# Comparative Study of Lifting Surface and CFD Methods in the Simulation of Morphing Process of Folding Wing

Authors: Hao Xu; Jinglong Han; Yong Xi; Guangshan Chen; Peng Cao
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

A folding wing aircraft can autonomously change its configuration in flight to respond to different flying environments. Hinge moment during morphing process is an important basis for driving mechanism design and structural strength check, and its calculation depends on the simulation of the morphing process. Existing studies mostly use the simplified lifting surface method for aerodynamic modeling. In this paper, the CFD-based simulation method is studied, and the results are compared with those by the lifting surface method. First, the unsteady aerodynamic modeling method of the folding wing based on the CFD method is studied, an effective description method is given to define the wall mesh motion caused by wing folding and aileron deflection, and an unfolding–refolding strategy is proposed to improve the quality of the internal mesh in the case of large folding angles. Then, the CFD aerodynamic model is coupled through a developed coupling calculation program with the flexible multibody structure model for the simulation of a flight-morphing process. The comparison with the results of the lifting surface method shows that hinge moments obtained according to the two aerodynamic models are markedly different. Analysis of the difference shows that airfoil thickness considerably affects aerodynamic loading distributions and hinge moments of folding wing aircraft. The lifting surface method ignores airfoil thickness, which will cause large simulation errors in hinge moments.


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## Body

## 1. Introduction

Folding wing, as a typical morphing scheme, can autonomously change the configuration in flight through folding and unfolding its wings to adapt to changes in flying environment [1–3]. Hinge moment during morphing process is an important basis for driving mechanism design and structural strength check and thus has become a research focus.As early as 2007 when the concept of the folding wing morphing aircraft was proposed, Lee and Weisshaar [4] performed static trim calculation on the folding wing based on the ZONA6 aerodynamic model and obtained hinge moments for different folding configurations. The results show that the hinge moment during the morphing process can reach a value as large as 7000 N·m. Subsequently, Reich et al. [5, 6] integrated the developed vortex lattice program and flexible multibody dynamic software to build an aeroelastic multibody morphing simulation tool to simulate the morphing process for a folding wing. Using this tool, Scarlett et al. [7] performed several wind tunnel simulations on the folding wing and studied the variations of hinge moments and load paths under specific motions. Xu et al. [8] deduced the time-domain unsteady aerodynamic influence coefficients suitable for the morphing process by the doublet lattice method and integrated this aerodynamic model with the flexible multibody structure model established in Adams to simulate the flight-morphing process of a folding wing. In the simulation, the general variations of the dynamic parameters, including the hinge moment, were investigated.The aforementioned studies are based on the lifting surface method, which is a linearized potential flow theory-based approximation technique [9]. According to the theory, flow around a thin wing can be expressed as the superposition of the three contributions of thickness, camber, and attack angle. The lifting surface method assumes that the flow around the wing caused by the thickness is symmetrical about the wing and does not generate lift and moment, so can be neglected in aerodynamic modeling. This type of aerodynamic model is widely applied in calculations of conventional fixed-wing aircraft [10] and extended to the design of folding wing. Actually, the approach of wing surfaces to each other will cause strong aerodynamic interference as the folding angle increases. The airfoil thickness determines the shape of the wing surface and plays an important role in the interference. That is, the applicability of the lifting surface method for the simulation of folding process remains to be verified.The CFD-based simulation method is investigated in this study, and the results are compared with those of the lifting surface method to verify the applicability of the lifting surface method. First, the unsteady aerodynamic modeling method of the folding wing based on the CFD method is studied. Then, the aerodynamic model is coupled with the flexible multibody structure model established in Adams to simulate the morphing process. Finally, the simulation results of lifting surface and CFD methods are compared.


## 2. Model of Folding Wing Aircraft

Figure1 shows the semimodel (right side) of the folding wing aircraft applied in the current work. The aircraft is a flexible multibody structure that includes the fuselage (I), inner (II) and outer (III) wings, and aileron (IV). These structures are made of aluminum and assume a NACA0006 airfoil shape. The aileron is located at 11–83% span and 80–100% chord of the outer wing. During folding, the outer wing retains parallel to the fuselage and the fuselage-inner wing angle is defined as the folding angle θ. Table 1 summarizes the main geometric parameters of the unfolding configuration. Additional details about the model are shown in Ref. [8].Figure 1
Model of the folding wing.
(a)
Internal and external structure(b)
Shape and dimensions(c)
Definition of the folding angleTable 1
Main geometric parameters of unfolding configuration.
PropertyValueMass,m (kg)823Center of gravity coordinate,xcg (m)2.45, 0.00, 0.00Moment of inertia,Iy (kg·m2)5475


## 3. Unsteady Aerodynamic Modeling of Folding Wing Based on the CFD Method

The morphing process of the folding wing is accompanied by the continuous dynamic change of the aerodynamic shape. Using the CFD method to model this process must address two problems. The first problem involves how to describe the wall mesh motion caused by wing folding and aileron deflection. The second one considers how to adjust the solving domain mesh according to the moved wall, especially for the large folding angle configuration.


