Abstract
To understand the nonlinear interaction between unsteady aerodynamic forces and the kinematics of structures, we theoretically and numerically investigated the characteristics of lift coefficients produced by a flapping thin flat plate controlled by the rotation axis position. The flat plate was placed in a 2-D incompressible flow at a very low Reynolds number (Re = 300). We showed that the behavior of the unsteady aerodynamic forces suggests the existence of a limit cycle. In this context, we developed a Reduced Order Model (ROM) by resolving the modified van der Pol oscillator using the Taylor development method and computational fluid dynamics (CFD) solutions. A numerical solution was obtained by integrating the differential equation of the modified van der Pol oscillator using the fourth-order Runge–Kutta method (RK4). The model was validated by comparing this solution with the reformulated equation of the added mass lift coefficient. Using CFD and ROM solutions, we analyzed the dependency of the unsteady lift coefficient generation on the kinematics of the flapping flat plate. We showed that the evolution of the lift coefficient is influenced by the importance of the rotation motion of the Leading Edge (LE) or Trailing Edge (TE), according to the position of the rotation axis. Indeed, when the rotation axis is moved towards the LE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the TE and the rotation axis. However, when the rotation axis is moved towards the TE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the LE and the rotation axis.
Highlights
Symmetry plays a pivotal role in aerodynamics at a low Reynolds number [1,2,3]
We studied unsteady flow around a flapping flat plate at a very low Reynolds number (Re = 300)
We studied temporal profiles and the harmonics in the spectra of the unsteady lift for three different rotation axis positions: Xc = 0.25c; Xc = 0.425c, and Xc = 0.65c
Summary
Symmetry plays a pivotal role in aerodynamics at a low Reynolds number [1,2,3]. In recent years, industry has been involved in the production of biomimicking flying and swimming robots as micro air and underwater vehicles at low Reynolds numbers. Brunton et al (2013) [23] numerically and experimentally analyzed the dynamics of pitching and plunging airfoils at a low Reynolds number They developed reduced-order models for the unsteady lift over a range of attack angles. Khalid et al (2017) [30] revealed that aerodynamic forces produced by oscillating NACA-0012 airfoils are independent of the initial kinematic conditions suggesting the existence of a limit cycle Using numerical simulations, they analyzed the shedding frequencies close to the excitation frequencies and modeled the unsteady lift force with a modified van der Pol oscillator. Hafien et al (2019) [31] developed a reduced order model for the lift coefficient of an airfoil equipped with extrados and/or trailing edge flexible flaps at low Reynolds number This model was obtained by resolving the modified van der Pol oscillator using the Taylor development method. We used the ROM and CFD results to study and analyze the effect of modifying the rotation axis of the flapping flat-plate on vortex-shedding, and on the unsteady lift coefficient
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