Modal Phase Study on Lift Enhancement of a Locally Flexible Membrane Airfoil Using Dynamic Mode Decomposition

Abstract

The dynamic mode decomposition serves as a useful tool for the coherent structure extraction of the complex flow fields with characteristic frequency identification, but the phase information of the flow modes is paid less attention to. In this study, phase information around the locally flexible membrane airfoil is quantitatively studied using dynamic mode decomposition (DMD) to unveil the physical mechanism of the lift improvement of the membrane airfoil. The flow over the airfoil at a low Reynolds number (Re = 5500) is computed parametrically across a range of angles of attack (AOA = 4°–14°) and membrane lengths (LM = 0.55c–0.70c) using a verified fluid–structure coupling framework. The lift enhancement is analyzed by the dynamic coherent patterns of the membrane airfoil flow fields, which are quantified by the DMD modal phase propagation. A downstream propagation pressure speed (DPP) on the upper surface is defined to quantify the propagation speed of the lagged maximal pressure in the flow separation zone. It is found that a faster DPP speed can induce more vortices. The correlation coefficient between the DPP speed and lift enhancement is above 0.85 at most cases, indicating the significant contribution of vortex evolution to aerodynamic performance. The DPP speed greatly impacts the retention time of dominant vortices on the upper surface, resulting in the lift enhancement.

1. Introduction

Inspired by the morphological characteristics of bat’s wings, membrane wings are proven to attain aerodynamic advantages by passively adapting their cambered shapes to unsteady flow conditions and promoting the membrane oscillation to delay the flow separation [1]. However, the underlying interaction of the flexible structure and unsteady flow, particularly the coupling patterns associated with the aerodynamic performance, should be explored further.

Numerical and experimental studies have emphasized the significant role of the coupling between dynamic vortex shedding and structural response in the aerodynamic advantages observed in membrane wings. The pioneering work of Smith and Shyy [2] used computational fluid dynamics for the flow past the membrane wing by solving the Navier–Stokes equations. The work found that the lift coefficient is decreased as a result of a laminar separation bubble (LSB) generated near the leading edge. The results show a great difference from a potential-flow-based solution. Gordiner [3] computationally investigated two-dimensional membrane airfoils with a robust sixth-order Navier–Stokes solver. They highlighted a close coupling between unsteady vortex shedding and the dynamic structural response, resulting in stall delay and lift improvement. Molki et al. [4] observed a uniform distribution of separation bubbles on the membrane wing surface, consequently reducing the recirculation region. Serrano-Galian et al. [5] employed direct numerical simulation (DNS) to study the dynamics of fluid–membrane coupled systems. Their findings indicate that membrane airfoils can improve aerodynamic performance near stall conditions. Kang et al. [6,7,8] proposed a locally flexible airfoil model, in which the flexible structure passively interacts with low Reynolds number flow. In this model, the flexible skin is reduced to the leading edge region $(0,0.1c)$, where c is the airfoil chord. It shows that the aerodynamic performance is enhanced by the consecutive moving separation bubbles induced by the oscillation of the locally flexible skin. Arif et al. [9,10] presented a NACA0012 airfoil model with a localized elastic membrane skin mounted on the suction surface, showing a positive effect on the aerodynamic performance of the airfoil. Sun et al. [11] applied the compliant materials to the NACA0012 airfoil skin and found that the thickness of the membrane and Young’s modulus significantly impact the aerodynamic characteristics near the stall angle of attack. Though the studies on the locally flexible membrane airfoil show the positive impact of the membrane on the aerodynamic performance, how to choose the location of the locally flexible skin is yet unclear.

From the experimental aspect, Song et al. [12] investigated the behavior of compliant membrane wings at different flow speeds. They found that the camber and vibration mode of the membrane wing are dependent on the angles of attack. Rojratsirikul et al. [13] discovered that the vibration-induced separation bubbles on the wing surface are the crucial factors for the lift enhancement. Their experimental work further confirmed the relationship between flow structure and vibration modes of the membrane wing in the chordwise and spanwise directions [14,15]. AçıKEL et al. [16] investigated a wind turbine airfoil with a partially flexible surface experimentally. They found that the presence of the membrane suppressed the formation of LSBs on the surface and constrained the wake region, subsequently resulting in enhanced lift and reduced drag. They also found that the partially flexible membrane interacted with the LSB. Bleischwitz et al. [17] conducted wind tunnel experiments for the aerodynamics of flexible membrane wings. The results indicate that the dynamic motions of the membrane wings maintain the leading-edge vortices to improve the lift.

The active flow control techniques can also be applied to suppress the LSB [18]. Shahrokhi et al. [19] investigated the aerodynamic performance of a double-slotted morphed flap airfoil using CFD simulations. The results show significant improvements in lift and drag reduction, with a maximum lift coefficient of 3.03 at a 30° deflection angle. The morphed flap design effectively delays flow separation, enhancing overall efficiency. Taleghani et al. [20] explored using tangential blowing to boost the effectiveness of an aircraft’s vertical tail. Results indicated an 8.5% increase in side force with a single slot and a 4%increase with discrete slots, which also use 70% less air. In addition, many intelligent materials have been used for the active flow control such as shape memory alloy (SMA), dielectric elastic material (DE) and piezoelectric macro-fiber composite (MFC) material. Dong et al. [21] investigated a morphing airfoil using SMA springs to adapt its shape for better aerodynamic performance. The SMA springs effectively control skin deformation, showing potential to enhance aircraft efficiency. Compared with SMA, DE and MFC are of particular interest due to their advantages of simple structure and high actuation frequency [22,23].

