Experimental and theoretical study of gravity-driven falling plates

Abstract

This research investigates the fluid-structure interaction phenomenon of gravity-driven falling rigid plates through a combination of experimental and theoretical approaches. Plates of varying dimensions and densities are systematically examined to explore the influence of non-dimensional parameters, including the Reynolds number ( $Re$ ) and dimensionless moment of inertia ( $I*$ ), on the falling patterns. High-speed photography is employed to extract plate trajectories and posterior kinematics calculations. In the range of relatively high Reynolds numbers ( $Re>800$ ), our study identifies three distinct falling modes: periodic fluttering, periodic tumbling, and marginal chaotic motion. The falling trajectories of the plates are analyzed and compared with their corresponding kinematic behaviors. By integrating theoretical analyses with experimental findings, we develop a semi-analytical model capable of calculating the real-time hydrodynamic forces and moments acting on falling plates. This model facilitates the prediction of falling trajectories for quasi-2-dimensional plates with arbitrary material and dimension combinations. Comparisons between model predictions and experimental results demonstrate good agreement in the fluttering and tumbling modes.

Introduction

Unsteady fluid-structure interaction phenomena are prevalent in nature and have long intrigued researchers due to their complex dynamics. Gravity-driven motion, observed in phenomena such as the dispersal of plant seeds by wind and the trajectories of falling leaves or paper strips, has garnered attention across various disciplines including aerospace engineering, meteorology, and biomechanics.^1,2 Despite their common occurrence, the dynamics of falling objects under the influence of both gravity and fluid flow remain intricate and not fully understood. Observations indicate that due to their large surface area and relatively light weight, falling leaves or paper strips rarely follow straight vertical trajectories. Instead, their motion is characterized by small perturbations that quickly amplify, leading to unsteady aerodynamic forces and significant changes in motion. Three standard modes of motion are typically observed depending on factors such as density ratio and object dimensions: periodic fluttering, periodic tumbling, and marginal chaotic motion.

The study of falling plates in fluid is a critical area of research with significant implications for the design and operation of fixed-wing micro air vehicles (MAVs). These MAVs, often deployed from larger unmanned air vehicles (UAVs) at high altitudes, rely on passive descent to conserve energy before activating their propulsion systems near the target area. The dynamics of a falling plate, which can be likened to the wing of an MAV, involve complex interactions between aerodynamic forces and the plate’s motion, leading to phenomena such as flutter, tumbling, and gliding. Understanding these behaviors is essential for optimizing MAV design to ensure stability and control during descent.

Experimental studies have been instrumental in uncovering the underlying flow physics of this problem. Willmarth et al. ³ conducted systematic investigations into the behavior of freely falling disks in different fluid mediums and dimensions, identifying two distinct unstable motions: periodic pitching (later termed fluttering) and tumbling, both strongly dependent on the dimensionless moment of inertia ( $I*$ ). Belmonte et al. ⁴ conducted quasi-2D experiments using solid thin flat strips, highlighting the transition from fluttering to tumbling at a critical Froude number, which has a similar definition to $I*$ . Further experimental studies on steady tumbling of paper cards in air suggested a scaling law linking autorotation frequency with density ratio and card dimensions. ⁵ Zhong et al. ⁶ observed three distinctive trajectories for freely falling disks with low $I*$ : planar zigzag, transitional, and spiral motion.

Numerical studies and analytical models have also significantly contributed to understanding the complexities of the problem. Tanabe and Kaneko ⁷ analyzed 2-D falling plates using Kutta-Joukowski’s theorem, identifying five distinct falling modes: periodic rotation, chaotic rotation, chaotic fluttering, periodic fluttering, and simple perpendicular descent. Jones ⁸ and Jones and Shelley ⁹ developed a vortex sheet model based on inviscid flow assumptions to investigate the dynamics of prescribed and freely falling plates. Andersen et al.^10,11 utilized numerical simulations and analytical models to successfully predict hydrodynamic forces and moments applied to falling plates.

Advancements in computational capabilities have enabled more comprehensive numerical studies to predict the diverse falling modes of rectangular plates. ¹² Wang et al. ¹³ applied an immersed boundary-lattice solver to study rectangular plates, revealing relationships between fluttering frequencies and the Froude number. Wan et al. ¹⁴ conducted direct numerical simulations (DNS) on elliptical cross-section falling plates, showing qualitative relationships between vortex development and force/moment generation. Lau et al. ¹⁵ and Kushwaha and De ¹⁶ demonstrated that initial conditions, such as the angle of release and separation between plates, strongly influence the transition from fluttering to tumbling mode. Furthermore, Lau and Huang ¹⁷ introduced flexibility into falling plates within a 2D viscous environment, showcasing transitions from tumbling to fluttering with decreasing stiffness.

The purpose of this study is to develop an accurate model for predicting the falling mode and trajectory of 2-D falling plates through a combination of theoretical analysis and experimental validation. High-speed photography is employed to extract and examine plate kinematics. Aerodynamic forces and coefficients are determined using theoretical calculations and experimental data fitting, enforcing quasi-two-dimensional conditions for falling trajectories and induced flow. A regime map is constructed to categorize plate motion based on the range of $Re$ and $I*$ considered, which are relevant for MAV designs. Additionally, the accuracy of the semi-empirical model in predicting fluttering and tumbling motion is validated through comparison with experimental data.