### 3.1. Description of the Wall Mesh Motion

To facilitate the description of the motion caused by wing folding and aileron deflection, we divide the aircraft surface into seven parts, namely, the fuselage (I), the inner wing (II), the inner section of the outer wing (III), the outer section of the outer wing (IV), the aileron (V), the inner blended region (VI), and the outer blended region (VII), as shown in Figure2. The two blended regions are used to smooth the deflected aileron and main wing [11].Figure 2
Division of the folding wing.First, only the folding of the wings and the deflection of the ailerons are considered. Taking the body-fixed coordinate system as a reference, the grid coordinate of the aircraft surface is assumed to bex0,y0,z0 when the wing folding and rudder deflection angles are all zero, and grid coordinate is x,y,z when the folding and deflection angles are θ and δ, respectively. Then, the following formula is developed:For part I:(1)x=x0,y=y0−y0l1z0tanθ2,z=z0.For part II:(2)x=x0,y=l1+y0−l1cosθ−z0tanθ2,z=z0+y0−l1sinθ.For part III:(3)x=x0,y=y0−l2−l2cosθ−l1+l2+l3−y0l3z0tanθ2,z=z0+l2sinθ.For part IV:(4)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ.For part V:(5)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδ.For part VI:(6)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδy0−yayb−ya.For part VII:(7)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδyd−y0yd−yc,where l1, l2, and l3 are the spans of the fuselage, inner wing, and inner section of the outer wing, respectively, and xa,ya, xb,yb, xc,yc, and xd,yd are the coordinates of the four corners of the blended region stated in the body-fixed coordinate system.Figure3 shows the wall mesh with a folding angle of 90° and a deflection angle of 10° obtained according to Equations (1)–(7). The description method given by this paper can effectively describe the wing folding and aileron deflection, obtain a suitable aerodynamic shape, and facilitate the solution of the flow field.Figure 3
Wall mesh with a folding angle of 90° and a deflection angle of 10°.The flexible deformation of the structure, as well as the heaving and pitching motions of the aircraft, is then added, and the coordinates are converted to the global coordinate system. The final coordinate transformation formula is(8)x′=x+Δxcosα+z+Δzsinα,y′=y+Δy,z′=−x+Δxsinα+z+Δzcosα+h,where α is the angle of attack and h is the altitude. x,y,z is the grid coordinate of the wall surface considering the folding of the wings and the deflection of the aileron, calculated by Equations (1)–(7). Δx,Δy,Δz is the flexible deformation stated in a body-fixed coordinate system obtained by interpolation of the structure deformation, and the thin-plate spline method is used in the interpolation [12]. x′,y′,z′ is the final grid coordinate expressed in the global coordinate system, which considers the wing folding, aileron deflection, rigid motion, and structure deformation.


### 3.2. Mesh in Solving Domain

At each time step, after updating the wall mesh according to Equation (8), dynamic mesh technology is needed to achieve the deformation of the internal mesh. In this paper, the solution of the flow field is based on Fluent software, and the unique diffusion-based smoothing of Fluent is adopted as the dynamic mesh update method. Compared with spring-based smoothing, diffusion-based smoothing is better at dealing with large-scale wall motion, especially for large-angle rotation. In addition to the smoothing method, the local remeshing method is used to improve the quality of the internal mesh at large folding angles, which marks and remeshes the cells whose quality is less than the minimum requirement [13, 14].Figure4 shows the internal mesh of several intermediate configurations. When the folding angle is less than 67.5°, a high-quality internal mesh can be obtained through smoothing and remeshing. When the folding angle reaches 90°, the quality of the internal mesh decreases significantly, affecting the convergence of numerical computation.Figure 4
Internal mesh of intermediate configurations.
(a)
Folding angle of 22.5°(b)
Folding angle of 45°(c)
Folding angle of 67.5°(d)
Folding angle of 90°To improve the quality of the internal mesh in the case of large folding angles, an unfolding–refolding strategy is proposed in this paper. Figure5 shows the unfolding–refolding strategy. First, the 50° folding angle configuration is selected as the basic configuration for meshing. Then, the basic configuration is unfolded into the 0° folding angle configuration, and the simulation of the folding process is started from the unfolded configuration. Figure 6 shows the internal mesh of 90° folding angle configuration obtained by this strategy, and the mesh quality has been significantly improved.Figure 5
Unfolding–refolding strategy.Figure 6
Internal mesh at 90° folding angle.


