Numerical Aeroelastic Analysis of a High-Aspect-Ratio Wing Considering Skin Flexibility

Abstract

Aeroelastic deformation of the high-aspect-ratio wing from a solar-powered UAV will definitely lead to the difference of its performance between design and actual flight. In the present study, the numerical fluid–structural coupling analysis of a wing with skin flexibility is performed by a loosely coupled partitioned approach. The bidirectional coupling framework is established by combining an in-house developed computational fluid dynamics (CFD) code with a computational structural dynamics (CSD) analysis solver and a time-adaptive coupling strategy is integrated in it to improve the computational stability and efficiency of the process. With the proposed method, the fluid–structure interactions between the wing and fluid are simulated, and the results are compared between the deformed wing and its rigid counterpart regarding the aerodynamic coefficients, transition location, and flow structures at large angles of attack. It can be observed that after deformation, the laminar transition on the upper surface is triggered earlier at small angles of attack and the stall characteristic becomes worse. The calculated difference in aerodynamic performance between the deformed and the designed rigid wing can help designers better understand the wing’s real performance in the preliminary stage of design.

1. Introduction

Solar-powered airplanes exhibit a huge potential for long endurance flights because of the unlimited supply of solar energy, and hence research on them has received continuous support and made great progress in the past few decades. Since the 1970s, projects such as the Sunrise [1,2], the Environmental Research Aircraft and Sensor Technology (ERAST) [3], the Zephyr [4], and the Solar Impulse [5] have been successively conducted for the exploration of solar-powered UAV with better performance. The solar-powered airplanes, with the advantages of low maintenance cost, low risk, environmental protection, and pollution-free, have a wide application prospect in military and civilian fields [6], such as uninterrupted relay communication [7], persistent reconnaissance, border patrol [8], environment and disaster monitoring [9], agriculture remote sensing [10], and so on. However, limited by the energy balance constraints during flight, all parts of the solar-powered aircraft, including energy, aerodynamic, structure, propulsion, and control system, must be optimized to meet the design requirements of high altitude and long endurance [11,12,13]. Nowadays, increasing the lift–drag ratio at cruise by using laminar airfoils and reducing the wing weight by implementation of lighter flexible skin are two commonly used approaches in the aerodynamic and structure systems to make the newly designed solar-powered UAV with higher flight efficiency. However, in practical applications, the elastic deformation of the partially flexible membrane will have a significant influence on aerodynamic characteristics of the wing which directly leads to the difference of its performance between design and actual flight. Therefore, there is still lots of work that needs to be done in the accurate prediction of real aerodynamic performance of high-aspect-ratio wing considering laminar transition and skin flexibility under flight conditions.

From the perspective of analysis and simulation, this non-traditional low Reynolds number laminar flow over flexible surfaces poses a severe challenge due to the following key factors. Difficulties arise due to the presence of laminar separation bubble, trailing edge separation, and other complex flow phenomena. The accurate prediction of flow transition is particularly important as it determines the range of laminar region and the estimated value of drag. What is more, there exists a strong coupling of the aerodynamics and structural response, which requires advanced multi-physical coupling methods. Therefore, the practical application of solar-powered aircraft still needs the development of related technologies and further studies of the simulation methods.

In recent years, a large number of experimental and computational studies have been carried out on the fluid–structural coupling characteristics of thin membrane wings. Alioli et al. [14] studied the response of membrane wings subjected to a variety of steady and unsteady flow conditions through experimental measurements, analytical computations, and tightly coupled fluid–structure co-simulations. The effect of elastic deformation on aerodynamic coefficients and flow structures is predicted and discussed in detail. An experimental study was conducted by Hu et al. [15] to assess the benefits of using flexible-membrane airfoil/wing at low Reynolds number compared with the rigid one. The particle image velocimetry measurements of flowfield elucidated that the flexible-membrane airfoils were capable of suppressing flow separation on the upper surface and therefore delaying stall at high angles of attack by changing their camber to adapt incoming flows. The influence of flexible-membrane skin deformation on the transient behavior of vortex and turbulent flow structures were proposed, and the associated underlying fundamental physics were revealed. In addition, investigations on numerical methods and flow mechanism have also been performed on aeroelastic analysis of general low-speed airfoils with partially flexible skins. Shyy et al. [16] used an extended XFOIL solver to simulate oscillating flow around two-dimensional membrane-top airfoil profiles in low Reynolds number environment, and a number of computations were done to establish the possible benefits of this configuration. The obtained results indicate that the flexible airfoil exhibits less sensitivities to oscillations in the freestream and yields overall more favorable aerodynamic performance than a similar rigid profile. Lian and Shyy [17] further coupled a RANS solver with a structural code to investigate the performance of a flexible airfoil based on SD7003 configuration. The e^N method, which was derived from the linear stability analysis and Orr–Sommerfeld equations, was selected as the transition model. The impacts of upper surface flexible membrane on the laminar separation and transition process were investigated in detail. It can be concluded that the separation and transition positions were obviously affected by the self-excited flexible surface vibration, while the time-averaged lift and drag coefficients were close to those of the rigid airfoil. Dong et al. [18] performed a two-way fluid–structure interaction method to investigate four kinds of segmented flexible airfoils with membrane material on the upper surface. The research focused on influence of flexible deformation on the aerodynamic and flow separation characteristics of those foils at high incidence with a Reynolds number of 1.35 × 10⁵. The transition process was considered in this simulation to improve the numerical calculating accuracy by utilizing the $\gamma -\overline{R{e}_{\theta t}}$ model proposed by Menter [19]. The results show that the segmented flexible airfoils perform a higher maximum lift coefficient and can effectively delay the stall.

All the above-mentioned works are mainly focused on the aerodynamic characteristics of flexible-membrane airfoils/wings applied in micro air vehicles (MAVs) and most of the problems studied are limited to the interactions between flexible surface and fluid for two-dimensional airfoils or simple wing segments. In the present paper, the numerical aeroelastic analysis of a high-aspect-ratio wing from a solar-powered UAV is performed to investigate the compounding effects of structural and flexible skin deformation on the aerodynamic, laminar and flow separation characteristics of the wing. A loosely coupled fluid–structural analysis framework is established by the integration of two separated solvers, as well as the proper combination of several suitable techniques. As a widely adopted algorithm, the loosely coupled approach is simple to implement, computationally inexpensive, and can provide good accuracy in realistic applications. The method just joins the well-developed CFD and CSD codes together and is applicable for a wide range of fluid–structural coupling problems. The aerodynamic force and laminar–turbulent transition location are obtained by employing an in-house CFD code to solve the RANS equations while the deformation of the flexible surface is conducted by a finite element structural solver. The radial basis function (RBF) method is used to transfer data between different physical fields, and the elasticity-based method with high robustness is performed for mesh deformation.

The remainder of this paper is organized as follows. Section 2 proposed the numerical methodologies involved in the two-way fluid–structural coupling process. Two validation cases of the transition prediction method and CFD/CSD coupling approach are carried out in Section 3. Section 4 shows the results and discussions of numerical investigation on the aeroelastic deformation and aerodynamic characteristics of a high-aspect-ratio wing with skin flexibility. Finally, the main conclusions are drawn in Section 5.