Methodology

The current study examines the quasi-two-dimensional fluid-structure interactions of rigid falling plates. Water, with a density of ${\rho }_f=998\mathrm{kg}/{\mathrm{m}}^3$ , was selected as the working fluid. The density of the plates ( $\rho$ ) was chosen such that the density ratio, defined as $\gamma =\rho /{\rho }_f$ , exceeds 1 for all plates. The trajectory of the plates was recorded using high-speed photography. Subsequently, the kinematics of the plates were analyzed and quantified utilizing an in-house MATLAB program. This section provides detailed specifications of the equipment and plates employed in the study.

Plates

The plates used in the experiments have rectangular cross-sections. The plates were manufactured from different materials, including brass, aluminum, transparent glass, and plastic to assess the influence of density ratio. The density ratio ranged from 1.2 to 8.4 and all plates shared a uniform width of 25 mm. The aspect ratio of the plate, denoted as the thickness-to-length ratio $\beta =T/L$ , is one of the parameters affecting the falling paths of plates. However, it turns out that the effects of the aspect ratio and density ratio can be combined into the dimensionless moment of inertia $I*$ , as introduced by Willmarth et al. ³ It is defined as the ratio of moment of inertia of the plate ( $I$ ) to the moment of inertia of a rigid cylinder of the fluid with diameter equals to the plate length, L: $$I*=\frac{I}{I_f}=\frac{\frac{1}{12}\rho TL(L^2+T^2)}{\frac{\pi }{32}{\rho }_f{L}^4}\approx \gamma \beta ,$$ (1)A total of 52 plate iterations were considered in the study, Table 1 summarizes the parameters for 10 representative samples.

Table 1.

Properties of plate samples and corresponding non-dimensional parameters.

Sample	$L$ (mm)	$T$ (mm)	$\beta$	$\gamma$	$I*$
1	29.22	1.61	0.055	1.19	0.066
2	24.30	1.62	0.067	1.23	0.082
3	29.17	2.12	0.073	1.23	0.090
4	24.89	2.47	0.099	1.23	0.122
5	14.30	1.03	0.072	2.50	0.180
6	10.21	2.03	0.199	2.50	0.498
7	12.00	4.25	0.354	2.50	0.887
8	24.08	1.02	0.042	2.76	0.117
9	12.55	2.00	0.159	2.70	0.430
10	12.55	2.00	0.159	2.76	0.441

Setup and high-speed photography

To ensure quasi-two-dimensional falling motion, a thin tank was designed and fabricated to restrict motion in the z-direction (see Figure 1a). The inner length and width of the tank are 584 mm and 27 mm, respectively The water level, measured from the bottom of the tank, was set to 432 mm, providing sufficient height for the plate motion to be developed. The width of the plates was set to 25 mm, resulting in a 1 mm gap between the plate ends and the inner tank walls. With these specific tank and plate dimensions, the descending motion was constrained to three degrees of freedom: two translational and one rotational. An electromagnet-based release system was designed and installed on the top of the tank (see Figure 1b) to release the plates from quiescence with controllable initial angles, thereby minimizing initial flow perturbations.

Figure 1.

Experimental setup. (a) Schematic of the tank and (b) electromagnet-based releasing system with a plate.

Because the boundary layer near the wall slightly influences the flow and plate dynamics, the setup is considered quasi-two-dimensional. In some runs, the plate made contact with the tank wall, altering its trajectory and causing it to flip. These runs were deemed flawed and excluded from analysis. The presented results are from successful runs where the plate did not touch the tank walls, thus maintaining the quasi-two-dimensional assumption.

A high-speed CMOS camera (SA3, Photron) with a resolution of $1024\times 1024$ pixels was employed to record the plates trajectory. The camera was positioned perpendicular to the x-y plane (as defined in Figure 2), capturing the region of interest along the trajectory of the plate. Images were recorded at a sampling frequency ranging from 250 to 800 Hz, depending on the falling mode, allowing for the reconstruction of time-resolved plate motion. To process the recorded images and extract plate kinematics, an in-house MATLAB program was developed. This program determined the mass-center location and orientation of the plates at each time instant based on the images. Subsequently, velocity and acceleration were computed through numerical differentiation of the plate’s position and rotation.

Figure 2.

Sketch of the two coordinate systems for the analytical modeling.

A semi-empirical model based on Kutta-Joukowski theorem and quasi-steady thin airfoil theory was developed to calculate the instantaneous hydrodynamic forces and moments on the falling plates. Coordinate system definition and governing equations are introduced in this section, details of the analytical model are discussed further in Section “Semi-empirical model for quasi-two-dimensional gravity-driven plate”.

Definition of two coordinate systems

As sketched in Figure 2, two coordinate systems were defined: an inertial system fixed to the laboratory, and a moving system fixed to the falling plate. The inertial coordinate system, represented by $x-y$ axes, has its origin fixed to the top-left corner of the region of interest. The orientation of the plate, $\theta$ , is defined in this system and refers to the angle formed by the long primary axis of the plate and the $x$ axis. The angle increases when the plate rotates counterclockwise. The plate-fixed system, represented by the $x^{\prime }-y^{\prime }$ axes, has its origin fixed to the mass center of the plate. The $x^{\prime }$ and $y^{\prime }$ axis align with the major and minor axis of the plate, respectively.