## 3.1. Description of the Wall Mesh Motion

To facilitate the description of the motion caused by wing folding and aileron deflection, we divide the aircraft surface into seven parts, namely, the fuselage (I), the inner wing (II), the inner section of the outer wing (III), the outer section of the outer wing (IV), the aileron (V), the inner blended region (VI), and the outer blended region (VII), as shown in Figure2. The two blended regions are used to smooth the deflected aileron and main wing [11].Figure 2
Division of the folding wing.First, only the folding of the wings and the deflection of the ailerons are considered. Taking the body-fixed coordinate system as a reference, the grid coordinate of the aircraft surface is assumed to bex0,y0,z0 when the wing folding and rudder deflection angles are all zero, and grid coordinate is x,y,z when the folding and deflection angles are θ and δ, respectively. Then, the following formula is developed:For part I:(1)x=x0,y=y0−y0l1z0tanθ2,z=z0.For part II:(2)x=x0,y=l1+y0−l1cosθ−z0tanθ2,z=z0+y0−l1sinθ.For part III:(3)x=x0,y=y0−l2−l2cosθ−l1+l2+l3−y0l3z0tanθ2,z=z0+l2sinθ.For part IV:(4)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ.For part V:(5)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδ.For part VI:(6)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδy0−yayb−ya.For part VII:(7)x=x0,y=y0−l2−l2cosθ,z=z0+l2sinθ−x0−y0−yayd−yaxd−xa+xatanδyd−y0yd−yc,where l1, l2, and l3 are the spans of the fuselage, inner wing, and inner section of the outer wing, respectively, and xa,ya, xb,yb, xc,yc, and xd,yd are the coordinates of the four corners of the blended region stated in the body-fixed coordinate system.Figure3 shows the wall mesh with a folding angle of 90° and a deflection angle of 10° obtained according to Equations (1)–(7). The description method given by this paper can effectively describe the wing folding and aileron deflection, obtain a suitable aerodynamic shape, and facilitate the solution of the flow field.Figure 3
Wall mesh with a folding angle of 90° and a deflection angle of 10°.The flexible deformation of the structure, as well as the heaving and pitching motions of the aircraft, is then added, and the coordinates are converted to the global coordinate system. The final coordinate transformation formula is(8)x′=x+Δxcosα+z+Δzsinα,y′=y+Δy,z′=−x+Δxsinα+z+Δzcosα+h,where α is the angle of attack and h is the altitude. x,y,z is the grid coordinate of the wall surface considering the folding of the wings and the deflection of the aileron, calculated by Equations (1)–(7). Δx,Δy,Δz is the flexible deformation stated in a body-fixed coordinate system obtained by interpolation of the structure deformation, and the thin-plate spline method is used in the interpolation [12]. x′,y′,z′ is the final grid coordinate expressed in the global coordinate system, which considers the wing folding, aileron deflection, rigid motion, and structure deformation.


## 3.2. Mesh in Solving Domain

At each time step, after updating the wall mesh according to Equation (8), dynamic mesh technology is needed to achieve the deformation of the internal mesh. In this paper, the solution of the flow field is based on Fluent software, and the unique diffusion-based smoothing of Fluent is adopted as the dynamic mesh update method. Compared with spring-based smoothing, diffusion-based smoothing is better at dealing with large-scale wall motion, especially for large-angle rotation. In addition to the smoothing method, the local remeshing method is used to improve the quality of the internal mesh at large folding angles, which marks and remeshes the cells whose quality is less than the minimum requirement [13, 14].Figure4 shows the internal mesh of several intermediate configurations. When the folding angle is less than 67.5°, a high-quality internal mesh can be obtained through smoothing and remeshing. When the folding angle reaches 90°, the quality of the internal mesh decreases significantly, affecting the convergence of numerical computation.Figure 4
Internal mesh of intermediate configurations.
(a)
Folding angle of 22.5°(b)
Folding angle of 45°(c)
Folding angle of 67.5°(d)
Folding angle of 90°To improve the quality of the internal mesh in the case of large folding angles, an unfolding–refolding strategy is proposed in this paper. Figure5 shows the unfolding–refolding strategy. First, the 50° folding angle configuration is selected as the basic configuration for meshing. Then, the basic configuration is unfolded into the 0° folding angle configuration, and the simulation of the folding process is started from the unfolded configuration. Figure 6 shows the internal mesh of 90° folding angle configuration obtained by this strategy, and the mesh quality has been significantly improved.Figure 5
Unfolding–refolding strategy.Figure 6
Internal mesh at 90° folding angle.


## 4. Coupling Simulation Platform

Simulation of the morphing process involves the modeling and calculation of multibody systems and unsteady aerodynamic forces. Although corresponding commercial software has been developed in both fields, no code can effectively consider these two issues at the same time. This paper develops coupled calculation programs based on shared memory technology to realize the cosimulation between structural and aerodynamic solvers. The coupling simulation flow is shown in Figure7, as follows:
(1)
Shared memory is created based on the file-mapping functions, enabling multiple software running on a computer to share data with each other. Data exchange is realized by reading/writing the memory, greatly improving the data transfer rate(2)
The aerodynamic and structural codes are solved independently according to their convergence criterion with one data exchange at each time step through loose coupling. The aerodynamic code in the loose coupling process is solved first, and the converged aerodynamic forces are outputted to the structural solver. The structural code is then solved, and the obtained displacements are outputted to the aerodynamic solver for the calculation of the next time step(3)
The process of each software is controlled by creating/deleting a logo file. At each time step, before one party starts to calculate, it deletes its own logo file and generates that of the other party. Given that the own logo file cannot be found, the program will be in a waiting state until the other party completes the calculation and regenerates the logo file(4)
The spline matrix is used to achieve the physical field interpolation between structure and aerodynamic nodes. The spline matrix is the displacement conversion matrix from structure nodes to aerodynamic nodes, and its transpose is the force conversion matrix from aerodynamic nodes to structural nodes [12](5)
Each software uses the secondary development function to access the coupled program. It outputs the calculation results of the current time step to the shared memory through the user subroutine and reads the calculation conditions of the next time stepFigure 7
Coupling simulation flow.On the basis of the coupled calculation program developed in this paper, the computational multibody dynamic software Adams and the computational fluid dynamic software Fluent are integrated to build a flight morphing simulation platform, as shown in Figure8. The platform has the modeling capabilities of flexible multibody dynamics, unsteady aerodynamics, and flight control and can be applied to the simulation of the morphing process for a folding wing. The specific modeling process is as follows:Figure 8
Flight morphing simulation platform.


