Fig. 2

Tumbling and fluttering motions of buoyant spheres in turbulence. Drawings representative of a typical trajectory traced by the a high MoI sphere and the b low MoI sphere in a turbulent flow at Re _λ  ≈ 300. The zoomed-in views of the rectangular windows pqrs and jklm are also shown. The high MoI sphere tumbles in the flow, where the direction of rotation does not change much during the motion. This tumbling motion induces a mean horizontal drift for the particle. The low MoI sphere flutters in the flow, where the directions of rotation and translation undergo frequent reversals at a rate comparable to that of vortex-shedding frequency. This fluttering is thought to stabilize its motion to remain vertical to the mean. c Sphere motion transformed into a moving TNB coordinate system, where \({{\mathbf T}}\)—the direction of the particle’s instantaneous velocity \({\mathbf{v}}_{\mathrm{p}}\), \({{\mathbf N}}\)—direction of curvature of the particle trajectory, and \({{\mathbf B}}\)—the binormal vector is defined such that \({{\mathbf N}} = {{\mathbf B}} \times {{\mathbf T}}\). The lower figure shows the TNB coordinate system, where the orientation of an arbitrary vector \(\mathbf{ f}\) can be expressed in terms of the elevation θ and azimuth ϕ angles. d, e Normalized histograms of the angle between the angular velocity vector \({\mathbf{\upomega}}\) and its time derivative \({\mathrm{d}}{\mathbf{\upomega}} {\mathrm{/}}{\mathrm{d}}t,\) expressed in terms of Δθ and Δϕ for the d high and e low MoI spheres, respectively. Supplementary Movies 1 and 2 compare the two types of motion. The typical amplitude of flutter and tumble is 1–2 sphere diameters during an oscillation