The instantaneous angle of attack, $\alpha$ , is defined as the angle formed by the $x^{\prime }$ axis and the translational velocity of the plate, $\mathit{\text{V}}$ . Conventional airfoil theory states that two hydrodynamic forces act upon the plate: lift ( $F_l$ ) and drag ( $F^{\nu }$ ) forces. As depicted in Figure 2, these forces act perpendicular and parallel to $\mathit{\text{V}}$ , respectively.

The mass center location and orientation of the plate are measured in the inertial coordinate system. Consequently, corresponding velocities and accelerations are numerically derived in this system based on experimental measurements. Thus, the components of the translational velocity in the moving system must be computed using the transformation: $$\left[\begin{array}{c}{u}^{\prime }\\ v^{\prime }\end{array}\right]=\left[\begin{array}{cc}cos(\theta )& sin(\theta )\\ -sin(\theta )& cos(\theta )\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$$ (2)where $u$ and $v$ are the velocity components in the inertial system, while $u^{\prime }$ and $v^{\prime }$ are the components in the moving system. A similar relation is applied to the acceleration components. The angular velocity and acceleration remain unchanged in both coordinate systems, requiring no transformation.

Governing equations for plate motion

In this two-dimensional problem, the falling plates exhibit three degrees of freedom: translation in the $x$ and $y$ directions, and rotation in the $\theta$ direction. The interactions exerted by the surrounding fluid on the plate are represented by forces and moments in the momentum equations that govern the motion of the plate. The resulting governing equations are as follows: $$(m+m_1)a_{x^{\prime }}=F_{b{x}^{\prime }}+F_{l{x}^{\prime }}-F_{x^{\prime }}^{\nu },$$ (3) $$(m+m_2)a_{y^{\prime }}=F_{b{y}^{\prime }}+F_{l{y}^{\prime }}-F_{y^{\prime }}^{\nu },$$ (4) $$(I+I_a)\ddot{\theta }=\tau +{\tau }^{\nu }.$$ (5)Equations (3) and (4) represent the linear momentum equations in the global inertial system but in $x^{\prime }$ and $y^{\prime }$ directions, respectively, while equation (5) is the moment-of-momentum equation for the rotational motion of the plate. In equations (3) and (4), $a$ , $F_b$ , $F_l$ and $F^{\nu }$ denote the translational acceleration, the body force, the hydrodynamic lift force and the drag force, respectively. Their components in $x^{\prime }$ and $y^{\prime }$ directions are accordingly denoted by the subscripts $x^{\prime }$ and $y^{\prime }$ . In equation (5), $\tau$ and ${\tau }^{\nu }$ refer to the driving moment and frictional moment, respectively. In the present study, the mass ( $m$ ) and moment of inertia ( $I$ ) of the plates vary depending on the material density ( $\rho$ ) and plate dimensions. Note that $m_1$ and $m_2$ represent the added-mass effects in $x^{\prime }$ and $y^{\prime }$ direction, respectively, while added moment of inertia is denoted by $I_a$ . According to Sedov, ¹⁸ the added mass and added moment of inertia are calculated as: $$\begin{array}{rl}& m_1=\frac{\pi }{4}{\rho }_f{T}^2,\\ & m_2=\frac{\pi }{4}{\rho }_f{L}^2,\\ & I_a=\frac{\pi }{128}{\rho }_f(L^2-T^2{)}^2,\end{array}$$ (6)where ${\rho }_f$ is the density of working fluid, while $T$ and $L$ represent the thickness and length of the plate, respectively. In equations (3) and (4), the body force $F_b$ , is the combination of gravity and buoyancy forces, and can be calculated with: $${\mathbf{F}}_b=LT(\rho -{\rho }_f)\mathbf{g},$$ (7)

where $\mathbf{g}$ is the gravitational acceleration, directed along the $-y$ direction in the inertial coordinate system with a magnitude of $9.8{\mathrm{m}/\mathrm{s}}^2$ .

According to the Kutta-Joukowski theorem, lift force $F_l$ is related to the velocity circulation $\mathrm{\Gamma }$ as: $$F_l=-{\rho }_f\mathrm{\Gamma }V,$$ (8)

where $V$ is the magnitude of the translational velocity of the plate. In the context of the falling plate problem, the plate not only translates but also rotates within the $x-y$ plane. Consequently, the circulation consists of two components: one arising from translational motion and the other from rotational motion, expressed as: $$\mathrm{\Gamma }={\mathrm{\Gamma }}_T+{\mathrm{\Gamma }}_R,$$ (9)

where ${\mathrm{\Gamma }}_T$ and ${\mathrm{\Gamma }}_R$ are the velocity circulation generated from the translation and rotation of the plate, respectively. Similarly, the self-rotation of the plate induces a rotational lift force ( $F_{l_R}$ ), contributing to the overall lift force ( $F_l$ ) as follows: $$F_l=F_{l_T}+F_{l_R},$$ (10)

where the translational component ( $F_{l_T}$ ) can be determined theoretically with $F_{l_T}={\rho }_f{V}^{2}L\pi \alpha$ when the angle of attack ( $\alpha$ ) is small, and experimentally for large angles of attack. In the present mopdel, $F_{l_R}$ is computed based on the Kutta-Joukowski theorem: $$F_{l_R}=-{\rho }_f{\mathrm{\Gamma }}_{R}V,$$ (11)

where the rotational circulation is proportional to the angular speed of the plate ( $\dot{\theta }$ ) by a constant coefficient $C_R$ , as represented by: $${\mathrm{\Gamma }}_R=\frac{C_R{L}^2}{2}\dot{\theta }.$$ (12)