### 4.1. Modeling of Flexible Multibody Structures

In structural modeling, the capabilities of Nastran and Adams are integrated. Thus, multibody dynamics and finite element models are combined for the simulation of complex flexible multibody systems. First, through Nastran software, the classic Craig–Bampton modal synthesis method is applied to perform modal analysis on the folding wing [15] and the modal neutral files of each substructure are generated. Then, each substructure is imported into Adams/View and assembled using revolute joints. To control wing folding and unfolding, a time-varying drive (rotational motion) is applied to the joint.Furthermore, to realize the loading of aerodynamic force, GFORCE-generalized forces are set at structure nodes and the definition method is selected as SUBROUTINE, meaning that its magnitude is assigned through a user-written subroutine. The subroutine employs SYSFNC and SYSARY macros to read structural displacements from Adams/Solver and outputs them to the shared memory. At the same time, the aerodynamic loads are read from the shared memory and fed back to Adams/Solver for structure solution of the next time step.The Adams model of the folding wing aircraft is shown in Figure9 (to ensure the clarity of the model, only part of the generalized forces is shown). We use a few elements for structural finite element division, except for the region near the hinge, to reduce the computational cost of subsequent morphing simulations. The final divided structural finite element model comprises 5842 CQUAD4 shell elements. A modal analysis is performed every 15° folding angle based on the flexible multibody dynamic model established in this paper. The natural frequencies of this model are compared with those obtained from the finite element model, which is built by Nastran, as shown in Figure 10. The natural frequencies calculated by Adams software are in good agreement with that of Nastran software, and the maximum error of the first four modal frequencies is 1.8%, which verifies the accuracy of the structural modeling method in this paper.Figure 9
Adams model of the folding wing aircraft.Figure 10
Comparison of natural frequencies.


### 4.2. Aerodynamic Modeling Based on the CFD Method

ICEM software is used in this paper to mesh the folding wing. Figure11 shows the mesh of the basic configuration with a 50° folding angle, which is tested for mesh convergence, and contains a total of approximately 940,000 tetrahedral cells.Figure 11
Mesh of the basic configuration with a 50° folding angle.
(a)
Far field of the computational domain(b)
Wall meshThe flow field is considered inviscid and solved by Fluent. Each time step calls the user subroutine to (1) read the aerodynamic loads on the wall and interpolate to the structure nodes and output to the shared memory and (2) read the structural displacements from the shared memory, modify the positions of the wall nodes according to Equation (8), and return them to Fluent/Solver.


### 4.3. Modeling of the Flight Control System

On the basis of the Adams built-in control module, a longitudinal stabilization control system is developed. The system uses the pitch rate and the plunging velocity as feedback signals to generate a stabilization control law based on the PID control method [16] and drive the deflection of aileron. The control law is as follows:
(9)δt=WϑθKpϑθϑ̇t+Kiϑθ∫0tϑ̇tdt+Kdϑθdϑ̇tdt+WhθKphθḣt+Kihθ∫0tḣtdt+Kdhθdḣtdt,where ϑ̇ is the pitch rate; ḣ is the plunging velocity; δt is the aileron deflection angle; Wϑθ and Whθ are the weights of the pitch and heave channels, respectively; Kpϑθ, Kiϑθ, and Kdϑθ are the parameters of the PID controller of the pitch channel; and Kphθ, Kihθ, and Kdhθ are the parameters of the PID controller of the heave channel.The control parameters are consistent with Ref. [8] and tuned based on different configurations and then interpolated according to the folding angle. For a specific configuration, we put the folding wing aircraft into operation in the nontrim state; adjust the PID parameters of the pitch and heave channels, respectively; and then, adjust the weights of the two channels until the aircraft can quickly reach a stable state without significant overshoot.


