Experimental study of kinematics and fluid structure interaction of gravity driven falling plates

When leaves and strips of paper fall in air, their trajectories can be quite complicated. This gravity-driven motion has drawn attention for ages in various research fields, e.g. aerospace engineering, meteorology, and biomechanics in breed dispersal, etc [1, 2]. Due to the geometry of leaves or papers, straight vertical falling path was very rare because tiny perturbation will be amplified quickly and results in aerodynamic force and moments which will change the falling motion significantly. Depending on the density ratio and dimensions of the falling plates, two types of motions were observed. The first is that the plate slides from side to side periodically while falling, which is called fluttering; the other type is called tumbling, in which case the plate maintains unidirectional rotation with respect to one of its horizontal primary axes and falls downward to one inclined direction. The first systematic study of the falling plate phenomenon dates back to Maxwell in 1940 [3]. He generalized some preliminary experiment results and qualitatively summarized the mechanism for different types of motion of falling paper strips. In 1964, Willmarth et al. systematically conducted the freely falling disk experiments with various types of circular disks [4]. With various combination of Reynolds number and non-dimensional moment of inertia, the falling disks’ motion can be stable or unstable. Two types of unstable motions were observed: periodic pitching (fluttering) and tumbling. The falling modes are characterized in a phase diagram with these two non-dimensional parameters. It is also found that, in case of fluttering motion, the pitching frequency is linearly related to the non-dimensional moment of inertia I∗, indicating its influence on the falling motion when Reynolds number is high (Re ∼ 104). In 1971, Smith et al. [5] performed experimental study about auto-rotation of the fixed wing to uncover the mechanism and dynamics of the autorotation phenomenon with different Reynolds and Strouhal numbers. Also, some preliminary experiments about freely falling autorotation disks are conducted to study the inner mechanism in the tumbling motion. In 1998, Belmonte et al. [6] conducted quasi-2-dimensional experiments with thin flat strips to study the transition from fluttering to tumbling. In their research the transition is found to be determined by the Froude number (similar physical meaning as the non-dimensional moment of inertia) and the transition occurs at Fr = 0.67±0.05. In addition, a phenomenological model related to the Froude number similarity is proposed to simulate the inertial lift and drag forces and successfully reproduced the motion. Recently in 2011, experimental studies on the 3 dimensional trajectory of the falling disks with low I∗ are conducted by Zhong et al. [7]. They observed 3 types of falling motions, which are governed by I∗ and Reynolds number, named as planar zigzag, transitional and spiral motion, respectively. Numerous theoretical models are also proposed and developed to predict the trajectories of falling plates. Unsteady nonlinear coupling of fluid and structure has brought inevitable complexity. In 1994, Tanabe et al. [8] modified the classical Kutta-Joukowski’s theorem for the 2-D falling plate problem and successfully revealed 5 types of paths for the falling plates: periodic rotation (PR), chaotic rotation (CR), chaotic fluttering (CF), periodic fluttering (PF) and simple perpendicular fall (SPF). In this model, different values of friction coefficient has been used. In addition, Mahadevan, [9] in 1999, summarized his experimental results about the steady tumbling paper cards in air and discovered that the autorotation frequency of the tumbling cards are related to its dimension by the law Ω T 1/2L−1, i.e. Ω = Ω(L/T 1/2), where Ω is the tumbling frequency, and T and L are the thickness and length of the paper card respectively. In 2005, Anderson, et al. [10] built a theoretical quasi-steady model by extending the Kutta-Joukowski’s theorem and used data fitting method to reproduce the dynamics response of plates, which yielded qualitative agreement with his experiment measurements. ∗PhD Candidate, Department of Mechanical and Aerospace Engineering †Assistant Professor, Department of Mechanical and Aerospace Engineering