The drag force ${\mathbf{F}}^{\nu }$ is characterized by the drag coefficient of the plate submerged in a uniform incoming flow. In this study, we adopted the model developed by Andersen et al. ¹¹ : $${\mathbf{F}}^{\nu }={\rho }_f\frac{L}{2}(A-B\frac{{u^{\prime }}^2-{v^{\prime }}^2}{{u^{\prime }}^2+{v^{\prime }}^2})\sqrt{{u^{\prime }}^2+{v^{\prime }}^2}\left[\begin{array}{c}{u}^{\prime }\\ v^{\prime }\end{array}\right],$$ (13)where $A$ and $B$ relate to the drag force coefficients for the translating flat plate at $0$ and $90\circ$ angle of attack. Details on the calculation of $A$ and $B$ are presented in Section “Semi-empirical model for quasi-two-dimensional gravity-driven plate”.

Equation (5) governs the rotational motion of the falling plate. According to thin airfoil theory, for small $\alpha$ , the dynamic pressure center is located at a quarter chord length from the leading edge of the airfoil, resulting in a driving moment given by: $$\tau =-\frac{F_{l_T}L}{4}cos(\alpha ).$$ (14)A similar expression is employed in the current model. Moreover, a frictional moment ${\tau }^{\nu }$ is induced due to the viscous force acting on the plate, which decelerates the rotation. Thus, ${\tau }^{\nu }$ is estimated as a simplified second-order polynomial of the angular velocity $\dot{\theta }$ as: $${\tau }^{\nu }=-\frac{C_{\mu }\pi }{16}{\rho }_f{L}^4\dot{\theta }|\dot{\theta }|,$$ (15)where $C_{\mu }$ is a constant coefficient to be determined from experimental results.

Upon substituting each term into equations (3)–(5), the governing equations for the plate motion are as follows: $$\begin{array}{rl}(m+m_1)a_{x^{\prime }}=& -LT(\rho -{\rho }_f)gsin(\theta )\\ & +F_{l_T{x}^{\prime }}+{\rho }_f{v}^{\prime }{\mathrm{\Gamma }}_R-F_{x^{\prime }}^{\nu },\end{array}$$ (16) $$\begin{array}{rl}(m+m_2)a_{y^{\prime }}=& -LT(\rho -{\rho }_f)gcos(\theta )\\ & +F_{l_T{y}^{\prime }}+{\rho }_f{u}^{\prime }{\mathrm{\Gamma }}_R-F_{y^{\prime }}^{\nu },\end{array}$$ (17) $$(I+I_a)\ddot{\theta }=-C\frac{F_{l_T}L}{4}cos(\alpha )-\frac{C_{\mu }\pi }{16}{\rho }_f{L}^4\dot{\theta }|\dot{\theta }|.$$ (18)These equations are similar to the governing equations in Anderson’s paper, ¹¹ however, some force and moment terms are modeled differently. The hydrodynamic force/moment terms and their corresponding coefficients in equations (16)–(18) are determined experimentally. Details on the calculation of these terms are outlined in Section “Semi-empirical model for quasi-two-dimensional gravity-driven plate”.

Experimental results

In the falling plate problem, six variables serve as the independent physical parameters: the density of the plate ( $\rho$ ), its dimensions including length ( $L$ ) and thickness ( $T$ ), the density ( ${\rho }_f$ ) and kinematic viscosity ( $\nu$ ) of the fluid, and the gravitational acceleration ( $g$ ). By combining these variables, three dimensionless parameters are derived to characterize the plates and their descending trajectories. The first two parameters are the aspect ratio ( $\beta =T/L$ ) and the density ratio ( $\gamma =\rho /{\rho }_f$ ), while the third parameter is the Reynolds number ( $Re=U_{0}L/\nu$ ). Here, $U_0=\sqrt{2(\rho /{\rho }_f-1)Tg}$ represents the characteristic velocity, estimated under the assumption that the buoyancy-corrected gravity force balances the vertical dynamic drag force in a vertically steady descending state.

The aspect ratio and density ratio can be combined into dimensionless moment of inertia, $I*$ , as definded in equation (1). $I*$ represents the dimensionless moment of inertia of the plate, approximately equal to the ratio between the moments of inertia of the plate and that of a fluid cylinder with a diameter equal to the length of the plate. In this study, the Reynolds number ranged from 800 to 15,000, and $I*$ ranged from 0.04 to 1.5. Within these ranges, the observed trajectories all fell within the unstable domain, consistent with previous findings by Willmarth et al. ³ Three types of descending modes were identified and analyzed: periodic fluttering, periodic tumbling, and marginal chaotic motion. The time evolution of the plate dynamics with different falling modes is summarized and discussed in this section.

