Falling Paper: Navier-Stokes Solutions, Model of Fluid Forces, and Center of Mass Elevation

We investigate the problem of falling paper by solving the two dimensional Navier-Stokes equations subject to the motion of a free-falling body at Reynolds numbers around ${10}^3$. The aerodynamic lift on a tumbling plate is found to be dominated by the product of linear and angular velocities rather than velocity squared, as appropriate for an airfoil. This coupling between translation and rotation provides a mechanism for a brief elevation of center of mass near the cusplike turning points. The Navier-Stokes solutions further provide the missing quantity in the classical theory of lift, the instantaneous circulation, and suggest a revised model for the fluid forces.

Figure 1

Rise of a falling journal cover under windless conditions, selected frames from a footage filmed at $300\mathrm{\text{frames}}/\mathrm{s}$.

Figure 2

Navier-Stokes solution of a tumbling ellipse at $\mathrm{R}\mathrm{e}=1100$, $I^{*}=0.17$, and $e=0.125$. (a) Body-fixed coordinate system and kinematic variables. (b) Trajectory and orientation of the chord (major axis of the ellipse) over five periods of motion. (c) The history of the chord and the force vector, equally spaced in time for the first period of the trajectory in (b). The chords numbered from 1 to 4 correspond to the frames shown in (d) and to the times marked with dots on the force history of Fig. 3. (d) Vorticity field at four instants during a full rotation. The frames display an area of $5\times 2.5$ chords and they are $4a/u_t$ time units apart. The vorticity field is color coded on a logarithmic scale.

Figure 3

The fluid force and torque on the tumbling ellipse shown in Fig. 2. Solid lines are computed forces, with the total force shown as thin lines and the pressure force as thick lines. The corresponding dashed lines are the best fits of the data based on the quasisteady model of Eqs. (3, 4, 5). The added mass tensor $m_{11}=0.53m$ and $m_{12}=3.1m$, $m_{22}=1.5m$, $m_{21}=0.56m$, and the circulation around the ellipse $\Gamma =2.6a^2\Omega +0.49a|\mathbf{v}|\sin (2\alpha )$ are obtained from the pressure force. These $m_{ij}$ differ from those predicted by inviscid theory ($m_{11}^{\mathrm{i}\mathrm{n}\mathrm{v}}=m_{21}^{\mathrm{i}\mathrm{n}\mathrm{v}}=0.0491m$ and $m_{22}^{\mathrm{i}\mathrm{n}\mathrm{v}}=m_{12}^{\mathrm{i}\mathrm{n}\mathrm{v}}=3.14m$). The viscous corrections are modeled with the expansions $F_x^{\nu }=-{\nu }_{11}u-{\nu }_{12}{u}^2$, $F_y^{\nu }=-{\nu }_{21}v-{\nu }_{22}{v}^2$, and ${\tau }^{\nu }={\nu }_{31}uv-{\nu }_{32}\Omega -{\nu }_{33}\Omega |\Omega |$, with ${\nu }_{11}=0.18$, ${\nu }_{12}=0.0075$, ${\nu }_{21}=0.070$, ${\nu }_{22}=0.054$, ${\nu }_{31}=-0.31$, ${\nu }_{32}=-0.05$, and ${\nu }_{33}=0.16$.

Figure 4

Circulation as a function of time for the falling ellipse of Fig. 2. The thick line corresponds to the value of the circulation found by integrating the vorticity field up to a radius of $3/2$ chords from the center of the ellipse. The thin line corresponds to the circulation obtained from fitting the pressure forces, the dashed line to the contribution of the rotational term of Eq. (6) alone. The peaks of negative circulation not captured by the fit correspond to the vortices shed by the ellipse at the turning points of its trajectory.

Figure 5

Trajectories obtained from ordinary differential Eqs. (3, 4, 5) for different values of the translational and rotational lift coefficients $c_L$ and $c_R$: (a) $c_R=2.6$, $c_L=0.49$, from the fit of Fig. 3; (b) $c_R=0$, $c_L=1.5$, as in classical translational lift. Center of mass elevation occurs in (a) but is absent in (b).