Bulletin of the American Physical Society
73rd Annual Meeting of the APS Division of Fluid Dynamics
Volume 65, Number 13
Sunday–Tuesday, November 22–24, 2020; Virtual, CT (Chicago time)
Session Q02: Aerodynamics: Fluid Structure Interactions, Membranes, Flutter (3:55pm  4:40pm CST)Interactive On Demand

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Q02.00001: Aeroelastic Mode Transition in SelfInduced Vibration of Flexible Membrane Wings Guojun Li, Boo Cheong Khoo, Rajeev Jaiman Strong fluidstructure coupling of the unsteady flow with a flexible membrane wing excites intertwined aeroelastic modes, thus forming the complex selfexcited vibration characteristics. Using fullyresolved numerical simulations, the unsteady coupled dynamics and the aeroelastic mode transition phenomenon are examined as a function of Reynolds number and mass ratio for a 3D flexible membrane wing. To gain further insight into the aeroelastic mode transition, we adopt an effective global mode decomposition method to isolate the frequencyranked dominant aeroelastic modes from the nonlinear coupled fluidmembrane fields. The contribution of each mode to the overall membrane responses is quantitatively calculated with the aid of the proposed mode decomposition method. Based on the aeroelastic modal analysis, we discover that the dominant aeroelastic modes are strongly influenced by mass ratio and Reynolds number. We propose an interaction cycle to establish a direct connection between the unsteady flow features and the membrane vibrations. As the change of mass ratio and Reynolds number, the original mode synchronization process is interrupted, and then the aeroelastic mode transition is triggered to form a new mode synchronization state. [Preview Abstract] 

Q02.00002: Eigenvalue analysis of membrane stability in inviscid flow Christiana Mavroyiakoumou, Silas Alben We study the stability of a thin membrane with a vortex sheet as a nonlinear eigenvalue problem in the parameter space of membrane mass and pretension. When both membrane ends are fixed, the stability boundary is fairly simple: light membranes become unstable by a divergence instability and heavy membranes appear to lose stability by flutter and divergence, which occurs for a pretension value that increases with the membrane mass. With the leading edge fixed and trailing edge free, or both edges free, the membrane eigenmode shapes become more complicated and eigenmodes transition in shape across the stability boundary. We compare the eigenvalue analysis with simulations of the corresponding initial value problem in the smallamplitude (growth) regime. [Preview Abstract] 

Q02.00003: Nonlinear Stability Boundaries of Elastically Mounted Pitching Swept Wings Yuanhang Zhu, Kenneth Breuer We study the nonlinear stability boundaries of cyberphysically mounted pitching swept wings in a water tunnel. As the wing sweep, $\Lambda$, increases from $0^\circ$ to $10^\circ$, the flowinduced limitcycle oscillations (LCOs) become more stable and can sustain at higher pitching frequencies. In contrast, as $\Lambda$ further increases to $25^\circ$, the LCOs become less stable and annihilate at lower pitching frequencies. We attribute this nonmonotonic behavior to the competition between two mechanisms: (a) the stabilization of leadingedge vortices (LEVs) by the wing sweep, which promotes LCOs and thus destabilizes the system, and (b) the increasing fluid damping brought by the wing sweep, which damps out LCOs and thus stabilizes the system. Because the flowinduced LCOs are near sinusoidal, we use prescribed sinusoidal motions to map out the energy transfer between the wings and the ambient fluid over a large range of pitching amplitudes and frequencies. We observe that the amplitudes and frequencies of flowinduced LCOs match remarkably well with the neutral energy transfer curve generated by prescribed motions. Lastly, we compare flow fields and LEV dynamics for different swept wings. [Preview Abstract] 
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Q02.00004: PhysicsInformed Neural Networks for the Modelling of FluidStructure Interactions Elijah Ang, Bing Feng Ng This paper proposes the use of physicsinformed neural networks (PINN) to overcome the large computational overheads in FluidStructure Interaction (FSI) simulations that couples Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) modules. Rather than using the difference between predicted and targeted outputs which is common in conventional neural networks, PINNs uses residuals of governing equations as the loss function for the network. As a proof of concept, a PINN was trained to solve for flow over a flat plate, and the results were able to replicate Blasius similarity solution. The neural networks were trained by computing loss functions defined as the norm of the residuals calculated at randomly sampled collocation points. A gradient descent algorithm was subsequently used to minimize the loss by adjusting the weights and bias. By optimizing the residuals, the PINN acted as a function approximator for the solution of the partial differential equations which govern the physics of the problem. The use of PINN was further extended to solve more complex flow problems and to model structural dynamics, which are then coupled to solve FSI problems. [Preview Abstract] 