Repeatability of falling mode and trajectory

It is important to acknowledge the inherent nonlinearity of the problem, as even minor perturbations to the plate or flow can lead to significant amplification, rendering precise experiment replication unattainable. Despite meticulous control over all experimental variables, it is evident that trajectories do not repeat. This observation is illustrated in Figure 3, where four flutter trajectories for the same plate under identical initial conditions exhibit non-coinciding paths. Moreover, within a single trajectory, noticeable subtle variations occur between different periods. However, the different trajectories share a high degree of similarity, albeit not being identical. For instance, kinematic characteristics such as oscillating magnitudes in both $x$ and $y$ directions remain approximately consistent. Further analysis confirms that falling and gliding speeds are closely aligned. Because the experiments cannot be perfectly repeated, it is challenging to conduct ensemble averaging on the experimental data. All of the experimental results presented are original, without filtering or ensemble averaging. A representative example of each mode is selected and presented in subsequent sections when characterizing different falling trajectory modes.

Figure 3.

Four trials of the same plate result in different but similar trajectories.

Characteristics of a periodic fluttering plate

The falling trajectory of a periodically fluttering plate is depicted in Figure 4, where the sequence of white rectangular cross-sections represents the real-time locations of the falling plate. The plate utilized in the test corresponds to Sample 3 as described in Table 1 ( $I*=0.090$ ), with $Re=2,852$ . During its descent along the trajectory, the plate oscillates periodically from side to side while descending vertically. Notably, the oscillation frequency in the $y$ -direction doubles that in the $x$ -direction.

Figure 4.

Superimposed image of the falling plate with a time interval of 80 ms between every two successive snapshots. Plastic plate, $Re=2852$ and $I*=0.090$ .

The kinematics results for this trajectory are included in Figure 5. Clear periodicity is observed in both the orientation angle of the plate ( $\theta$ ) and the $x$ -location of its mass center, as depicted in Figure 5(a). This periodic behavior is further evident in Figure 5(b), where the corresponding translational and angular velocities in the $x$ , $y$ , and $\theta$ directions are presented. The displacement of the plate mass center in the $y$ -direction appears to be a combination of constant falling speed and periodic oscillation, as evidenced by the $V_y$ history depicted in Figure 5(b). The accelerations in the $x$ , $y$ , and $\theta$ directions are denoted by $a_x$ , $a_y$ , and $ϵ$ in Figure 5(c), respectively. Approximately four fluttering periods were recorded along this falling path. Turning points are defined as the points where the plate motion reverses in the $x$ direction.

Figure 5.

Kinematic features of the fluttering trajectory shown in Figure 4. (a) Location of mass-center and angular orientation; (b) translational and angular speeds; (c) translational and angular accelerations.

For example, points A and B in Figure 5(a) represent two such turning points. Between two successive turning points, the plate undergoes a gliding phase, which is hence referred to as the gliding section. As depicted in Figure 5(b), both translational and angular speeds are very small around the turning points. During the gliding section, the magnitude of $V_x$ generally surpasses that of $V_y$ , suggesting a predominant influence of $V_x$ on the translational velocity. The translational velocity generates lift force, which is applied to the pressure center of the plate situated closer to the leading edge. The non-coincidence of the pressure center and mass center results in a driving moment, responsible for the rotation of the plate. This phenomenon results in a phase relationship between $\omega$ and $V_x$ , which is evident in Figure 5(b). A similar behavior, albeit less pronounced, is observed between $a_x$ and $ϵ$ in Figure 5(c).

Characteristic of a periodic tumbling plate

The falling trajectory of a periodic tumbling plate is presented in Figure 6. The plate utilized corresponds to Sample 6 as described in Table 1 ( $I*=0.498$ ), with $Re=2,494$ . The tumbling plates were particularly sensitive to disturbances and usually could only persist the tumble motion for 3-4 turns. In Figure 6, recording starts 1 period after the initial release; after 4 tumbling periods, the plate flipped about its minor axis, indicating the failure of 2-D approximation. The falling path of the plate’s mass center is approximately an inclined straight line that goes from the top right to the center of the image, indicating that $\omega$ and $V_x$ are unidirectional and negative in this case.

Figure 6.

Superimposed snapshots of the falling plate with time interval of 30 ms. Glass plate, $Re=2,494$ and $I*=0.498$ .

The kinematics for this case are plotted in Figure 7. $V_x$ is consistently negative and its magnitude changes periodically. During brief intervals within each tumbling period, the plate exhibits a positive $V_y$ , indicating the lift exerted by the surrounding fluid.

Figure 7.

Kinematic features of the tumbling trajectory shown in Figure 6. (a) Location of mass-center and angular orientation; (b) the translational and angular speeds; (c) translational and angular accelerations.

This upward motion quickly diminishes, resulting in a local maximum $y$ value, followed by a negative $V_y$ . The point at which the plate is vertical is referred to as the turning point in this falling mode. Between two successive turning points, the plate glides within the fluid similarly to the fluttering case. Two distinguishing characteristics of the tumbling plate are that $\theta$ monotonically decreases (or increases, depending on initial conditions) throughout its entire trajectory, and $\omega$ remains unidirectional due to the large moment of inertia of the plate. For this case, $V_x$ and $V_y$ exhibit comparable magnitudes, thus both play significant roles in generating lift force and driving moment, and no clear phase alignment is observed with $\omega$ .