Recent studies have further revealed the frequency lock-in phenomenon in the coupling between vortex shedding and membrane vibration. Gursul et al. [24] found that the self-excited vibrations of flexible wings can excite boundary layer instabilities to delay stall and increase lift as the frequencies of the wing structure and fluid flow evolution are closely matched. Tiomkin et al. [25,26] found that substantial aerodynamic benefits are achieved when the reduced frequency of the membrane locks onto the frequency of the flow. Li and Khoo [27] found that the onset and shift of the membrane vibration modes depend on the frequency lock-in between the natural frequency of the tensioned membrane and the vortex shedding frequency or its harmonics. He et al. [28] confirmed the occurrence of frequency lock-in in the flexible membrane wings at various Reynolds numbers. However, there exists frequency mismatch at Re = $6\times {10}^4$, where the dominant frequency of the flow is lower than the vibration frequency of the membrane, because of the interaction between the flows from leading and trailing edges. The frequency synchronization between unsteady flow and the membrane structure is also observed in the flexible wings with active flow control techniques [29,30]. In most of the studies above, the frequency lock-in phenomena is referred to as the relations of the first-order frequency. The authors’ work further demonstrates the second-order frequency lock-in also imposes great impact on aerodynamic performance for low-Reynolds-number airfoil [31]. The lift coefficient can be further boosted as the occurrence of the frequency lock-in with the first-order and second-order frequencies. It also indicates different lift enhancement effects are observed in the cases of the primary and secondary frequency lock-ins. It implies an unsolved connection between the lock-in frequency and flow characteristic structures for the membrane wings to account for the different lift enhancement effects.

Modal decomposition [32,33] can effectively extract the main characteristics of the flow and elucidate the underlying mechanisms of critical flow features, particularly those related to characteristic frequencies. This is especially relevant for unsteady flows influenced by both low and high frequencies [34]. The proper orthogonal decomposition (POD) method is a typical modal decomposition method which has been favored by researchers due to its simplicity and effectiveness. However, it is incapable of the accurate identification of the modal growth rate and frequency characteristics of the dynamic systems. As a follow-up method, the spectral POD method improves its frequency identification capability with a specific frequency range. The dynamic mode decomposition (DMD) method, on the other hand, makes up for the shortcomings of POD in frequency recognition. It can globally extract characteristic modes of the flow field with the individual flow frequency and mode growth rate [35,36]. It can also highlight the dominant features and essential components from complex evolution. Hence, it has been extensively used in flow analysis for the transonic buffet [37,38], shear layer characteristics [39], and combustion instabilities [40]. Mohan et al. [41] studied the inherent physics behind the phenomena of dynamic stall in plunging airfoils using dynamic mode decomposition. They employed DMD to identify a dominant dynamic mode featuring a large recirculation region over the airfoil and four dynamic modes’ harmonics. Tiomkin et al. [26] found that the dominant mode shape of the membrane oscillations behaves like the second structural mode, with a frequency lower than the structural frequency. However, they ignored the impact of fluid–structure interaction and assumed that the membrane oscillations result from the inherent instability of the membrane wing, other than vortex shedding in the flow. Zhang et al. [42] investigated a flow past a forced-oscillation cylinder. The flow modes resolved by the DMD illustrate the vortex shedding mode and the pattern alternation of the shedding vortices behind the cylinder. These studies mainly focus on the DMD modal amplitude of the unsteady flow. While DMD excels in extracting modal frequencies and growth rates, phase information for capturing spatiotemporal relationships has often been under-emphasized. It is critical in describing spatial propagation and synchronized behaviors in dynamical systems, enhancing insights into the interactions of spatiotemporal modes. In this study, the information of the modal phase is used to quantitatively investigate the impact of the perturbation from the fluid–structure interaction of the membrane related to the lift enhancement.

In this paper, the flow past the locally flexible membrane airfoil is simulated by a fluid–structural coupling solver. The fluid–structure interaction is further analyzed by the DMD method to provide profound insight on the relationship between phase pattern and the aerodynamic performance of the airfoil. The structure of this study is organized as follows: The numerical methodology and the DMD method are presented in Section 2. The computational model and aeroelastic responses of the elastic membrane airfoil are illustrated in Section 3. The DMD modal shape and phase results are analyzed in Section 4. The DMD phase-based coherent pattern is investigated in Section 5. Finally, the main conclusions are drawn in Section 6.

2. Methodology

2.1. Computation Model and Setup

In our study, the computation model and the geometry of the fluid and structure domains are depicted in Figure 1. Rojratsirikul et al. [43] investigated the fluid–structure interaction of the membrane wing in three dimensions. They discovered that the chordwise vibrational modes are dominant at high angles of attack for a low Reynolds number flow, which is consistent with the two-dimensional flow over the membrane airfoils. Further, the dominant vibrating frequencies of the membrane exhibit strong similarities with those of two-dimensional membrane airfoils. Gordnier [3] also stated that the fluid flow and structure response can be assumed to be two-dimensional at low Reynolds number. Therefore, the current study adopts the 2D assumption for the research of the membrane wing. An airfoil with the flexible membrane skin is proposed. A two-dimensional, laminar flow is assumed in the low Reynolds number range in light of the work from Gordnier [3]. The far field of the fluid domain is 40 times the chord length of the airfoil. A no-slip condition is adopted for the surface of the airfoil in the fluid solver. The Reynolds number based on the chord of the airfoil is Re = 5500, where the Reynolds number is defined as $Re={\displaystyle \frac{{\rho }_{\infty }{V}_{\infty }c}{{\mu }_{\infty }}}$. The velocity of the freestream is 1 m/s.

