Wing in Ground Effect Lab (3D)

What ground effect is

A wing makes lift by turning air downwards. Far from the surface the downwash it leaves behind also tilts the local flow at the wing, so the effective angle of attack is smaller than the geometric one and the lift vector leans back: that backward lean is the induced drag, CDi = CL^2 / (pi e AR). When the wing flies within about one span of a flat surface, the surface blocks the downwash. In potential flow this is equivalent to a mirror (image) wing below the surface whose vortices rotate the opposite way. Their upwash cancels part of the downwash at the real wing, so the same lift needs less incidence and produces less induced drag.

The model on this page

CDi = phi(h/b) CL^2 / (pi e AR) with the induced-drag factor phi = 33 (h/b)^1.5 / (1 + 33 (h/b)^1.5), a Hoerner and Borst type fit in the height ratio h/b (wing height over span). The older McCormick form phi = (16 h/b)^2 / (1 + (16 h/b)^2) is drawn dashed for comparison; it sits above the numerical solution between 0.1 and 0.5. The lift curve keeps the same factor: CL = a0 (alpha - alpha0) / (1 + a0 phi / (pi e AR)), so at fixed incidence the lift rises as the craft descends. Level flight needs L = 1/2 rho V^2 S CL = m g, and the power to hold it is P = D V with D = 1/2 rho V^2 S (CD0 + CDi).

Height stability

Because lift grows when the craft sinks and falls when it climbs, a wing in ground effect has a restoring force k = -dL/dh. Vertical speed also changes the angle of attack by -h'/V, which gives a damping force c = (dL/dalpha) / V. The heave equation integrated here is m h'' = L(h, alpha - h'/V) - m g. Linearised about the equilibrium: w0 = sqrt(k/m), zeta = c / (2 sqrt(k m)), damped period 2 pi / (w0 sqrt(1 - zeta^2)). Out of ground effect (h/b above about 1) k tends to zero and the craft has no height stability of its own; this is why such craft need an autopilot to climb or hold altitude, and why a gust near the water is forgiving but a wave crest is not.

Numerical checks on this page

(1) A Prandtl lifting-line code (48 horseshoe vortices with their mirror images) is solved with Gaussian elimination and its induced drag at the same CL, divided by the free-air value, is compared with the model fit for the current aspect ratio. (2) A 0.15 m nudge at trim is integrated with RK4 and the measured oscillation period is compared with the linear theory above. (3) A long integration from a displaced height must settle to the height found by bisection on L(h) = W.

Honest simplifications

The lift curve slope and the e = 0.9 efficiency are fixed numbers; real craft use cambered sections with a strong change of the pitching moment near the surface, which decides pitch stability and is not modelled (the wing is held at fixed incidence and only heave is dynamic). The fit for phi is a curve through lifting-line results: the real value depends on aspect ratio, taper, end plates and flap setting. The static-pressure (ram) contribution that ground-effect craft with end plates use at very small height, hull spray drag, wave-induced forces on the hull and compressibility are left out. Atmosphere density is 1.225 kg/m3. The craft is an invented generic shape, not a particular historical or current design.

What to try

• Load Cruise in ground effect and press Gust: the height oscillates and settles back to the green line on graph 3.
• Open Climbing out of ground effect: graph 2 shows the lift-to-drag ratio drop to the orange free-air value and the heave stiffness readout falls to almost nothing.
• Compare Slender wing (aspect ratio 8) with Stubby wing (aspect ratio 2): the same h/b gives nearly the same phi, but induced drag matters far more on the stubby wing.
• Raise the wave height and lower the craft: the hull touches the crest (splash) before the aerodynamic limit.
• Turn on Hold height and change airspeed: graph 4 shows the power needed at the current height against the free-air curve.