Instability and trajectories of buoyancy-driven annular disks: A numerical study

We investigate the stability of the steady vertical path and the emerging trajectories of a buoyancy-driven annular disk as the diameter of its central hole is varied. The steady and axisymmetric wake associated with the steady vertical path of the disk, for small hole diameters, behaves similarly to the one past a permeable disk, with the detachment of the vortex ring due to the bleeding flow through the hole. However, as the hole diameter increases, a second recirculating vortex ring of opposite vorticity forms at the internal edge of the annulus. A further increase in the hole size leads to the shrinking of these recirculating regions until they disappear. The flow modifications induced by the hole influence the stability features of the steady and axisymmetric flow associated with the steady vertical path. The fluid-solid coupled problem shows a nonmonotonic behavior of the critical Reynolds number for the destabilization of the steady vertical path, for low values of the disk's moment of inertia. However, for very large holes, with dimension approximately more than half of the external diameter, a marked increase of the neutral stability threshold is observed. The nature of the primary instability changes as the hole size increases, with large (small) amplitude oscillations of the trajectory at intermediate (very small and large) internal diameters. We then illustrate results obtained with fully nonlinear simulations of the time-dependent dynamics, together with a comparison of the linear stability analysis results. Falling styles, typically described as steady, hula-hoop, fluttering, chaotic, and tumbling, are shown to emerge as attractors for the nonlinear dynamics of the coupled fluid-structure system. The presence of a central hole does not always reduce the falling Reynolds number, and it may cause the transition from tumbling towards fluttering, from fluttering to hula-hoop, and from hula-hoop to steady, hence promoting trajectories with smaller lateral deviations from the vertical path. The observed trajectories and patterns agree well with linear stability analysis results, in the vicinity of the threshold of instability.

Figure 1

(a) Sketch of the flow configuration with the employed coordinate systems. (b) Some falling style of thin disks. The falling mode has a direct impact on the lateral distance $R$ covered, which vanishes in the case of steady vertical fall while it scales like $R\approx H_{i}tan\alpha$, where $H_i$ is the falling height, in the case of the tumbling mode.

Figure 2

$\varepsilon ={10}^{-3}$. Streamlines and isocontours of the streamwise relative velocity, for increasing values of $\delta$ and fixed $\text{Re}=50$.

Figure 3

$\varepsilon ={10}^{-3}$. Streamlines and isocontours of the streamwise relative velocity, for increasing values of $\delta$ and fixed $\text{Re}=50$.

Figure 4

$\varepsilon ={10}^{-3}$. Streamlines and isocontours of the axial relative velocity, for increasing values of $\delta$ (from the top to the bottom) and increasing values of $\text{Re}=25,50,100$ (from the left to the right).

Figure 5

$\varepsilon ={10}^{-3}$. Isocontours of (a) length of the recirculation region $L_R$ and (b) its distance from disk $X_R$, obtained by finding the zeros of the axial relative velocity on the axis, as functions of Re and $\delta$. (c) Isocontours of the nondimensional drag $D_0$ as a function of Re and $\delta$.

Figure 6

(a), (b) Nonoscillatory mode. (a) Real part of the streamwise component of the velocity eigenvector, rescaled with ${\vartheta }_{\pm }$, for $\text{Re}=144, \delta =0.1$, and $\varepsilon ={10}^{-3}$. The imaginary part is identically zero. (b) Critical Reynolds number of the nonoscillatory mode as a function of $\delta$. (c) Marginal stability curves in the $({\mathcal{I}}^{*},\text{Re})$ plane, for $\delta =0.1$ and $\varepsilon ={10}^{-3}$. The $(+)$ and $(-)$ help identify the regions with positive and negative real part of the eigenvalues, and the symbols $o$ and $x$ connect the neutral curves to the corresponding Strouhal number ones (on the bottom). The gray regions identify where the steady vertical path is linearly stable with respect to azimuthal perturbations.

Figure 7

$\varepsilon ={10}^{-3}$ and $\delta =0.1$. Real part of the streamwise component of the velocity eigenvector, rescaled with ${\widehat{\vartheta }}_{\pm }$, at the marginal stability, for different modes and values of ${\mathcal{I}}^{*}$.

Figure 8

Marginal stability curves in the $({\mathcal{I}}^{*},\text{Re})$ plane, for increasing values of $\delta$ and fixed $\varepsilon ={10}^{-3}$. The $(+)$ and $(-)$ help identify the regions with positive and negative real part of the eigenvalues, and the symbols $o$ and $x$ to connect the neutral curves to the corresponding Strouhal number ones. The gray regions identify where the steady vertical path is linearly stable with respect to azimuthal perturbations.

Figure 9

Same as Fig. 8 for (a) $\delta =0.7$ and (b) $\delta =0.8$.

Figure 10

$\varepsilon ={10}^{-3}$. Real (on the top) and imaginary (on the bottom) parts of the streamwise component of the velocity eigenvector, rescaled with ${\widehat{\vartheta }}_{\pm }$, for (a) ${\mathcal{I}}^{*}=0.001$ and (b) ${\mathcal{I}}^{*}=0.1$ and increasing values of $\delta$, from the top to the bottom. The plotted modes are identified as the first threshold encountered by the steady vertical path as Reynolds increases, for fixed $\delta$.

Figure 11

(a) Falling styles [different symbols, see also Fig. 1] from simulations of the fully nonlinear dynamics and marginal instability thresholds for steady vertical fall in the $({\mathcal{I}}^{*},G(1-\delta ))$ plane ($\varepsilon =1/60$).

Figure 12

Vertical velocity as a function of time, for different values of $G$ and $\delta$. For each value of $G$, velocities are rescaled with the nominal falling velocity of the full disk $U_0=U_g(\delta =0)$ ($\varepsilon =1/60$).

Figure 13

(a) Falling styles from simulations of the fully nonlinear dynamics and curves marking the instability thresholds for steady vertical fall ($\varepsilon =1/60$). (b) From the top to the bottom: tumbling: $G=1971, \delta =0$; chaotic motion: $G=1237, \delta =0.4$; flutter: $G=524, \delta =0$.

Figure 14

Trajectories described in the absolute reference frame as functions of time, following the colormap (on the top and on the center), and unwrapped inclination angles ${\vartheta }_{x1}$ and ${\vartheta }_{x2}$ around the fixed axes as functions of time rescaled with its maximum value (on the bottom). $G=280$ and (a) $\delta =0.25$ (flutter), (b) $\delta =0.4$ (hula-hoop); (c) $G=1237$ and $\delta =0$ (tumbling). The red line in panel (c) visualizes the normal-to-the-disk direction and helps in identifying the rotating motion of the disk in the tumbling regime.