## 4.1. Modeling of Flexible Multibody Structures

In structural modeling, the capabilities of Nastran and Adams are integrated. Thus, multibody dynamics and finite element models are combined for the simulation of complex flexible multibody systems. First, through Nastran software, the classic Craig–Bampton modal synthesis method is applied to perform modal analysis on the folding wing [15] and the modal neutral files of each substructure are generated. Then, each substructure is imported into Adams/View and assembled using revolute joints. To control wing folding and unfolding, a time-varying drive (rotational motion) is applied to the joint.Furthermore, to realize the loading of aerodynamic force, GFORCE-generalized forces are set at structure nodes and the definition method is selected as SUBROUTINE, meaning that its magnitude is assigned through a user-written subroutine. The subroutine employs SYSFNC and SYSARY macros to read structural displacements from Adams/Solver and outputs them to the shared memory. At the same time, the aerodynamic loads are read from the shared memory and fed back to Adams/Solver for structure solution of the next time step.The Adams model of the folding wing aircraft is shown in Figure9 (to ensure the clarity of the model, only part of the generalized forces is shown). We use a few elements for structural finite element division, except for the region near the hinge, to reduce the computational cost of subsequent morphing simulations. The final divided structural finite element model comprises 5842 CQUAD4 shell elements. A modal analysis is performed every 15° folding angle based on the flexible multibody dynamic model established in this paper. The natural frequencies of this model are compared with those obtained from the finite element model, which is built by Nastran, as shown in Figure 10. The natural frequencies calculated by Adams software are in good agreement with that of Nastran software, and the maximum error of the first four modal frequencies is 1.8%, which verifies the accuracy of the structural modeling method in this paper.Figure 9
Adams model of the folding wing aircraft.Figure 10
Comparison of natural frequencies.


## 4.2. Aerodynamic Modeling Based on the CFD Method

ICEM software is used in this paper to mesh the folding wing. Figure11 shows the mesh of the basic configuration with a 50° folding angle, which is tested for mesh convergence, and contains a total of approximately 940,000 tetrahedral cells.Figure 11
Mesh of the basic configuration with a 50° folding angle.
(a)
Far field of the computational domain(b)
Wall meshThe flow field is considered inviscid and solved by Fluent. Each time step calls the user subroutine to (1) read the aerodynamic loads on the wall and interpolate to the structure nodes and output to the shared memory and (2) read the structural displacements from the shared memory, modify the positions of the wall nodes according to Equation (8), and return them to Fluent/Solver.


## 4.3. Modeling of the Flight Control System

On the basis of the Adams built-in control module, a longitudinal stabilization control system is developed. The system uses the pitch rate and the plunging velocity as feedback signals to generate a stabilization control law based on the PID control method [16] and drive the deflection of aileron. The control law is as follows:
(9)δt=WϑθKpϑθϑ̇t+Kiϑθ∫0tϑ̇tdt+Kdϑθdϑ̇tdt+WhθKphθḣt+Kihθ∫0tḣtdt+Kdhθdḣtdt,where ϑ̇ is the pitch rate; ḣ is the plunging velocity; δt is the aileron deflection angle; Wϑθ and Whθ are the weights of the pitch and heave channels, respectively; Kpϑθ, Kiϑθ, and Kdϑθ are the parameters of the PID controller of the pitch channel; and Kphθ, Kihθ, and Kdhθ are the parameters of the PID controller of the heave channel.The control parameters are consistent with Ref. [8] and tuned based on different configurations and then interpolated according to the folding angle. For a specific configuration, we put the folding wing aircraft into operation in the nontrim state; adjust the PID parameters of the pitch and heave channels, respectively; and then, adjust the weights of the two channels until the aircraft can quickly reach a stable state without significant overshoot.


## 5. Numerical Examples

In this section, the CFD aerodynamic model is used to simulate the morphing process for a folding wing, and the results are compared with those by the lifting surface method. The flight altitude used in the simulation is 2 km, and the flight speed is 100 m/s.


### 5.1. Simulation Results Based on the CFD Method

The flight folding simulation starts from the trim state of unfolded configuration calculated by the CFD aerodynamic model. The wings start folding at 10 s and complete at 40 s. The folding law is shown in Figure12(a). Once wings are folded in place, the folded configuration of aircraft continues to maintain a 1 g level flight. The main dynamic parameters of the aircraft are depicted in Figures 12(b)–13(f).Figure 12
Simulation results based on the CFD aerodynamic model.
(a)
Folding angle(b)
Angle of attack(c)
Altitude(d)
Deflection angle of aileron(e)
Inner wing hinge moment(f)
Outer wing hinge momentFigure 13
Comparison of simulation results.
(a)
Folding angle(b)
Angle of attack(c)
Altitude(d)
Deflection angle of aileron(e)
Inner wing hinge moment(f)
Outer wing hinge momentThe figures show that the initial trim angle of attack is 1.46° and the deflection angle of the aileron is −0.41°. Wings start folding at 10 s. With the increase of folding angle, wing area continues to decrease and aircraft drops. For altitude recovery, ailerons are deflected upward to generate head-up torque, increasing attack angle to maintain lift balance. The aircraft completes its folding at 40 s and soon reached a new state of equilibrium, i.e., folded configuration trimmed state. Compared to the initial unfolded trimmed condition, the attack angle is increased to 3.27°, and ailerons are deflected upward to −3.31°.Using flight simulations, hinge moments during the morphing process can be monitored. In the initial unfolded state, hinge moments of the inner and outer wing are 2,614 and 883 N·m, respectively. As wings are folded, inner and outer wing hinge moments first decrease and then increase. After folding in place, hinge moments of the inner and outer wing are increased to 2,696 and 1054 N·m, respectively.


