vehicle equipped with a combined cycle engine can have horizontal takeoff and landing， reusability， and high-speed cruising in near space. This type of vehicle offers substantial advantages in terms of survivability and combat capabilities due to its high speed and altitude. It has the potential to overcome existing defense systems and can fulfill various mission requirements such as surveillance， strike operations， and transportation between space and Earth. Consequently， both domestic and international researchers have shown great interest in this technolog
Hypersonic vehicles have a broad range of altitude capabilities， operate at high speeds， and incorporate innovative engine systems. However， the environmental changes experienced during flight pose significant challenges in the aerodynamic configuration design of hypersonic vehicles. These challenges primarily manifest in the following aspec
When designing a vehicle capable of wide-speed range flight， the first challenge is that different types of vehicles have different aerodynamic configuration characteristics. The aerodynamic configuration design usually tends to meet the design requirements for the hypersonic flight phase. In the hypersonic phase， an aerodynamic configuration with a high slenderness ratio， large sweep angle， and small aspect ratio performs better. However， these design characteristics may result in insufficient lift during takeoff and landing.
Additionally， selecting airfoil profiles and using trailing edge flaps or other components can effectively delay flow separation， thereby increasing lift. However， as the flight transitions to transonic and higher speeds， the lift-to-drag ratio is predominantly influenced by the pressure difference between the compression on the windward side and the expansion on the leeward side of the incoming airflow rather than by a single component. Changes in the lift mechanism and the occurrence of shockwaves directly impact the vehicle’s aerodynamic configuration design. A significant swept-back wing can lower the critical Mach number， while a small aspect ratio design helps reduce wave drag. Furthermore， increasing the integration of blended wing-body design can minimize interference caused by wave systems. Considering the need to address external environmental changes and adapt the lift mechanism， designing for a high lift-to-drag ratio throughout the entire wide-speed range flight poses a substantial challenge.
In transonic flight， shock waves affect the lift-to-drag ratio and introduce challenges to the design of the aerodynamic configuration by influencing stability and control characteristics. The longitudinal stability of the vehicle is assessed through the static stability margin， which represents the relative position of the aerodynamic center in relation to the center of mass. This margin determines the direction of the resulting moment caused by external disturbances acting upon the vehicle. The aerodynamic center experiences significant shifts as the vehicle transitions from subsonic to transonic， supersonic， and hypersonic speeds. Initially， it shifts backward and then forward. These shifts in the aerodynamic center have a considerable impact on the longitudinal stability of the vehicle， thereby presenting another difficulty in the design of the aerodynamic configuration for wide-speed hypersonic vehicles.
The aerodynamic design of a hypersonic vehicle typically incorporates a small aspect ratio， which helps optimize its performance at high speeds and reduce wave drag. However， the disadvantage of this design is that the vehicle has poor heading and lateral stability. As the Mach number increases， the lateral and directional stability diminishes significantly. Moreover， in high-altitude and high Mach number flight conditions， the efficiency of control surfaces is compromised， making it more difficult to achieve proper trim and affecting the lift-to-drag ratio. Therefore， to address the conflicting requirements of lateral and directional stability across a wide range of speeds， it becomes necessary to introduce novel supplementary control surfaces in the design of wide-speed range vehicles. These additional control surfaces help resolve the stability design contradiction while maintaining effective control characteristics throughout the flight envelope.
Many traditional hypersonic vehicles utilize aerodynamic configurations like the revolution body， wave-rider body， and lifting body. These configurations primarily focus on optimizing aerodynamic performance during hypersonic flight and usually struggle to meet the aerodynamic requirements at subsonic， transonic， and supersonic speeds—deviation from the design point results in a sharp deterioration in aerodynamic performance. Therefore， morphing aerodynamic configuration design is crucial for the vehicle to adapt to flight conditions within a wide speed range. Vehicles have the potential to maintain excellent flight performance throughout the entire flight process by appropriately adjusting their aerodynamic configuration.
