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1. Reduced-Order Modeling and Experimental Studies of Two-Way Coupled Fluid-Structure Interaction in Flapping Wings

   https://doi.org/10.1115/DETC2019-98291  

   Schwab, Ryan K. ; Reid, Heidi E. ; Jankauski, Mark A. ( August 2019 , Volume 8: 31st Conference on Mechanical Vibration and Noise)

   Flapping, flexible wings deform under both aerodynamic and inertial loads. However, the fluid-structure interaction (FSI) governing flapping wing dynamics is not well understood. Conventional FSI models require excessive computational resources and are not conducive to parameter studies that consider variable wing kinematics or geometry. Here, we present a simple two-way coupled FSI model for a wing subjected to single-degree-of-freedom (SDOF) rotation. The model is reduced-order and can be solved several orders of magnitude faster than direct computational methods. We construct a SDOF rotation stage and measure basal strain of a flapping wing in-air and in-vacuum to study our model experimentally. Overall, agreement between theory and experiment is excellent. In-vacuum, the wing has a large 3ω response when flapping at approximately 1/3 its natural frequency. This response is attenuated substantially when flapping in-air as a result of aerodynamic damping. These results highlight the importance of two-way coupling between the fluid and structure, since one-way coupled approaches cannot describe such phenomena. Moving forward, our model enables advanced studies of biological flight and facilitates bio-inspired design of flapping wing technologies.

    

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2. Deformable Blade Element and Unsteady Vortex Lattice Fluid-Structure Interaction Modeling of a 2D Flapping Wing

   https://doi.org/10.1115/DETC2020-22638  

   Reade, Joseph ; Jankauski, Mark A. ( August 2020 , ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference)

   null (Ed.)

   Flapping insect wings experience appreciable deformation due to aerodynamic and inertial forces. This deformation is believed to benefit the insect’s aerodynamic force production as well as energetic efficiency. However, the fluid-structure interaction (FSI) models used to estimate wing deformations are often computationally demanding and are therefore challenged by parametric studies. Here, we develop a simple FSI model of a flapping wing idealized as a two-dimensional pitching-plunging airfoil. Using the Lagrangian formulation, we derive the reduced-order structural framework governing wing’s elastic deformation. We consider two fluid models: quasi-steady Deformable Blade Element Theory (DBET) and Unsteady Vortex Lattice Method (UVLM). DBET is computationally economical but does not provide insight into the flow structure surrounding the wing, whereas UVLM approximates flows but requires more time to solve. For simple flapping kinematics, DBET and UVLM produce similar estimates of the aerodynamic force normal to the surface of a rigid wing. More importantly, when the wing is permitted to deform, DBET and UVLM agree well in predicting wingtip deflection and aerodynamic normal force. The most notable difference between the model predictions is a roughly 20° phase difference in normal force. DBET estimates wing deformation and force production approximately 15 times faster than UVLM for the parameters considered, and both models solve in under a minute when considering 15 flapping periods. Moving forward, we will benchmark both low-order models with respect to high fidelity computational fluid dynamics coupled to finite element analysis, and assess the agreement between DBET and UVLM over a broader range of flapping kinematics.

    

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3. A Novel Fluid–Structure Interaction Framework for Flapping, Flexible Wings

   https://doi.org/10.1115/1.4044268  

   Schwab, Ryan ; Johnson, Erick ; Jankauski, Mark ( December 2019 , Journal of Vibration and Acoustics)

   Fluid–structure interaction (FSI) plays a significant role in the deformation of flapping insect wings. However, many current FSI models are high-order and rely on direct computational methods, thereby limiting parametric studies as well as insights into the physics governing wing dynamics. We develop a novel flapping wing FSI framework that accommodates general wing geometry and fluid loading. We use this framework to study the unilaterally coupled FSI of an idealized hawkmoth forewing considering two fluid models: Reynolds-averaged Navier–Stokes computational fluid dynamics (RANS CFD) and blade element theory (BET). We first compare aerodynamic modal forces estimated by the low-order BET model to those calculated via high fidelity RANS CFD. We find that for realistic flapping kinematics, BET estimates modal forces five orders of magnitude faster than CFD within reasonable accuracy. Over the range flapping kinematics considered, BET and CFD estimated modal forces vary maximally by 350% in magnitude and approximately π/2 radians in phase. The large reduction in computational time offered by BET facilitates high-dimensional parametric design of flapping-wing-based technologies. Next, we compare the contributions of aerodynamic and inertial forces to wing deformation. Under the unilateral coupling assumption, aerodynamic and inertial-elastic forces are on the same order of magnitude—however, inertial-elastic forces primarily excite the wing’s bending mode whereas aerodynamic forces primarily excite the wing’s torsional mode. This suggests that, via conscientious sensor placement and orientation, biological wings may be able to sense independently inertial and aerodynamic forces. 

