Angular motions of freely falling spheroidal hailstone models

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Abstract
The free fall behavior of rotating oblate spheroids with Reynolds numbers of 4 × 104 to 4 × 105 is studied by solving semiempirical Eulerian equations of motion which use a quasistatic approximation for aerodynamic forces and torques. The angular motions are classified in terms of solutions to the restricted set of equations in which the center of mass falls uniformly. One class of solutions, named symmetric gyration, is relevant for studies of the symmetric growth of hailstones because symmetrically equivalent points on the surface of a spheroid are equally exposed to the flow. For large hailstones with horizontal total angular momentum, critical frequencies sufficient for symmetric gyration are in the range of 2‐6 Hz (Strouhal numbers 0.005 to 0.01), they increase with increasing axis ratio, decrease with size, and are independent of the density ratio spheroid/air. Solutions to the full equations of motion show that the rotational energy of a symmetrically gyrating spheroid increases with time. This results in an increase in nutational amplitude which depends on the spheroid, its spin rate and its nutation amplitude. Such an effect occurs because the horizontal velocity, which tends to align with the horizontal major axis, causes the aerodynamic torque to have an oscillating vertical component in phase with the oscillating angular velocity about the vertical. A similar interaction prevents a steady helical motion.

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