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Three-dimensional fluid–structure interaction analysis of a flexible flapping wing under the simultaneous pitching and plunging motion

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Abstract

Recent advance in flapping-wing MAVs has led to greater attention being paid to the interaction between the structural dynamics of the wing and its aerodynamics, both of which are closely related to the performance of a flapping wing. In this paper, an improved computational framework to simulate a flapping wing is developed. This framework is established by coupling a preconditioned Navier–Stokes solution and a co-rotational beam analysis with a restrained warping degree of freedom. Validation of the present framework is performed by a comparison with examples from either earlier analyses or experiments. Further, a numerical analysis of a wing under simultaneous pitching and plunging motion is examined. The results are compared with those obtained with a wing under pure plunging motion, in order to assess the additional motion effect within a spanwise flexible wing. The comparison shows different aerodynamic characteristics induced by the flexibility of the wing, which can be beneficial.

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Abbreviations

\(\underline{\underline{R}}\) :

Rotational matrix

\(\underline{\underline{T}}_{s}\) :

Operator relating spatial and material angular variations

\(\underline{E}\) :

Topology of the configuration

\(\underline{x}\) :

Position vector

\(\underline{u}\) :

Nodal transverse displacement vector

\(l_{n}\) :

Deformed length of the element

\(\underline{\theta }\) :

Nodal rotational displacement vector

\(\alpha \) :

Warping degrees of freedom

\(\underline{\underline{B}}\) :

Transformation matrix

\(\underline{\underline{E}}\) :

CR transformation matrix

\(\underline{\underline{H}}\) :

Auxiliary matrix

\(\underline{q}\) :

Nodal displacement vector

\(\dot{\underline{q}}\) :

Nodal velocity vector

\(\ddot{\underline{q}}\) :

Nodal acceleration vector

V :

Virtual work

\(\varPhi \) :

Strain energy

\(\mathcal {K}\) :

Kinetic energy

\(\rho \) :

Material density

\(\underline{f}\) :

Elemental internal force vector

\(\underline{f}_{K}\) :

Elemental inertial force vector

\(\underline{f}_{e}\) :

Elemental external force vector

\(\underline{f}_{Km}\) :

Elemental inertial force vector with prescribed motion

\(\underline{F}_{\mathrm{pre}}\) :

Predictor

\(\underline{\underline{K}}\) :

Elemental stiffness matrix

\(\underline{\underline{M}}\) :

Elemental mass matrix

\(\underline{\underline{C}}_{K}\) :

Elemental gyroscopic matrix

\(\underline{\underline{K}}_{\mathrm{Dyn}}\) :

Elemental dynamic stiffness matrix

\((*)^{\alpha }\) :

Quantity including the warping DOF

\((*)_{\mathrm{G}}\) :

Quantity referring to the global frame

\((*)_{\mathrm{L}}\) :

Quantity referring to the local frame

\((*)^{\mathrm{n}}\) :

Time index

h :

Structural timestep

\(\alpha _{\mathrm{int}}, \gamma , \beta \) :

Constants in HHT-\(\alpha \) method

\(\underline{W}\) :

Conservative solution vector

\(\overrightarrow{\underline{F}}\) :

Inviscid flux vector

\(\overrightarrow{\underline{F}_{v}}\) :

Viscous flux vector

\(\underline{Q}_{p}\) :

Primitive solution vector

\(\underline{\underline{\varGamma }}_{a}\) :

Preconditioning matrix

\(C_{\mathrm{W}}\) :

Sectional warping coefficient

\(z_{\mathrm{tip}}\) :

Displacement at the tip

\(z_{m}\) :

Prescribed plunging motion

\(z_{o}\) :

Amplitude of plunging motion

\(\theta _{m}\) :

Prescribed pitching motion

\(\theta _{o}\) :

Amplitude of pitching motion

c :

Chord length

\(k_{G}\) :

Reduced frequency

\(f_{m}\) :

Physical frequency

\(U_{\infty }\) :

Flow velocity

\(C_{\mathrm{P}}\) :

Pressure coefficient

\(C_{\mathrm{L}}\) :

Lift coefficient

\(C_{\mathrm{T}}\) :

Thrust coefficient

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Acknowledgments

This research was supported by a grant to Bio-Mimetic Robot Research Center funded by Defense Acquisition Program Administration (UD130070ID) and also be by Advanced Research Center Program (No. 2013073861) through the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) contracted through Next Generation Space Propulsion Research Center at Seoul National University.

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Authors and Affiliations

  1. IAAT (Institute of Advanced Aerospace Technology), Seoul National University, Seoul, Republic of Korea

    Sang Joon Shin

  2. Seoul National University, Seoul, Republic of Korea

    Haeseong Cho

  3. Inha University, Incheon, Republic of Korea

    Namhun Lee & Seungsoo Lee

  4. Korea Aerospace Research Institute, Daejeon, Republic of Korea

    Jun Young Kwak

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  1. Haeseong Cho
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Correspondence to Sang Joon Shin.

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Cho, H., Lee, N., Kwak, J.Y. et al. Three-dimensional fluid–structure interaction analysis of a flexible flapping wing under the simultaneous pitching and plunging motion. Nonlinear Dyn 86, 1951–1966 (2016). https://doi.org/10.1007/s11071-016-3007-7

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