Q02.00005: Comparison of numerical methods for 3D FluidStructure Interaction problems at low Reynolds numbers Cayetano MartinezMuriel, Rauno Cavallaro, Rocco Bombardieri, Oscar Flores, Manuel GarciaVillalba The objective of this work is to compare two numerical methods to solve 3D FSI problems for flexible bodies. In both methods, the fluid phase is computed with the inhouse code TUCAN, that solves the NavierStokes equations of the incompressible flow, where the presence of the body is modelled by using the Immersed Boundary Method proposed by Uhlmann (2005; J. Comp. Phys. 209). For the structural solver, the first method uses a multibody algorithm (MB) based on the rigidbody dynamics algorithm proposed by Felis (2017; Auton. Robot. 41), modelling the flexibility of the body as a system of rigid bodies connected by flexible joints (i.e, springs). The second method employs an inhouse, nonlinear, finiteelement structural solver (AUGUSTO) to model the flexibility of the body. Timeintegration is performed using a $\beta$Newmark method with numerical damping on highorder modal spurious artifacts. The coupling between the fluid solver (TUCAN) and the structural solver (MB or AUGUSTO) is weak in both cases. Results will be presented for a flexible plate ($\pi_0 = O(10^010^2), \pi_1 = O(10^{4}  10^{2})$) immersed on a free stream ($Re = O(100)$), allowing a direct comparison between both structural solvers. [Preview Abstract] 

Q02.00006: Fluidstructure interaction simulations of flexible and elastic bodies Sujyesh Aanandh Manjunathan, Iman Borazjani Fluid structure interaction of flexible and elastic of high aspect ratio finds its application in areas such as propulsion of underwater vehicles, drag reduction and enhanced energy harvesting by utilizing the large scale flow induced deformation experienced by it. Such fluid structure interactions are investigated here using a partitioned approach by coupling curvilinear immersed boundary based fluid solver for incompressible flow with an opensource finite element solver for structures. The dynamic analysis of the elastic structure is approached using Newmark method of implicit direct integration that is unconditionally stable and validated using the method of manufactured solution. The fluid and structure domains are strongly coupled for numerical stability. The coupled solver after validation is used to investigate the influence of leeward attached flexible and elastic plate on agitation and subsidence of vortex induced vibration of circular cylinder. [Preview Abstract] 

Q02.00007: Resonances in pulsatile channel flow with an elastic wall Duo Xu, Matthias Heil, Thomas Seeböck, Marc Avila Fluid flows through elastic conduits are ubiquitous in engineering and physiology. For the case of pulsatile and oscillatory flows, such as blood flow in the arteries and air flow in the respiratory airways, the flow time scale interacts with the natural time scales of the vessels. We investigate this interaction in a channel flow driven by a pulsatile pressure difference. The bottom channel wall is rigid, whereas an elastic membrane is clamped between two rigid sections of the upper wall. Our simulations show that after a transient determined by the viscosity of the fluid, the membrane pulsates with the driving frequency. Interestingly, the amplitude of the oscillation varies nonmonotonously with the governing parameters and exhibits strong resonances. The membrane response is determined by multiple governing parameters, but we show that it can be quantitatively modeled with a harmonic oscillator equation (with nonconventional damping). All key features of the system are predicted by our model: oscillation amplitude, phase lag, resonance point and vanishing of the resonance when viscous damping dominates. [Preview Abstract] 

Q02.00008: Response of Flexible Cantilevered Cylinders in Wind Jennifer Cardona, John Dabiri We present an experimental investigation of cantilevered flexible cylinders rigidly mounted in a wind tunnel. Many examples of flowstructure interactions (including plants in the wind) may be represented more closely by flexible cantilevered cylinders than by the canonical elastically mounted rigid cylinder in a crossflow. However, cantilevered cylinders have been less often studied, and prior work analyzing the response of flexible cantilevered cylinders has focused mainly on cylinders of low mass ratio for which it is known that synchronization of structure vibration and vortex shedding occurs over a broader range of reduced velocities. Here, the response of flexible cantilever cylinders will be presented and compared to previously observed responses of cylinders in cross flow due to vortexinduced vibrations including cantilevered cylinders with lower mass ratios. Understanding of responses due to fluidstructure interactions like this may be useful toward applications such as visual anemometry, in which visual observations of a structure undergoing wind loading are used to approximate wind speed. [Preview Abstract] 

