Aeroelastic mode decomposition and mode selection mechanism in fluid-membrane interaction
AAeroelastic mode decomposition and mode selectionmechanism in fluid-membrane interaction
Guojun Li a, ∗ , Rajeev Kumar Jaiman b , Boo Cheong Khoo a a Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1,Singapore 117576 b Mechanical Engineering, University of British Columbia, Vancouver, BC Canada V6T 1Z4
Abstract
In this study, we present a global Fourier mode decomposition (FMD) frameworkto isolate and extract the aeroelastic modes from a coupled fluid-membrane sys-tem. Of particular interest is to build a direct connection between the dynamicmodes both in the fluid and structure fields and explore the underlying modeselection mechanism through the decomposed modes. To begin, a high-fidelitythree-dimensional (3D) variational fluid-structure interaction solver based onthe recently developed partitioned body-fitted formulation is employed to sim-ulate the fluid-membrane interaction (FMI) for a 3D flexible membrane at amoderate Reynolds number and different angles of attack. The proposed FMDmethod is applied to decompose the physical fields into frequency-ranked modesand identify the dominant modes based on the mode energy spectra. A compar-ison of the dynamic modes for a rigid wing, a rigid cambered wing and a flexiblemembrane is conducted to study the role of flexibility in membrane dynamicsfrom two aspects, namely (i) camber effect and (ii) flow-excited vibration. Basedon the mode decomposition analysis and the wavenumber-frequency spectra, wesuggest a feedback loop for flexible membrane wings undergoing synchronizedself-excited vibration. This feedback loop reveals that the aeroelastic modes areselected through the mode and frequency synchronization during FMI to exhibitsimilar modal shapes in the membrane vibration and the pressure pulsation. ∗ Corresponding author
Email address: [email protected] (Guojun Li)
Preprint submitted to Journal of Fluids and Structures August 28, 2020 a r X i v : . [ phy s i c s . f l u - dyn ] A ug eywords: Fluid-membrane interaction, Fourier mode decomposition,flexibility effect, self-excited vibration.
1. Introduction
During the past decades, morphing wings with elastic membrane componentshave received substantial attention from the aerospace engineering community inthe context of bio-inspired flying vehicles [1, 2, 3, 4]. A flexible membrane inter-acting with an unsteady flow forms a highly nonlinear coupled fluid-membranesystem, which exhibits a variety of vibrational modes and fluid modes with awide range of spatial and temporal scales. These aeroelastic modes and scalesare closely related to the aerodynamic performance of the membrane structureand play an important role in efficient flight and control strategies. Hence,identifying and isolating the most influential aeroelastic modes from the cou-pled system is essential to further understand the mechanism and control offluid-membrane interaction (FMI) problems.Numerous experimental and computational studies on FMI have been car-ried out in the past years. Song et al. [5] examined the aeromechanics of mem-brane wings as a function of aspect ratio, flexibility and pre-strain value. It canbe observed from the phase map of membrane mode that the dominant modeswitched from the first mode to higher modes as the angle of attack (AOA) andReynolds number increased. Recently, Rojratsirikul et al. [6, 7, 8] performeda series of experiments to study the dynamic behaviors of flexible membranewings at moderate Reynolds numbers. Different types of dominant vibrationalmodes and vortical structures have been observed in the FMI problem. Bleis-chwitz et al. [9] investigated the effect of aspect ratio on membrane dynamicsin wind tunnel experiments. The frequencies corresponding to the dominantmodes were correlated with the frequencies of the force fluctuations. In viewof the limitations of collecting physical data of interest in wind tunnel exper-iments, a high-fidelity numerical simulation method becomes an effective toolto gain further insight into the coupled mechanism of the flexible membrane.2ith the aid of advanced numerical simulations, Sun et al. [10, 11, 12, 13]systematically studied the nonlinear dynamic behaviors of flexible membranewings. The vibration of the flexible membrane excited by the unsteady flowgradually transited from the periodic state to the chaotic state as the changeof the relevant aeroelastic parameters. From the frequency spectra analysisof the membrane responses in the aforementioned literature, it was found thatthe membrane vibration usually exhibited multiple frequency peaks, which wereclosely related to the vortical structures with a variety of scales. These multi-modal mixed responses of the coupled fluid-membrane system pose a seriouschallenge to identify and isolate the aeroelastic modes of interest from the sys-tem. The understanding of the mechanism of how a specific aeroelastic mode isselected in the coupled system is limited.With respect to the dominant mode identification in a coupled fluid-membranesystem, some ingenious methods have been adopted to distinguish the most in-fluential modes. Standard deviation analysis has been widely applied in the FMIproblem to determine the dominant vibration modes [6, 8, 14]. As Bleischwitzet al. [9] pointed out, the excited modal shapes overlapped together, which in-creased the difficulty of isolating the dominant modal shapes from the coupledsystem. Although the traditional standard deviation analysis of the membranedeflection could reflect the dominant structural modal shape to some extent, theoccasionally appearing modes or overlapping modes with small energies may becovered by the dominant modes, which makes them hard to be identified fromthe overall membrane vibrations. Additionally, the frequency spectra and thevibration state analysis at a single point are not suitable to reflect the dynamiccharacteristics of the whole membrane structure [10]. Therefore, global modeidentification methods are naturally desirable to capture the dynamic behaviorsof the entire physical field of interest. The relevant dynamic information corre-sponding to each mode such as mode energy and mode frequency could help usgain further insight into the dynamic characteristics when performing the modeidentification.To overcome these apparent drawbacks in the study of FMI, a spatial de-3omposition method was proposed by Bleischwitz et al. [9] to isolate the spatial-temporal modal shapes of a one-dimensional (1D) membrane. The contributionof individual modal shape to the overall vibration was quantitatively calculated.Some recently developed mode decomposition techniques, including proper or-thogonal decomposition (POD) [15] and dynamic mode decomposition (DMD)[16, 17] were