2. Numerical Coupling Concept and Methodology

The interaction between flexible surface and fluid is a typical fluid–structural coupling problem. The deformation of the flexible surface changes the flow field in the fluid region, and the variation of pressure distribution on the surface in turn affects the deformation amplitude of the flexible part. In the process of solving fluid–structural coupling problems, the fluid solver, structure solver, and the data transfer scheme at coupling interface are three key factors which have a significant influence on the computational accuracy and efficiency.

2.1. CFD/CSD Coupling Framework

In aeroelastic computations, two typical coupling strategies are usually employed: the tightly coupled and loosely coupled methods. In the former methodology, the equations of fluid and structure are integrated together and solved simultaneously to update all the variables, while in the latter method, the fluid and structure equations are solved separately in each domain and then data transfer is performed on the interface between them. In the present research, a loosely coupled partitioned approach is adopted to solve the two-way fluid–structure interaction problems. Compared with tightly coupled partitioned or monolithic schemes, loosely coupled schemes are more computationally efficient and are better capable of integrating the well-developed computational solvers together. An in-house hybrid-unstructured code HuSolve [20] is proposed as the CFD solver for aerodynamics analysis in the coupling framework and a finite element analyzer is used as the structural solver. The data from each solver is transferred through the interface between the subdomain of fluid and structure by RBF interpolation method. The brief schematic of the fluid–structure analysis framework is demonstrated in Figure 1 and the detailed procedures of the framework are as follows:

(1)

The aerodynamic loads are obtained by employing the CFD module to solve the RANS equations based on the grid and aerodynamic parameters and the aerodynamic force vectors $F(t)$ are transferred to the structural mesh through the RBF interpolation method;

(2)

The structural analysis is conducted by the FEM module of CSD solver based on the CSD grid and the external forces on the boundary to obtain the deformation of the structure;

(3)

Then it comes to the judgement step: If the displacement of the structure converges, the calculation will be terminated. If the convergence is not reached, proceed to step (4);

(4)

The displacement distribution of the interface $q(t)$ is interpolated into the wall surface of the CFD grid by RBF method. The mapped displacement distribution is transformed into a coordinate matrix $x(t)$ to define the new boundary shape of the interface;

(5)

The new computational grid is updated by using the elasticity-based mesh deformation method;

(6)

After that, the iteration process restarts from step (1) with the obtained new boundary condition and CFD grid.

For the coupling problem studied in this paper, the characteristic time of fluid is several orders of magnitude smaller than the structural response. Therefore, when the fluid is disturbed by the structural deformation, it can be considered to reach a new stable state instantaneously. Based on this physical assumption, aeroelastic simulations are performed for a high-aspect-ratio wing in Ref. [21] and the computational results are analyzed in comparison with the experiment and others’ calculations. It is shown that the differences between the calculated and experimental results can be maintained within 10% in most cases, and the coupling analysis method based on the assumptions proposed here is able to provide calculation results with good accuracy. Therefore, a computationally cheaper and simplified coupling procedure for fluid–structural coupling calculation is proposed through steady-state flow field calculation and transient structure analysis in the present study. This strategy can achieve excellent balance between the computational efficiency and accuracy of coupling analysis, and its specific implementation is depicted in Figure 2. With further application of the adaptive coupling time step approach, a significant reduction in time spent on the calculation can be achieved. As shown in the previous research work, the time needed for the adaptive time step coupling calculation is less than 10% of that using the constant time step simulation [22,23]. In Figure 2, $N$ is the number of iteration steps for steady flow field calculation. $\Delta t_S{}^i (i=1, 2, 3,\dots )$ is the time step of transient structural calculation and $\Delta t_C{}^i (i=1, 2, 3,\dots )$ represents the coupling time step of fluid–structural coupling calculation. $Q_{FS}$ is the aerodynamic load distribution transmitted from the fluid domain to the structural solver, and $Q_{SF}$ is the node displacement distribution of the coupling interface interpolated from the structural domain to the fluid solver.

In general, the time step of structural calculation $\Delta t_S{}^i$ is set to a constant during coupling calculation. In the adaptive time step coupling calculation, $\Delta t_S{}^i$ varies dynamically according to the convergence of structural calculation and the corresponding judgment criteria. The coupling time step $\Delta t_C{}^i$ is also changed based on certain rules presented in Figure 3. The minimum coupling time step is given as $\Delta t_{Cmin}$ to avoid the problem of too many coupling iterations caused by the excessive reduction of structural calculation, and the upper limit of adaptive time step $\Delta t_{Cmax}$ is introduced here to prevent the loss in computational stability caused by the excessive increase of coupling time step. As demonstrated in Figure 3, there are three main conditions that exist for the variation of coupling time step in the calculation process.

(1)

When $\Delta t_S{}^i<\Delta t_{Cmin}$, the coupling time step $\Delta t_C{}^i=\Delta t_{Cmin}$ and the data transfer on the interface is not carried out until the sum of structural calculation time between two coupling points satisfies ${\displaystyle \sum \Delta t_S{}^i}>\Delta t_{Cmin}$;

(2)

If $\Delta t_{Cmin}<\Delta t_S{}^i<\Delta t_{Cmax}$, the coupling time step is set as $\Delta t_C{}^i=\Delta t_S{}^i$;

(3)

When the adaptive time step continues to increase to $\Delta t_{Cmax}$, the coupling time step is proposed as $\Delta t_C{}^i=\Delta t_S{}^i=\Delta t_{Cmax}$.

The adjustment strategy of coupling time step in the time-adaptive coupling calculation is given above. The criteria that controls the time increment of structural calculation is proposed as follows. The time increment size is directly associated with the convergence of Newton’s iterations and the time integration accuracy. If the measured half-increment residual $R^J$ in the current calculation is greater than the given tolerance $To{l}^J$, which means that the current time increment size is too large to meet the time integration accuracy requirement, the time increment size will be reduced as

$$\Delta t_S{}^i=D_A\left(\frac{To{l}^J}{R^J}\right)\Delta t_S{}^i$$

(1)

where, $D_A$ is usually given as 0.85.

