Experimetal study of a freely falling plate with an inhomogeneous mass distribution

A homogeneous thin plate often flutters while falling through a fluid under gravity. The center of gravity of the plate moves back-and-forth horizontally and the plate tilting angle oscillates symmetrically from the horizontal. Here we show that such a scenario is qualitatively changed for a plate with noncoinciding centers of gravity and buoyancy due to an inhomogeneous mass distribution. Mismatch of the centers causes an external torque that breaks the symmetry of rotational motion, shifts the mean tilting position from the horizontal, and leads to a net horizontal plate displacement. In laboratory experiments with a Reynolds number around 1500, we found that the net horizontal displacement scales linearly with the separation between the centers up to a critical value, beyond which the plate falls vertically in an edge-on configuration with the heavier side downward. Experimental results are compared to predictions of a quasi-steady numerical model. Our work demonstrates that motion of freely moving objects in a fluid depends sensitively on external torques, which potentially can be used as an effective control method.

Figure 1

Front view of the experimental system showing the falling plate, the illuminating light-sheet, and the imaging window. A laboratory reference frame (XYZ) is fixed in space with $Y$ axis pointing vertically and other two horizontally. The light-sheet lies in the $XY$ plane.

Figure 2

(a) Construction of the plates. Two materials (distinguished by colors) with different densities are used to construct the plate. Geometric parameters are defined in the text. (b) A comoving frame ($xy$) is fixed on the plate and follows the plate motion. The moving frame has its origin at the center of gravity of the plate, denoted as $O$, and its two axes are along and perpendicular to the plate orientation that is quantified by a tilting angle $\theta$ from the horizontal. The center of exterior geometry, i.e., the center of buoyancy, is denoted by $O^{\prime }$. The centers of gravity and buoyancy have velocities of $\vec{V}\text{and}\overrightarrow{V^{\prime }}$, respectively.

Figure 3

Experimental (left column) and simulated (right column) trajectories (dashed lines) of four plates from Set A with increasing offsets between the centers of gravity and buoyancy. Schematic drawings of plates are shown in the upper right corner of each panel in the left column. Instantaneous positions of plates are shown by short red lines on top of the trajectories of the center of gravity and blue dots denote the direction of the offset.

Figure 4

Temporal records for titling angle (top row), horizontal coordinate (middle row), and vertical coordinate (bottom row) for the centers of gravity for a plate with $e/a=0$ (left column) and a plate with $e/a=0.014$ (right column) from Set A. Experimental and quasi-steady model results are shown by solid and dashed lines, respectively. Simulations reproduce the overall features of experiments but they are different in details.

Figure 5

(a),(b) Temporally averaged tilting angle, $\langle -\theta \rangle$. (c),(d) Temporally averaged velocities of the center of gravity, $\langle u\rangle \text{(blue}\text{diamonds)}\text{and}\langle -v\rangle \text{(red}\text{circles)}$, as functions of the offset for plates in Set A in the left column and for plates in Set B in the right column. Experimental and model results are shown by symbols and lines, respectively. Crosses in (a) and (c) are results from repeated experiments to show experimental reproducibility. Solid triangles in (a) and (c) are measured with a plate that is 35 cm long in the $Z$ direction.

Figure 6

Instantaneous velocity and vorticity fields overlayed with trajectories of two plates from Set A: (a) $e/a=0$, (b) $e/a=0.014$ .