To adapt to different external environments and changes in flight states， adjusting the aerodynamic configuration of a vehicle is not a new concept. One example is the F-111 fighter developed by General Dynamics in 1965， which incorporates a variable swept-back win
Various morphing configuration schemes have been proposed to achieve a wide range of flight capabilities. These morphing configurations can typically be categorized into four groups： Wing deformation， fuselage deformation， dynamic system deformation， and hybrid deformation based on the deformed part. Further classification can be done based on characteristics like the deformation range and the deformation method. In addition to the design of aerodynamic configuration schemes， the effect of configuration deformation on aerodynamic performance under different flight conditions is also an important research content. Among these categories， large-scale wing deformation， such as variable span length， variable swept-back， and wing folding， has garnered substantial interest from researchers worldwide.
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Previous research has focused on exploring innovative deformation modes， evaluating aerodynamic performance， and optimizing configuration
To further understand the performance characteristics of different configurations of hypersonic morphing vehicles in the wide-speed range， this paper adopts a numerical simulation method to systematically study the aerodynamic performance and static stability characteristics of a morphing hypersonic vehicle in subsonic， transonic， supersonic， and hypersonic states for the first tim
Based on the existing research， many configuration schemes have been proposed for achieving wide-speed range flight in supersonic and hypersonic vehicles. The U.S. XB-7
The U.S. SR-7
Furthermore， the wing design of the SR-71 adopts a thin airfoil and incorporates strake wings. This design serves multiple purposes. Firstly， it provides vortex lift at subsonic speeds， addressing the challenge of insufficient lift during takeoff and landing for supersonic vehicle configurations. Secondly， the strake wing increases the sweep angle of the wing’s leading edge， raising the critical Mach number and effectively reducing wave dra
An aerodynamic configuration scheme of the morphing vehicle is proposed based on successful aerodynamic configuration cases of existing supersonic vehicles. The design process considers the application background of wide-speed range flight， combining the performance advantages of different shapes under different flight conditions by wing folding. This allows the vehicle to possess the external characteristics of both high-speed and low-speed vehicles and adjust aerodynamic performance by wing foldin
The vehicle features a large swept delta wing and a double vertical tail configuration regarding the fuselage. The fuselage is seamlessly integrated with the strake and large swept delta wings， forming a blended wing body. The wings are designed with a foldable outer wing component， allowing for adjusting the folding angle based on external conditions and mission requirements. The wing folding angle impacts key external parameters such as wing span， wing projected area， and aspect ratio.
Fig.1 Schematic diagrams of different wing folding angles of vehicle
| Parameter | Numerical value |
|---|---|
| Fuselage length/m | 11.8 |
| Wing span/m | 4.9—6.9 |
| Fuselage height/m | 0.88 |
|
Wing area range/ | 4.31—4.72—5.06 |
| Sweep angle of strake wing/(°) | 83 |
| Sweep angle of main wing/(°) | 41 |
| Aspect ratio range | 0.7—1.5 |
| Leading edge passivation radius/mm | 10 |
For flow simulation in this study， a hybrid mesh is employed. During the mesh generation process， the far field boundary is set at 15 times the length of the fuselage， and the boundary layer mesh near the wall undergoes refinement. The first layer mesh has a height of 0.005 mm， with a growth rate of 1.05 in the normal direction. Prismatic cells are used for meshing around the wall， while tetrahedral cells are used for far-field meshes away from the wall. The total volume mesh consists of approximately 17 million cells.
Fig.2 Vehicle surface meshing of vehicle
Fig.3 Volume mesh of vehicle
Fig.4 Mesh at symmetry plane and wall
This study utilizes an efficient in-house code for computational fluid dynamics（CFD） numerical simulation. The code is based on GPU acceleration and solves the Reynolds-averaged Navier-Stokes equations. It employs a finite-volume method， the k⁃ω SST turbulence model， and the AUSMPW+ upwind schem
Fig.5 Wind tunnel test model and geometry model
Fig.6 Comparison of calculated and experimental data
Fig.7 Comparison of density gradient between simulation and experiment
Based on the analysis above， it can be concluded that the numerical method employed in this study effectively captures the external flow field characteristics of hypersonic vehicles. The numerical simulation results exhibit a high level of confidence and reliability.
