The effect of transverse gusts on free-falling plates

The kinematics of a free-falling body in an incompressible homogeneous quiescent flow depends on at least two dimensionless groups such as, for example, the nondimensional mass M ≡ m / ρ L² and the Galilei number Ga ≡ (m g / ρ)^1/2 / ν , where m and L are the mass and length of the body, ρ and ν are the density and the kinematic viscosity of the fluid. At low M and Ga, a two-dimensional body such as a plate falls steadily, while at higher M and Ga it flatters, tumbles, or combinations of the above. First, we investigate the effect of the permeability of a rectangular plate on the kinematics by coupling the two-dimensional Navier-Stokes equations and Euler's equation of motion, and by modelling the plate permeability with the Darcy equation. We found that the permeability increases the upper limit of M and Ga at which the plate falls steadily. Secondly, we investigate the transient effect of a transverse (horizontal) gust on the falling (vertical) velocity. The gust is defined by a smooth increase (through a cosine function) of the horizontal velocity from zero to u_G over a period t_G. We restrict our investigation to plates that fall steadily in the absence of a gust. We found that, for any u_G and t_G , the time-averaged terminal velocity over the transient is lower than the terminal velocity in quiescent flow. Hence, on average, a plate falling in a gusty flow falls slower than a plate falling in a quiescent flow. The effect of the gust increases with increasing M and Ga, and with increasing u_G . Conversely, the effect of the gusts plateaus for decreasing t_G and vanishes for increasing t_G . Overall these results suggest that the effect of horizontal gusts can be significant on slowing down the descent of free-falling bodies and may inform dispersal models of plant seeds such as, for example, the dandelion, and meteorological models on the transport of, for example, snowflakes.