Fluttering and Tumbling of Two-Dimensional Concave/Convex Bodies

We study the orientation dynamics of two-dimensional concavo-convex solid bodies denser than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle Φ relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number Re_c⁽¹⁾  and a subcritical pitchfork bifurcation at a critical Reynolds number Re_c⁽²⁾ > Re_c⁽¹⁾ . For  Re < Re_c⁽¹⁾  , the concave-downwards orientation ( Φ = 0 ) is unstable and bodies overturn into the  Φ = π  orientation. For Re_c⁽¹⁾ < Re < Re_c⁽²⁾ , the falling body has two stable equilibria at Φ = 0  and  Φ = π  for steady descent. For Re > Re_c⁽²⁾ , the concave-downwards   Φ = 0  orientation is again unstable, and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable  Φ = π   orientation. The Re_c⁽²⁾ ≈ 15 at which the subcritical pitchfork bifurcation occurs is distinct from the Re or the onset of vortex shedding, which causes the  Φ = π  equilibrium to also become unstable, with bodies fluttering about  Φ = π .  The complex orientation dynamics of irregularly shaped bodies evidenced here is relevant in a wide range of physical scenarios.