Experimental findings [43] indicate that the membrane undergoes an essentially oscillatory deformation in the transverse direction. The membrane, which consists of a thin latex sheet, is stretched between two rigid mounts. Both ends of the membrane are subjected to fixed constraints. Uniform pressure of 1 atm is applied on the internal face of the membrane structure. The length of the membrane (the distance between two fixed constraints as shown in Figure 1) can be parametrically altered by changing the fore location of the membrane. The structural parameters of the membrane airfoil are outlined in

Table 1. Structural parameters of the elastic membrane airfoil.

Structural Parameters	Value
Chord (c)	300 mm
Length of membrane	0.55c–0.70c
Thickness (t)	0.2 mm
Elastic modulus (E)	34,150 Pa
Structural density (${\rho }_s$)	402.3 kg/m3
Pressure on the lower surface	1 atm

.

In the study, the impact of the length of the membrane for the locally flexible airfoil is investigated. The aeroelastic effects of the elastic membrane airfoil are analyzed by the parametric studies in the range of angle of attack AOA = 4°–14° and the length of membrane LM = 0.55c–0.70c. All the cases are computed using the high-Fidelity aeroelastic solver depicted in Section 2.2. The Courant number is set up as 1. Three different time step sizes are chosen for the time step sensitivity analysis, which are 0.005 s, 0.001 s, and 0.0005 s. The relative errors between the time-averaged lift coefficients with these three time steps are all less than 5%. Taking into account both the computational cost and accuracy, a time step size of 0.001 s is selected. The numerical solution is completed in 10 seconds, consisting of ${10}^4$ steps. The inner iteration between two consecutive time steps is terminated once the residuals of the fluid properties reach ${10}^{-10}$. All cases are required to meet this convergence criterion.

Mesh independence tests are conducted using three different grid sizes in

Table 2. Grid information for mesh independence tests and the related results.

Cases	Coarse-Size Grid	Middle-Size Grid	Fine-Size Grid
Grid size	125 × 65	254 × 81	402 × 161
Time-averaged lift coefficient	0.5627	0.5919	0.5942
1st-order frequency of the flow	3.50	3.40	3.40
1st-order frequency of the structure	3.40	3.40	3.50

[44]. These meshes in the near-wall region are depicted in Figure 2, Figure 3 and Figure 4, respectively. The simulations are performed at AOA = 10° and Re = 5500. Figure 5 demonstrates the time-dependent lift coefficients of the membrane for three different sizes of mesh. It is shown that the evolution of the lift coefficients for the three meshes is consistent. The relative error in the time-average lift coefficient between the coarse-size grid and the middle-size grid is 4.93% (based on the middle-size grid). The related error between the middle-size grid and the fine-size grid is significantly decreased to 0.39%. In addition, the first-order frequencies of structure and flow are nearly identical for all three grid sizes. Figure 6 gives the time-average deformation of the membrane for three different sizes of mesh. It shows that the deformation of the membrane is almost identical with all three grids. Consequently, for a balance between computational time and accuracy, the middle-size grid is selected for subsequent analysis.

2.2. Aeroelastic Model and Solver

The aeroelastic solver consists of two parts: the structure solver module and the fluid solver module. For the structural part [45], the constitutive relationship of the Neo-Hookean model for the hyperelastic material of the membrane is adopted. The geometric nonlinearity is considered in the model. A pin–pin boundary condition is applied on both ends of the membrane. The upper surface of the membrane is coupled with the unsteady flow, whilst the lower surface suffers a uniform pressure of 1 atm. The structural domains are discretized using second-order quadrilateral elements. The node displacement vector u in the unit can be expressed as

$$\mathit{u}=\Sigma {\mathit{u}}^A{N}^A$$

(1)

where the index represents the node of the element, ${\mathit{u}}^A$ is the node displacement, and $N^A$ is the shape function.

The Galerkin finite element method is used to obtain the structural equation in discrete form:

$$\mathit{M}{\ddot{\mathit{u}}}_i+{\mathit{K}}_i\Delta {\mathit{u}}_{i+1}={\mathit{F}}_{body}+{\mathit{F}}_{CFD}\left({\mathit{u}}_i\right)$$

(2)

$${\mathit{u}}_{i+1}={\mathit{u}}_i+\Delta {\mathit{u}}_{\mathit{i}+\mathbf{1}}$$

(3)

For the fluid solver, the unsteady Navier–Stokes equations are approached by the finite volume method for the low-Reynolds-number flow. The discrete spatial scheme is the ROE scheme with the third-order MUSCL interpolation method. The LUSGS implicit method is used for time advancing. The details of the method can be referred to in the authors’ work [46].

Figure 7 shows the aeroelastic coupling framework of the membrane airfoil. The loose coupling method shown in Figure 7b is used to deal with the fluid–structural interactions. A high-performance computing cluster is used to perform a multi-physics simulation.