Characteristic of marginal chaotic motion

Plates characterized by an intermediate value of $I*$ ( $0.3<I*<0.45$ ) did not exhibit a periodic fluttering or tumbling pattern during descent. Instead, their motion consisted of a random combination of fluttering and tumbling, termed marginal chaotic motion. An example of such a chaotic trajectory is depicted in Figure 8. The plate used in this particular test corresponds to Sample 9, as described in Table 1 ( $I*=0.43$ ), with $Re=3,296$ . The corresponding kinematic features for this trajectory are included in Figure 9.

Figure 8.

Superimposed snapshots of marginal chaotic motion with a time interval of 20 ms. Aluminum plate, $Re=3,296$ and $I*=0.43$ .

Figure 9.

Kinematic features of the marginal chaotic trajectory shown in Figure 8. (a) The mass-center location and orientation; (b) the translational and angular velocities; (c) translational and angular accelerations.

As shown in Figure 9(a), following three complete tumbling periods, the plate fluttered twice and resumed tumbling for the rest of the descending path. At the fluttering turning point, the plate was oriented almost vertically. The moment of inertia is insufficient to turn the plate. The transition between fluttering and tumbling occurs unrepeatable, resulting in varying trajectories for plates exhibiting this falling mode.

The motion during tumbling sections exhibits similar features to those observed in the periodic tumbling case, with $\theta$ monotonically decreasing and positive $V_y$ near the turning points (see Figure 9b). Conversely, during the fluttering period, kinematic features resemble those of the periodic fluttering mode, with $V_x$ and $\omega$ being in-phase. While the plate is capable of establishing a tumbling motion, $\omega$ is notably low near the turning points due to the insufficient $I*$ , thereby creating the possibility for a fluttering motion to initiate.

The marginal chaotic motions observed during testing showed high sensitivity to random perturbations and initial conditions. This resulted in unique motions characterized not only by trajectory but also by velocity and acceleration, highlighting the non-linear nature of the problem.

Regime map

Previous studies on freely falling disks^3,19 and auto-rotating wings ²⁰ have employed Reynolds number ( $Re$ ) and dimensionless moment of inertia ( $I*$ ) to generate regime maps of the falling modes. While the definitions of $Re$ and $I*$ may vary depending on the specific geometry under investigation, the regime map facilitates the observation of general relationships and trends. For instance, in the studies on falling disks and auto-rotating wings, researchers observed that unstable motion emerges after surpassing a certain threshold Reynolds number ( $(O)\sim {10}^2$ ). This unstable motion encompasses flutter, marginal chaotic, and tumbling falling modes. Once the motion becomes unstable, the specific mode is determined by the value of $I*$ , with tumbling occurring for higher $I*$ values independently of the geometry. However, it’s noteworthy that threshold values for $I*$ may vary across different geometries.

The regime map, illustrating the falling motion of the rectangular plates employed in the present study, is depicted in Figure 10.

Figure 10.

Regime map showing the falling mode related to $Re$ and $I*$ .

Within the considered Reynolds number range ( $Re>800$ ), an unstable motion is observed across all plates, with the majority exhibiting a fluttering trajectory. Tumbling motion is observed in a smaller subset of samples characterized by $I*>0.45$ , while marginal chaotic motion is seen in only four plates within the range $0.30<I*<0.5$ . There is some overlap in the transition limits between modes, particularly evident in the upper limit for marginal chaotic motion. This is attributed to the sensitivity of the mode to the initial release angle, as further examined via DNS by Lau et al. ¹⁵ Although the angle is carefully set, a small delay on either side of the release mechanism can potentially induce the mode transition, especially when $I*$ is relatively close to the upper and lower limits of the marginal chaotic mode.

Semi-empirical model for quasi-two-dimensional gravity-driven plate

The governing equations for the falling plate kinematics were introduced in Section “Governing equations for plate motion” (equations (16)–(18)). To numerically integrate these equations, five terms or coefficients must be determined: (1) translational lift force, $F_{l_T}$ ; (2) drag force, $F^{\nu }$ ; (3) rotational velocity circulation, ${\mathrm{\Gamma }}_R$ ; (4) frictional moment coefficient, $C_{\mu }$ ; and (5) driving moment coefficient, $C$ . In this section, we derive the terms and coefficients based on our current experimental data. Furthermore, we discuss and validate an empirical model based on modified thin airfoil theory to predict the trajectories of fluttering and tumbling plates.

Lift force

When an airfoil or flat plate is submerged in a uniform incoming flow, it generates an aerodynamic lift force perpendicular to the flow velocity. In the case of an inviscid steady flow and small angles of attack, this lift force ( $F_l$ ) can be expressed as a function of fluid density ( ${\rho }_f$ ), incoming flow velocity ( $V$ ), chord length ( $L$ ), and the angle of attack ( $\alpha$ ) using the equation: $$F_l=c_l(\alpha )\frac{1}{2}{\rho }_f{V}^{2}L,$$ (19)where $c_l(\alpha )$ represents the lift coefficient as a function of the angle of attack. According to the thin airfoil theory, for small angles of attack ( $\alpha <12\circ$ ), the lift coefficient is linearly proportional to the angle of attack: $$c_l=2\pi \alpha .$$ (20)However, at large angles of attack, the relationship between $c_l$ and $\alpha$ is no longer linear. Unfortunately, there is limited literature available on systematic measurements of $c_l$ for flat plates across a wide range of $\alpha$ . Considering that the thickness-to-length ratio of the majority of samples in the study is relatively close to 0.12, we employ $c_l$ measurements conducted by Sheldahl and Klimas ²¹ for the NACA0012 airfoil to model $F_{l_T}$ in equations (16)–(18).

