Mass and Moment of Inertia Govern the Transition in the Dynamics and Wakes of Freely Rising and Falling Cylinders

In this Letter, we study the motion and wake patterns of freely rising and falling cylinders in quiescent fluid. We show that the amplitude of oscillation and the overall system dynamics are intricately linked to two parameters: the particle’s mass density relative to the fluid $m^{*}\equiv {\rho }_p/{\rho }_f$ and its relative moment of inertia $I^{*}\equiv I_p/I_f$. This supersedes the current understanding that a critical mass density ($m^{*}\approx 0.54$) alone triggers the sudden onset of vigorous vibrations. Using over 144 combinations of $m^{*}$ and $I^{*}$, we comprehensively map out the parameter space covering very heavy ($m^{*}>10$) to very buoyant ($m^{*}<0.1$) particles. The entire data collapse into two scaling regimes demarcated by a transitional Strouhal number ${\text{St}}_t\approx 0.17$. ${\text{St}}_t$ separates a mass-dominated regime from a regime dominated by the particle’s moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical two-single ($2S$) vortex mode to a two-pair ($2P$) vortex mode. Thus, autorotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.

Figure 1

(a) Oscillatory rise trajectory of a buoyant cylinder ($m^{*}=0.1$), with a constraint on rotation ($\mathrm{\Omega }=0$). (b) Oscillatory rise trajectory of the same cylinder with a low moment of inertia ($I^{*}=0.1$). Here, $\mathrm{\Omega }$ is the angular velocity, $D$ is the diameter of the particle, and $\lambda$ is the wavelength of the oscillation. (c),(d) Vorticity snapshots for the cylinders in (a) and (b), respectively. The angular velocity $\mathrm{\Omega }$ is zero for the rotationally constrained cylinder. The red-white-blue color map is used for the vorticity. Allowing autorotation changes the wake pattern dramatically.

Figure 2

Contour map of the (a) translational and (b) rotational amplitudes of motion for buoyant ($m^{*}<1$) and heavy cylinders ($m^{*}>1$) in the [$m^{*}-I^{*}$] parameter space at $\mathrm{Ga}\approx 500$. The shaded regions mark the very high values of $I^{*}$, which are primarily of mathematical interest. (c) Vorticity plots for the points marked as (A), (B), (C), and (D) in (a) and (b). The vorticity is scaled to the [$-1$, 1] range. The corresponding movies are given in the Supplemental Material [34].

Figure 3

(a) Oscillation amplitude $A/D$ versus $1/(m^{*}{\text{St}}^2)$ for buoyant and heavy cylinders. The dark blue curve shows the scaling relation $A/D\propto 1/(m^{*}{\text{St}}^2)$, which holds for $A/D\le 0.6$. The inset to (a) shows the compensated plot; the plateau for small $1/(m^{*}{\text{St}}^2)$ demonstrates the robustness of this scaling. (b) $A/D$ versus $1/{\text{St}}^2$ for buoyant and heavy particles. For $A/D>0.6$, the oscillation amplitude scales as $A/D\propto 1/{\text{St}}^2$, as is also demonstrated by the plateau for large $1/{\text{St}}^2$ in the compensated plot in the inset to (b). (c) Strouhal number contour map covering the [$m^{*}-I^{*}$] parameter space for the full family of buoyant and heavy cylinders from the present study. A transitional Strouhal number ${\text{St}}_t\approx 0.17$ separates the two scaling regimes shown in (a) and (b). The dashed inclined line in (c) denotes the homogeneous cylinder cases, while the shaded region of high $I^{*}$ values is primarily of mathematical interest.

Figure 4

Vortex-shedding time ${\tau }_v^{*}$ versus the inverse of the relative speed $1/U_{\text{rel}}^{*}$ for all heavy and buoyant cylinders from the present study. The dashed line suggests a nearly linear dependence. The inset shows the mean rise speed $U_{\text{rise}}^{*}$ and the mean rotational speed $U_{\text{rot}}^{*}\equiv ⟨|\mathrm{\Omega }|⟩D/2U_g$ versus ${\tau }_v^{*}$, where $\mathrm{\Omega }$ is the instantaneous angular velocity of the cylinder, $D$ is the cylinder diameter, and $U_g\equiv \sqrt{gD|1-{\rho }_p/{\rho }_f|}$ is a gravitational velocity scale [20].