2.3. CFD/CSD Coupling Verification

The geometry of the verification case is from the experimental setup by Rojratsirikul [13]. The model is a membrane wing, fixed at both ends. The relevant parameters of fluid and structure are shown in

Table 3. Parameters of the fluid and structure in the verification case.

Structural Parameters		Flow Parameters
Chord (c)	136.6 mm	Freestream velocity	1.0 m/s
Thickness (t)	0.2 mm	Fluid density	1.0 kg/m3
Elastic modulus (E)	34,150 Pa	Re	2500
Structural density (${\rho }_s$)	402.3 kg/m3	Angle of attack	4°

. Figure 8 and Figure 9 show the aeroelastic results of the membrane wing when the angle of attack (AOA) is 4° and $Re=2500$. The time-averaged deformation of the membrane and the airfoil surface pressure coefficient are in reasonable agreement with the results from Buoso et al. [47]. The relative maximum deviation of the time-averaged deformation of the structure is 2.75%, which shows the reliability and accuracy of the proposed model. It is seen that the deformation of the membrane is under-predicted due to the relatively low pressure distribution over the membrane airfoil shown in Figure 9.

2.4. Dynamic Mode Decomposition

The dynamic mode decomposition (DMD) is a data-driven matrix decomposition that is capable of extracting the spatial–temporal coherent structures arising in dynamical systems. Consider a discrete-time dynamical system from the CFD discrete form:

$${\mathit{X}}_{\mathit{k}+\mathbf{1}}=\mathit{F}\left({\mathit{X}}_{\mathit{k}+\mathbf{1}}\right)$$

(4)

where k is a timestep at time $t=k$.

Then, a set of snapshots of flow fields is collected from CFD simulation, and rewritten in the matrices $\mathit{X}$ and ${\mathit{X}}^{\prime }$ given by

$$\begin{array}{c}\hfill \mathit{X}=[x_1,x_2,\cdots ,x_{m-1}]\\ \hfill {\mathit{X}}^{\prime }=[x_2,x_3,\cdots ,x_m]\end{array}$$

(5)

where data matrix $\mathit{X},{\mathit{X}}^{\prime }\in {\mathbb{C}}^{n\times m-1}$, m is the snapshots number, and n is the dimension of the collected data.

Define a linear operator A that satisfies

$${\mathit{X}}^{\prime }=\mathit{A}\mathit{X}$$

(6)

which yields the operator $\mathit{A}={\mathit{X}}^{\prime }{\mathit{X}}^{†}$, where ${\mathit{X}}^{†}$ denotes the Moore–Penrose inverse of $\mathit{X}$.

In the fluid flow, since the CFD state vector is typically high-dimensional, the eigen-decomposition of $\mathit{A}$ has to be approximated. One of the commonly used techniques is to leverage a singular value decomposition (SVD) of the snapshot matrix X to obtain the low-rank approximation of $\mathit{A}$, i.e.,

$$\begin{array}{c}\hfill \mathit{X}=\mathit{u}\Sigma {\mathit{V}}^{*}\\ \hfill {\mathit{X}}^{\prime }=\mathit{A}\mathit{u}\Sigma {\mathit{V}}^{*}\end{array}$$

(7)

where $\mathit{u}\in {\mathbb{C}}^{n\times n},\mathit{V}\in {\mathbb{C}}^{m-1\times m-1}$, and $\Sigma \in {\mathbb{C}}^{n\times m-1}$. Note that ${\mathit{V}}^{*}$ is the conjugate transpose of $\mathit{V}$. Truncate the SVD by selecting the r dominant singular values and their corresponding modes. Define a low-rank r linear operator $\tilde{\mathit{A}}$:

$$\tilde{\mathit{A}}={\mathit{u}}^{*}{\mathit{X}}^{\prime }\mathit{V}{\Sigma }^{-\mathbf{1}}$$

(8)

where $\tilde{\mathit{A}}\in {\mathbb{C}}^{r\times r}$ has some of the eigenvalues of $\mathit{A}$ and the $ith$ eigenvalue ${\mu }_i$ is given by

$$\tilde{\mathit{A}}{\omega }_i={\mu }_i{\omega }_i$$

(9)

where ${\omega }_i$ is the corresponding eigenvector of ${\mu }_i$. The dynamic mode and the amplitude are defined as

$$\begin{array}{cc}\hfill \Phi & =\mathit{u}\mathit{W}\hfill \\ \hfill b={\mathit{W}}^{-\mathbf{1}}& {\mathit{u}}^{*}{x}_1={\Phi }^{-\mathbf{1}}{x}_1\hfill \end{array}$$

(10)

Then, the snapshots of the flow field are the function of the modes as follows:

$$\begin{array}{cc}\hfill [x_1,x_2,\cdots ,x_m]=\Phi b& {\mathbf{\Pi }}_{\mathit{vander}}=[{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_r]\hfill \\ \hfill \left[\begin{array}{ccc}{b}_1& & \\ & \cdots & \\ & & b_r\end{array}\right]& \left[\begin{array}{cccc}1& {\mu }_1& \cdots & {\mu }_1^{m-1}\\ \cdots & \cdots & \cdots & \cdots \\ 1& {\mu }_r& \cdots & {\mu }_r^{m-1}\end{array}\right]\hfill \end{array}$$

(11)

It is observed that the response of the dynamical system depends on the dynamic mode $\Phi$, dynamic amplitude b, and the eigenvalues. Each eigenvalue represents the evolution process of the corresponding dynamic modes. ${\displaystyle \frac{Re\left[log\left({\mu }_i\right)\right]}{\Delta t}}$ is the growth rate of the mode, and ${\displaystyle \frac{Im\left[log\left({\mu }_i\right)\right]}{\Delta t}}$ is the frequency of the mode, where $\Delta t$ is the interval of the snapshots.