applied in the extraction of the modal shapes or the coherent wakestructures for flexible membrane wings. A global mode decomposition methodbased on the discrete Fourier transform (DFT) technique, namely Fourier modedecomposition (FMD), was proposed by Ma et al. [18] to decompose the con-cerned physical field into frequency-ranked dynamic modes along with the cor-responding amplitude and phase information. Chen et al. [19] mathematicallydemonstrated that the connection between DMD and DFT and DMD will bereduced to DFT when the fluctuations of the physical variables are performed inthe mode decomposition process. Serrano-Galiano et al. [20] applied the FMDmethod to identify the coherent vortex structures formed in FMI. These studieshave successfully addressed the dominant mode identification problem in themulti-modal mixed responses. However, regarding the mode decomposition ofthe fluid-structure interaction problems, these fluid-only or structure-only ap-proaches usually focus on one of the fluid and structure fields and treated themseparately in the data analysis, which loses the inherent connection betweenfluid modes and structure modes. Only a handful of literature can be found tobuild a bridge between the dynamic modes in both fields to explore the coupledmechanism. Recently, Goza et al. [21] developed a framework based on PODand DMD for the mode decomposition of flapping flags immersed in an un-steady flow. In this combined formulation, the collected data in both fluid andstructure fields was decomposed in a unified matrix, which naturally ensuredthe inherent correlation between the dynamic modes in both fields.In this paper, we propose an easy-to-implement and effective frameworkbased on the FMD method for the aeroelastic mode decomposition in the FMIproblem. Of particular interest is to present physical insight into the underly-ing mechanism of how the unsteady turbulent flow interacts with the extensible4hree-dimensional (3D) membrane to excite particular wake patterns and selectspecific vibrational modes. We employ the Fourier-decomposed modes of thecoupled system to address the following challenges in FMI and answer the spe-cific key questions that are relevant to the membrane aeroelasticity: (i) Whichtypes of membrane vibrations and wake patterns are dominant during FMI andhow do we identify these dominant aeroelastic modes in the coupled system?(ii) What is the aeroelastic mode selection mechanism during FMI? To address(i), following the idea of [21], we extend the original FMD method for fluid-onlyanalysis to the FSI system. The physical variables of interest in both the fluidand structure fields are decomposed via FMD. Meanwhile, the contribution ofthe dominant aeroelastic mode to the overall membrane dynamics is calculatedquantitatively. The obtained mode energy spectrum is utilized to identify thedominant aeroelastic modes by detecting the frequency peaks. We then presentthe correlated dynamic modes in the fluid and structure fields. A compari-son of the dynamic modes between a rigid wing, a rigid cambered wing and aflexible membrane wing is conducted to investigate the role of flexibility in themembrane aeroelasticity. Based on the mode decomposition analysis and thewavenumber-frequency spectra of the membrane responses, we suggest a feed-back loop to reveal the underlying mode selection mechanism for the coupledsystem. To our knowledge, this is the first time to utilize the proposed FMDframework to extract the aeroelastic modes for 3D FMI problems. Besides,there has not been a specialized investigation to build a direct connection be-tween the fluid and structure modes to explore the role of flexibility and explainthe underlying mode selection mechanism during FMI.The rest of this paper is organized as follows. In section 2, the governingequations for the coupled fluid-structure system and the algorithm of FMD aswell as its application in FMI are introduced. The description of the FMI prob-lem and the verification for the proposed FSI simulation framework is providedin section 3. We present the dynamic behaviors of the flexible membrane atdifferent angles of attack in section 4. The application of the proposed FMDframework to FMI and the exploration of the mode selection mechanism are5hen discussed. In section 5, the main conclusions of the aeroelastic mode de-composition in FMI are summarized.
2. Numerical methodology
The governing equations for the incompressible unsteady viscous flow withan arbitrary Lagrangian-Eulerian (ALE) reference frame and the DDES modelare discretized via stabilized Petrov-Galerkin variational formulation [22, 23].The generalized- α method is utilized to integrate the ALE flow solution in timedomain and it can ensure unconditionally stable and second-order accuracyfor linear problem [24]. A user-defined parameter named the spectral radius ρ ∞ is employed to control the high-frequency damping desired for a coarserdiscretization in time and space. The flow variables are updated based on thegeneralized- α scheme given as: u f,n +1 = u f,n + ∆ t∂ t u f,n + γ f ∆ t ( ∂ t u f,n +1 − ∂ t u f,n ) (1) ∂ t u f,n + α fm = ∂ t u f,n + α fm ( ∂ t u f,n +1 − ∂ t u f,n ) (2) u f,n + α f = u f,n + α f ( u f,n +1 − u f,n ) (3) u m,n + α f = u m,n + α f ( u m,n +1 − u m,n ) (4)where ∂ t represents the time partial derivative of a physical variable and ∆ t denotes the time step size. u f,n and u m,n are the fluid and mesh velocitiesfor each spatial point in the fluid domain x f ∈ Ω f ( t ) at the time step n . Thegeneralized- α parameters α f , α fm and γ f are defined as α f = 11 + ρ ∞ , α fm = 12 (cid:18) − ρ ∞ ρ ∞ (cid:19) , γ f = 12 + α fm − α f (5)Suppose S f,h is the space of the trial solutions and V f,h denotes the spaceof test functions. The variational formulation of the fluid equation can be writ-ten as: find the velocity and pressure fields [ u f,n + α f h , p f,n +1 h ] ∈ S f,h such that6 [ φ fh , q h ] ∈ V f,h (cid:90) Ω f ( t ) ρ f ( ∂ t u fh + ( u fh − u mh ) · ∇ u fh ) · φ fh dΩ+ (cid:90) Ω f ( t ) σ fh : ∇ φ fh dΩ + (cid:90) Ω f ( t ) σ ddes fh : ∇ φ fh dΩ+ n fel (cid:88) e =1 (cid:90) Ω e τ m ( ρ f ( u fh − u mh ) · ∇ φ fh + ∇ q h ) · R m dΩ e − (cid:90) Ω f ( t ) ∇ · u fh q h dΩ + n fel (cid:88) e =1 (cid:90) Ω e ∇ · φ fh τ c ∇ · u fh dΩ e − n fel (cid:88) e =1 (cid:90) Ω e τ m φ fh · ( R m · ∇ u fh )dΩ e − n fel (cid:88) e =1 (cid:90) Ω e ∇ φ fh : ( τ m R m ⊗ τ m R m )dΩ e = (cid:90) Ω f ( t ) b f ( t n + α f ) · φ fh dΩ + (cid:90) Γ h h f · φ fh dΓ (6)where ρ f represents the fluid density, σ fh denotes the Cauchy stress tensor fora Newtonian fluid and σ ddes fh is the turbulent stress term. φ fh and q h are thetest functions of the fluid velocity u fh and pressure p . The terms in the firstline denote the Galerkin terms for the momentum