Kite flight under varying wind - 2D side view

Forces on the kite

The kite is a flat plate of area A on a taut line of length L at elevation φ. The air moves past it at the relative velocity v_rel = wind − kite velocity, and the angle of attack α is the angle between the sail plane and v_rel. The aerodynamic force has lift L = ½ ρ v_rel² A C_L perpendicular to v_rel and drag D = ½ ρ v_rel² A C_D along it, here with the flat-plate coefficients C_L = 1.2 sin 2α and C_D = C_D0 + 2.4 sin²α. Gravity m g and the line tension T complete the picture.

Equations

Because the line is inextensible, only the swing along the circle of radius L is free: m L φ'' = F_tangential and the line tension is T = F_radial + m L φ'². At equilibrium the net tangential force is zero and the line carries the whole load. The sail plane is held at a fixed bridle angle ψ to the line, so as the kite climbs its attitude changes, which makes the equilibrium self-correcting: a stronger wind lifts the kite higher until the pull is aligned with the line again.

Scaling

Aerodynamic forces scale with the square of the wind speed, so in a light-kite limit (m → 0) the equilibrium angle does not depend on the wind speed at all while the tension grows 4 times for a doubled wind. Weight breaks this scaling and is why a heavy kite sags in light winds.

Honest simplifications: a point-mass kite with a flat-plate polar and no stall hysteresis, a straight massless line (no sag, no drag of the line, no stretch), a two-dimensional side view with the wind in the plane, and a horizontal wind with no height gradient. Gusts are a sine plus an Ornstein-Uhlenbeck turbulence term. Slack-line episodes are flagged but the line is still treated as taut.

• Use the Gusty day preset and watch the elevation and tension traces follow the wind.
• Increase the kite mass: the equilibrium angle drops in light winds.
• Change the bridle angle ψ and read the equilibrium graph: a steeper sail gives a higher kite and a larger pull.
• Choose the Light-kite preset and compare tensions at two wind speeds: they differ by the factor of the square of the speed ratio.