The aerodynamic wall

Abstract

We study the trajectory of dense projectiles subjected to gravity and drag at large Reynolds number. We show that two types of trajectories can be observed: if the initial velocity is smaller than the terminal velocity of free fall, we observe the classical Galilean parabola: if it is larger, the projectile decelerates with an asymmetric trajectory first drawn by Tartaglia, which ends with a nearly vertical fall, as if a wall impeded the movement. This regime is often observed in sports, fireworks, watering, etc. and we study its main characteristics.

Edge effect: Liquid sheet and droplets formed by drop impact close to an edge

“…Owing to air drag, the horizontal speed decreases with time and x(t ) reaches an asymptotic value = lim t→∞ x called the aerodynamic wall [73]. Equation (B1) is nondimensionalized with characteristic timescale T and length scale L defined in the limit of small Reynolds number, T = 2ρr 2 /(9μ a ), L = gT 2 , y = x/L, and τ = t/T from which we obtain …”

“…We here consider the horizontal direction of ejection e x and compute x(t ), the distance traveled horizontally since ejection, regardless of direction θ v . Owing to air drag, the horizontal speed decreases with time and x(t ) reaches an asymptotic value = lim t→∞ x, the aerodynamic wall [73].…”

Far-field properties of aerated water jets in air

“…From the above analysis, air resistance is important for aerated jets in air. With air resistance, the following equation can be written for a projectile (Cohen et al, 2013):…”

“…2, in which C D = 0.47 is used. Cohen et al (2013) used a C D value of 0.44 in modeling trajectory of water jets in air from fire hoses. Although this modeling approach is rather crude, satisfying agreement is obtained between the modeling results and measurements of the jet upper boundary.…”

Physics of ball sports

“…Parabola and Tartaglia are both characterized by a maximal range, x max , which can be calculated [5] and which imposes a physical constraint to the size of sports fields. Indeed, it's no use playing on a 100 m-long field, if the maximal range of the ball is 10 m. Figure 4(d) shows a strong correlation between the size of sports fields and the maximal range, x max , of the associated balls, from table tennis to golf.…”