### 5.2. Comparison with the Results of the Lifting Surface Method

Figure13 compares the simulation results calculated by the CFD and lifting surface methods. The lifting surface method is consistent with Ref. [8] and is developed on the basis of the doublet-lattice method. The results of the two different aerodynamic models are markedly different, especially for the hinge moments.The difference is first manifested in the changing trends of hinge moments. The results calculated by the lifting surface method show that the inner wing hinge moment increases first and then decreases, whereas the outer wing hinge moment increases with wing folding. However, the simulation results based on the CFD method show that hinge moments of the inner and outer wings tend to decrease first and then increase.The difference is also reflected in the magnitude. The hinge moments of the inner and outer wings calculated by the CFD method are smaller than those calculated by the lifting surface method. Taking the CFD results as a reference, Figure14 shows the calculation differences of the hinge moments at various folding angles (the value of the 90° folding angle configuration is the maximum value after 40 s). For inner wing hinge moment, the maximum calculated difference is 20%, which occurs at a folding angle of approximately 45°. For outer wing hinge moment, the maximum difference is 46% and occurs at a 90° folding angle.Figure 14
Calculation differences of the hinge moments.
(a)
Inner wing hinge moment(b)
Outer wing hinge momentThe above comparison shows that the hinge moment values obtained using the two aerodynamic models are markedly different. In general, the aerodynamic results calculated by the CFD method are relatively reliable. The application of the lifting surface method to the calculation of folding wing hinge moment may have major defects. The specific analysis will be given in Section5.3.


### 5.3. Influence of Airfoil Thickness and Deficiency of the Lifting Surface Method

In a common fixed-wing aircraft, thickness-induced flows are symmetric about the wing and do not produce lift or lift moment because no pressure differences exist at the corresponding locations of the upper and lower surfaces. So in aerodynamic modeling, airfoil thickness can be neglected. The resulting model is called the lifting surface model. The vortex lattice method, unsteady vortex lattice method, and double-lattice method all belong to the lifting surface model category [10]. This type of aerodynamic model is widely applied in steady and unsteady aerodynamic calculations of conventional fixed-wing aircraft and has obtained numerous successful results.Unlike conventional fixed-wing aircrafts, a folding wing morphing aircraft should complete wing folding and unfolding in flight. When the folding angle is large, the wings are close to each other, aerodynamic interference occurs between the surfaces of the wing, and two low-pressure areas are generated between inner and outer wings and between inner wing and fuselage. The pressure nephogram (Figure15) of 90° folding angle configuration calculated by the CFD method shows this aerodynamic interference phenomenon. The NACA63006 symmetrical airfoil is adopted in the calculation, and the Mach number and angle of attack are 0.3 and 0°, respectively. According to the linearized potential flow theory, these values represent a typical thickness problem.Figure 15
Low-pressure areas caused by airfoil thicknessθ=90∘.The formation of the low-pressure areas is described in the following. Two channels of airflow are formed between wings and fuselage and narrow at the middle section because of the thickness of airfoil. When passing through the two channels, the airflow accelerates, forming two low-pressure areas. The low-pressure areas are asymmetrical about the wing, that is, the flow caused by the airfoil thickness will affect the aerodynamic load and the hinge moments of the folding wing.The low-pressure areas affect the hinge moments of the folding wing in two ways, as shown in Figure16. The first is to generate a moment directly on the hinge through the normal aerodynamic loads on the wings caused by the low-pressure areas. The second is to produce additional angles of attack and aileron deflection through the impact on the overall aerodynamic performance of the aircraft, in turn affecting the load distribution of the wings and indirectly producing a moment on the hinge. Figure 17 shows the magnitude of the influence on the hinge moments through the two ways. Among them, the first subscript “1” or “2” of △M11, △M21, △M12, and △M22 means the first or second way. The second subscript “1” or “2,” respectively, indicates an influence on inner or outer wing hinge moments. M1 and M2 are inner and outer wing hinge moments calculated by the CFD method in Section 5.1, respectively.Figure 16
Two ways in which the low-pressure areas affect the hinge moments.Figure 17
Influence of two ways on the hinge moments.
(a)
△M11/M1 and △M21/M1(b)
△M12/M2 and △M22/M2The influence of the two ways can be clearly compared from Figure17.
(1)
For inner wing hinge moment, the thickness of airfoil increases inner wing hinge moment through the first way and decreases it through the second way. The effect of the second way is greater than that of the first way. The first way has a maximum effect of 7.8%, while the second way has a maximum effect of 20.0%(2)
For outer wing hinge moment, airfoil thickness will reduce outer wing hinge moment through the two ways. Similarly, the effect of the second way is greater than that of the first way. The maximum effect of the first way is 15.3%, while the maximum effect of the second way is 30.1%Finally,△M11+△M21/M1 and △M12+△M22/M2 are calculated to investigate airfoil thickness total effect on hinge moments and compare with the simulation difference in Figure 14. The results are shown in Figure 18.
(1)
For inner wing hinge moment, the influence of the two ways will be offset, and the maximum influence occurs at a folding angle of approximately 60°, which is 20.1%. For outer wing hinge moment, the influence of the two ways will be superimposed, and the maximum influence occurs at the 90° folding angle, which is 45.4%(2)
The difference of simulation between the two aerodynamic models is consistent with airfoil thickness effect, indicating that simulation difference is mainly caused by the ignorance of the airfoil thickness in the lifting surface methodFigure 18
Comparison of simulation differences and thickness effects.
(a)
Inner wing hinge moment(b)
Outer wing hinge momentThe above analysis shows that the thickness of airfoil considerably affects aerodynamic loading distribution and hinge moments of folding wing aircraft. The lifting surface method ignores airfoil thickness, which will cause large simulation errors in hinge moments.Table2 lists the calculation time required for the lift surface and CFD methods. Among these methods, the solution of the lift surface model adopts a single core, while that of the CFD model adopts 16 cores (the CPU is Intel Xeon E5-2650 with a dominant frequency of 2.00 GHz). Compared with the lifting surface method, the CFD method is computationally expensive. Therefore, developing the thickness correction based on the lift surface method in subsequent work is necessary.Table 2
Comparison of the calculation time.
Aerodynamic modeling methodCalculation timeLifting surface method0.6 hCFD method (solving the Euler equation)151 h


