Viscous rebound of a quasi-two-dimensional cylinder on a solid wall

The purpose of the present study is to extend the simple concept of apparent coefficient of restitution, widely approached in the literature for the case of a single contact point between a sphere and a wall, to the case of bouncing whose complexity is increased due to the shape of the contacting object. For this purpose, experiments are carried out with a finite-length cylinder, freely falling in a liquid at rest. A rigid tail is attached to the cylinder, allowing one to maintain a vertical trajectory and to keep the axis of the cylinder parallel to the bottom wall down to a small gap between them. Yet, more complex 3D motions of the cylinder with respect to the wall occur during bouncing, including multiple-contact-point bouncing between the cylinder and the bottom as well as cavitation. When the Stokes number St (the ratio between characteristic inertial forces experienced by the particle compared to viscous forces in the fluid) is increased, the experimental results suggest that the ratio of the apparent coefficient of restitution to the solid one increases from 0 (at low St) to 1 (at large St) with a critical Stokes number ${\mathrm{St}}_c$ below which no bouncing is observed, as usually obtained in the literature for a single-contact-point bouncing sphere. In a conceptual approach to understanding the observed experimental features using a cutoff length scale prior to contact and a contact timescale, we investigated numerical modeling of an idealized situation, a 2D infinite cylinder falling parallel to the wall. To this end, we carried out numerical 2D simulations where the fluid equations of motion were coupled to the particle equation of motion through an immersed boundary method. The particle equation of motion was coupled to an elastic force to model bouncing. This numerical model requires parametrization of the cutoff length (interpreted as a roughness) and the contact time (associated with the contact elasticity) used here to capture the experimental observations. The simulations confirmed that (1) the departure of the coefficient of restitution from 0 is strictly dependent on the apparent roughness and (2) the coefficient of restitution depends on the contact time. Finally, in an effort to rationalize the experiments and the simulations for such a conceptual approach, we modeled the coefficient of restitution as the product of two contributions to the mechanical loss of energy: the collision-to-terminal velocity ratio ($V_c/V_t$) of the approach phase and the rebound-to-collision velocity ratio ($-V_r/V_c$) of the contact phase. We then interpreted the experimental measurements in light of this model, showing evidence that the assumption of a global relationship between the contact time and an apparent roughness (all being linked to the bouncing complexity including multiple-contact-point and cavitation in experiments) leads to a reasonably good prediction of the coefficient of restitution in the intermediate regime in St. This suggests the relevance of lumping the complex details of physical phenomena involved during contact into a simple concept based on the contact apparent roughness and elasticity.

Figure 1

(a) Design and picture of the four cylinders used. (b) Schematic of the experiment; the delimited area in red is the field of view of the camera. Schematic and definitions of the extracted quantities in the (c) $\left(x\text{−}y\right)$ plane and (d) $\left(y\text{−}z\right)$ plane, and (e) from real images.

Figure 2

Series of images to illustrate rebounds with (a) one contact (front contact) for cylinder A in salt water, and (b) three-contact (rear/front/rear front contact) case for cylinder B in salt water. The frames with contacts are indicated in red; a dashed-dot white circle shows the front face of the cylinder. (c) Sorting of rebound cases based on $\delta z/R$, the nondimensional contact time, and number of contacts for cases in UCON/salt water (blue) and salt water (red). Cylinders are sorted by symbols.

Figure 3

Cylinder D in water. (a) Position of the front and rear faces of the cylinder (as well as the average of the two) with time, with the inset zooming on the $1\text{ms}$ around the rebound. (b) Series of images from the camera (every $200\mu \text{s}$) with symbols indicating the corresponding times in the inset in (a); the red arrow indicates the cavitation bubble when visible.

Figure 4

(a) Distribution of the observed angles in air and in salt water at rebound of the cylinder itself ${\varphi }_{\mathrm{in}}$, the initial orientation ${\psi }_{\mathrm{in}}$, and the change in the orientation of its trajectory with respect to a perfect rebound with no friction ${\psi }_{\mathrm{in}}-\theta /2$. (b) Coefficient of restitution $e_y$ in air, as a function of the rotation at rebound, sorted by the nature of the cylinder (symbols); the filled red symbols with error bars are the estimates of $e_{y_0}$ since $\text{St}>{10}^5$ for all cases.

Figure 5

Coefficient of restitution as a function of the Stokes number for experiments corresponding to nearly normal rebound. Colors of symbols indicate the fluid considered, while the shape refers to the properties of the cylinders. The critical value for rebound, ${\mathrm{St}}_c=75\pm 25$, is indicated by the black dashed line with gray shaded area.

Figure 6

(a) 2D numerical simulation setup, including the grid distribution and boundary conditions. Evolution in time of (b) the separation gap between the cylinder surface and the wall and (c) the cylinder velocity. The inset in (b) shows a zoom at small gaps. The colors correspond to different Stokes numbers indicated in (c). The black dots indicate the collision velocity $V_c$. The elasticity parameter used for those simulations is $\alpha =4$.