equation and the secondline represents the viscous and turbulent stress terms. The Petrov-Galerkinstabilization terms for the momentum equation are presented in the third line.The fourth is the Galerkin and the Galerkin/least-squares stabilization termsfor the continuity equation, respectively. The two residual terms in the fifthline are introduced as the approximation of the fine scale velocity on elementinteriors based on the multiscale argument. The terms in the sixth line of theright-hand side of Eq. (6) denote the body forces and the Neumann boundaryconditions. The stabilization parameters τ m and τ c are the least-squares metricsadded to the element level integrals in the fully discretized formulation and R m represents the element-wise residual of the momentum equation.The motion equations for a flexible structure are discretized using finite ele-ment method and the variational statement is expressed as: find the structural7isplacement d sh ∈ S s,h such that ∀ φ sh ∈ V s,h (cid:90) t n +1 t n (cid:32)(cid:90) Ω si ρ s ∂ d sh ∂t · φ sh dΩ + (cid:90) Ω si σ sh : ∇ φ sh dΩ (cid:33) d t = (cid:90) t n +1 t n (cid:32)(cid:90) Ω si b s · φ sh dΩ + (cid:90) Γ i h s · φ sh dΓ (cid:33) d t (7)where ρ s is the structural density and φ sh denotes the test function for thestructural displacement d sh , σ sh is the stress tensor and h s = σ sh · n s representsthe Neumann condition at the boundary Γ sN,i . b s is the body force on theflexible structures Ω si . The kinematic joints are considered as constrains on thedisplacement field.The fluid and structural motion equations are coupled by satisfying thevelocity and traction continuity along the coupling interface Γ fsi , written as u f ( ϕ s ( x s , t ) , t ) = u s ( x s , t ) ∀ x s ∈ Γ fsi (8) (cid:90) ϕ s ( γ,t ) σ f ( x f , t ) · n f dΓ + (cid:90) γ σ s ( x s , t ) · n s dΓ = 0 ∀ γ ∈ Γ fsi (9)where Γ fsi = Ω f (0) ∩ Ω si is the fluid-structure interface for the i th componentat t = 0. u s is the structural velocity at time t . ϕ s is the function that mapsthe initial Lagrangian point x s ∈ Ω si to its deformed position at time t . n f and n s represent the outer normals to the deformed fluid and the undeformedstructural interface boundaries, respectively. γ denotes any part of the interfaceΓ fsi and ϕ s ( γ, t ) is the associated fluid domain at time t .The fluid equations and the multibody structural equations are integrated ina partitioned manner and the coupled governing equations are solved based ona typical predictor-corrector scheme. In the fluid-elastic structure coupled pro-cedure, the fluid forces and the structural displacements are transferred alongthe fluid-solid interface with non-matching meshes by satisfying the energy con-servation via the compactly-supported RBF method. An efficient remeshingalgorithm based on the extended RBF method is employed to update the body-fitted fluid meshes in spaces by following the motion of the pitching flexible foils.The fluid forces are corrected based on the recently developed nonlinear inter-8ace force correction (NIFC) scheme [22] at each iterative step to ensure the nu-merical stability for fluid-structure interaction problems with significant addedmass effect. The developed fluid-multibody structure interaction framework hasbeen validated for flexible flapping wings [23] and flow-induced vibration of 2Dflexible membrane foils [25]. To identify and extract the global spatial modes at the specific frequencyof interest, the Fourier mode decomposition (FMD) projects the entire physicaldata from the spatio-temporal domain to the spatio-frequency domain based onthe discrete Fourier transform (DFT). In this section, the algorithm of FMDis introduced briefly and its application procedure to the FMI problems is pre-sented.The physical data Y ( x , t ) ∈ R M × N is collected from experiments and numer-ical simulations in a time-discrete way. These data collected at M discretizedspatial points x and N sampling time constants is stored in an M × N matrix.The fluctuation components Y (cid:48) ( x , t ) reflect the perturbation in the time-varyingglobal physical field. Following the idea of Fourier series, the fluctuation func-tion Y (cid:48) ( x , t ) can be written as a Fourier series in an exponential form: Y (cid:48) ( x , t ) = N − (cid:88) k =0 c k ( x ) e i πkN n (10)The Fourier complex coefficient c k ( x , f k ) at the discretized spatial point x and the discretized frequency f k is expressed as: c k ( x , f k ) = F ( Y (cid:48) ( x , t )) = 1 N N − (cid:88) n =0 Y (cid:48) ( x , t ) e − i πkN n (11)In the global FMD analysis, we perform the discrete Fourier transform(DFT) to transfer the fluctuation function from the time domain to the fre-quency space f = [ f , f , . . . , f K ] ∈ R × K to obtain the global spatial modesequence C = [ c ( x , f ) , c ( x , f ) , . . . , c K ( x , f K )] ∈ R M × K . The fast Fouriertransform (FFT) algorithm is used to speed up the Fourier transform. The9lobal spatial mode sequence C consists of the decomposed spatial mode c k ( x , f k )at different frequencies f k . The Fourier-transformed coefficients at each meshpoint contain the information of the intensity and the initial phase of the spatialmode in the frequency domain. The global amplitude spectrum A k ( x , f k ) andthe global phase spectrum θ k ( x , f k ) can be written as: A k ( x , f k ) = | c k ( x , f k ) | and θ k ( x , f k ) = − tan − Im( c k ( x , f k ))Re( c k ( x , f k )) (12)In order to identify the dominant modes from the entire decomposed modesequence, we calculate the sequence of the global power spectrum s k and themode energy spectrum e k , which are expressed as s k = (cid:107) A k ( x , f k ) (cid:107) and e k = s k (cid:80) Kk =1 s k (13)The mode energy spectrum e k represents the contribution of individual spa-tial mode to the overall physical field. The real and imaginary parts of thespatial mode c k ( x , f k ) show an explicit physical significance for the spatial dis-turbance structures (modal shapes) and their intensity as well as initial phasedifference. The dominant modes can be determined as the modes with mostmode energies in the mode energy spectrum. Subsequently, these dominantmodes are extracted from the frequency-ranked global mode sequence at the se-lected frequencies with relatively large mode energies. The global FMD methodavoids the limitation of the traditional Fourier transform (FT) analysis at a sin-gle point and provides a global view to reflect the correct dynamics of the entirephysical field by containing the information of the decomposed modes at eachspatial point [18]. A summary of the FMD algorithm is presented in Algorithm1. ALGORITHM 1: FMDInput: Storage of physical variables in the spatial field expressed as Y ( x , t ) ∈ R M × N ( M , number of spatial