## 5.1. Simulation Results Based on the CFD Method

The flight folding simulation starts from the trim state of unfolded configuration calculated by the CFD aerodynamic model. The wings start folding at 10 s and complete at 40 s. The folding law is shown in Figure12(a). Once wings are folded in place, the folded configuration of aircraft continues to maintain a 1 g level flight. The main dynamic parameters of the aircraft are depicted in Figures 12(b)–13(f).Figure 12
Simulation results based on the CFD aerodynamic model.
(a)
Folding angle(b)
Angle of attack(c)
Altitude(d)
Deflection angle of aileron(e)
Inner wing hinge moment(f)
Outer wing hinge momentFigure 13
Comparison of simulation results.
(a)
Folding angle(b)
Angle of attack(c)
Altitude(d)
Deflection angle of aileron(e)
Inner wing hinge moment(f)
Outer wing hinge momentThe figures show that the initial trim angle of attack is 1.46° and the deflection angle of the aileron is −0.41°. Wings start folding at 10 s. With the increase of folding angle, wing area continues to decrease and aircraft drops. For altitude recovery, ailerons are deflected upward to generate head-up torque, increasing attack angle to maintain lift balance. The aircraft completes its folding at 40 s and soon reached a new state of equilibrium, i.e., folded configuration trimmed state. Compared to the initial unfolded trimmed condition, the attack angle is increased to 3.27°, and ailerons are deflected upward to −3.31°.Using flight simulations, hinge moments during the morphing process can be monitored. In the initial unfolded state, hinge moments of the inner and outer wing are 2,614 and 883 N·m, respectively. As wings are folded, inner and outer wing hinge moments first decrease and then increase. After folding in place, hinge moments of the inner and outer wing are increased to 2,696 and 1054 N·m, respectively.


## 5.2. Comparison with the Results of the Lifting Surface Method

Figure13 compares the simulation results calculated by the CFD and lifting surface methods. The lifting surface method is consistent with Ref. [8] and is developed on the basis of the doublet-lattice method. The results of the two different aerodynamic models are markedly different, especially for the hinge moments.The difference is first manifested in the changing trends of hinge moments. The results calculated by the lifting surface method show that the inner wing hinge moment increases first and then decreases, whereas the outer wing hinge moment increases with wing folding. However, the simulation results based on the CFD method show that hinge moments of the inner and outer wings tend to decrease first and then increase.The difference is also reflected in the magnitude. The hinge moments of the inner and outer wings calculated by the CFD method are smaller than those calculated by the lifting surface method. Taking the CFD results as a reference, Figure14 shows the calculation differences of the hinge moments at various folding angles (the value of the 90° folding angle configuration is the maximum value after 40 s). For inner wing hinge moment, the maximum calculated difference is 20%, which occurs at a folding angle of approximately 45°. For outer wing hinge moment, the maximum difference is 46% and occurs at a 90° folding angle.Figure 14
Calculation differences of the hinge moments.
(a)
Inner wing hinge moment(b)
Outer wing hinge momentThe above comparison shows that the hinge moment values obtained using the two aerodynamic models are markedly different. In general, the aerodynamic results calculated by the CFD method are relatively reliable. The application of the lifting surface method to the calculation of folding wing hinge moment may have major defects. The specific analysis will be given in Section5.3.


