Bird Wing Flight

What is modelled

A pigeon-sized bird is held in a wind tunnel and flaps its wings. Each wing is cut into 24 spanwise strips. For every strip and every instant of the flap cycle the model works out the air velocity seen by the strip (the tunnel speed plus the strip's own up-and-down speed), the angle of attack, and the lift and drag of that strip. Summing the strips and averaging over a flap cycle gives the mean lift, the mean thrust and the mechanical power, which are compared with the bird's weight and body drag.

Equations

• Planform: elliptic chord c(y) = c0 √(1 − (y/s)2) with half-span s = b/2 and area S = b2/AR (aspect ratio 7); the strip chords are the exact averages over each strip, so the area is exact.
• Flapping: φ(t) = Φ sin(2π f t); the strip at distance r moves vertically at vz = r φ̇ cos φ. The air arrives at the strip with velocity (−U, −v_z), so α = θ + atan(−vz/U), where θ is the wing incidence plus twist.
• Section forces: dL = ½ ρ |w|2 c dr CL(α), dD = ½ ρ |w|2 c dr CD. Lift is perpendicular to the relative flow, so on a downstroke it tilts forward and gives thrust: dFx = dL (−vz/|w|) cosφ − dD U/|w|, dFz = dL (U/|w|) cosφ − dD vz/|w|.
• Coefficients: CL = a α with the finite-wing slope a = 2π AR/(AR+2) up to 17 deg, then a flat-plate roll-off; CD = CD0 + CL2/(π e AR) plus a separation term after stall.
• Upstroke flex: on the upstroke the span is shortened by up to 55 %, the feathers open (lift slope down by up to 45 %) and the hand wing twists nose-down by up to 8 deg.
• Body drag ½ ρ U2 CDb Sb with C_Db = 0.1 and S_b = 0.00813 m^2/3 (m in kg), the default of Pennycuick's flight program. Power is the work done against the aerodynamic force, −Fz vz, summed over both wings.
• Strouhal number St = f A / U with A the peak-to-peak tip excursion.

Checks run in the page

• A wing held still reproduces the closed form ½ ρ U2 S a α for lift and the matching induced-drag formula, to rounding error.
• With no incidence, no twist and no flex, a symmetric flap cancels the vertical force exactly yet still pushes the bird forward (the Knoller-Betz effect).
• The cycle means do not change with finer time steps (64 against 1024) or finer strips (12 against 192).
• The level-flight trim solver returns mean lift equal to weight and mean thrust equal to body drag for the pigeon, and the page reports the Strouhal number it implies, which is lower than the 0.2 to 0.4 band observed for cruising birds (Taylor, Nudds and Thomas 2003) - see the simplifications.

Honest simplifications

• The trimmed Strouhal number comes out near 0.1, below the 0.2 to 0.4 of real cruising birds: with no unsteady losses, wing inertia or fuselage interaction the model needs far less flapping than a real bird, which is a known weakness of quasi-steady theory, not evidence that birds flap too much.
• The model is quasi-steady: it has no leading-edge vortex, wake capture, added mass or unsteady wake, which matter at low speed, so slow-flight numbers are only indicative. The lift slope, drag coefficients, the 17 deg stall angle, the flex factors and the body drag coefficient are textbook-order choices, not fitted to a particular species.
• The wing is a flat strip with a single twist law; real wings are cambered, with separating primaries that act as slotted tip surfaces. The wing flaps in a vertical plane with no stroke-plane tilt, no sweep in the force law, and the lateral force components cancel by symmetry.
• The bird is held fixed (a wind tunnel): body pitching, tail forces and the bird's own acceleration are not simulated. Net thrust above body drag would accelerate the bird; the trim button instead removes the surplus by reducing the amplitude.
• Aerodynamic power ignores the inertia of the wings and the metabolic efficiency of muscle (roughly 20 to 25 %), so the numbers are mechanical power, not the food a bird needs.
• The airflow streaks and the force arrows are drawn from the model values for illustration; the streaks do not bend around the bird.

• Press Set incidence for lift = weight, then Trim level flight: the solver finds the wing incidence and flap amplitude where mean lift equals weight and mean thrust equals body drag.
• Set Upstroke flex to 0 (Wings stay open): the upstroke now pushes the bird down and costs lift, which the flexed wing avoids.
• Watch the blade-element diagram: on the downstroke the relative wind comes from below, the lift vector tilts forward and gives thrust.