However, if the following Equation (2) is satisfied in $M$ consecutive time increments

$$\Delta t_S{}^i{\left(\frac{R^J}{\Delta t_S}\right)}_k<W_{G}To{l}^J,k=i-M+1,i$$

(2)

Then the next time increment will be increased to

$$\Delta t_S{}^{i+1}=\min \left(D_G\Delta t_p,D_M\Delta t_S{}^i\right)$$

(3)

where, the default value of each variable is generally set as $M=3$, $W_G=0.75$, $D_G=0.8$; $D_M$ is defined as the time increment growth factor, and the value is 1.25 for most structural dynamic response problems. In the present method, $\Delta t_p$ can be expressed as

$$\Delta t_p=M\left(\frac{To{l}^J}{{\displaystyle {\sum }_{k=1}^M{(R^J/\Delta t_S)}_k}}\right)$$

(4)

2.2. Computational Fluid Dynamics

Fluid dynamics computations are performed by using an in-house developed CFD code HuSolve (Hybrid-unstructured Solver), which solves the RANS equations by adopting a vertex-centered finite volume approach. This flow solver is designed for both the two-dimensional and three-dimensional CFD simulations and can accommodate multiple grid types, including hexahedrons, prisms, tetrahedrons, and pyramids. The conservation form of the governing equations can be written as follows:

$$\frac{\partial }{\partial t}{\displaystyle \underset{V}{\iiint }Q}dV+{\displaystyle \underset{S}{∯}{F}_i(Q)}·\mathit{n}dS-{\displaystyle \underset{S}{∯}{F}_{\nu }(Q)}·\mathit{n}dS=0$$

(5)

where $Q={[\rho ,\rho u,\rho v,\rho w,e]}^T$ is the vector of conservative variables with $\rho ,u,v,w,e$ representing the fluid density, velocity components in Cartesian system and the total internal energy per unit volume. $V$, $S$, and $n$ are the control volume, the boundary of the control volume, and the corresponding normal vector of the boundary face, respectively. The convective and viscous flux vectors are represented by $F_i(Q)$ and $F_v(Q)$.

HuSolve can employ the incompressible and compressible Navier–Stokes equations for different flow situations. In order to obtain the upwind scheme, the Roe’s flux-difference splitting scheme [24] is adopted in the current study to calculate the inviscid fluxes at interfaces between neighboring control volumes surrounding each node using an approximate Riemann solver based on the values on either side of the interface. For second-order accuracy, interface values are obtained by extrapolation of the control volume centroidal values, based on gradients computed at the mesh vertices using an unweighted least-squares technique. The full viscous fluxes are discretized by a finite-volume formulation in which the required velocity gradients on the dual cell faces are computed according to the Green–Gauss theorem. The solution is updated with an implicit LU-SGS (Lower-Upper Symmetric Gauss–Seidel) pseudo time iteration scheme and local time-step scaling is employed to accelerate convergence to steady state. Various turbulence models, including the one-equation Spalart–Allmaras (SA) model and two-equation k-ω Shear Stress Transport (k-ω SST) models, are available in the code for the simulation of turbulent flows.

For the prediction of flow transition, the Langtry–Menter correlation-based transition model [25] is solved together with the k-ω SST turbulence model [26] in the RANS computations. The present transition model is built on two transport equations: one is for intermittency $\gamma$ and the other is for transition onset momentum thickness Reynolds number $\overline{R{e}_{\theta t}}$. The transport equations for $\gamma$ and $\overline{R{e}_{\theta t}}$ can be written as Equations (6) and (7), respectively:

$$\frac{\partial (\rho \gamma )}{\partial t}+\frac{\partial (\rho U_j\gamma )}{\partial x_j}=P_{\gamma 1}-E_{\gamma 1}+P_{\gamma 2}-E_{\gamma 2}+\frac{\partial }{\partial x_j}\left[(\mu +\frac{{\mu }_t}{{\sigma }_{\gamma }})\frac{\partial \gamma }{\partial x_j}\right]$$

(6)

$$\frac{\partial (\rho \overline{R{e}_{\theta t}})}{\partial t}+\frac{\partial (\rho U_j\overline{R{e}_{\theta t}})}{\partial x_j}=P_{\theta t}+\frac{\partial }{\partial x_j}\left[{\sigma }_{\theta t}(\mu +{\mu }_t)\frac{\partial \overline{R{e}_{\theta t}}}{\partial x_j}\right]$$

(7)

The intermittency function is used to trigger transition locally and to control the production and destruction terms in the SST $k$ transport equation while the function for $\overline{R{e}_{\theta t}}$ is proposed to capture the non-local influence of the turbulence intensity and relate the empirical correlation to the onset criteria in the intermittency equation.

The interaction between the transition model and the k-ω SST turbulence model is performed by modified kinetic energy production and destruction terms as:

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho U_{j}k)}{\partial x_j}=\widetilde{P_k}-\widetilde{D_k}+\frac{\partial }{\partial x_j}\left[(\mu +{\sigma }_k{\mu }_t)\frac{\partial k}{\partial x_j}\right]$$

(8)

$$\widetilde{P_k}={\gamma }_{eff}{P}_k$$

(9)

$$\widetilde{D_k}=\min (\max ({\gamma }_{eff},0.1),1.0)D_k$$

(10)

$${\gamma }_{eff}=\max (\gamma ,{\gamma }_{sep})$$

(11)

where $P_k$ and $D_k$ are the original production and destruction terms for the SST model. The effective intermittency ${\gamma }_{eff}$ is introduced here to also include separation effects into the formulation. Further details of the transition model can be seen in reference [25,27].

2.3. Computational Structural Dynamics

A finite element solver developed based on the open source code Calculix [28] is adopted to achieve the geometric nonlinear response of the structure. The structural dynamic equation can be expressed as:

$$M \ddot{\mathit{q}}+C \dot{\mathit{q}}+\mathit{K}\mathit{q}=\mathit{F}(t)$$

(12)

where $\ddot{\mathit{q}}$, $\dot{\mathit{q}}$, and $\mathit{q}$ represent the acceleration, velocity, and displacement vectors of the structural nodes in turn. $M$, $C$, and $K$ are the structural mass matrix, the structural damp matrix, and the structural stiffness matrix, respectively. $\mathit{F}(t)$ is the transient aerodynamic force acting on the structure. In this study, an implicit time integration scheme based on Newmark method is used to calculate the instantaneous dynamic or quasi-static response of the structural systems. The nonlinear equations are solved by a combination of load incremental procedures and Newton–Raphson iterative method.

For the high-aspect ratio wing investigated in this research, the deformation of wing beam can be directly obtained through the CSD analysis while the membrane deformation needs to be captured by post-processing of the results. By deducting the bending and torsional deformation from the deformed wing shape, a configuration with only membrane deformation is produced. The displacements of the structural nodes in the membrane region can be obtained by subtracting the coordinates of this new configuration from those of the original rigid wing.

