Reynolds-Number-Dependent Drag and the Free-Kick Trajectory of a Spinning Soccer Ball

Reynolds-Number-Dependent Drag and the Free-Kick Trajectory of a Spinning Soccer Ball Simulation code, output data, and figures accompanying the manuscript: Dutta, K. (2026). Reynolds-Number-Dependent Drag and the Free-Kick Trajectory of a Spinning Soccer Ball. Submitted to Sports Engineering. Contents soccer_drag_crisis_trajectory.py simulation script (self-contained; regenerates all data and figures below) figs/ Cd-vs-Re.png Cd(Re) interpolant used by the model trajectory-3d.png 3D trajectory: constant-Cd vs variable-Cd(Re(t)) Re-Cd-timeseries.png Re(t) and Cd(t) along the representative flight lateral-swing.png lateral deflection vs downrange distance sensitivity.png deflection bias vs initial strike speed data/ cd_re_dataset.csv (u, Re, Cd) reference points used to build Cd(Re) trajectory_constant_cd.csv full time series, constant-Cd flight trajectory_variable_cd.csv full time series, variable-Cd(Re(t)) flight sensitivity_deflection_vs_speed.csv lateral deflection at 20 m downrange vs launch speed summary.csv single-row table of key scalar outputs summary.txt human-readable version of summary.csv What the script does Integrates the 3D equations of motion of a spinning soccer ball, m dv/dt = m g − (1/2) Cd ρ A |v| v + S (ω × v), for a representative free kick (u0 = 25 m/s, 15° elevation, 10 rev/s spin) and for a sensitivity sweep of launch speeds (14–32 m/s, same elevation and spin), under two treatments of the drag coefficient: Constant Cd (Cd = 0.25, fixed for the whole flight): the conventional approach used in most existing trajectory models. Variable Cd(Re(t)): a Reynolds-dependent drag coefficient, built from a monotone cubic (PCHIP) interpolant through 13 reference points reproducing the classical smooth-sphere drag-crisis curve of Achenbach (1972, 1974), recomputed at every integration step from the ball's instantaneous speed. Both cases use the same Magnus force law, F_l = S(ω × v), with S calibrated from a reference lift coefficient C_l = 1.23. Integration uses SciPy's solve_ivp (Dormand–Prince 5(4), dense output, rtol = 1e-8, atol = 1e-10). Ground impact and fixed-downrange quantities are located by root-finding on the solver's dense interpolant, not read off the plotting grid. Reproducing the outputs python soccer_drag_crisis_trajectory.py Regenerates figs/ and data/ in the working directory. No command-line arguments; parameters are set at the top of the script (U0, ANGLE, SPIN, TARGET_X). Dependencies: Python 3, NumPy, SciPy (solve_ivp, PchipInterpolator), Matplotlib. Data provenance and limitations The Cd(Re) curve is a re-tabulated, simplified representation of Achenbach's smooth-sphere drag-crisis data, not a direct digitization of the original wind-tunnel records. It does not represent a real match ball's panels, seams, or drag-crisis shift; see the accompanying manuscript (Sections 2.2 and 5) for the ball-specific literature and the consequences of this simplification. Spin decay, wind, and ground effects are not modeled. License Code and data are released under CC-BY 4.0. Please cite the manuscript above if you use this material.