Q02.00009: A datadriven framework for the isolation, tracking and aerodynamic load estimation of distinct vortex structures Karthik Menon, Rajat Mittal This work presents a physicsbased and datadriven computational framework for the analysis of vortexdominated fluidstructure interaction problems. The dynamics of such problems are typically dictated by multiple distinct, forceproducing vortical structures. However, accurately estimating the aerodynamic loads induced by each of these vortex structures in complex viscous flows remains an open question. In the analysis framework presented here, a rigorous force and moment partitioning method is used in conjunction with clustering techniques to simultaneously isolate, track, and quantify the forceproduction due to several distinct vortex structures in complex, unsteady flowfields. This flexible, automated framework allows us to precisely compute the force and moment induced by each vortical structure on an immersed body, and also correlate the spatiotemporal evolution of each structure to its dynamical influence on the problem. [Preview Abstract] 

Q02.00010: Thrust Estimation for a Flapping Membrane Foil Using Control Volume Analysis Gali Alon Tzezana, Varghese Mathai, Kenneth Breuer Analytic models have shown that the use of a compliant membrane may improve or impair the propulsive performance of a flapping wing, depending on the ratio between elastic, inertial, and fluid forces. However, such models are limited by assumptions of small deformations and potential flow. Therefore, it is of interest to experimentally measure the thrust produced by flapping membranes. The experimental evaluation of thrust of a flapping membrane is challenging, due to the difficulty to separate inertial forces from the small aerodynamic (or hydrodynamic) forces. Here, we use control volume analysis to estimate the thrust force generated by a heaving membrane in a water flume. Particle Image Velocimetry (PIV) measurements provide velocity data in the wake, and the velocity fluctuations are used to estimate the downstream pressure profile. We compare the results with direct force measurements, where inertial forces are subtracted, and discuss the strengths and limitations of both methods. While the mean velocity profile contains much of the information needed to predict the propulsive performance, the contribution of unsteady effects is not negligible. As the membrane compliance increases, a transition is observed in the mean wake structure, accompanied by thrust reduction. [Preview Abstract] 

Q02.00011: Prediction of the Onset of Aeroelastic Flutter using Complex Networks Sombuddha Bagchi, Vishnu R. Unni, Abhishek Saha We use complex networkbased methods to unravel the dynamic characteristics of strain rate fluctuations of a structure suspended in turbulent flow to examine the viability of using network parameters as precursors for onset of aeroelastic flutter. The dynamic transition from low amplitude chaotic regime to the high amplitude limit cycle oscillations can be detected by measures that quantify the topology of the network. Strain rate time series contain a range of information regarding the dynamical states of the system. The network construction technique determines the information extracted from time series. We use quantification measures corresponding to 3 types of networks, modified recurrence network (MRN), visibility network (VN) and synchronization network (SN). The information on periodicities in the system dynamics is provided by the MRN; VN characterizes local patterns in the time series and thus provides information on temporally local interactions, and the SN embodies information about the attractor geometry and correlations at larger time scales. We show that degree, clustering and characteristic path length (CPL) for MRN; degree, CPL and closeness centrality for VN; and degree, CPL and betweenness centrality for SN can be used as precursors for the onset of flutter. [Preview Abstract] 

Q02.00012: Relationship between the wake and the flexibility of a falling plate Edwin M. Lau, WeiXi Huang, JunDuo Zhang, YongXia Jia The natural free fall of a flat body has received considerable attention through experimental and theoretical analyses modelling as a rigid body. However, the motion of a leaf in its fall appears to remain incomprehensible. A falling leaf conforms and its motion may be affected. Here, free falling is investigated using flexible, flat plates in particle image velocimetry. The experiment depicts a plate deformation and the corresponding flow fields. Plate flexibilities in the chordwise direction are controlled through various mixtures of PDMS and polymer materials. A water tank is prepared with glass beads for laser reflections. The plates are released in free falls within the tank. The movement of the flow is captured at about one thousand frames per second using a high speed camera. As the plate’s chordwise deformation is observed, it is accompanied by a downwashing flow. This flow is produced by the vortical wake generated as the plate traverses across the still fluid as it falls. A relationship between the plate’s flexibility and the flow is drawn, suggesting that downwash affects deformation. In turn, this deformation would lead to the change in wake proximity and the configuration that produces the downwash itself, and would ultimately change the motion of the plates. [Preview Abstract] 