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4. Quasi three-dimensional deformable blade element and unsteady vortex lattice reduced-order modeling of fluid–structure interaction in flapping wings

   https://doi.org/10.1063/5.0129128  

   Schwab, R. ; Reade, J. ; Jankauski, M. ( December 2022 , Physics of Fluids)

   Flapping, flexible insect wings deform under inertial and fluid loading. Deformation influences aerodynamic force generation and sensorimotor control, and is thus important to insect flight mechanics. Conventional flapping wing fluid–structure interaction models provide detailed information about wing deformation and the surrounding flow structure, but are impractical in parameter studies due to their considerable computational demands. Here, we develop two quasi three-dimensional reduced-order models (ROMs) capable of describing the propulsive forces/moments and deformation profiles of flexible wings. The first is based on deformable blade element theory (DBET) and the second is based on the unsteady vortex lattice method (UVLM). Both rely on a modal-truncation based structural solver. We apply each model to estimate the aeromechanics of a thin, flapping flat plate with a rigid leading edge, and compare ROM findings to those produced by a coupled fluid dynamics/finite element computational solver. The ROMs predict wing deformation with good accuracy even for relatively large deformations of 25% of the chord length. Aerodynamic loading normal to the wing's rotation plane is well captured by the ROMs, though model errors are larger for in-plane loading. We then perform a parameter sweep to understand how wing flexibility and mass affect peak deflection, mean lift and average power. All models indicate that flexible wings produce less lift but require lower average power to flap. Importantly, these studies highlight the computational efficiency of the ROMs—compared to the convention modeling approach, the UVLM and DBET ROMs solve 4 and 6 orders of magnitude faster, respectively.

    

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5. Analysis of passive flexion in propelling a plunging plate using a torsion spring model

   https://doi.org/10.1017/jfm.2018.736  

   Arora, N. ; Kang, C.-K. ; Shyy, W. ; Gupta, A. ( December 2018 , Journal of Fluid Mechanics)

   We mimic a flapping wing through a fluid–structure interaction (FSI) framework based upon a generalized lumped-torsional flexibility model. The developed fluid and structural solvers together determine the aerodynamic forces, wing deformation and self-propelled motion. A phenomenological solution to the linear single-spring structural dynamics equation is established to help offer insight and validate the computations under the limit of small deformation. The cruising velocity and power requirements are evaluated by varying the flapping Reynolds number ( $20\leqslant Re_{f}\leqslant 100$ ), stiffness (represented by frequency ratio, $1\lesssim \unicode[STIX]{x1D714}^{\ast }\leqslant 10$ ) and the ratio of aerodynamic to structural inertia forces (represented by a dimensionless parameter $\unicode[STIX]{x1D713}$ ( $0.1\leqslant \unicode[STIX]{x1D713}\leqslant 3$ )). For structural inertia dominated flows ( $\unicode[STIX]{x1D713}\ll 1$ ), pitching and plunging are shown to always remain in phase ( $\unicode[STIX]{x1D719}\approx 0$ ) with the maximum wing deformation occurring at the end of the stroke. When aerodynamics dominates ( $\unicode[STIX]{x1D713}>1$ ), a large phase difference is induced ( $\unicode[STIX]{x1D719}\approx \unicode[STIX]{x03C0}/2$ ) and the maximum deformation occurs at mid-stroke. Lattice Boltzmann simulations show that there is an optimal $\unicode[STIX]{x1D714}^{\ast }$ at which cruising velocity is maximized and the location of optimum shifts away from unit frequency ratio ( $\unicode[STIX]{x1D714}^{\ast }=1$ ) as $\unicode[STIX]{x1D713}$ increases. Furthermore, aerodynamics administered deformations exhibit better performance than those governed by structural inertia, quantified in terms of distance travelled per unit work input. Closer examination reveals that although maximum thrust transpires at unit frequency ratio, it is not transformed into the highest cruising velocity. Rather, the maximum velocity occurs at the condition when the relative tip displacement ${\approx}\,0.3$ . 

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