## 5.3. Influence of Airfoil Thickness and Deficiency of the Lifting Surface Method

In a common fixed-wing aircraft, thickness-induced flows are symmetric about the wing and do not produce lift or lift moment because no pressure differences exist at the corresponding locations of the upper and lower surfaces. So in aerodynamic modeling, airfoil thickness can be neglected. The resulting model is called the lifting surface model. The vortex lattice method, unsteady vortex lattice method, and double-lattice method all belong to the lifting surface model category [10]. This type of aerodynamic model is widely applied in steady and unsteady aerodynamic calculations of conventional fixed-wing aircraft and has obtained numerous successful results.Unlike conventional fixed-wing aircrafts, a folding wing morphing aircraft should complete wing folding and unfolding in flight. When the folding angle is large, the wings are close to each other, aerodynamic interference occurs between the surfaces of the wing, and two low-pressure areas are generated between inner and outer wings and between inner wing and fuselage. The pressure nephogram (Figure15) of 90° folding angle configuration calculated by the CFD method shows this aerodynamic interference phenomenon. The NACA63006 symmetrical airfoil is adopted in the calculation, and the Mach number and angle of attack are 0.3 and 0°, respectively. According to the linearized potential flow theory, these values represent a typical thickness problem.Figure 15
Low-pressure areas caused by airfoil thicknessθ=90∘.The formation of the low-pressure areas is described in the following. Two channels of airflow are formed between wings and fuselage and narrow at the middle section because of the thickness of airfoil. When passing through the two channels, the airflow accelerates, forming two low-pressure areas. The low-pressure areas are asymmetrical about the wing, that is, the flow caused by the airfoil thickness will affect the aerodynamic load and the hinge moments of the folding wing.The low-pressure areas affect the hinge moments of the folding wing in two ways, as shown in Figure16. The first is to generate a moment directly on the hinge through the normal aerodynamic loads on the wings caused by the low-pressure areas. The second is to produce additional angles of attack and aileron deflection through the impact on the overall aerodynamic performance of the aircraft, in turn affecting the load distribution of the wings and indirectly producing a moment on the hinge. Figure 17 shows the magnitude of the influence on the hinge moments through the two ways. Among them, the first subscript “1” or “2” of △M11, △M21, △M12, and △M22 means the first or second way. The second subscript “1” or “2,” respectively, indicates an influence on inner or outer wing hinge moments. M1 and M2 are inner and outer wing hinge moments calculated by the CFD method in Section 5.1, respectively.Figure 16
Two ways in which the low-pressure areas affect the hinge moments.Figure 17
Influence of two ways on the hinge moments.
(a)
△M11/M1 and △M21/M1(b)
△M12/M2 and △M22/M2The influence of the two ways can be clearly compared from Figure17.
(1)
For inner wing hinge moment, the thickness of airfoil increases inner wing hinge moment through the first way and decreases it through the second way. The effect of the second way is greater than that of the first way. The first way has a maximum effect of 7.8%, while the second way has a maximum effect of 20.0%(2)
For outer wing hinge moment, airfoil thickness will reduce outer wing hinge moment through the two ways. Similarly, the effect of the second way is greater than that of the first way. The maximum effect of the first way is 15.3%, while the maximum effect of the second way is 30.1%Finally,△M11+△M21/M1 and △M12+△M22/M2 are calculated to investigate airfoil thickness total effect on hinge moments and compare with the simulation difference in Figure 14. The results are shown in Figure 18.
(1)
For inner wing hinge moment, the influence of the two ways will be offset, and the maximum influence occurs at a folding angle of approximately 60°, which is 20.1%. For outer wing hinge moment, the influence of the two ways will be superimposed, and the maximum influence occurs at the 90° folding angle, which is 45.4%(2)
The difference of simulation between the two aerodynamic models is consistent with airfoil thickness effect, indicating that simulation difference is mainly caused by the ignorance of the airfoil thickness in the lifting surface methodFigure 18
Comparison of simulation differences and thickness effects.
(a)
Inner wing hinge moment(b)
Outer wing hinge momentThe above analysis shows that the thickness of airfoil considerably affects aerodynamic loading distribution and hinge moments of folding wing aircraft. The lifting surface method ignores airfoil thickness, which will cause large simulation errors in hinge moments.Table2 lists the calculation time required for the lift surface and CFD methods. Among these methods, the solution of the lift surface model adopts a single core, while that of the CFD model adopts 16 cores (the CPU is Intel Xeon E5-2650 with a dominant frequency of 2.00 GHz). Compared with the lifting surface method, the CFD method is computationally expensive. Therefore, developing the thickness correction based on the lift surface method in subsequent work is necessary.Table 2
Comparison of the calculation time.
Aerodynamic modeling methodCalculation timeLifting surface method0.6 hCFD method (solving the Euler equation)151 h


## 6. Conclusions

Hinge moment is an important basis for driving mechanism design and structural strength verification of folding wing, and its calculation depends on the simulation of the morphing process. The CFD-based simulation method is investigated in this study, and the results are compared with those of the widely used lifting surface method. The results show the following:(1)
The description method given by this paper can effectively describe the wing folding and aileron deflection, obtaining a suitable aerodynamic shape. The adopted dynamic mesh technology and unfolding–refolding strategy can ensure the quality of the internal mesh(2)
The comparison of the simulation results based on the CFD method (solving the Euler equation) and the lifting surface method shows that the two aerodynamic models have considerable differences in the calculation of the folding wing’s hinge moments(3)
The thickness of airfoil significantly affects aerodynamic loading distributions and hinge moments of folding wing aircraft. The lifting surface method ignores airfoil thickness, which will cause large simulation errors in hinge moments



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*Source: 2476196-2022-02-22.xml*