Q02.00013: An Experimental Study of Surface Topology Effect on the Aerodynamics of Rectangular Cylinders Kian Kalan, Alireza Safaripour, Ahmed Naguib, Manoochehr Koochesfahani The motivation behind this study is to provide a fundamental understanding of the effects of surface topology on the instability of rectangular cylinders to galloping. Galloping is a form of aeroelastic instability that elasticallymounted noncircular cylinders may experience and it can lead to largeamplitude selfsustained transverse oscillations of the cylinders (usually normal to the flow direction). Here, the motivating application is the possible galloping of suspension cables of Precision Airdrop Systems, which could potentially impair their ability to land their cargo on target. The crosssection of these cables is nominally rectangular with fully rounded corners and a braided surface topology, which we represent by a twodimensional Fourier expression. The effects of surface topology and Reynolds number on the aerodynamics of these cylinders are investigated through complementary singlecomponent molecular tagging velocimetry and direct force measurements performed in a closedloop water tunnel facility. These results are compared to those of a baseline smoothsurface cylinder to provide insight into the links between the flow behavior around the cylinders and their aerodynamic load characteristics. [Preview Abstract] 

Q02.00014: Why do nonlinear springs give higher energy extraction efficiencies during Vortex Induced Vibrations? Rameez Badhurshah, Rajneesh Bhardwaj, Amitabh Bhattacharya Vortex Induced Vibrations (VIV) of a springmass system immersed in uniform fluid flow can be used for hydrokinetic energy generation. Several researchers have found that introducing nonlinearity in the spring potential dramatically increases the range of reduced velocity over which structure synchronizes (locks in) with the vortex shedding, and also yields high energy extraction efficiency. To understand how spring nonlinearity affects lockin, a theory is formulated, in which the rate of energy generation via vortex shedding is balanced by the rate of energy dissipated via damping. The theory implies that a universal ``Equilibrium Constraint" exists between oscillation amplitude and structure frequency. Lockin occurs when the natural frequency versus amplitude curves intersect the EC curves. As a result, nonlinearity in springs can widen the span of reduced velocity over which lockin occurs. We find that the EC is dependent on the ratio of damping coefficient to reduced velocity, which in turn explains the higher energy extraction efficiencies seen in VIVbased hydrokinetic energy generators in the presence of nonlinear springs. Numerical simulations, based on Immersed Boundary Method, have been used to validate the above theoretical analysis. [Preview Abstract] 

Q02.00015: Coupled dynamics and stability of cantilever beams at low Reynolds number: application to whisker dynamics Shayan Heydari, Thomas L. Janssen, Neelesh A. Patankar, Mitra J. Z. Hartmann, Rajeev Jaiman Fluidstructure interaction between a flexible cantilever beam and the surrounding flow is ubiquitous in nature and engineering systems. For instance, a rat's whisker  approximated as a cantilever beam with a constant circular crosssection  is shown to undergo oscillatory motion even at very low Reynolds numbers ({\$}Re{\$}). To study the coupled dynamics and selfexcited stability of a flexible cantilever beam, we carry out highfidelity 3D numerical experiments at Reynolds numbers {\$}Re\textless 50{\$} for varying angles of attack and aspect ratios. Our goal is to investigate the exact origin of the selfexcited oscillations at low Reynolds numbers and highlight critical fluidstructure interaction aspects of the coupled system. In this study, we attempt to answer four key questions: (i) How can a flexible beam undergo a synchronized highamplitude oscillation below critical Reynolds number (i.e., no periodic vortex shedding)? (ii) What is the intrinsic relationship between the steady wake flow and the oscillatory modes of the beam? (iii) How do the structural and/or wake nonlinearities contribute to the onset of fluidelastic instability? and (iv) What is the optimal range of bending stiffness to sustain the fluidelastic instability? [Preview Abstract] 