2.4. Interface Coupling and Data Transfer Method

Recently, benefitting from the efforts of Wendland, Becker [29], and Lombardi et al. [30] over the years, the RBF method has become widely applied in interpolation methods within the fields of CFD and CAE. One of its major advantages is that the interpolation method can work well on different kinds of meshes regardless of mesh types and topology. Therefore, the RBF interpolation scheme is proposed to handle the two-way data transfer between the non-matching meshes. The general form of the RBF interpolation functions can be written as [31,32]:

$$f\left(\mathit{r}\right)={\displaystyle \sum _{i=1}^{N_R}{w}_i\varphi \left(‖\mathit{r}-{\mathit{r}}_i‖\right)}+\varphi \left(\mathit{r}\right)$$

(13)

where ${\mathit{r}}_i$ is the location of the supporting center for the RBF labeled with index $i$. $‖\mathit{r}-{\mathit{r}}_i‖$ is the so-called Euclidean distance and $\varphi$ is defined as the basis function with respect to it. $N_R$ in the summation symbol denotes the number of RBFs involved in the interpolation and $f\left(\mathit{r}\right)$, consisting of all basis functions, represents the function value to be evaluated at location $\mathit{r}$. The weight coefficients $w_i$ of those basis functions are determined by fulfilling the given values $s_i$ at the $N_R$ centers in Equation (14) and meeting the additional requirements in Equation (15) at the same time.

$$f\left({\mathit{r}}_i\right)={\mathit{s}}_i, i=1, 2, ···, N_R$$

(14)

$${\displaystyle \sum _{i=1}^{N_R}{w}_i\psi \left(r_i\right)}=0$$

(15)

Here, Equation (15) holds for all polynomials $\psi$ with a degree less or equal to that of polynomial $\phi$. Additionally, the minimal degree of polynomial $\phi$ directly depends on the choice of the basis function.

A large variety of basis functions are provided for data interpolation under different conditions. In this study, Wendland’s C² function is selected as the basic function [33]. The formulation of this function can be written as:

$$\varphi \left(\eta \right)={\left(1-\eta \right)}^4\left(4\eta +1\right)$$

(16)

where $\eta =‖\mathit{r}-{\mathit{r}}_i‖/d$ and $d$ denotes the supporting radius of the RBF series. The maximum value of $\eta$ is limited to 1, which gives a zero value to an RBF at a large distance $d$.

As depicted in Figure 4, the fluid and solid domains are coupled through a moving interface defined as $\Gamma (t)$. The spatial coordinates of the grid nodes and basis functions in RBF are used to determine the transformation and coefficients matrices in the process of data interpolation. In order to guarantee the accuracy of data interpolation through the non-matching meshes and bidirectional coupling calculation between the solvers, several coupling relations and compatibility conditions must be satisfied on the interface [34]. The compatibility conditions are proposed to adjust the coefficients matrix in RBF interpolation and provide rules for data exchange between the fluid and solid domains. The specific forms of the compatibility conditions are as follows:

$${\mathit{\sigma }}_S\cdot {\mathit{n}}_{\Gamma }=-p\cdot {\mathit{n}}_{\Gamma }$$

(17)

$${\mathit{x}}_F(t)-{\left.{\mathit{x}}_F(t)\right|}_{t=0}={\mathit{u}}_S$$

(18)

$$\frac{\partial {\mathit{x}}_F}{\partial t}=\frac{\partial {\mathit{u}}_S}{\partial t}={\mathit{v}}_F$$

(19)

Here, Equation (17) states that the structural stress tensor ${\mathit{\sigma }}_S$ must be equal to the fluid pressure $p$ along the normal direction ${\mathit{n}}_{\Gamma }$ at the fluid–solid interface. ${\mathit{x}}_F$, ${\mathit{v}}_F$, ${\mathit{u}}_S$ represent the time-dependent position vector of a fluid point, the fluid velocity field, and the structural displacement vector, respectively. Therefore, Equations (18) and (19) demonstrate that the structural displacements and velocities of the moving boundary must be compatible with the boundary motion of the fluid domain.

After the displacement information is transferred from the solid domain, mesh deformation method is performed to accommodate the new boundary shape and update the flow field mesh. Boundary layer mesh with high quality is necessary for the accurate prediction of flow transition. Hence, the elasticity-based method, which is considered as a mesh deformation method with high quality and strong robustness, is proposed here to ensure the good density distribution and orthogonality of the mesh during flexible skin deformation. The method is first proposed by Tezduyar [35] in dealing with the incompressible flow around a cylinder, and its main idea is to treat the mesh deformation problem as analogous to an assumed elasticity problem [36]. In the present study, an improved elasticity-based mesh deformation method, of which the elastic modulus $E$ is taken as the function of the dual-cell volume and the distance from the nearest solid boundary, is used during the coupling calculation. The modulus of an arbitrary grid element can be expressed as:

$$E_i=\beta \times (1/L_i)+(1-\beta )\times (1/V_i)$$

(20)

Here, $L_i$ is the distance from the element to the nearest solid boundary, $V_i$ is the dual-cell volume, and $\beta$ represents the weight coefficient of the two factors. The elastic modulus of an arbitrary grid element is set to be inversely proportional to the distance from the nearest moving boundary so that the cells near the boundary are stiffer than those far away. Therefore, the shape and quality of the mesh in and around the boundary layer can be well maintained during the deformation process [37]. What is more, the elastic modulus is also related to the cell volume to make the deformation near the boundary easier to propagate to the flow field.

3. Validation Cases

The accurate calculation of elasticity mainly depends on the implementation of reliable aerodynamic computation and data transfer between fluid and solid domain in the coupling process. Therefore, in order to validate the fluid–structural analysis framework used in this study, two cases are used to verify the reliability and capability of aerodynamic calculation and coupling analysis methods.

3.1. Validation of the Transition Prediction Method

The Aerospatiale A airfoil [38] was designed in 1986 at Aerospatiale in France and was tested in the 1.5 × 3.5 m wind tunnel of the ONERA F1, as well as the 1.4 × 1.8 m wind tunnel of the ONERA F2. In the experiment, the laminar boundary layer was observed to grow from the leading edge and end at a laminar separation bubble near the suction peak, located at approximately 12% of the chord length. The separation bubble directly caused a separation-induced transition and lead to the development of a downstream turbulent boundary layer. At the trailing edge of the airfoil, the turbulent boundary layer eventually separated due to a large inverse pressure gradient.

In the present study, a set of computational mesh with different mesh level was set up to study the effect of circumferential grid refinement on the calculation results. The number of circumferential grid points and mesh size of the five meshes are depicted in

Table 1. Details of the meshes.

Mesh Level	Extra Coarse	Coarse	Medium	Fine	Extra Fine
Number of circumferential grid points	80	160	240	320	400
Number of elements (×104)	10	13	16	19	22

. The grid spacing of the first layer is set to 1 × 10⁻⁶ m to ensure y⁺ ≈ 1 and the wall normal expansion ratio is given as 1.15 globally for all the meshes. The two-dimensional quadrilateral mesh is first generated and then directly stretched 0.2 chord length along the span direction. The computational mesh with extra fine mesh size is shown in Figure 5, and it can be seen that the mesh near the airfoil surface has been properly refined to better capture the complex flow phenomena in the boundary layer.