mesh points; N , number of totaltime samples) 10utput: Selected FMD modes C = [ c , c , . . . , c K ] and mode energies E = [ e , e , . . . , e K ](i) Extract the fluctuation physical variable matrix from time-varyingglobal physical field: Y (cid:48) ( x , t ) = Y ( x , t ) − Y ( x , t )(ii) Calculate the global Fourier coefficients c k ( x , f k ) by applying FMDapproach(iii) Find the mode energy e k for its corresponding spatial mode(iv) Determine the dominant spatial modes by detecting the peaks ofthe mode energy distributions as a function of frequencyBefore we perform the FMD analysis for the FMI problem, there are severalpre-processing steps to be completed. We first collect the time-varying physicalvariables of interest both in fluid and structural domains at each discretizedmesh point with a sampling frequency of f s . Considering the deformation ofthe flexible membrane under aerodynamic loads, the positions of the body-fittedfluid mesh are updated by following the motion of the membrane at each iter-ative step in the developed FMI framework. In the present study, the physicalfield is decomposed via FMD method at fixed mesh points. We perform aninterpolation for the collected physical variables from the moving fluid mesh tothe fixed reference mesh based on an RBF interpolation method. Subsequently,all the physical data is stored in a snapshot way for FMD analysis. The globalFMD method is utilized to decompose the fluid and structural physical fieldsinto frequency-ranked spatial modes based on Eq. (11). The primary advan-tage of the global FMD analysis is to establish correct correspondences of thedecomposed fluid and structural Fourier modes by choosing the modes in bothdomains at the same selected frequency. This approach avoids the drawbacksof energy-ranked POD modes, that is, the decomposed fluid modes cannot belinked with the structural modes directly due to the multiple frequency contentsin the modes. Since the FMD analysis is based on the DFT of signals at everysingle spatial position, there is less constraints on the form and dimensionality11f the collected data, which avoids to manipulate the complex matrices. Thisfeature of FMD makes it easy to calculate in parallel, or even separately cal-culate the collected time-varying data at different spatial positions on differentdistributed computing devices at the same time to save computing resources.Benefiting from the superiority of FMD, it is helpful to build an intrinsic rela-tionship between the flow-induced vibrations and the coherent flow structuresto reveal the physical mechanism of FMI because of the explicit physical inter-pretation of the FMD results.
3. Problem set-up and verification
In this study, we explore the mode selection mechanism for FMI problemsthrough aeroelastic mode decomposition with the aid of the FMD technique.For that purpose, we perform a series of numerical simulations for a 3D flexiblemembrane wing at different angles of attack to gain additional insight on thecoupled mechanism. This 3D membrane wing with supporting rigid frame wasconducted in the wind tunnel experiments by Rojratsirikul et al. [8]. Thegeometry information and the section of the supporting frame is presented infigure 1. The membrane has a chord length of c = 68 .
75 mm and an aspectratio of AR = 2. The thickness of the membrane is h = 0 . ρ s = 1000kg/m and the Young’s modulus of E = 2 . d = 5 mm and the diameter of therod is 2 r = 2 mm. In the current study, the membrane wing is simulated atthe same Reynolds number of Re = 24300 as that in the experiment. Themembrane aeroelasticity at several angles of attack are compared against theresults obtained from the experiments to validate the FSI framework and thenthey are investigated in detail to reveal the coupled mechanism in the followingsections. 12 c Frame Membrane d rh FrameMembrane
Figure 1: Membrane wing geometry and section of supporting frame.
L B HbZ Y X Γ in | u f | = U ∞ Γ slip u f · n f = 0 , σ f · n f = 0Γ slip u f · n f = 0 , σ f · n f = 0 Γ out σ f · n f = 0 ∇ ˜ ν · n f = 0Γ no − slip ( u f = , ˜ ν = 0) α (a)(b)Figure 2: Three-dimensional computational set-up for fluid-membrane interaction: (a)schematic diagram of the computational domain and the boundary conditions and (b) repre-sentative mesh distribution in (Y,Z)-plane in the fluid domain. Figure 2 (a) depicts the three-dimensional computational domain and bound-ary conditions for a 3D flexible membrane immersed in an unsteady flow with a13xed AOA α . The length, width and height of the computational domain are allset to 50 c . A stream of oncoming flow with uniform velocity of u f = ( U ∞ , , in . The slip-wall boundary con-dition is applied on four side boundaries (Γ slip ). The boundary condition on themembrane surface (Γ no − slip ) is set to the no-slip boundary condition. The outletboundary Γ out has a traction-free boundary condition. In the numerical simu-lation, all the degrees of freedom of the rigid frame are fixed, and the passivedeformation of the flexible membrane is allowed under aerodynamic loads.To evaluate the aerodynamic characteristics of the membrane wing, we inte-grate the surface traction for the first layer of elements on the membrane surfaceto obtain the instantaneous lift, drag and normal force coefficient, which are de-fined below: C L = 1 ρ f U ∞ S (cid:90) Γ ( ¯ σ f · n ) · n z dΓ (14) C D = 1 ρ f U ∞ S (cid:90) Γ ( ¯ σ f · n ) · n x dΓ (15) C N = 1 ρ f U ∞ S (cid:90) Γ ( ¯ σ f · n ) · n c dΓ (16)where U ∞ is the freestream velocity and ρ f represents the air density. The areaof the membrane surface is denoted as S = bc . n x and n z are the projection ofthe unit normal n to the membrane surface on the x -axis and z -axis, respectively.The unit normal vector n c is perpendicular to the membrane chord. ¯ σ f denotesthe fluid stress tensor. The deformation of the flexible membrane is mainlydriven by the pressure acting on its surface, and the pressure coefficient is givenas: C p = p − p ∞ ρ f U ∞ (17)where p and p ∞ represent the local pressure and the far-field pressure, respec-tively. 14 .2. Mesh convergence and verification To choose a proper mesh with sufficient resolution for the following numericalsimulation, we conduct a mesh convergence study for the 3D flexible membraneby designing three sets of meshes namely M1, M2 and M3. The unstructuredfinite element is adopted to discretize the 3D fluid domain and the structure do-main is discretized by the structured finite element. These three sets of meshesconsist of 341821, 823864 and 1,304282 eight-node brick elements in the fluiddomain and the corresponding element numbers in structure domain are 160,228 and 352, respectively. A stretching ratio of ∆ y j +1 / ∆ y j = 1 .
25 is set withinthe boundary layer mesh to maintain y + < .