The mode selection is critical for the reflection of the dynamic behaviors. The typical choice is based on the magnitude of the amplitudes b. However, the amplitude criterion is not perfect because there exist modes with very high amplitudes but which are very fast damped. Hence, an energy criterion for which the amplitude of the mode is weighted by its temporal coefficient from Tissort [48] is used in this study for mode selection:

$$E_i={\parallel {\varphi }_i{b}_i{\left({\mathbf{\Pi }}_{\mathit{vander}}\right)}_i\parallel }_2$$

(12)

where ${\left({\mathbf{\Pi }}_{\mathit{vander}}\right)}_i$ represents the $ith$ row of the matrix ${\mathbf{\Pi }}_{\mathit{vander}}$. The dynamic mode is selected until the modal energy converges to 99% of the whole flow field.

3. Coupling Response

3.1. Aerodynamic Performance Comparison

Figure 10 presents the variation of the time-average lift coefficient between the locally flexible membrane airfoil with respect to the rigid airfoil at different AOAs and LMs. The results show that the lift coefficients of all membrane airfoils are increased compared to the rigid airfoil. The minimum lift enhancement is 0.08% at AOA = 4° and LM = 0.70c. It is seen that the lift enhancement effect is amplified with increasing AOA for the same LM. Such effect reaches its maximum at AOA = 12°. The lift coefficient of the membrane airfoil increases by 92.61% compared to the rigid airfoil at AOA = 12° and LM = 0.70c. As the AOA is 14°, the lift coefficient decreases by about 22% for different LMs.

The change of the LM is used to alter the inherent mode pattern of the structural oscillation, and further influence the fluid–structure interaction. The lift enhancement effect shows a noticeable difference for different LMs at the same AOA, especially at AOA = 6°, 8°, 10°. The difference between the maximum and minimum lift enhancement at these three AOAs all exceed 10%. In particular, for AOA = 8°, the lift enhancement is 0.35% at LM = 0.60c, while this value sharply reaches 14.47% at LM = 0.65c. This sharp change of the lift enhancement can also be found at AOA = 10°. At this AOA, the lift enhancement is 13.79% at LM = 0.60c, which is much lower than the values observed at LM = 0.55c and LM = 0.65c. Such a small change in the LM can lead to a significant aerodynamic improvement of the membrane airfoil, which implies that there might be a flow pattern transition within the membrane airfoil flow between these cases. It is also noted that the variations in lift enhancement for different LMs at AOA = 12° and AOA = 14° are relatively small. The variance of the lift coefficients is more stable at these two AOAs compared to the smaller AOAs.

Based on the results above, the lift enhancement exhibits considerable variation as the AOAs range from 6° to 12° and LM ranges from 0.60c to 0.65c. It suggests that the flow state over the membrane airfoils may undergo significant changes within these parameter ranges. Therefore, these specific cases are selected as typical cases for the further analysis.

3.2. Structural Responses

Figure 11 and Figure 12 present the time-averaged deformation of the membrane at LM = 0.60c and LM = 0.65c for various AOAs, respectively. For LM = 0.60c, a positive deflection can be observed for all the cases. This deflection increases with the angle of attack since the coupling becomes stronger at larger AOAs. As AOA is lower than 8°, the membrane undergoes a small change in the mean deflection. The maximum deformation at AOA = 4° and AOA = 6° is less than 0.007 in both cases. However, as AOA increases from 10° to 14°, the mean deflection increases dramatically due to the strong interaction between the fluid and structure. The mean deflection of the membrane at AOA = 14° is nearly twice that at AOA = 8°. Membrane deformation at other LMs across various AOAs consistently follows the same trend observed at LM = 0.60c.

The standard deviation of the transverse displacement of the membrane is used to quantify the coupling strength of the membrane vibration, as depicted in Figure 13 and Figure 14. For the case of LM = 0.60c, a weak aeroelastic effect is characterized by a harmonic membrane’s oscillation with a small normalized amplitude of less than 0.0005 at AOA = 4° and 6°. This value increases to 0.0017 at AOA = 8° and further amplifies to 0.0018 at AOA = 10°. The results also indicate the choice of the LM can potentially change the oscillation pattern. As the AOA increases from AOA = 10° to 12°, the peak of the dominant oscillation shifts rearward along the membrane. An asymmetric vibrational pattern is observed for the case of LM = 0.60c. However, this asymmetric pattern occurs as the AOA increases from AOA = 8° to 10° for LM = 0.65c.