The flow conditions in the computation are as follows: the freestream Mach number is 0.15, the angle of attack is 13.3°, and the Reynolds number is 2.1 × 10⁶. The $\gamma -\overline{R{e}_{\theta t}}$ method is performed here for the prediction of laminar transition with a freestream turbulence intensity of 0.2%. Figure 6 shows the comparison of the calculated pressure coefficient distributions and the skin friction coefficient distributions on the suction side of the airfoil with the experimental data. As can be seen from the figure, the pressure coefficient distribution curve obtained by extra coarse mesh fluctuates obviously in the back half of the chord, indicating that the convergence of the calculation using this mesh is relatively poor. From the partial enlargement of the pressure distribution curve in Figure 6a and the distribution of the surface friction drag coefficient in Figure 6b, it can be seen that both calculations with the extra coarse and coarse meshes fail to capture the separation bubbles near the leading edge of the airfoil. As a result, the flow transition position predicted by the extra coarse and coarse mesh locate at 5% and 8% of the chord respectively which are significantly earlier than the experimental value. The mesh with medium size is able to predict the leading edge separation bubble and laminar flow transition phenomenon well, and the calculated pressure and skin friction coefficient distribution are also in good agreement with the experimental results. Then, the calculated results basically do not change significantly when the number of circumferential grid points increases beyond 240, which means that the mesh convergence is achieved for this case. Through the analysis above, a conclusion can be drawn that enough grid points in the streamwise direction are needed in order to properly resolve the flow separation and transition phenomenon in the boundary layer.

Based on the medium mesh, the aerodynamic characteristics of the foil were investigated in detail by the RANS method combined with kω SST and $\gamma -\overline{R{e}_{\theta t}}$ transition model. The surface pressure and skin friction distribution obtained from full turbulence and free transition calculations were compared with the experimental data in Figure 7. Obviously, the results predicted by the free transition method are in better agreement with the experimental values. In contrast, the absolute value of the pressure coefficient suction peak at the leading edge calculated by the turbulence model is only about 3.8, which is significantly lower than the experimental value of about 4.2. The differences of skin friction distribution caused by the two calculation methods can be seen more clearly in Figure 7b and Figure 8. In the free transition calculation, laminar flow with low skin friction can be maintained along the upper surface of the foil until 12% of the chord. Then, flow transition is triggered by the appearance of laminar separation bubble, causing the skin friction coefficient to increase steeply and exceed the values predicted in the full turbulence calculation.

Comparisons of the aerodynamic coefficients obtained from full turbulence and free transition computations with experimental values are given in

Table 2. Comparison of calculation results with experimental data.

Aerodynamic Coefficient	Experimental Data	Full Turbulence		Free Transition
		Value	$\mathit{\Delta }$	Value	$\mathit{\Delta }$
Lift	1.562	1.447	−7.4%	1.570	0.5%
Drag	0.0208	0.0315	51.4%	0.0222	6.7%

. The lift and drag coefficients obtained by the medium mesh with the transition model are 1.570 and 0.0222 respectively, which compare quite well with the experimentally measured values of 1.562 and 0.0208, yielding a difference of only 0.5% and 6.7%. In contrast, the lift and drag coefficients under full turbulence condition are 7.4% lower and 51.4% higher than the tested results.

The comparisons of the above results fully demonstrate that the transition prediction method used here is able to capture the laminar separation phenomenon, accurately predict the transition point, well characterize the changes in velocity distribution and flow properties within the boundary layer, and can better reflect the real aerodynamic performance of a low-speed airfoil.

3.2. Validation of the CFD/CSD Coupling Method

In order to validate the present fluid–structural coupling method, the structural response of a flexible membrane airfoil during aerodynamic excitation is simulated in this study. The validation model used here is simplified from the experimental model of Rojratsirikul et al. [39]. As shown in Figure 9, part A represents the two rigid ends of the airfoil which are fixed during the calculation, while part B represents the flexible membrane. The membrane is made from a transparent plastic sheet with a thickness of 0.2 mm, elastic modulus of E = 2.2 MPa and density of 1 g/cm³. A total of 16 monitoring points were placed in a uniform distribution along the chord length of part B to monitor the structural deformation during the coupling calculation.

In order to simplify the calculation, both the two-dimensional CFD mesh and FEM mesh were generated and directly stretched 0.1m along the span direction. The unstructured CFD mesh is composed of 59,700 nodes and 29,502 hexahedral elements. A hexahedral FEM mesh with 5522 nodes and 2500 elements in total is adopted in the computation. The freestream velocity is 7.5 m/s and the corresponding Reynolds number is 79,700. As the characteristic time of fluid is orders of magnitude smaller than the structural responses, instantaneous structural deformation calculation is performed with a constant coupling time step size of $\Delta t_C=0.002s$, while steady state flow computation with 500 steps is proposed at every time interval. Time evolution of the structural deformation of the membrane airfoil and the flow field streamlines at initial and final state are depicted in Figure 10. Before 0.5 s, the deformation and vibration of the structure is more pronounced and the shape of the flexible membrane changes significantly with time. After 0.5 s, the overall deformation tends to be stable, and there is only small amplitude of vibration exists. The comparison of flow field streamlines before and after deformation demonstrates that the flexible membrane changes its camber to adapt incoming flows to balance the pressure differences on the upper and lower side of the airfoil, therefore suppressing severe flow separation on the leeward side of the airfoil.

Figure 11 compares the predicted time-averaged deformation of the membrane airfoil with the experimental value at 16°. It can be seen that the calculated airfoil deformation is slightly smaller than the experimental value in the first half of the chord length, while the displacement of the elements between 0.6 c to 1.0 c fits well with the tested data. The variation of displacement with time at monitoring point 8, located near the mid-point of the chord, is depicted in Figure 12. The displacement at this point oscillates periodically with very small amplitude and it can be observed from the figure that there are about 4 oscillation periods between 1.891 s and 1.986 s. Hence, the calculated vibration frequency is 42.1 Hz compared to the experimental value of 40.5 Hz. The calculated results are in good agreement with the experimental values, and the numerical method of fluid–structure coupling used in this paper can be applied to the study of flexible membranes.

Figure 13 shows the variation of lift and drag coefficient with time during the 2 s. As the camber of the foil increases obviously after deformation, the lift coefficient increases greatly from 1.3 to about 1.45. The change of the coefficients above fully reflects the effect of the change in airfoil camber on the aerodynamic performance.

4. Results and Discussion

4.1. Geometry and Computational Model

A typical solar-powered UAV configuration is selected for the present research in this section. The UAV is actually a MALE (Medium Altitude Long Endurance) solar aircraft which is working as a communication relay between the HALE UAVs or low altitude satellites and the ground. Its maximum flight altitude is expected to reach 8000 m or even 10,000 m. As depicted in Figure 14, the configuration is mainly composed of a high-aspect-ratio wing, two vertical tail, one horizontal tail, and two connecting rods which connect the main wing with the vertical tails. The wing is divided into two trapezoidal wing segments named Section A and Section B with different taper ratio, and their relative positions in the wing are highlighted with different colors in Figure 15. The main geometric parameters of each part of the solar plane are shown in

Table 3. Main geometric parameters of the solar plane.

Component	Parameter	Value
Wing	Reference area	12.338 m2
	Mean aerodynamic chord	0.73 m
	Span	16.64 m
	Aspect ratio	22.8
	Span (Section A)	5.92 m
	Span (Section B)	2.4 m
	Mounting angle	0°
	Dihedral angle (Section A)	0°
	Dihedral angle (Section B)	3°
Horizontal wing	Chord	0.38 m
	Span	3.11 m
	Mounting angle	−3°
Vertical wing	Chord	0.638 m
	Span	1.11 m

.