0. The representative mesh distri-bution in the (Y,Z)-plane at the mid-chord position is presented in figure 2 (b).The non-dimensional time step is set to ∆ tU ∞ /c = 0 . Re = 24300 with a corresponding freestream velocity of U ∞ = 5 m/s in the numerical simulation. Table 1: Mesh convergence of a 3D rectangular flexible membrane wing at Re =24300 and α = 15 ◦ . Mesh M1 M2 M3Fluid elements 341 821 823 864 1 304 282Structural elements 160 228 352Mean lift C L C D C (cid:48) Lrms C (cid:48) Drms St α = 15 ◦ with obvious vortex sheddingphenomenon is considered for the mesh convergence study. Table 1 summarizesthe aerodynamic force and vortex shedding frequency statistics for the three15ets of meshes. We calculate the percentage difference for M1 and M2 withrespect to the finest mesh M3 to evaluate the mesh convergence error. It can beobserved that the error of the mean lift and mean drag is less than 1% and themaximum difference of the force fluctuation is 2 . α ( ◦ ) ¯ k d k m a x / c PresentExperiment (a) α ( ◦ ) C F l e N − C R i g N PresentExperiment (b)Figure 3: Flow past a 3D flexible membrane wing: (a) comparison of the magnitude of time-averaged normalized maximum membrane deformation ( ¯ (cid:107) d (cid:107) max /c ) for the present numericalsimulation and experiment [8], (b) comparison of the time-averaged normal force coefficientdifference ( C Fle N − C Rig N ) between the flexible membrane wing and rigid wing for the presentnumerical simulation and experiment [8]. The 3D membrane immersed in the unsteady flow at different angles of attackis simulated to compare against the experimental data for validation purpose.Figure 3 presents the magnitude of the mean maximum membrane deformationand mean normal force coefficient difference between the flexible membrane andthe rigid wing obtained from the present simulation and the experiment [8].It can be observed that the overall trend is well predicted by our numericalsimulations. We also compare the flow field and the streamlines around themembrane to the available results of the experiment at α = 10 ◦ and α = 23 ◦ ,respectively. Good agreements can be seen for the distribution of the meanvelocity magnitude and the size as well as the location of the separation flow.16 a) (b)(c) (d) : 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5 k U k /U ∞ Figure 4: Time-averaged normalized velocity magnitude and streamlines on the mid-spanplane obtained from: (a) present simulation at α = 10 ◦ , (b) experiment [8] at α = 10 ◦ , (c)present simulation at α = 23 ◦ , (d) experiment [8] at α = 23 ◦ .
4. Results and discussion
In this section, we present the fluid-membrane coupled dynamics at differentAOAs. To identify the dominant aeroelastic modes, the global mode decomposi-tion method based on the FMD technique is adopted to decompose the physicalvariables into frequency-ranked modes and extract these modes at the selectedenergetic frequencies. The flow features and the decomposed modes of a rigid17ing, a rigid cambered wing and a flexible membrane are compared to explorethe role of flexibility and further reveal the aeroelastic mode selection mecha-nism.
Before proceeding with further investigation on the connection between theflow-excited membrane vibration and the induced wake dynamics via the pro-posed mode decomposition method, an overview of the membrane aeroelasticresponses is displayed to provide a brief impression of the fluid-membrane inter-action problems. Figure 5 shows the standard deviation analysis of the normal-ized membrane displacement over several cycles for the 3D membrane to reflectthe dominant structural vibration modes both in the chord-wise and span-wisedirections. A typical chord-wise second mode is discovered for the elastic mem-brane at α = 15 ◦ . However, the dominant structural mode of the 3D membranewing cannot be distinctly identified from the standard deviation analysis at α = 20 ◦ and 25 ◦ . XYZ δ sdn /c (a) (b) (c)Figure 5: Standard deviation analysis of normalized membrane displacement normal to thechord δ sdn /c for U ∞ =5m/s at: (a) α =15 ◦ , (b) α =20 ◦ and (c) α =25 ◦ . YZ -1.5 -1.325 -1.15 -0.975 -0.8 -0.625 -0.45 -0.275 -0.1 C upperp − C lowerp (a) XYZ -0.001 -0.00075 -0.0005 -0.00025 0 0.00025 0.0005 0.00075 0.001 δ n − δ n c (b) XYZ -1.8 -1.6375 -1.475 -1.3125 -1.15 -0.9875 -0.825 -0.6625 -0.5 C upperp − C lowerp (c) XYZ -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 δ n − δ n c (d) XYZ -2.5 -2.2875 -2.075 -1.8625 -1.65 -1.4375 -1.225 -1.0125 -0.8 C upperp − C lowerp (e) XYZ -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 δ n − δ n c (f)Figure 6: Flow past a 3D rectangular membrane wing: pressure coefficient difference betweenthe upper and lower surfaces (a,c,e) and fluctuation of membrane displacement (b,d,f) at fourdifferent instantaneous times within one period at (a,b) α =15 ◦ , (c,d) α =20 ◦ and (e,f) α =25 ◦ . Tregidgo et al. [26] found the standard deviation contour for a disturbedmembrane in the gusty flow demonstrated the first mode but the instantaneousfluctuating displacement indicated the span-wise second and third modes withinone cycle actually. The traditional standard deviation analysis is not a suitableindicator to reflect the correct dominant modes of the whole membrane vibra-tions with overlapping modal shapes. The instantaneous pressure coefficient19ifference between the upper and lower surfaces and the fluctuation contours ofthe membrane displacement within one cycle for α ∈ [15 ◦ , ◦ ] are presented infigure 6 to reflect the evolution of the membrane aeroelasticity. All the pressuredifference distributions on the membrane surface show complex evolutions overtime and overlapping modal shapes in space. The instantaneous membrane dis-placement fluctuations at α = 15 ◦ behave an obvious chord-wise second modeand varied span-wise modes. We can observe the chord-wise second and thirdmodal shapes that appear occasionally for α = 20 ◦ and 25 ◦ shown in figures6 (d) and (f). Considering the irregular displacement fluctuation under thepressure pulsations, the dominant structural motion will be covered up in thestandard deviation contours due to the time-averaged sense of the second andthird modes. (a) α = 15 ◦ (b) α = 20 ◦ (c) α = 25 ◦ : 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5 k U k /U ∞ Figure 7: Flow past a 3D rectangular membrane wing: time-averaged velocity magnitude onfive slices along the span-wide direction at (a) α =15 ◦ , (b) α =20 ◦ and (c) α =25 ◦ . Figure 7 and figure 8 present the time-averaged velocity magnitude and theturbulent intensity on five equispaced slices along the span-wise