It is observed that the maximum values of $Y_{STD}$ at AOA = 4° and AOA = 6° for all LMs are all less than 0.001. The coupling in these two cases is relatively weak. As AOA increases further, a larger maximum value of $Y_{STD}$ is obtained with a proportional relationship with AOA for the same LM. However, for the same AOA, the relationship between the maximum value of $Y_{STD}$ and LM is not straightforward, especially at AOA = 8° and 10°. At AOA = 8°, the maximum value of $Y_{STD}$ first decreases from 0.0021 at LM = 0.55c to 0.0017 at LM = 0.60c, then increases, reaching a maximum at LM = 0.70c. At AOA = 10°, the maximum value of $Y_{STD}$ reaches a minimum at LM = 0.60c and a maximum at LM = 0.65c. It is noted that the lift enhancement of the membrane airfoil at AOA = 8° and AOA = 10° both show unsatisfactory performance at LM = 0.60c (see Figure 10). It indicates that the coupling strength of the membrane vibration has a significant impact on the aerodynamic performance of the membrane airfoil. However, this impact comes essentially from altering the coherent flow patterns of the membrane flow field.

4. Dynamic Mode Decomposition Results and Analysis

In this section, the dynamic mode decomposition (DMD) method is employed to gain valuable insights into the flow pattern in the coupling. The pressure snapshots are chosen in the analysis due to their significance in revealing modal phase information, as demonstrated by Poplingher et al. [37]. The modal phase of pressure modes can be used to identify the pressure propagation in the unsteady flow and explore the lift enhancement mechanism. The pressure information is obtained from the instantaneous flow field solution taken at equally spaced time intervals with $\Delta t=0.001s$. A total of ${10}^4$ snapshots are used, including 20–25 flow cycles of the unsteady flow. The selection of DMD modes is based on the energetic criterion in Section 2.4, and all oscillatory modes are derived as complex conjugate pairs.

In Figure 15 and Figure 16, the modal energy distribution of the first four mode pairs at LM = 0.60c and LM = 0.65c is depicted, respectively. For the case at LM = 0.60c, the first-order mode pair accounts for 99.23% energy in the flow field at AOA = 4°. At AOA = 8°, the contribution of the first-order mode pair decreases to 81.96%, while the second-order mode pair possesses 8.66% of the energy. As AOA is greater than 8°, the third- and fourth-order mode pairs are involved. The order increase in mode pairs results from flow separation and membrane vibration at higher AOAs. As AOA reaches 14°, the first-order mode pair once again dominates the unsteady flow, accounting for 88.39% of the modal energy. The energy contribution of the first-order mode initially experiences a decrease, reaches its minimum value at AOA = 8°, and subsequently increases gradually. The shift in energy contribution of the first-order mode implies the transition of flow modes, associating with the complex coupling patterns.

4.1. Modal Shape Results and Analysis

Figure 17 presents the first-order DMD modal shapes of pressure around the elastic membrane airfoil with various AOAs at LM = 0.60c. At AOA = 6°, it is found that the flow mode represents Kármán vortex street in the wake. At AOA = 8° and 10°, the pressure extreme points in the modal contour appear on the upper surface, indicating the shedding vortex is formed upstream over the airfoil. As the AOA increases further, the shear layer moves away from the membrane surface, and a large-size vortex is formed on the upper surface at AOA = 12°.

The first-order DMD modal shapes of the membrane airfoil with various AOAs at LM = 0.65c are presented in Figure 18. The evolution of the vortex shedding shares a similar trend with the cases at LM = 0.60c. However, the discrepancy between these two membrane lengths is observed at AOA = 8° and 10°. At AOA = 8°, the flow mode at LM = 0.65c exhibits more vigor than that at LM = 0.60c. Only one vortex is observed on the upper surface of the membrane airfoil at LM = 0.60c, other than three vortices at LM = 0.65c. The existence of these vortices corresponds to the low-pressure regions in the pressure field. When a sequence of vortices is formed on the upper surface at LM = 0.65c, the low-pressure region can cover a large part of the upper surface, which is beneficial to the aerodynamic performance of the membrane airfoil. Accordingly, it can be inferred that the number of vortices may play a significant role in lift enhancement. It is consistent with the fact that the lift enhancement at LM = 0.65c (14.47%) is greater than that at LM = 0.60c (0.35%) at AOA = 8°, as shown in Figure 10.

4.2. Modal Phase Results and Analysis

The modal phase of pressure modes can be used to identify the pressure propagation in the unsteady flow to study the coupling feature. Figure 19 and Figure 20 present the first-order DMD modal phase for different AOAs at LM = 0.60c and LM = 0.65c, respectively. The modal phase propagation on the membrane airfoil surface is also included. The direction of pressure propagation is identified from the variation of the DMD phase values, i.e., a negative change of phase value reflects a lagged maximal pressure value at a corresponding spatial location.

It is observed that the pressure propagation is moving upstream along the lower surface in all cases. In the case of AOA = 6° and LM = 0.60c, a step change from −180° to 180° is found at x = 0.9401c on the lower surface. This phase change is due to the fact that the phase range of a complex number spans from -180° to 180°, which does not affect the direction of propagation. On the upper surface, however, two pressure propagation branches are observed in most cases. One branch is a downstream pressure propagation(DPP), which is identified by the black arrows as shown in Figure 19 and Figure 20. The pressure propagates to the trailing edge along the upper surface. The other branch is an upstream pressure propagation (UPP), which is identified by the red arrows. The boundary between the DPP and the UPP is marked by the dashed lines. The two-branch region exhibits differently with the AOAs and LMs. At AOA = 6°, the UPP region covers about 75% of the upper surface. With the increasing in AOA, the boundary is moving upstream and the UPP region is decreasing. The DPP starts to dominate the pressure fluctuation on the upper surface of the airfoil.

