Orientation dynamics of two-dimensional concavo-convex bodies

We study the orientation dynamics of two-dimensional concavo-convex solid bodies that are 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 $\varphi$ relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number $R{e}_c^{\left(1\right)}$ and a subcritical pitchfork bifurcation at a Reynolds number $R{e}_c^{\left(2\right)}$. For $Re<R{e}_c^{\left(1\right)}$, the concave-downwards orientation of $\varphi =0$ is unstable and bodies overturn into the $\varphi =\pi$ orientation. For $R{e}_c^{\left(1\right)}<Re<R{e}_c^{\left(2\right)}$, the falling body has two stable equilibria at $\varphi =0\text{and}\varphi =\pi$ for steady descent. For $Re>R{e}_c^{\left(2\right)}$, the concave-downwards orientation of $\varphi =0$ is again unstable and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable $\varphi =\pi$ orientation. The $R{e}_c^{\left(2\right)}\approx 15$ at which the subcritical pitchfork bifurcation occurs is distinct from the $Re$ for the onset of vortex shedding, which causes the $\varphi =\pi$ equilibrium to also become unstable, with bodies fluttering about $\varphi =\pi$. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to the settling of mollusk shells.

Figure 1

The idealized concavo-convex body used here. The four elliptical sections are shown in different colors: blue is a semiellipse with semimajor axis $1+\delta$ and unit semiminor axis, orange is a semiellipse with semimajor axis $1-\delta$ and semiminor axis $b=0.5$, and red and green are semicircular sections with radius $\delta$. Here, $\delta =0.05$. The $\times$ denotes the location of the center of mass of the body.

Figure 2

The vorticity ${\omega }_y$ shown in color and the outline of the concavo-convex body shown as the solid black line. (a) For $Re=10<R{e}_c^{\left(2\right)}$, the $\varphi =0$ orientation is a stable fixed point. (b) For $Re=12$, the initial orientation ${\varphi }_0=\pi /12$ is outside the basin of attraction of the fixed point at $\varphi =0$. As the oscillations increase in amplitude, they are reflected in the deviations of the dividing streamline from the vertical, and vortices are shed. (c) For $Re=17>R{e}_c^{\left(2\right)}$, the $\varphi =0$ orientation is unstable and all initial orientations ${\varphi }_0$ lead to steady descent at $\varphi =\pi$. Note, however, that this $Re$ is smaller than required for the onset of vortex shedding. (d) For $Re=25$, in the concave-up orientation, an incipient Karman vortex street is seen and the system oscillates with a finite amplitude.

Figure 3

The orientation evolution $\varphi \left(t\right)$ of the bodies (see Fig. 1). The final orientation is determined by the combination of the Reynolds number $Re$ and the initial orientation ${\varphi }_0$. For small Reynolds number, $Re=2.5<R{e}_c^{\left(1\right)}, \varphi$ increases monotonically with $t$ for all initial orientations ${\varphi }_0$, overshooting $\varphi =\pi$ slightly before reaching $\varphi =\pi$. For $R{e}_c^{\left(1\right)}<Re=12<R{e}_c^{\left(2\right)}$, the trajectories are oscillatory; for small ${\varphi }_0=0.07,\varphi \left(t\right)$ decays to zero as the body reaches a state of steady descent, while for ${\varphi }_0=\pi /12$, the body tumbles into the $\varphi =\pi$ orientation before descending steadily. For $Re>R{e}_c^{\left(2\right)}\approx 15$, the body flips to the $\varphi =\pi$ orientation for all ${\varphi }_0$, and the time in flight before the body tumbles decreases with increasing $Re$ and increasing ${\varphi }_0$.

Figure 4

Phase space plots of the angular velocity $\dot{\varphi }$ versus the angle $\varphi$ for Reynolds numbers $Re=2.5,10,17,25,100$. For $Re=2.5$, the orientation $\varphi =0$ is unstable and the trajectory $\dot{\varphi }\left(\varphi \right)$ towards $\varphi =\pi$ is non-oscillatory. For $Re=10$, the orientation $\varphi =0$ is a stable spiral and the trajectory spirals inwards. For $Re=17$, perturbations about $\varphi =\pi$ decay to zero and the trajectory is a decaying spiral around $(\varphi =\pi ,\dot{\varphi }=0)$. For $Re=25$, the system is overstable about the $\varphi =0$ orientation, and the trajectory spirals outwards, reaching the $\varphi =\pi$ orientation where it becomes a limit cycle of finite radius. A limit cycle of larger radius is reached for $Re=100$.

Figure 5

Bifurcation diagram for $(\varphi ,\dot{\varphi })=(0,0)$ as the initial orientation ${\varphi }_0\ge 0$ and the Reynolds number $Re$ are varied. Blue circles denote initial conditions that are attracted to $(\varphi ,\dot{\varphi })=(0,0)$, while black squares denote initial conditions that are attracted to $(\varphi ,\dot{\varphi })=(\pi ,0)$. The dot-dashed lines separating the circles from the squares are separatrices drawn by hand. For $R{e}_c^{\left(1\right)}<Re<R{e}_c^{\left(2\right)}$, two stable fixed points, $\varphi =0$ and $\varphi =\pi$, exist. At the subcritical pitchfork bifurcation at $Re=R{e}_c^{\left(2\right)}$, the $\varphi =0$ ($\varphi =\pi$) fixed point becomes unstable (remains stable). At a higher $Re$ associated with vortex shedding, the $\varphi =\pi$ fixed point becomes unstable and a limit cycle in $(\varphi ,\dot{\varphi })$ appears (Fig. 4).

Figure 6

The angular velocity (blue, left axis) and the circulation ${\mathrm{\Gamma }}_0-{\mathrm{\Gamma }}_{0.05}$ (red, right axis) for $Re=10$ and $Re=20$ with initial angles ${\varphi }_0$ to the vertical. The latter curves are displaced vertically for clarity. The former (latter) initial condition is stable (unstable) in the concave-downwards orientation.