Q02.00016: Compliant membranes exhibit enhanced drag due to membrane fluctuations. Asimanshu Das, Varghese Mathai, Kenneth Breuer We study the kinematics and dynamics of a highly compliant membrane disk placed headon in a uniform flow. With increasing flow velocity, the membrane disks deform nonlinearly into increasingly parachutelike profiles. The experiments were carried out in a closedloop lowspeed wind tunnel with Reynolds number in the range of {\$}10\textasciicircum 4 10\textasciicircum 5. Remarkably, these aerodynamically sustained membrane disks show a higher flow resistance (drag) than similarly shaped rigid concave bodies. We model the steady structural response of the membranes using a nonlinear aeroelastic model. The predictions of the model agree well with the mean deformations of the membrane disks for the full range of experimental parameters studied. Through a simultaneous quantification of the unsteady membrane kinematics and forces, we detect the onset of large amplitude membrane fluctuations, match with the observed drag modulation and have their origins in the resonance between the flow structures and the membrane's natural frequency. A drum model with anisotropic springstiffness is proposed, which quantitatively captures the observed resonant response. Further, PIV experiments are being conducted to yield deeper insights into the steady and transient fluidstructure interactions. [Preview Abstract] 

Q02.00017: Fluid structure interactions of an oscillating compliant membrane hydrofoil Varghese Mathai, Gali Alon Tzezana, Kenneth Breuer We study the fluidstructure interactions of a compliant membrane hydrofoil undergoing heaving and pitching oscillations in a uniform flow. Experiments were conducted in a low speed, circulating water channel with a mean flow velocity U$_\infty$ = 0.3 m/s, and chordbased Reynolds number, Re = 3 \times 10$^4$. Simultaneous measurements of the forces, membrane deformation and the flow field we conducted using a 6axis load cell, highspeed imaging, and particle image velocimetry (PIV), respectively. The membrane foil passively adapts its shape and camber during each oscillation cycle, which enables the leadingedgevortex to remain attached, thus contributing to lift enhancements and better power production when compared to a rigid symmetric hydrofoil. We model the instantaneous membrane deformation using the YoungLaplace equation, by considering the instantaneous angle of attack, in good agreement with the membrane tracking measurements for a range of pitch angles. The total lift force generated by the oscillating membrane is decomposed into a lumped model that includes a steady lift contribution and a vortex lift. The force measurements and PIV results will be used to explain the origin of these contributions. [Preview Abstract] 

Q02.00018: Addedmassinduced beating between a submerged pendulum and a free surface wave mode Timothy T. K. Chan, Varghese Mathai, Sander Wildeman One would expect a pendulum released inside a tank of quiescent liquid to oscillate with a monotonically decreasing amplitude until it finally comes to rest. Here we present a surprising observation where a submerged heavy pendulum starts up again after coming to a halt. We demonstrate that this effect arises from the twoway coupled interactions between the pendulum and the sloshing wave mode generated in the tank of liquid. A strong coupling is observed when the pendulum’s natural frequency matches that of the wave mode. We model this behavior using a lumped variational approach that treats the pendulum and the wave mode on equal footing. The beating behavior is shown to resemble that of the wellknown physics demonstration of two oscillating pendula coupled by a weak spring. While it is not immediately obvious how such an interaction arises inside a liquid, where there are no springs or other elastic forces, we show that the coupling between the pendulum and the liquid originates from pure added mass effects. Our work demonstrates how accelerationinduced forces such as “kinetic buoyancy” (known from e.g. the pioneering work of Bjerknes)  commonly used to describe the behavior of submerged bodies  can strongly influence the motion of the bulk surrounding fluid. [Preview Abstract] 

Q02.00019: Computational study of active control of aeroelastic flutter using synthetic jets Hansong Liu, JungHee Seo, Rajat Mittal Aeroelastic flutter is an important phenomenon in a variety of applications including aircraft design and hydrokinetic energy harvesting. Active flow control is one possible approach to suppress or enhance aeroelastic flutter. Zeronetmass flux synthetic jet have unique properties that enable a variety of applications in flow control such as suppression of flow separation and drag reduction and in the current study, we show that synthetic jets also have the potential to control aeroelastic flutter. We use computational modelling with a sharpinterface immersed boundary method to model the air flow over a pitching airfoil and to investigate the effects of synthetic jet actuation on aeroelastic flutter of the airfoil. The concept of “energy maps” that quantify the energy exchange between the elastic structure and the flow, are used to guide and optimize the synthetic jets for flutter suppression. [Preview Abstract] 