direction atthree AOAs with obvious vortex shedding phenomena, respectively. It can beobserved from figure 7 that the low-velocity region is larger on the slice of themid-span plane than those on the slices near the wingtip. Similarly, the un-steady flow near the mid-span location shows higher turbulent intensity. Fromfigure 5 and figure 6, we see that the region close to the mid-span location of20he membrane has the largest vibration amplitude. Due to the displacementconstraints of the membrane at the wingtip, the vibration amplitude near thewingtip becomes smaller. Thus, the flow fluctuations contributed by the mem-brane vibration is weaker at the wingtip than those near the mid-span location.As the AOA increases, both the low-velocity region and the high turbulent in-tensity region expand further. (a) α = 15 ◦ (b) α = 20 ◦ (c) α = 25 ◦ : 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 r u ′ + w ′ /U ∞ Figure 8: Flow past a 3D rectangular membrane wing: turbulent intensity on five slices alongthe span-wide direction at (a) α =15 ◦ , (b) α =20 ◦ and (c) α =25 ◦ . Evolutions of the streamlines around the membrane on the mid-span planeat different AOAs are presented in figure 9. We observe that large-scale vorticesare produced near the LE and shed into the wake alternatively at α = 15 ◦ . Thetime-varying pressure difference between the upper and lower surfaces causedby the formation and shedding of the vortices produces the periodic pressureexcitations applied on the elastic membrane, which leads to the membrane vi-bration in the second mode. The scale of the vortices is enlarged continuously asthe AOA increases to α = 20 ◦ and α = 25 ◦ . The secondary vortex is observedon the membrane surface, making a contribution to the occasional chord-wisethird mode. From the discussions above, we find the membrane vibrations andthe flow features around the flexible membrane exhibit multi-modal mixed re-sponses both in the temporal and spatial domains. The aeroelastic response isoverlapped together in the instantaneous plots, which restricts us to isolate the21ibration and the flow pattern of interest from the coupled system. (a) α = 15 ◦ (b) α = 20 ◦ (c) α = 25 ◦ : 1 0.85 0.7 0.55 0.4 0.25 0.1 0.05 0.2 0.35 0.5 C p Figure 9: Flow past a 3D rectangular membrane wing: instantaneous streamlines on the mid-span plane colored by pressure coefficient within one cycle at (a) α =15 ◦ , (b) α =20 ◦ and (c) α =25 ◦ . The mode decomposition technique is capable of providing a new perspec-tive for the coupled mechanism of flexible membrane undergoing self-excitedvibrations. In the current study, we apply the proposed global FMD method todecompose the coupled system into frequency-ranked aeroelastic modes. Thedominant modes are identified by detecting the frequency peaks. For simplic-ity, we first demonstrate the decomposition process of the FMD method for theflexible membrane at the AOA of α = 15 ◦ . The detailed explanations of thedecomposed aeroelastic modes are then provided. Finally, we summarize themost influential modes for the flexible membrane at other AOAs.22 .5 1 1.5 2 2.551015202530 St ( % ) M o d ee n e r g y δ ′ n C ′ p ω ′ y St = 0 . St = 1 . (a)(b) (c)(d) (e)Figure 10: Aeroelastic mode decomposition of 3D flexible membrane at α =15 ◦ : (a) modeenergy spectra of the surface displacement fluctuations, the pressure coefficient fluctuationsand the Y -vorticity fluctuations based on the FMD analysis; the decomposed membranedisplacement modes at St = (b) 0.99 and (c) 1.96 and the surface pressure difference modesat St = (d) 0.99 and (e) 1.96. The membrane displacement, the pressure and the vorticity along the span-wise direction of 512 equispaced time-varying samples are collected at eachspatial point of the reference fixed mesh in the spatio-temporal physical field.These physical variables are stored in a matrix form for mode decomposition.The global mode energy spectra calculated from Eq. (13) are presented in figure230 (a). Two obvious frequency peaks at St = 0 .
99 and 1.96 are observed in thecomputed mode energy spectra for the mode decomposition. It is noticed thatthe energetic frequencies are consistent for the decomposed structure and fluidFourier modes. This indicates that the membrane vibrations and the flow fluctu-ations are excited in a frequency synchronized way. The decomposed aeroelasticmodes colored by the real part of the Fourier transform coefficients based on thedisplacements and the pressure difference distributions of the membrane surfaceat the selected dominant frequency of St = 0 .
99 and 1.96 are plotted in figure 10(b-e), respectively. We notice that a typical chord-wise second mode is excitedat St = 0 .
99 and a high-order mode both in the chord-wise and span-wise direc-tions is observed at a higher frequency of St = 1 .
96. Except for the decomposedsurface pressure modal shapes near the LE, the overall modes present similarmodal shapes as the decomposed surface displacement modes at both energeticfrequencies.To study the spatial flow structures induced by the membrane vibration,we extract the dynamic Fourier modes in the spatial pressure and Y -vorticityfields on the mid-span plane. The real part Re ( F ( C p − ¯ C p )) and the ampli-tude (cid:12)(cid:12) F ( C p − ¯ C p ) (cid:12)(cid:12) of the decomposed pressure fluctuation fields at the non-dimensional frequency of St = 0 .
99 corresponding to the chord-wise secondmode are shown in figure 11 (a,b), respectively. The real part of the trans-formed coefficients reflects the spatial structures of the mode. The amplituderepresents the intensity distributions of the decomposed physical variables. Twosmall-scale pressure fluctuation regions are observed near the LE on the uppermembrane surface. These pressure fluctuations are mainly caused by the rolledup vortices at the LE in figure 12 (a). Two larger pressure fluctuation regions onthe upper surface are generated during the periodic leading edge vortex (LEV)shedding process along the path in figure 12 (a). From the amplitude contourof the decomposed pressure field in figure 11 (b), the large-scale pressure pul-sations with high values are noticed on the upper surface. The severe vorticityfluctuations are mainly formed at the periodic vortex shedding regions near theLE and TE in figure 12 (b). 24s the increase of the non-dimensional frequency to St = 1 .
96 with thehigh-order mode, the pressure wavelength and the flow scales become smaller.The high-intensity pressure pulsations still keep close to the membrane surfacein figure 11 (d). However, the amplitude values in this region are far less thanthose at St = 0 .