To study the aerodynamic characteristics of the folding wing vehicle at different flight speeds， this research considers four flight conditions： Subsonic， transonic， supersonic， and hypersonic. These conditions encompass a range of altitudes from 0 to 24 km and Mach numbers from 0 to 5， covering the typical conditions and flight states encountered by comprehensive speed range vehicles. Furthermore， to analyze various modes during the wing folding process， three wing folding angles （0°， 45°， and 90°） are selected for numerical simulation， encompassing the start， stop， and intermediate stages of morphing. In addition to changes in flight altitude and Mach number， the variation range of the angle of attack and sideslip angle is 0°—8° with an interval of 2°. The performance parameters analyzed and studied are lift coefficient， drag coefficient， and static stability. Detailed operating conditions are summarized in
| θ/（°） | H/km | Ma | α/（°） | β/（°） |
|---|---|---|---|---|
| 0/45/90 | 0 | 0.3 | 0/2/4/6/8 | 0/2/4/6/8 |
| 0/45/90 | 5 | 0.8 | 0/2/4/6/8 | 0/2/4/6/8 |
| 0/45/90 | 15 | 2 | 0/2/4/6/8 | 0/2/4/6/8 |
| 0/45/90 | 24 | 5 | 0/2/4/6/8 | 0/2/4/6/8 |
Numerical simulations are conducted in the subsonic condition of Ma 0.3 to explore how the vehicle’s lift， drag， longitudinal stability， and directional stability vary with the angle of attack and sideslip angle. Subsequently， the numerical results are utilized to compare the performance across different wing folding angle configurations.
According to
Fig.8 Variation of lift-drag characteristics with angle of attack (H = 0 km, Ma = 0.3, β = 0°)
Fig.9 Streamline diagrams at x=10.9 m slice (Ma=0.3, α=8°)
Before researching a vehicle’s stability， it is necessary to define the relevant reference values， coordinate systems， and directions. The moment coefficient is referenced to a point at （7.0， -0.008， 0）， assumed to be the center of mass （with a mean value of ）. Additionally， the positive direction for the pitching moment is defined as upward. The subsequent research on the vehicle’s stability is based on the same definitions， which will not be repeated.
Based on the curve of the pitching moment coefficient Cmz， it can be observed that in the subsonic speed state （Ma = 0.3）， as the angle of attack increases， the downward moment of all three wing folding angle configurations increases （<0）. This indicates that vehicles in different wing folding states are all longitudinally statically stable. Under the subsonic speed condition， the primary lift generation is attributed to the wing. The longitudinal center of pressure typically lies close to the wing， generally at the vehicle’s rear body. A larger folding angle corresponds to a smaller wing area and span， resulting in a decreased lift coefficient. Consequently， the aerodynamic contribution from the wing becomes relatively smaller. The pressure center coefficient Xcp and the aerodynamic center coefficient Xac move forward， approaching the center of mass. Therefore， by considering the slope of the pitch moment curve and the static stability margin presented in
Fig.10 Pitching moment coefficient (H = 0 km, Ma = 0.3, β = 0°)
Fig.11 Coefficients of pressure center and aerodynamic center (H = 0 km, Ma = 0.3, β = 0°)
From
Fig.12 Yawing moment coefficient (H = 0 km, Ma = 0.3, β = 0°)
Under transonic conditions， the aerodynamic characteristics of the morphing vehicle are similar to those observed under subsonic conditions. The lift and drag coefficients are higher， but their variation trends with angle of attack are consistent. Different folding angle configurations also exhibit differences in lift and drag characteristics.
Fig.13 Variation of lift and drag characteristics with angle of attack (H = 5 km, Ma = 0.8, β = 0°)
Fig.14 Streamline diagram at x = 10.9 m slice (Ma = 0.8, α = 8°)
As the flight speed increases from subsonic to transonic， the lift coefficient increases， and the lift loss caused by wing folding also increases， resulting in a more significant difference in lift coefficient between the 90° folding angle and the other two folding angle configurations. While experiencing lift loss， wing folding also reduces the induced drag of the vehicle. The drag coefficients of the 0° and 45° folding angle configurations are similar but significantly more extensive than that of the 90° folding angle configuration. Considering the combined effect of lift coefficient and drag coefficient， the loss of lift caused by outer wing folding is greater than the reduction of drag. Under the subsonic condition （Ma=0.8）， the vehicle’s lift-to-drag ratio decreases as the wing folding angle increases.