Drag force

As introduced in Section “Governing equations for plate motion”, the drag force is calculated as $${\mathbf{F}}^{\nu }=-c_d(\alpha )\frac{1}{2}{\rho }_f{V}^2\frac{\mathbf{V}}{V},$$ (21)where $V$ denotes the magnitude of translational velocity, and $c_d(\alpha )$ represents the drag coefficient as a function of $\alpha$ . The drag force primarily arises from viscous shear applied to the plate surfaces at small angles of attack, while at larger angles of attack, it is primarily generated due to pressure differences on both sides of the plate. According to Andersen et al., ¹¹ the drag coefficient $c_d(\alpha )$ is expressed as: $$c_d(\alpha )=A-B[co{s}^2(\alpha )-si{n}^2(\alpha )]=A-B\frac{{u^{\prime }}^2-{v^{\prime }}^2}{{u^{\prime }}^2+{v^{\prime }}^2}.$$ (22)Combining equations (21) and (22), the drag force can be modeled as: $${\mathbf{F}}^{\nu }={\rho }_f\frac{L}{2}(A-B\frac{{u^{\prime }}^2-{v^{\prime }}^2}{{u^{\prime }}^2+{v^{\prime }}^2})\sqrt{{u^{\prime }}^2+{v^{\prime }}^2}\left[\begin{array}{c}{u}^{\prime }\\ v^{\prime }\end{array}\right],$$ (23)

in which the coefficients $A$ and $B$ need to be determined. Data fitting based on the kinematics derived from high-speed visualization gives $A=0.9$ and $B=0.65$ .

Rotational velocity circulation

The rotation of the plate induces additional circulation, referred to as rotational circulation and denoted as ${\mathrm{\Gamma }}_R$ . This circulation is anticipated to be a function of the angular speed: ${\mathrm{\Gamma }}_R={\mathrm{\Gamma }}_R(\dot{\theta })$ . If the plate is substituted with a cylinder of diameter $L$ , the rotational circulation would be expressed as ${\mathrm{\Gamma }}_R=\pi L^2\dot{\theta }/2$ , representing the upper limit of circulation achievable by a plate of length $L$ . In the model, we define the rotational circulation as ${\mathrm{\Gamma }}_R=C_R{L}^2\dot{\theta }/2$ , where $C_R$ is a coefficient determined from experimental data. Following the analogy of the cylinder, this coefficient is expected to be less than $\pi$ . Through data fitting of plate kinematics measurements, we find that $C_R$ is approximately $2.1$ , and this value holds consistent for both fluttering and tumbling cases.

Frictional moment coefficient

Friction generated during plate rotation produces a resistant moment referred to as the frictional moment, and denoted as ${\tau }^{\nu }$ . To estimate the frictional moment, a simplified approach is adopted. When fluid impinges normally on a plate ( $\alpha =90\circ$ ), the pressure force distributed across the plate integrates to determine the drag force. Similarly, when a plate rotates in fluid, fluid can be considered to be impinging on the plate perpendicularly with a speed of $r\dot{\theta }$ , thus generates local pressure force with a magnitude of $0.5c_d(\alpha =90\circ ){\rho }_f(r\dot{\theta }{)}^2$ . The frictional moment ${\tau }^{\nu }$ opposes the angular velocity, and its magnitude can be estimated using the drag coefficient when the flow is normal to the plate: $${\tau }^{\nu }=-c_d(\alpha =90\circ ){\rho }_f{\int }_0^{\frac{L}{2}}r\dot{\theta }|r\dot{\theta }|rdr,$$ (24)utilizing $c_d(\alpha =90\circ )=1.8$ from Sheldahl and Klimas, ²¹ equation (24) simplifies to: $${\tau }^{\nu }=-\frac{1.8}{64}{\rho }_f{L}^4\dot{\theta }|\dot{\theta }|.$$ (25)By comparing equation (25) with equation (15), we determine $C_{\mu }$ to be approximately 0.15.

Driving moment coefficient

According to the thin airfoil theory, the lift center is located approximately at 1/4 of the chord length from the leading edge, resulting in the generation of a lift moment that influences the orientation of the plate. This lift moment, combined with the frictional moment, governs the angular motion of the plate. For small angles of attack, the lift moment can be computed using the formula: $$\tau =\frac{1}{4}{F}_{l}Lcos(\alpha ),$$ (26)where $F_l$ represents the lift force. With a higher angle of attack, a modification factor $C(\alpha )$ should be introduced to adjust the lift moment as follows: $$\tau =C(\alpha )\frac{1}{4}{F}_{l_T}Lcos(\alpha ),$$ (27)where $F_{l_T}$ is the translational lift force calculated using equation (19).