Figure 7

Contour plot (in natural log scale) of the softness dimensionless number $S=\frac{\mu V_c/\left(\eta R\right)}{E_{el}/{\alpha }^{2.5}}$ for all numerical simulations used for this study. The value of this parameter ranges between ${10}^{-3}$ (the darkest color) and $O\left(10\right)$ (the lightest color). The dots indicate the parameters used in the different simulations carried out with $\eta =0.01$.

Figure 8

(a) Collision-to-terminal velocity ratio $V_c/V_t$ and (b) rebound-to-collision velocity ratio $-V_r/V_c$ as a function of the Stokes number. $V_c$ is the particle velocity at the collision onset, for $\eta =0.01$. The symbols correspond to numerical simulations. The solid line in panel (a) predicts $\frac{V_c}{V_t}$ from Eq. (7). In panel (b) different symbols correspond to different $\alpha$: stars, plus signs, up triangles, down triangles, and squares correspond to $\alpha =0.5$, 1, 4, 7, and 10, respectively (from rigid to soft cylinder). The solid lines in the inset correspond to $-\frac{V_r}{V_c}$ obtained from Eq. (9) using a collision time $\tau =\pi \sqrt{\frac{m^{*}}{k_n}}$ and $\alpha =1$, 4, 7, and 10 (from top to bottom).

Figure 9

Coefficient of restitution as a function of the Stokes number; the symbols are from numerical simulations and the dashed lines correspond to $e_{RB}$ [Eq. (11)] assuming that the cylinder is infinitely rigid and that the near-wall hydrodynamic force during settling is dominated by lubrication. Panel (a) shows evidence of the role of apparent surface roughness: the blue plus and empty red circles correspond to $\eta =0.01$ and 0.1, respectively, obtained with the contact elasticity parameter $\alpha =1$. Panel (b) evidences the role of elasticity: the stars, plus signs, up triangles, down triangles, and squares correspond to simulations carried out with $\alpha =0.5$, 1, 4, 7, and 10, respectively and contact roughness parameter $\eta =0.01$. The dashed line is reported from panel (a). The solid line corresponds to $e_{Iz}$ [Eq. (16)] assuming a balance between elastic deformation and lubrication forces during contact. The dotted lines are calculated from $e=-\frac{V_c}{V_t}\times \frac{V_r}{V_c}$, where $\frac{V_r}{V_c}$ is obtained from Eq. (9) with $\tau =\pi \sqrt{\frac{m^{*}}{k_n}}$ like in Fig. 8. These dotted lines correspond to $\alpha =1$, 4, 7, and 10, from top to bottom (from rigid to soft contact).

Figure 10

(a), (c) The coefficient of restitution normalized by its value in air as a function of the Stokes number and (b), (d) the nondimensional contact time as a function of the Stokes number scaled by ${\mathrm{St}}_c$. Symbols in (a) and (b) are obtained from numerical simulations, and similar to Fig. 9, obtained with different contact elasticity parameters $\alpha$. Red symbols in (c) and (d) correspond to experiments as in Fig. 5. Vertical error bars for experiments in (c) are associated to uncertainties in $e_y^0$ and horizontal ones to uncertainties in ${\mathrm{St}}_c$ (not reproduced in panel d). Empty (resp. filled) symbols are for one-contact (resp. two-contact) rebounds. The black dashed line in (a) and (c) is for $e_{RB}$. In (a) colored lines correspond to model (13) with constant dimensionless contact time $t_c{V}_t/D$ obtained from (b) (lines, same color legend). In (c) and (d), the blue solid line is for $e_{Iz}$ and ${\tau }_{Iz}$ [Eqs. (16) and (3)] with $\eta \in [3.6–15]\times {10}^{-4}$, while the green and cyan lines correspond to models $e_{mC1}$ and $e_{mC2}$ [Eqs. (14) and (15)] with ${\mathrm{St}}_c=75$ ($\gamma =2$) and apparent nondimensional contact time being constant at $(6\pm 3)\times {10}^{-2}$ or fitted by $(8\pm 4)\times {10}^{-3}\sqrt{\text{St}/{\mathrm{St}}_c-1}$, respectively. Shaded areas around the models describe the uncertainties on the fitting parameters or the value of ${\mathrm{St}}_c$.

Figure 11

Configuration used for the calculation of the viscous lubrication force in the gap between an infinite circular cylinder (of radius $R$) and a rigid plane.

Figure 12

Same as in Fig. 10 but with different values of the threshold for rebounds in models $e_{mC1}$ and $e_{mC2}$ with $\gamma =2$: (a) ${\mathrm{St}}_c=50$ and (b) ${\mathrm{St}}_c=100$.