From the curve of the pitching moment coefficient with angle of attack in
Fig.15 Pitching moment coefficient (H = 5 km, Ma = 0.8, β = 0°)
Fig.16 Pressure and aerodynamic center position coefficient (H = 5 km, Ma = 0.8, β = 0°)
Similarly， during the transonic （Ma=0.8） phase， the effect of wing folding on the vehicle’s directional stability follows a pattern similar to that in the subsonic phase. The three wing folding states progressively increase the restorative moment as the sideslip angle increases， indicating that the vehicle maintains static stability. The downward folding of the outer wing causes a pressure difference between the inner and outer surfaces， resulting in a yawing moment generated by the lateral airflow.
Fig.17 Yawing moment coefficient (H = 5 km, Ma = 0.8, β = 0°)
Fig.18 Comparison of pressure distribution on the lower surface at Ma = 0.8
In the supersonic phase， numerical simulation research was conducted under flight conditions with a Mach number of 2.
Fig.19 Variation of lift and drag characteristics with angle of attack （H = 15 km, Ma = 2, β = 0°）
Fig.20 Contour of pressure distribution at symmetry plane and x = 11.9 m slice (Ma = 2, α = 4°)
When the folding angle changes from 0° to 45°， the high-pressure airflow overflowing from the wing’s lower surface moves away from the upper surface， resulting in a significant reduction in the low-pressure region compared to the 0° folding angle. At the same time， the downward folding of the wing suppresses the outward leakage of high-pressure airflow from the lower surface， expanding the area of the high-pressure region on the lower surface of the wing， thus exhibiting a similar wave-riding effec
When the wing folding angle reaches 90°， the pressure difference between the upper and lower surfaces of the wing is converted into the pressure difference between the inner and outer sides of the vertical wing， which cannot contribute to lift. Therefore， the lift coefficient of the 90° folding angle configuration is smaller than the other two configurations.
The drag experienced by the vehicle under supersonic conditions mainly comes from shock waves， and the influence of the fuselage on the drag is more significant compared to the low-speed stat
In terms of lift-to-drag ratio， the lift loss still outweighs the drag reduction， resulting in a smaller lift-to-drag ratio as the folding angle increases. However， under supersonic conditions， the difference in lift is no longer solely dependent on the size of the wing’s effective lift area. The increased pressure on the lower surface due to wing folding compensates for the lift loss and reduces the difference in the lift-to-drag ratio.
Fig.21 Pitching moment coefficient （H = 15 km, Ma = 2, β = 0°）
Fig.22 Pressure and aerodynamic center coefficients （H = 15 km, Ma = 2, β = 0°）
Looking at the relationship between the positions of the aerodynamic center， it can be noted that a smaller folding angle results in better longitudinal stability. Based on the analysis of lift and drag characteristics， it is known that wing folding restricts the spanwise flow of the airflow over the lower surface of the vehicle， expanding the high-pressure region on the lower surface. However， due to the presence of the folding angle， the folding wing section only provides the longitudinal component of the lifting force when generating the restoring moment. When the folding angle is 90°， and the wing is vertically raised， the outer wing cannot generate an effective longitudinal restoring momen
Additionally， due to the changes in the fuselage’s forces， the aerodynamic center of the three wing folding configurations moves forward during the supersonic phase compared to the low-speed phase.
Fig.23 Yawing moment coefficient （H = 15 km, Ma = 2, β = 0°）
In the hypersonic phase， numerical simulations are conducted under a Mach number of 5.
Fig.24 Variation of lift and drag characteristics with angle of attack （H = 24 km, Ma = 5, β = 0°）
From the curves， it can be observed that both the lift coefficient and drag coefficient increase with increasing angle of attack， and there is minimal difference in the aerodynamic performance among the three wing folding configurations. Under subsonic conditions， the maximum difference in lift-to-drag ratio is 1.09； under hypersonic conditions， the maximum difference is reduced to 0.07.