The time-averaged flow field of the membrane airfoil for different AOAs at LM = 0.60c and LM = 0.65c are presented in Figure 21 and Figure 22 to provide the physical link with the phase propagation of the pressure. As the AOA increases, the flow separation point gradually moves upstream, and a strong recirculation zone is developed on the upper surface of the membrane airfoil. The movement of the flow separation point exhibits similar behavior to the boundary between the UPP and the DPP. The increase in the DPP region is consistent with the augment of the recirculation zone. It is inferred that the DPP can reflect the pressure fluctuation of the separation flow. Since the pressure distribution on the upper surface, particularly in the separation flow zone, has a significant impact on the aerodynamic performance of the airfoil, the DPP region can be a key indicator for the physical interpretation for the lift enhancement of the membrane airfoil.

5. DMD Phase-Based Coherent Pattern Analysis

According to the analysis of the pressure propagation above, the DPP region of the membrane airfoil can be a useful tool to unveil the mechanism of the lift enhancement. In order to quantitatively describe the dynamic characteristics of the pressure propagation, a pressure propagation speed is defined by the modal phase in the DPP region51. Given the variation of the modal phase in a certain direction, the pressure propagation speeds in that direction are computed by

$$U_p={\displaystyle \frac{2\pi f}{\Delta \varphi /\Delta x}}$$

(13)

where $U_p$ is the propagation speed in m/s, f is the modal frequency in Hz, and $\Delta \varphi /\Delta x$ is the slope of the modal phase with respect to the spatial coordinate in rad/m.

In this section, the modal phase variation in the DPP region for the first-order DMD mode pair is calculated. The propagation speeds of the DPP are compared with the lift enhancement of the membrane airfoils as shown in Figure 23. At AOA = 4°, the DPP speed increases from 0.4410 m/s to 0.9518 m/s as the LM grows. The lift enhancement is found to decrease from 7.01% to 0.8% as the LM grows. However, this relationship changes at AOA = 6° and LM = 0.65c, with the DPP speed and lift enhancement both increasing as the LM grows from 0.65c to 0.70c. At AOA = 8°, the DPP speed initially decreases before sharply increasing to 0.5807 m/s. A similar positive correlation in the relation between DPP speed and lift enhancement is observed across all LM cases at AOA = 8°. As the AOA increases from 10° further to 14°, the DPP speed gradually decreases from 0.3968 m/s to 0.3900 m/s as the LM grows. An inverse relation between DPP speed and lift enhancement is found as the LM grows.

In order to further explain the inherent relation between the DPP speed and lift enhancement, the instantaneous flow of the membrane airfoil within one flow cycle at LM = 0.60c and 0.65c is illustrated in Figure 24 and Figure 25, respectively. Due to the phase change of DPP, the DPP speed can quantify how fast the lagged maximal pressure propagates through the flow separation zone. Hence, it is useful to track the path of the center point of the separation bubble on the upper surface within one flow cycle to understand the relationship between DPP speed and lift enhancement. The separation bubble traces are marked by the red dashed lines as shown in Figure 24 and Figure 25. In addition, the lift, drag, lift–drag coefficients in a flow cycle at different AOAs and LMs are presented in Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31.

At AOA = 6° and LM = 0.60c, a trailing separation bubble is developed near the rear of the airfoil on the upper surface. The moving length of the separation bubble within one flow cycle is about 0.17c along the upper surface. A similar vortex structure is observed at AOA = 6° and LM = 0.65c. In this case, the separation bubble becomes similar in scale to that at LM = 0.60c. The corresponding narrow separation region leads to a relatively small DPP region. At AOA = 8° and LM = 0.60c, the separation bubble is formed at x = 0.7317c initially and moves downstream. Only one separation bubble on the upper surface is observed at LM = 0.60c while there exist two separation bubbles at LM = 0.65c. The instantaneous surface pressure coefficients($C_p$) at AOA = 8° and LM = 0.60c/0.65c are given in Figure 32, where the black asterisk marks the peak $C_p$ value at each instance. One pressure propagation region is identified by the peak values in a flow cycle at LM = 0.60c, while two pressure propagation regions are observed at LM = 0.65c. The pressure propagation regions correspond to the separation bubble traces in Figure 24 and Figure 25. The DPP speed on the upper surface at LM = 0.65c (0.5807 m/s) is greater than that at LM = 0.60c (0.4150 m/s). It implies the peak pressure at LM = 0.65c propagates faster than that at LM = 0.60c. The faster propagation induces more separation bubbles to generate on the upper surface. It is consistent with the fact that more negative $C_p$ peaks are found on the upper surface at LM = 0.65c, as it is depicted in Figure 32. Furthermore, since the vortex strength and size at LM = 0.60c are comparable to those at LM = 0.65c, the number of $C_p$ peaks on the upper surface becomes a decisive factor for the aerodynamic performance at AOA = 8°. Consequently, greater lift enhancement is observed at LM = 0.65c.