As demonstrated in Figure 15, the wing adopts a high-lift airfoil with a relative thickness of 13.8% and the same foil is used from the wing root to the tip along the span without any geometric twist. NACA0012 symmetric airfoil with a relative thickness of 12% is adopted for both horizontal and vertical tail, and a mounting angle of −3° is set for the horizontal tail. Based on the configuration above, three sets of hexahedral meshes with different densities are produced for the computations of the solar plane’s aerodynamic characteristics. The detailed parameters of the meshes are listed in

Table 4. Details of the meshes in different level.

Mesh Level	L1	L2	L3
Number of elements (×106)	465	867	1519
Number of nodes (×106)	455	852	1494
yplus	2	1	0.5

, and the corresponding far field and surface mesh for L2 are displayed in Figure 16.

A finite element model based on the main structural components of the high-aspect-ratio wing model is presented in Figure 17. The model mainly contains the beams, the wing ribs, the leading and trailing edges of the wing, and the upper and lower surfaces. The leading and trailing edges of the wing each occupy 10% of the chord length, while the left 80% of the chord in the middle include the lower surface composed of hard skin and the upper surface formed by flexible membrane. All three wing beams are formed by I-shaped plastic core with carbon fiber material laid on the surface. The cross-sectional shape of them is shown in Figure 17, but there are significant differences in the specific parameters between them. The finite element model of the wing contains tetrahedral shell elements and beam elements, and the shell elements on the surface is refined to better simulate the elastic deformation of the flexible skin.

Table 5. Material properties of the wing.

Material	Elastic Modulus/MPa	Poisson’s Ratio	Density/(kg/m3)
T700	350,000	0.3	1590
CCM40J	210,000	0.3	1590
RS50	114	0.36	51.3
Membrane	1000	0.29	1350

shows the properties of all materials used in the wing finite element model, and

Table 6. Parameters and materials of the shell components in the wing.

Thin shell	Thickness	Material
Upper surface	0.06 mm	Membrane
Lower surface	0.4 mm	T700
Wing front	0.4 mm
Wing trailing edge	0.4 mm
Ribs	4 mm	CCM40J, RS50

depicts the specific parameters and materials of the structural components. T700 and CCM40J are two kinds of carbon fiber composite materials and RS50 is a kind of PMI foam plastic. The membrane in Table 5 represents the flexible plastic film applied on the upper surface of the wing. For the beams, foam plastic is used as the base material, and the carbon fiber is laid on its surface in a certain direction. Table 6 shows the parameters and materials of the shell components and

Table 7. Parameters and materials of the beam components in the wing.

Beam	Section Profile (a, b)	Number of Carbon Fibre Layers (0.1 mm/Layer)	Material
$\mathrm{Beam}1 (y\in [0,5.92]$)	(3, 5) mm	20	T700, RS50
$\mathrm{Beam}1 (y\in [5.92,8.32]$)	(2, 5) mm	10
$\mathrm{Beam}2 (y\in [0,5.92]$)	(6, 8) mm	50
$\mathrm{Beam}2 (y\in [5.92,8.32]$)	(2, 5) mm	10
$\mathrm{Beam}3 (y\in [0,5.92]$)	(3, 5) mm	20
$\mathrm{Beam}3 (y\in [5.92,8.32]$)	(2, 5) mm	10

demonstrates the parameters of section profiles, the materials and the number of carbon fiber layers of the beams. After assigning the materials to all structural components, the averaged mass distribution of the finite element model along the wing span is obtained and given in Figure 18. The weight of wing ribs is evenly distributed over the wing segments, and it is shown in the mass distribution that most of the weight is concentrated in Section A of the wing. Assuming that the upper surface of the wing adopts the same material and thickness as the lower surface, the total weight of the half wing will reach about 19.55 kg. By replacing the upper surface with flexible thin membrane, the total weight is reduced by nearly 28.4% to the value of 14 kg. The amount of weight reduction by applying the lighter flexible membrane on the upper surface is an important factor in assessing the overall performance of the wing.

4.2. Aerodynamic Performance of the Solar Plane

The mesh convergence analysis of the CFD numerical calculation is carried out for the solar-powered UAV model. The UAV has a cruise altitude of H = 6000 m, a cruise flight speed of V = 12.5 m/s, and a lift coefficient of C_L = 1.2 at cruise state. Based on L1, L2, and L3 meshes with different densities, RANS method combined with transition model is adopted to conduct numerical calculation and analysis at 4° and 14° angle of attack, respectively, to clarify the influence of mesh density on the basic aerodynamic calculation results of the solar plane with rigid wings. Figure 19 shows the variation of basic aerodynamic coefficients of the UAV configuration with mesh density at 4° and 14° angle of attack. The UAV achieves a lift coefficient of 1.2 at 4° angle of attack, mainly because the wing of the plane is equipped with a high-lift foil with a relative thickness of 13.8%. At a large angle of attack, the aerodynamic forces will fluctuate inevitably due to the appearance of flow separation on the wing. Hence, the coefficients C_L, C_D, and C_m at 14° are all averaged through several stable fluctuation periods. It can be seen that there is a great difference between the absolute values of aerodynamic coefficients calculated by L1 and L2 meshes, but with the mesh further refined to L3, the lift, drag, and torque coefficients all tend to converge to a certain value. In order to strike a balance between the computational accuracy and efficiency, the L2 mesh is used for all the subsequent numerical calculations.

Based on the L2 mesh, numerical calculations of the solar plane in the range of 0° to 18° incidence angles are carried out using the RANS method combined with the k-ω SST turbulence model and $\gamma -\overline{R{e}_{\theta t}}$ transition model, respectively. A comparison of the aerodynamic coefficients obtained from the two calculations is shown in Figure 20. For angles of attack above 12°, the coefficients C_L, C_D and C_m are all averaged through several fluctuation periods. The lift coefficients from the free transition calculation are obviously higher than those obtained from the full turbulence calculation. As the slope of the lift curve is larger in free transition calculation, the difference between lift coefficients increases linearly with the increment of incidence angle from 0° to 10°. The maximum lift coefficient in free transition calculation reaches 2.0, which is 0.1 higher than that from the full turbulence calculation. Apart from the above differences, the two calculation results also have many similarities. The flow nonlinearity both appears at 10°, and the stall characteristics are similar with the same stall angle of attack. As the laminar flow on the wing surface is taken into account in the free transition calculation, the predicted drag coefficients of the solar plane are significantly smaller than those from full turbulence simulation. Therefore, the lift drag ratio at 4° reaches 33.8 in the free transition case, which is 6.7 higher than that of the full turbulence case. In addition, the longitudinal static stability turns out to be smaller at the cruise angle in the free transition calculation due to the change of aerodynamic load distribution and magnitude.