Based on the numerical results， the aerodynamic forces and moment coefficients of the vehicle under hypersonic conditions are not significantly affected by the folding of the wing. Therefore，
Fig.25 Contour of pressure distribution at y = 0.01 m slice (Ma = 5, α = 4°)
Fig.26 Contour of pressure distribution on the lower surface and around wing (Ma = 5, α = 4°)
In the numerical simulation results under supersonic conditions， it is concluded that wing folding can restrict the spanwise flow of high-pressure airflow on the lower surface and enhance the compression of the airflow on the lower surface. Therefore， the pressure difference between the upper and lower surfaces of the wing increases as the folding angle increases. This phenomenon becomes more pronounced when the flight speed grows to a hypersonic state （Ma=5）. The accumulation of high-pressure airflow on the lower surface significantly affects the pressure distribution on the vehicle surface.
Additionally， the pressure difference between the upper and lower surfaces， particularly in the region affected by folding， becomes more extensive. Therefore， during the wing folding process， although the effective windward area of the vehicle is reduced， there is no significant change in the lift and drag coefficients of the vehicl
Fig.27 Pitching moment coefficient （H = 24 km, Ma = 5, β = 0°）
Fig.28 Pressure and aerodynamic center coefficient （H = 24 km, Ma = 5, β = 0°）
Fig.29 Yawing moment coefficient （H=24 km, Ma= 5, β = 0°）
This article initially establishes the aerodynamic configuration of a hypersonic morphing vehicle， focusing on the wing folding morphing scheme as the research subject. The research compares the aerodynamic characteristics， including lift， drag， and static stability， of three different wing folding configurations under various flight condition
（1） Lift and drag characteristics： Under the exact flight conditions， the vehicle exhibits a higher lift coefficient， drag coefficient， and lift-to-drag ratio when the wing folding angle is smaller. A smaller wing folding angle at low-speed flight conditions results in a larger wing windward area， leading to a higher lift-to-drag ratio and significant lift advantages， although accompanied by increased drag. As the flight speed increases， the lift mechanism changes， and the compressive effect of the folding outer wing on the lower surface high-pressure airflow becomes more prominent. A larger folding angle leads to a higher pressure differential between the upper and lower surfaces， gradually reducing the difference in lift-to-drag ratio compared to cases with smaller folding angles.
（2） Longitudinal static stability： Under the exact flight conditions， larger wing folding angles result in a forward shift in the aerodynamic center position and a smaller longitudinal static stability margin. However， the aerodynamic center always remains behind the center of mass， and the vehicle maintains longitudinal static stability throughout a wide range of speeds. During the transition from subsonic to hypersonic flight speeds， the changes in forces on the fuselage and wings cause the aerodynamic center to shift rearward and forward. However， the relative position relationship remains unchanged， and the law of better longitudinal stability with smaller folding angles remains valid within the Mach number of 0.3—5. The differences in longitudinal static stability between different wing folding angles initially decrease and then increase.
（3） Directional static stability： The vehicle exhibits better directional stability under the exact flight conditions as the wing folding angle increases. The downward folding of the outer wing compensates for the stabilizer’s function， limiting the forward shift of the lateral pressure center and providing a restoring moment. As the flight speed increases， the states with wing folding angles of 0° and 45° gradually transition from directional stability to instability. The state with a 90° folding angle consistently maintains directional stability. With higher flight speeds， the differences in directional stability become more pronounced.
A comprehensive understanding of the advantages and limitations of different wing folding configurations in different flight phases is achieved by comparing the performance differences under various flight conditions. The ability to adjust flight performance based on flight conditions and task requirements has been validated by applying aerodynamic configuration morphing design within a wide-speed range backgroun
Contributions Statement
Prof. LUO Shibin planned the research content and guided the numerical simulation research. Mr. YUE Hang carried out the numerical simulation research and wrote the manuscript. Dr. LIU Jun contributed to the analysis of the results and manuscript writing. Dr. SONG Jiawen contributed to the data processing. Dr. CAO Wenbin provided the code for CFD numerical simulation. All authors commented on the manuscript draft and approved the submission.
Acknowledgements
This work was supported by the foundation of National Key Laboratory of Science and Technology on Aerodynamic Design and Research （No. 614220121020114） and the Key R&D Projects of Hunan Province （Nos.2021GK2011， 2023GK2022）.
Conflict of Interest
The authors declare no competing interests.
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