Day 1Flown 0 kmFat 0 gMode —Altitude 0 mFormation saving 0 %FPS 0

Drag to orbit, wheel or pinch to zoom. Glowing columns are thermals: the flock circles up in one, then glides in V to the next. Blue lines are the trailing wing-tip vortices of each bird. Map (bottom right): upwash (warm) and downwash (cool) at the flight level.

Crane Flock Aerodynamics: V-Wake Upwash and Thermal Relay Soaring

Every bird sheds two wing-tip vortices. Fly in the rising side of the wake of the bird ahead and you borrow its lift. Then add thermals and compare the energy per kilometre.

Formation geometry and wake model

Flock and flight

Thermals, route and day

Time

Live readouts

Induced-power saving of each bird

Sink polar and MacCready glide

Thermal updraft and the circling radius

Fat reserve along the route

Experiment: flap in V, soar, or both?

Runs the whole route three ways on the same day cycle (fast-forward) and compares days, energy per kilometre and fat left.

Independent numerical checks

Controls

Drag / one finger: orbit the camera around the lead crane; wheel / pinch: zoom.
Formation buttons set the layout; the sliders change stagger and wing-tip gap. "Optimise wingtip gap" scans the gap for the largest flock-average saving.
Strategy: flap in V all day, soar only (wait when thermals are too weak), or soar when a thermal gives at least the minimum climb rate and flap otherwise.
Time scale = simulated hours per real second.

The science

Wake model. A bird of weight mg flying at speed V with elliptic loading sheds a vortex pair of strength Γ₀ = 4 m g / (π ρ V b) with spacing s = π b / 4. Each vortex is treated as a straight semi-infinite line, with a Rankine-type core and no decay: w(x, y) = Σ ± Γ₀/(4π) · d/(d² + r_c²) · (1 + x/√(x² + d²)), positive upward outside the pair and negative (downwash) between the vortices. A follower feels the lift-weighted mean of that field over its own span, and its induced power falls by L w̄ / P_ind with P_ind = 2 (m g)² / (π ρ b² V). The leader gains nothing; a bird directly behind another is in downwash and pays extra.

Honest limits. The ideal wake overestimates real savings (roll-up, decay, gusts and wing-beat unsteadiness are ignored; real V's are loose and birds keep adjusting). The wake realism slider (default 0.6) scales the upwash; field studies of birds in formation report savings far smaller than the ideal-wake value. Phase-locking of the flap is not modelled, so the effect is for steady flapping or gliding only and the benefit is switched off while circling in a thermal.

Glide polar. Gliding sink is w_s = (A/v + B v³)/(m g) with A = 2 (m g)²/(π ρ b²) (induced, multiplied by the load factor n² = 1/cos²φ when banked) and B = ½ ρ S_b C_D (body drag). Best glide ratio at v = (A/B)^1/4, minimum sink at (A/3B)^1/4.

Thermals. The updraft is Gaussian, w(r) = w_t exp(−(r/R)²). Circling at bank φ and speed v has radius r = v²/(g tan φ) and a climb rate w(r) − w_s(v, φ); the page scans bank and speed for the best value. Between thermals the cross-country speed (MacCready) is V_xc = v c / (c + w_s(v)); its optimum is where the line from (0, c) is tangent to the polar, which the page checks numerically. Thermal strength follows a sine curve through the day; spacing, inversion height and the minimum usable climb rate are sliders.

Energy. Flapping costs mechanical power / 0.23 + basal metabolism, with the usual induced, parasite and profile terms; soaring is charged at 2.3 × basal metabolism. Fat is 39.3 MJ/kg and feeding happens in the morning and evening. All numbers are illustrative, not measurements of a particular crane population. Real cranes also use wind, ridge lift and weather fronts that are not modelled.

The 3D valley is a close-up loop over generic wetlands and thermal columns; physics runs on the whole route. Models built with Blender (crane, valley terrain, trees, reeds, standing cranes) and rendered with Three.js.