99 due to the weaker mode energy of the high-order mode.Meanwhile, the small-scale vortices originating from the LE move backwards tomerge with the TEVs behind the membrane in figure 12 (c). It can be observedfrom figure 12 (d) that the high-intensity vorticity fluctuation region shrinksand the amplitude value in this region is reduced, compared to the decomposedvorticity field at the dominant frequency of St = 0 . : 0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 Re ( F ( C p − ¯ C p )) (a) : 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 (cid:12)(cid:12)(cid:12) F ( C p − ¯ C p ) (cid:12)(cid:12)(cid:12) (b) : 0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008 0.01 Re ( F ( C p − ¯ C p )) (c) : 0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.03 (cid:12)(cid:12)(cid:12) F ( C p − ¯ C p ) (cid:12)(cid:12)(cid:12) (d)Figure 11: Aeroelastic mode decomposition of 3D flexible membrane at α =15 ◦ : contours ofthe real part (a,c) and the amplitude (b,d) of the Fourier transform coefficients of the pressurecoefficient fluctuation field corresponding to the non-dimensional frequency of St = (a,b) 0 . . The aeroelastic responses at α = 20 ◦ and α = 25 ◦ are also decomposedvia the proposed FMD technique. The aeroelastic mode energy spectra of the25embrane displacement, the spatial pressure and Y -vorticity are presented infigure 13. It can be observed that the membrane vibration is excited in a widefrequency range. The global mode energy spectra of the decomposed structureand fluid fields exhibit similar frequency peaks. A summary of the decomposeddynamic modes at the energetic frequencies is provided in figure 14. The oc-casional chord-wise second and third modes are successfully identified from thecoupled system. Both the dynamic modes in the fluid and structure domains areconnected directly in figure 14. Due to the existence of the rolled up vortices atthe LE, a small pressure perturbation region is observed. Despite this difference,the modal shapes for the fluid and structural modes show high correlations. LEV shedding pathWake merging zone : -30 -27 -24 -21 -18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18 21 24 27 30 Re ( F ( ω y − ¯ ω y )) (a) : 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 |F ( ω y − ¯ ω y ) | (b) LEV shedding pathTEV shedding pathWake merging zone : -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Re ( F ( ω y − ¯ ω y )) (c) : 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 |F ( ω y − ¯ ω y ) | (d)Figure 12: Aeroelastic mode decomposition of 3D flexible membrane at α =15 ◦ : contoursof the real part (a,c) and the amplitude (b,d) of the Fourier transform coefficients of the Y − vorticity fluctuation field corresponding to the non-dimensional frequency of St = (a,b)0 .
99 and (c,d) 1 . a) α = 20 ◦ (b) α = 25 ◦ Figure 13: Aeroelastic mode energy spectra of the membrane displacement fluctuations, thesurface pressure fluctuations and the spatial Y -vorticity fluctuations at (a) α = 20 ◦ and (b) α = 25 ◦ . St=0.752St=1.450St=2.819
Surface displacement Surface pressure difference Space pressure Space Y-vorticity (a) α = 20 ◦ St=0.053St=0.189St=0.776St=2.872
Surface displacement Surface pressure difference Space pressure Space Y-vorticity (b) α = 25 ◦ Figure 14: Aeroelastic mode decomposition and dominant mode identification of fluid-membrane interaction at (a) α = 20 ◦ and (b) α = 25 ◦ . .3. Aeroelastic mode selection mechanism From the observation of the aeroelastic responses and the spatial flow struc-tures of the flexible membrane, the intertwined modes are excited through FMI.The results indicate that flexibility plays an important role in selecting partic-ular aeroelastic modes via an underlying mechanism. To explore the role offlexibility in membrane aeroelasticity, we further simulate a rigid flat wing anda rigid cambered wing at the AOA of α = 15 ◦ with obvious shedding vortices.The rigid flat wing has the same wing geometry as the undeformed geometry ofthe flexible membrane. The rigid cambered wing shares the same wing shapeas the mean wing shape of the flexible membrane under aerodynamic loads.Figure 15 presents the comparison of the pressure coefficient difference be-tween the upper and lower surfaces, the turbulent intensity on the mid-spanplane and the local effective AOA ( α eff ) between these three wings. The localeffective AOA is defined as the angle between the oncoming freestream flow andthe tangent of the wing shape at the LE. The flexibility affects the membranedynamics from two aspects, namely (i) camber effect and (ii) flow-excited vi-bration. Compared to the rigid flat wing, the low-pressure area near the LEenlarges and the turbulent intensity is reduced dramatically near the surface forthe rigid cambered wing. This can be attributed to the reduced α eff from 15 ◦ to10 . ◦ as the wing is cambered up. When the additional membrane vibrationis allowed for the flexible membrane, α eff is altered as a function of time, whichhas a mean value of 10 . ◦ . This time-varying α eff leads to the extended suctionregion at the LE and the excited high-intensity turbulent fluctuation near themembrane surface.To gain further insight into the role of flexibility, we further analyze thedecomposed fluid fields of the rigid flat wing and the rigid cambered wing with512 time-varying samples and compare them with those of the flexible wing.Figure 16 presents the frequency spectra of the global mode energy of the surfacepressure coefficient fluctuation C (cid:48) sp and the spatial Y -vorticity fluctuation ω (cid:48) y .The frequency spectra of the rigid flat wing show multiple frequency peakswithin St ∈ [0 , . St = 0 . St = 0 . (a)(b) α eff ( ◦ ) t / T Rigid flat Rigid cambered Flexible (c)Figure 15: Comparison of flow features between a rigid flat wing, a rigid cambered wing anda flexible membrane wing at α =15 ◦ : (a) time-averaged pressure coefficient difference betweenthe upper surface and the lower surface, (b) turbulent intensity on the mid-span plane and(c) local effective angles of attack at the LE on the mid-span location. igure 16: Comparison of global mode energy spectra of the fluctuations of the surface pressurecoefficient fluctuation and the spatial Y -vorticity fluctuation between a rigid flat wing, a rigidcambered wing and a flexible membrane at α =15 ◦ . St=0.0805St=0.5102
Space pressureReal part Space pressureAmplitude Space Y-vorticityReal part Space Y-vorticityAmplitude (a)
St=0.0805St=0.4297
Space pressureReal part Space pressureAmplitude Space Y-vorticityReal part Space Y-vorticityAmplitude (b)Figure 17: Comparison of decomposed pressure and Y -vorticity fields at α =15 ◦ for: (a) arigid flat wing, (b) a rigid cambered wing.