The factor $C(\alpha )$ is dependent on the angle of attack $\alpha$ and can also be influenced by flow unsteadiness, such as the shedding of leading-edge and trailing-edge vortices. For the sake of simplicity, a constant value of $C=0.4$ is utilized in this model for both fluttering and tumbling plates. This value of $C$ has been found to yield good agreement between the predicted kinematics and experimental results for both types of motion.

Model equations and validation with experimental results

After determining the coefficients that fully define equations (16)–(18), scale analysis is employed to simplify the governing equations. It is observed that both the rotational lift component in the $x^{\prime }$ -direction ( ${\rho }_f{v}^{\prime }{\mathrm{\Gamma }}_R$ ) and the drag force component in the $y^{\prime }$ -direction ( $F_{y^{\prime }}^{\nu }$ ) have a smaller order of magnitude compared to the other terms. Consequently, these terms are neglected in the model, resulting in the following simplified equations: $$\begin{array}{rl}(m+m_1)a_{x^{\prime }}=& -LT(\rho -{\rho }_f)gsin(\theta )\\ & +F_{l_T{x}^{\prime }}-F_{x^{\prime }}^{\nu },\end{array}$$ (28) $$\begin{array}{rl}(m+m_2)a_{y^{\prime }}=& -LT(\rho -{\rho }_f)gcos(\theta )\\ & +F_{l_T{y}^{\prime }}+{\rho }_f{u}^{\prime }{\mathrm{\Gamma }}_R,\end{array}$$ (29) $$(I+I_a)\ddot{\theta }=C\frac{F_{l}L}{4}cos(\alpha )-\frac{C_{\mu }\pi }{16}{\rho }_f{L}^4\dot{\theta }|\dot{\theta }|,$$ (30)The set of ODEs is numerically integrated using a 4th order Runge-Kutta scheme to determine the trajectory of the plates. Predicted kinematics, including locations and accelerations, are then compared with experimental measurements to validate the model. It is important to note that the model does not incorporate random disturbances and is not suited for predicting marginal chaotic trajectories. Therefore, this section exclusively presents cases involving fluttering and tumbling plates.

Model-predicted kinematics of representative samples for each falling mode, along with the corresponding experimental data, are presented in Figures 11 and 12. The plates employed in these cases correspond to Sample 2 (for fluttering) and Sample 10 (for tumbling), as described in Table 1. $I*$ and $Re$ for each case are included in the figure captions for reference. For the fluttering plate, five complete periods are compared (See Figure 11). The predicted trajectory exhibits the same oscillation frequency as the one extracted from experimental data, with only minor differences in magnitude observed. Similarly, translational and angular accelerations show good agreement with results from experiments.

Figure 11.

Model-predicted kinematics compared to experimental data for fluttering plate ( $Re=1,887$ and $I*=0.082$ ): (a) $a_x$ ; (b) $a_y$ ; (c) $ϵ$ ; and (d) falling trajectory.

Figure 12.

Model-predicted kinematics compared to experimental data for tumbling plate ( $Re=3,296$ and $I*=0.441$ ): (a) $a_x$ ; (b) $a_y$ ; (c) $ϵ$ ; and (d) falling trajectory.

Due to setup limitations discussed in Section “Characteristic of a periodic tumbling plate”, a fewer number of tumbling periods are compared (See Figure 12). Nonetheless, the model predicts the trajectory and instantaneous accelerations accurately. Small discrepancies are observed within the last tumbling period, which are attributed to the quasi-two-dimensional assumption possibly no longer being valid in the experiment. Overall, the model effectively predicts fluttering and tumbling trajectories with accuracy, provided the material and dimensions of the plates, as well as the initial conditions, are known. Furthermore, predictions across a broad range of $I*$ indicate that the transition from periodic fluttering to tumbling motion occurs at $I*=0.3$ . This transition value is consistent with observations from the regime map presented in Figure 10.

Conclusions

In the Reynolds number range considered ( $800<Re$ $<15,000$ ), the falling plates displayed unstable trajectories, exhibiting three distinct modes: periodic fluttering, periodic tumbling, and marginal chaotic motion. The classification of these modes was confirmed by the regime map, which highlighted the predominant influence of $I*$ on the falling behavior:

Plates exhibited periodic fluttering for $I*\le 0.3$ .

Marginal chaotic motion was observed for plates with $0.3<I*<0.45$ .

Periodic tumbling occurred when $I*\ge 0.45$ .

High-speed image acquisitions allowed for the derivation of instantaneous velocities and accelerations, revealing characteristic periodicity for fluttering and tumbling modes. Plates exhibiting marginal chaotic motion displayed high sensitivity to initial conditions and random disturbances, resulting in unique trajectories characterized by a combination of fluttering and tumbling motions.

A semi-empirical model, developed based on thin airfoil theory, laminar boundary layer theory, and experimental data fitting, accurately predicted kinematics for fluttering and tumbling plates. The model predictions aligned well with recorded trajectories and instantaneous accelerations, including the transition from fluttering to tumbling at $I*=0.3$ , consistent with experimental observations. However, it is important to note that the model lacks suitability in predicting marginal chaotic motion and instead bypasses it into the tumbling mode.