Q02.00020: Piezoelectric energy harvesting from fluidstructure interaction Abinayaa Dhanagopal, Thomas Ward Piezoelectric energy harvesters have risen in popularity due to their ability to convert ambient vibrational energy into power. Small scale harvesters built on this principle can potentially charge batteries and power microscale devices. Experiments were conducted inside a lowspeed wind tunnel (0 to 3.6 m/s) using commercial piezoelectric transducers subjected to uniform flow with the goal of maximizing the resultant voltage. Rectangular bluff bodies with varying aspect ratios (0.53.0) were loaded onto flexible piezoelectric transducers. The cantileveredtip mass system was arranged in a parallel configuration to aid in vibration frequency control. Images at high speed were captured to observe the rise in voltage generated by coupled bluffbody and cantilever tip vibration as a function of incident wind speed. This allowed for the assessment of the flutter/vortexinduced vibration potential of the system. The operating limit for each bluff body of a given B/D ratio was established in terms of the wind speed, Reynolds number (Re), Strouhal number (St), and reduced velocity. Beyond the operating limit, no positive effect on the resultant voltage was observed. These results will aid the design and development of vibrationbased energy harvesters. [Preview Abstract] 

Q02.00021: On the fluidstructure interaction of synthetic leaves. Wing Lai, Dan Troolin, Sun Jiao, Shyuan Cheng, Leonardo Chamorro The dynamics of a series of synthetic leaves and induced turbulence were studied experimentally in a wind tunnel for Cauchy numbers resembling those observed in nature. We use digital image correlation, highframerate particle image velocimetry, and a highresolution load cell to track the structure motions, flow field, and forces on the synthetic leaves. Here, we will discuss some features of the techniques and distinct behavior of the structures and wake flow in response to various incoming flows. Particular emphasis is placed on characterizing flow instability, the role of leave shape on the structure motions; for this purpose, we used loworder decomposition to uncover modulating mechanisms.. [Preview Abstract] 

Q02.00022: The oscillatory dynamics of whiskers: nonlinear modal coupling vs. fluid coupling Thomas L. Janssen, Shayan Heydari, Rajeev Jaiman, Mitra J. Z. Hartmann, Neelesh A. Patankar Previous work has shown that rats use their whiskers to localize airflow sources. A single whisker bends primarily in the direction of flow and vibrates around its deflected position. The vibrations have components both in and out of the bending plane. This dynamic response is surprising because galloping or vortex induced vibrations are not expected for thin beams with circular crosssections at low Reynolds numbers (Re). We hypothesized that the dynamics could result either from geometric nonlinearities that lead to coupled structural bending modes, or from nonlinear fluidstructure interactions at low Re, or from a combination of both effects. To investigate the effect of coupled bending modes, we developed a model for a thin, long, and stiff cantilever beam under the influence of fluid loading at low Re. We include terms to capture hardening nonlinearity due to stored potential energy in bending, inertia nonlinearity due to kinetic energy of axial motion, as well as a quasistatic low Re drag model to provide simplified fluid forcing. To separately investigate the effects of nonlinear fluid coupling, we consider a second model in which the fluid is fully resolved but the beam is simplified by considering only linear modes.~A comparison between these two studies will be presented. Ultimately, our goal is to combine the two techniques discussed here to fully resolve fluidstructure interactions that capture nonlinearities due to both structural and fluid phenomena. [Preview Abstract] 

Q02.00023: Invertedvibrating cantilever in a uniform flow Veera Sajjanapu, Thomas Ward We present an experimental study of the interaction between flexible structures and a flowing fluid medium using a soap film. The vertically falling soap film is capable of attaining speeds ranging from 1.5  3 m/s with an operating test section width of 8 cm yielding us a Reynolds number $Re$ $>$ 5000. Experiments were conducted for flexible cantilever beams of lengths L $\leq$ 10 mm and of thicknesses ranging from 0.03  0.08 mm that were placed at angles of attack ranging from 0$^{\circ}$  180$^{\circ}$. We visualize the beam displacements, vibrations and wake with a highspeed camera. Assuming small vibrational amplitudes, we consider the EulerBernoulli beam theory to understand aspects of the dynamics. Data will be presented for Strouhal number versus inclination angles for the full range of angles. We observe a flutter behavior which seems to be dependent on the cantilever rigidity modulus. [Preview Abstract] 
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