Orientation dynamics of two-dimensional concavo-convex bodies

Abstract

We study the orientation dynamics of two-dimensional concavo-convex solid bodies more dense than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle $\phi$ relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number $Re_{c}^{(1)}$, and a subcritical pitchfork bifurcation at a Reynolds number $Re_{c}^{(2)}$. For $Re<Re_{c}^{(1)}$, the concave-downwards orientation of $\phi=0$ is unstable and bodies overturn into the $\phi=\pi$ orientation. For $Re_{c}^{(1)}<Re<Re_{c}^{(2)}$, the falling body has two stable equilibria at $\phi=0\text{ and }\phi=\pi$ for steady descent. For $Re>Re_{c}^{(2)}$, the concave-downwards orientation of $\phi=0$ is again unstable, and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable $\phi=\pi$ orientation. The $Re_{c}^{(2)}\approx15$ at which the subcritical pitchfork bifurcation occurs is distinct from the $Re$ for the onset of vortex shedding, which causes the $\phi=\pi$ equilibrium to also become unstable, with bodies fluttering about $\phi=\pi$. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to settling of mollusk shells.