Figure 21 demonstrates the comparison of surface pressure coefficient and skin friction coefficient distribution contour at different angles of attack in the two cases. The pressure on the upper surface is lower in the free transition case, which directly leads to a higher lift coefficient of the plane as compared in Figure 20. Meanwhile, significant discrepancies can also be observed in the distribution of skin friction. In the free transition case, a large range of laminar flow region with low skin friction exists on the wing surface. With the increase of incidence angle, the flow transition is triggered earlier, and the length of laminar flow region is obviously shortened. At 12° angle of attack, the laminar flow region on the wing surface only occupies 25% of the chord length, and the horizontal wing is nearly all covered by turbulent flow except a small laminar zone near the leading edge.

4.3. Coupling Calculation by the Time-Adaptive Coupling Strategy

As the main lifting component of the solar-powered UAV, the aerodynamic characteristics of the wing have a very important impact on the overall flight performance of the plane. Therefore, in order to simplify the problem, other components are omitted, and only the high-aspect-ratio wing is selected as the research object in the present study. The aeroelastic simulation of the wing with flexible skin is performed using CFD/CSD coupling method. The time-adaptive coupling strategy is used to adjust the time step during the calculation and improve the convergence. Three computational cases with details presented in

Table 8. Details of the different computational cases.

Case	Case A	Case B	Case C
Method	Constant	Adaptive	Adaptive
Initial time step of CSD computation (s)	0.001	0.001	0.001
$\Delta t_{Cmin}$	0.05	0.05	0.05
$\Delta t_{Cmax}$	0.05	0.2	0.5
Number of coupling steps	50	25	16
$\mathrm{Maximum} \mathrm{wing} \mathrm{tip} \mathrm{displacement} \mathrm{at} \alpha =4^{\circ }$ (mm)	190.29	190.31	190.35
CPU time (min)	400	186	112

are performed in this section. Case A is calculated by the method with constant coupling time step size, while Case B and Case C are solved using the time-adaptive coupling analysis approach with different $\Delta t_{Cmax}$. Instantaneous structural deformation calculation is performed with a time step of $\Delta t_S$, while steady state flow computation with 100 steps is proposed at every time interval. The results show that the maximum wing tip displacements at 4° obtained from the three cases are consistent with each other. The fluid–structural coupling simulations with both constant and adaptive time step are performed on a DELL TOWER with 1 CPU and 10 cores, and the CPU time spent on each computational case is listed in Table 8. By comparison, Case C only takes 28.0% time of Case A, which obviously shows that the adaptive coupling time stepsize approach can greatly reduce the number of coupling steps and computational expense.

The adaptive variation process of structural calculation and fluid–structural coupling time step is given in Figure 22. The structural calculation time step is adjusted according to the convergence of deformation during the whole calculation process. The time step increment tends to decrease if the deformation is difficult to converge and increase significantly when the convergence is easier to achieve in order to reduce the computational effort. The calculated displacement distribution of beam2 along the span at 4° in Case C is compared with the measured data in Figure 23. The measured data were derived from a full-scale static load test for aircraft structure. In this test, the deformation data of the beam structure was recorded by sensors equipped on the beam. Due to the limited test conditions, there was not much detailed data available for comparative analysis. The calculated displacement distribution is less bent compared to the measured results, and the calculated deformation at the wing tip is only 5% lower than the measured value. The results indicate that the time-adaptive coupling analysis approach can well reflect the actual deformation state of the wing structure. The wing skin generally has a role in transferring the surface pressure loads to the beams and ribs. The change of wing skin has minimal effect on the overall bending and torsional deformation of the wing. The aeroelastic simulation of the wing with “hard skins” is performed at 4° and the maximum wingtip displacement is 188.20 mm, which is only slightly less than that of the wing with flexible membrane. Hence, it can be concluded that the wing structure is still on the safe side after replacing the hard upper surfaces with flexible membrane.

4.4. Aeroelastic Analysis of the Wing with Skin Flexibility

Numerical calculations of the two-way fluid–structure coupling were carried out in the range of 0° to 18° angle of attack to obtain the deformation characteristics of the wing structure and flexible surface under different incidence angle. Figure 24a shows the comparison of aerodynamic shape of the deformed wing at different angles of attack, while Figure 24b depicts the displacement and torsional deformation of the wing tip through detailed data. Airflow orientation and the indication of LE (Leading edge node) and TE (Trailing edge node) are also demonstrated in Figure 24a. It can be seen that the displacement and torsional deformation of the wing tip both increase linearly with the increment of incidence angle before 10°. Then, the displacement deformation of the leading and trailing edge reaches the maximum value of 300.24 mm and 290.77 mm, respectively, at 12°. When the angle of attack increases further, the wing tip displacement decreases gradually. The wing tip torsion angle compared with the rigid case is negative before 4°, and then gradually increases until it reaches the maximum value of 1° at 14° angle of attack. The variation of wing deformation with angle of attack is directly related to the flow conditions on the wing. When the angle of attack reaches 10°, small local flow separation appears at the trailing edge of the wing and the aerodynamic load begins to increase non-linearly. Then, when the incidence angle comes to 12°, the aerodynamic load and deformation of the wing both reach their maximum values. After that, the loss of lift due to flow separation exceeds the lift increment brings by the increase of incidence angle. As the angle of attack continues to rise, the flow separation zone at the trailing edge gradually expands to the front and directly leads to the subsequent decrease in the aerodynamic load and wing deformation.

For the high-aspect-ratio wing with skin flexibility, the wing profile is subject to displacement, torsional deformation, and changes in the geometric shape of the flexible skin under aerodynamic loads during aeroelastic calculation. A comparison of the deformed wing profiles at 4 sections along the span under various incidence angles is given in Figure 25 after removing the bending deformation. The specific changes in the aerodynamic performance of the local wing profiles can be reflected in the comparisons. x and z are the coordinates of the wing profile on the x and z axes respectively and c represents the local chord length of the wing profile. The black dot located at the middle of the upper side of the wing profile represents the position where the upper surface is connected with the middle beam of the wing. Taking the location of the black dot as the reference point to translate the deformed profiles so that all reference points are overlapped at the same position. At this time, the overall torsional deformation of the wing profile at each section and the elastic deformation of the upper surface flexible part can be clearly observed. As can be seen from the figure, the geometric deformation of the flexible skin is dominant at the section located at 15% of the half-wingspan, and the maximum deformation position gradually moves forward with the increase of incidence angle. As this section is very close to the symmetry plane, almost no torsional deformation is observed for this profile. When the section further moves outward along the span, the amount of torsional deformation gradually increases.

