  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.