30e summarize the decomposed Fourier modes both in the pressure and Y -vorticity fields for the rigid flat wing and the rigid cambered wing in figure17. The dynamic modes of the rigid cambered wing show weaker pressure andvorticity fluctuations near the surface, compared to the rigid flat wing. Thecamber effect induced by the passive deformation alters the vortex sheddingpatterns. When the flexible membrane is excited by the pressure pulsation tovibrate in particular modes, it can be observed from figure 11 and figure 12 thatboth the pressure wavelength and the wake pattern are changed to synchronizewith the corresponding vibration mode at the same coupled frequency. Theobserved mode synchronization phenomenon indicates that there is an inherentmechanism to select aeroelastic modes in the coupled system. To quantitativelyconnect the membrane vibration and the pressure pulsation, we further calculatethe wavelength of each Fourier mode by projecting the mode from the spatial-frequency space to the wavenumber-frequency space. Considering the significantvariation of the modal shapes along the chord-wise direction, the membrane vi-bration and the pressure fluctuation signals along the membrane chord at themid-span location are analyzed. The wavelength corresponding to a specificvibrational frequency can be computed based on the double Fourier transform(DFT). The wavenumber-frequency spectrum displays the relationship betweenthe wavenumber and the frequency of a waveform and it offers detailed informa-tion on how the wave travels. This spectrum is calculated based on the DFT toconvert the physical signals y ( x, t ) into both wavenumber and frequency space.The equation of DFT is expressed as F ( y ( x, t ))( f, λ ) = 1 N M N − (cid:88) t =0 M − (cid:88) x =0 y ( x, t ) e − i πft e − i πλ x (18)where λ is the wavelength and f represents the frequency. F ( y ( x, t ))( f, λ ) isthe double Fourier-transformed coefficient.We extract the time-varying membrane vibration and surface pressure sig-nals at 256 equispaced points at the mid-span location along the chord-wisedirection of the membrane wing for 512 non-dimensional time instants. Thesedata samples are stored in a time-space matrix. The mean values are removed31rom the full signals and the fluctuations of the analyzed signals are mappedinto frequency-wavenumber space through the DFT. The wavenumber-frequencydiagrams of the membrane deflection fluctuations and the pressure differencefluctuations are plotted in figure 18. These 2D diagrams are colored by theamplitude of the double Fourier transformed coefficients of the analyzed signalfunctions. The x -axis is the wavenumber of the unit chord length c/λ and the y -axis indicates as the Strouhal number St = f c/U ∞ . (a) (b)Figure 18: Wavenumber-frequency spectra of: (a) membrane deflection fluctuations and (b)pressure coefficient difference fluctuations along the flexible membrane at the mid-span loca-tion. The dominant frequency of St = 0 .
99 and its harmonics with obviouslyconcentrated energies are observed in both diagrams. The excitation frequencycontents are consistent for the membrane vibration and the pressure pulsation.In the wavenumber-frequency spectrum of the membrane deflection fluctuations(figure 18 (a)), the wavelength of the membrane vibration at St = 0 .
99 isapproximately one chord length of the wing. The wavelength reduces to 0.66 ofthe wing chord length at St = 1 .
96 for the high-order mode. Different from thepure modal shape of the structural vibration, the decomposed pressure modesin figure 10 (b) show shorter wavelengths due to the existence of the rolled-upLEVs in the proximity of the LE. Although the LEVs moving backward induce a32mall portion of the pressure fluctuations at the LE, the remaining modal shapesof the pressure field exhibit similar characteristics to the membrane vibration.The predominant wavelength of the surface pressure fluctuations at St = 0 . c . For the high-order mode at St = 1 .
96, thepressure wavelength becomes smaller to 0.625 c . ParticularvibrationalmodeSpatial vortexshedding Surface pressurefluctuations
Exciterolled upvortices Induce flow fluctuationsas vortices movebackwardDrive synchronizedvibrations by similarpressure wavelengths
Figure 19: Illustration of a feedback loop of fluid-membrane coupled mechanism.
The investigation of the decomposed aeroelastic modes suggests a feedbackloop shown in figure 19 to reveal the mode selection mechanism for FMI prob-lems with obvious vortex shedding phenomenon. In this highly coupled system,the membrane vibration in particular vibrational modes excites the separatedshear layer to sooner roll up, and then form large-scale vortices. As these vor-tices detach from the membrane surface and are convected downstream, strongsurface pressure fluctuations are induced by the passing-by vortical structures.Subsequently, the flexible membrane is synchronously driven by the pressurepulsations to excite particular vibrational modes with similar modal shapes.Eventually, the unsteady flow and the membrane vibration enter a strong cou-pled state and the frequency synchronization to select the particular aeroelasticmodes. 33 . Conclusions
We presented an aeroelastic mode decomposition framework based on theglobal Fourier mode decomposition technique to extract and identify the dy-namic modes of interest both in the fluid and structure fields. The FMD methodwas first applied to a thin 3D flexible membrane with multi-modal mixed vi-brations excited by the unsteady flow at a moderate Reynolds number. Thecorrelated dominant fluid and structure modes were successfully extracted fromthe coupled system according to the global mode energy spectra. To explorethe role of flexibility in membrane aeroelasticity, we further compared the flowfeatures and the dynamic modes of a rigid flat wing, a rigid cambered wingand their flexible counterpart at a moderate AOA with obvious vortex sheddingphenomenon. We found the camber effect caused by the passive deformationreduced the local effective AOA and altered flow features. The flow-inducedvibration excited vortices at the LE in a frequency synchronized way. Basedon the modal analysis and the wavenumber-frequency spectra, we suggested afeedback loop between the membrane vibration mode, the vortex shedding pro-cess and the surface pressure fluctuations. This feedback loop revealed that themembrane flexibility acted as a coordinator between the flexible membrane andthe unsteady flow to form a mode synchronization and sustain the membrane vi-bration. This mode decomposition method have the potential to be extended tothe data analysis of other fluid-structure interaction problems with thin flexiblestructures. It is able to provide new insight into the coupled mechanism of thesystem with intertwined modes by connecting them directly. The FMD methodavoids the complex matrix manipulation of the POD and DMD techniques andrelaxes restrictions. Considering the features of the FMD method, it is naturallysuitable for parallel computation to save computing resources for large amountsof data analysis. However, it is not ideal and suitable for all contexts. A com-bined application with other mode decomposition techniques could offer a betterand comprehensive understanding of fluid-structure interaction problems.34 cknowledgements
The first author wishes to acknowledge supports from the National Uni-versity of Singapore and the Ministry of Education, Singapore. The secondauthor would like to acknowledge the support from the University of BritishColumbia and the Natural Sciences and Engineering Research Council of Canada(NSERC).
Declaration of interests
The authors report no conflict of interest.
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