Monitoring points are set on the upper surface to obtain the deformation of the flexible membrane under aerodynamic load. As shown in Figure 26, monitoring points M1, M2 are set at 45% of the wing span and M3, M4 are set at 75% of the wing span. Among them, M1 and M3 are located at 30% of the local chord while M2 and M4 are at the place 70% of the local chord length away from the leading edge. The deformation of the monitoring points at different angles of attack presented in the figure show that the deformation of flexible surface around 75% of the span is smaller than that at 45%. The deformation of M2 at all angles of attack is the largest among the four monitoring points and the maximum displacement is about 4.27 mm. For M1 and M3, the displacements increase linearly from 0° to 10°, and then decreases monotonically. For M2, the displacement increases only up to 8°, and then decreases rapidly from the maximum value of 4.27 mm at 10° to the minimum value of 3.16 mm at 14°. After that, the wing surface deformation increases linearly between 14° and 18° due to the local pressure reduction caused by the large flow separation on the surface near the trailing edge. The variation of deformation with angle of attack at M4 is basically consistent with M2, but the overall deformation amplitude is smaller than that of M2. The maximum displacement is 3.26 mm at 6° and the minimum value is 2.06 mm at 14°. The deformation of membrane surface is influenced by flow separation on the upper side of the wing. For the high-aspect-ratio wing in this research, flow separation firstly appears at the trailing edge of the trapezoidal section with no dihedral angle, and then expands towards the front and wingtip. As a result, M2 is the first to be affected when local flow separation occurs at 10°. Afterwards, the flow separation area gradually expands with the angle of attack and begins to impact the membrane deformation near M1 and M3 at 12°, resulting in a gradual reduction of the structural node displacements after 12°.

Figure 27 compares the aerodynamic coefficients of the wing before and after deformation of the whole wing structures. It should be noted that the deformed wing is found to have a slightly larger slope of the lift coefficient curve than the rigid wing between 2° and 10° angle of attack. The lift coefficient increases linearly with the incidence angle changing from 2° to 10°, and finally reaches the maximum value of 0.046 at 10°. After the wing deformation, the stall characteristics of the wing at high angle of attack are significantly deteriorated. The stall angle of attack is reduced from 14° to about 12°, and the maximum lift coefficient also faces a reduction from 1.92 to 1.89. As indicated in Figure 27a, the deformed wing produces a substantial lift reduction over the rigid case particularly at angles of attack exceeding 12°, and the maximum decrease occurs at 18° with a value of 0.146. In addition, the elastic deformation of the wing also results in the enhancement of drag coefficient at low angles of attack. At 4°, the drag coefficient increases from 0.0320 to 0.0337 with an increment of 5.3%, and thus directly leads to a reduction of lift–drag ratio by 1.9. For high angles of attack above 12°, a considerable increment of drag coefficient is observed in Figure 27b due to the aggravation of flow separation on the deformed wing by the elastic deformation.

The change of wing surface pressure distribution after elastic deformation is directly related to the deformation of each wing section. The influence of deformation on the pressure distribution is mainly manifested in two aspects: First, the torsional deformation of the wing profile changes the local incidence angle of flow; Secondly, the deformation of the upper flexible wing surface changes the geometric and local curvature of the profile, and then affects the velocity distribution in the boundary layer. The comparison of pressure distributions at various sections of the wing before and after deformation under different angles of attack is shown in Figure 28. It can be clearly seen that the pressure distribution on the upper surface at 2° and 4° angles of attack is mainly influenced by the deformation of the flexible skin. The flexible surface protrudes outward under the action of aerodynamic load, which accelerates the surface flow to a certain extent, and therefore resulting in the reduction of local pressure. When the angle of attack comes to 8°, the influence of wing torsional deformation on pressure distribution begins to appear. For sections located at 45%, 75%, and 95% of the span, the positive torsion of the wing profile directly leads to the increase of local flow angle and suction peak near the leading edge of the upper surface.

The deformation of wing surface will cause the change of transition position. Figure 29 shows the transition location on the upper surface of the section located at 30% of the span before and after wing deformation under various incidence angles, and the differences between them are highlighted by the blue line with diamond symbols. At low angles of attack, the length of laminar region on the surface is relatively long, and the deformation will cause the laminar–turbulent transition to be triggered earlier. Then, as the incidence angle increases, the influence of wing deformation on the transition position decreases rapidly. After the angle of attack reaches 12°, the transition on the upper surface of the rigid wing is triggered at the location very close to the leading edge. At this time, the deformation of the wing surface will hardly affect the transition position anymore. Presented in Figure 30 is the contour of wing turbulent kinetic energy at 4° and 8° angle of attack and the enlarged area shows the details of the wing segment between 25% and 35% of the span. The results show that the transition position on the wing surface changes from 51.2% to 42.5% of the chord length after deformation. However, when the angle of attack comes to 8°, no significant change is observed on the transition position. It is indicated that the flexible surface deformation is supposed to be the main factor affecting the laminar–turbulent transition location on the upper surface of the wing.

The change of stall characteristics is closely related to the variation of separated flow above the wing before and after deformation. The streamlines of the flow field at four sections along the span at 14° and 18° angle of attack are shown in Figure 31, where the red dots in the figure represent the starting position of the wing surface flow separation. The comparison in the figure shows that the starting point of flow separation on the upper side of wing profile at 15% section tends to move forward after deformation, and the size of separation zone above the trailing edge is obviously expanded. The deterioration of flow separation characteristics here is mainly caused by wing surface deformation because there is no significant torsional deformation at this section, as demonstrated in Figure 25a. For wing profiles at 45% and 75% of the span, only small differences can be observed in the trailing edge separation at 14°, while the flow separation is greatly intensified with the expansion of separation zones at 18° angle of attack. Here, the change in flow structures is primarily attributed to the combined effect of torsional and flexible surface deformation of the wing. At 95% section near the wing tip, the flow around the rigid and deformed wing profile are basically similar to each other.

5. Conclusions

In this paper, a loosely coupled partitioned approach integrated with a time-adaptive coupling strategy is developed and the fluid–structural coupling analysis of a high-aspect-ratio wing from a solar-powered UAV is performed to investigate the compounding effects of structural and flexible skin deformation on the aerodynamic, laminar, and flow separation characteristics of the wing. Based on the comparison and analysis of the results, the underlying flow physics are revealed, and several important conclusions are drawn as follows:

(1)

The time-adaptive coupling approach takes only 28.0% of the time spent by the method with constant coupling time step size for a single case. It obviously shows that the adaptive coupling time stepsize approach can greatly reduce the number of coupling steps and computational expense on the premise of ensuring calculation accuracy;

(2)

Obvious bending, torsion, and flexible surface deformation are observed on the wing at various incidence angles. The maximum displacement and torsion angle of the wing tip is 300.24 mm and 1°, respectively. The surface deformation depends on its position on the wing and the maximum displacement absolutely exceeds 4 mm;

(3)

The elastic deformation degrades the aerodynamic performance of the wing. The lift–drag ratio around the cruise angle is slightly reduced and the stall characteristic becomes worse after deformation;

(4)

The laminar–turbulent transition location on the upper surface is triggered earlier at small angles of attack, mainly due to the deformation of flexible skin. The expansion of flow separation at large angles of attack is found to be the combined effect of the flexible surface and torsional deformation of the wing.

The conclusions above can help designers get a better understanding of the differences in aerodynamic performance between the deformed and the designed rigid wing. The time-adaptive loosely coupled analysis method developed in this paper is capable of providing a reliable and efficient prediction for the real aerodynamic performance of a high-aspect-ratio wing in the design stage.