At AOA = 10° and LM = 0.60c, two separation bubble traces are observed. The vortex structure in this case is similar to that at AOA = 8° and LM = 0.60c. At AOA = 10° and LM = 0.65c, two separation bubbles are formed at t = 0T. However, these bubbles merge into a larger separation bubble at t = 0.5T. Only one complete bubble trace is identified on the upper surface in this case. The instantaneous surface pressure coefficients at AOA = 10° for LM = 0.60c and LM = 0.65c are depicted in Figure 33. The DPP speed on the upper surface at LM = 0.60c (0.4292 m/s) is larger than that at LM = 0.65c (0.3348 m/s). Consistent with the previous analysis, the number of low-pressure regions at LM = 0.60c on the upper surface is greater than that at LM = 0.65c, as shown in Figure 33. However, at t = 0.5T, it is noted that the peak $C_p$ value at LM = 0.65c is almost twice as large as that at LM = 0.60c, which results from the merging of the separation bubbles. This large-scale low-pressure region caused by the large-scale separation bubble apparently has a significant impact on the aerodynamic performance. Accordingly, in order to obtain better aerodynamic performance, the DPP speed for the large scale vortex should be as low as possible to maintain the separation bubble on the upper surface. At AOA = 12°, the vortex structures at LM = 0.60c and LM = 0.65c are similar. A single large-scale separation bubble trace is observed on the upper surface. The DPP speed at LM = 0.60c (0.3764 m/s) is marginally higher than that at LM = 0.65c (0.3724 m/s). Consequently, the lift enhancement at LM = 0.65c (88.61%) is marginally greater than that at LM = 0.60c (88.29%), attributed to the lower DPP speed.

Based on the above analysis, the lift enhancement mechanism of the membrane airfoil with different AOAs and LMs is summarized herein. As the AOA is below 8°, the coupling becomes a static aeroelastic problem. The influence of the DPP region appears to be weak because of the small AOA and weak membrane vibration as indicated in Table 2. In this AOA range, the aerodynamic performance is predominantly influenced by the camber effect of the membrane airfoil. At AOA = 8°, a larger DPP speed induces more vortices with similar strength and scale on the upper surface and leads to a greater number of low-pressure regions. Therefore, at AOA = 8°, the larger DPP speed implies greater lift enhancement. When the AOA exceeds AOA = 8°, the flow separation region is increased due to the increased AOA. Because of the merging of the vortices on the upper surface, a large-scale vortex on the upper surface has a dominant impact on the aerodynamic performance. The low DPP speed for such vortex maintains the low-pressure region on the upper surface of the airfoil. Accordingly, the smaller DPP speed reflects greater lift enhancement as the AOA exceeds 8°.

6. Conclusions

A parametric study on the aerodynamic performance of the locally flexible membrane airfoil is carried out in the range of angles of attack (AOA = 4°–14°) and the length of the membrane (LM = 0.55c–0.70c). The modal features and pressure propagation characteristics of the membrane airfoil flow are analyzed using dynamic mode decomposition. The main conclusions are drawn as follows:

(1)

The elastic membrane airfoil shows different lift enhancement at various AOAs and LMs. The lift coefficients of all membrane airfoils are increased compared to the rigid airfoil. The lift coefficient of the elastic membrane airfoil increases by 92.61% compared to the rigid airfoil at AOA = 12° and LM = 0.70c. Moreover, the lift enhancement exhibits considerable variation as AOA ranges from AOA =68° to 12° and LM ranges from 0.60c to 0.65c.

(2)

Two spatial features of the unsteady flow are found to relate to the lift improvement. At AOA = 8°, more vortices, indicating more low-pressure regions, are observed on the upper surface of the membrane airfoil at LM = 0.65c compared to the case at LM = 0.60c. This result is consistent with the fact that the lift enhancement at LM = 0.65c (14.47%) is greater than that at LM = 0.60c(0.35%) at AOA = 8°. However, the number of vortices is not a decisive factor on the aerodynamic performance as the AOA increases to 10°. The lift enhancement at LM = 0.65c is about five times larger than that at LM = 0.60c while more vortices are observed at LM = 0.60c.

(3)

The modal phase information based on DMD is used to identify the pressure propagation in the unsteady flow to study the coupling dynamics. Two kinds of pressure propagation region are identified on the upper surface of the membrane airfoil, i.e., the upstream pressure propagation (UPP) and the downstream pressure propagation (DPP). The boundary between the UPP and the DPP exhibits behavior similar to that of movement of the flow separation point, which suggests that the DPP region can serve as a key indicator for the physical interpretation of lift enhancement in the membrane airfoil.

(4)

The DPP speed is used to quantify the propagation speed of the lagged maximal pressure in the flow separation zone and unveil the relation between the evolution of the separation bubbles and the lift enhancement of the membrane airfoil. The lift, drag, lift–drag coefficients of the membrane airfoil at different AOAs and LMs are analyzed. The results show that two different mechanisms are found with opposite trend of the DPP speed with respect to the LMs. (I) The positive correlation between DPP speed and lift enhancement is found at AOA = 8°. The faster DPP induces more vortices, indicating more low-pressure regions at LM = 0.65c compared to the other cases, which lead to a higher lift enhancement. (II) An inverse relation between DPP speed and lift enhancement is found as the LM grows when the AOA exceeds 8°. In these cases, the separation bubbles are merged into a large-scale separation bubble. The lower DPP speed can maintain this large-scale separation bubble on the upper surface of the airfoil, which benefits the aerodynamic performance. The membrane airfoil investigated in this study is primarily applicable to low Reynolds number flows. In ongoing research, the flow control of locally flexible airfoil will aim for high Reynolds number flows. Furthermore, all the models in this study are two-dimensional. It will be worth exploring the three-dimensional flow structure induced by the coupling of the membrane structures.