Formation Flight: Induced Drag Savings in a V

Prandtl lifting lines with mutual coupling, developed here from Biot–Savart and Kutta–Joukowski rather than fitted to a fuel-saving percentage. Write Γ = W/(ρVb) Σ aₙ sin(nθ), y/b = −cos(θ)/2. Lift trim fixes a₁ = 4/π; zero rolling moment fixes a₂ = 0. Each wing's unknown higher harmonics, incidence and linear differential-control twist follow from 2Γ/(a₀c) − w = V(α + τy/b). Self-induced downwash uses the exact Fourier operator; other wings contribute their finite bound vortex and semi-infinite straight trailing sheet, integrated by midpoint quadrature with a finite core. All wings are solved simultaneously with LU decomposition.

Induced drag is Dᵢ = −ρ∫Γw dy, integrated on the Fourier collocation grid. The isolated wing is solved with the identical planform and resolution. The lateral sweep repeats the coupled solve at each spacing. Equal-load and zero-roll integrals are checked independently on 320 points. Electrical power is V(Dᵢ + qSCᴅ,other)/η + Pₕ, and endurance is usable battery energy divided by power. These equations follow classical lifting-line theory; Lissaman & Shollenberger (1970) and Hummel (1983) motivate the formation application, not numerical calibration.

Every spacing sample is screened against the section lift and trim-angle limits. The recommended sample is recomputed at the selected geometry resolution. Selected-resolution harmonic and wake-quadrature refinement errors are computed on every run.

Identical, rigid, straight, unswept horizontal wings; rectangular or elliptic chord distribution. Steady incompressible attached flow and small induced angles; specified linear section lift slope. All wings support their own weight at the same speed. An ideal linear control twist cancels rolling moment; its actuator drag, authority and power are not modeled. Wakes remain parallel to the freestream without roll-up, descent, decay or meander. Other drag and propulsion efficiency stay unchanged in formation. Batteries have equal usable energy and constant mass; role rotation has no transition cost.

Use aspect ratios 4–30, Mach ≤ 0.3 and section lift coefficients within the entered airfoil limit. Low-Reynolds-number separation and flapping birds need additional data and unsteady analysis. This is a straight-sheet, finite-core lifting-line model; the wake-core radius is a sensitivity input, not a validated wake-age correlation. Tip overlap and vertical proximity are especially core-sensitive. Every run checks the selected resolution against a solve with twice the harmonics and a separate solve with twice the wake quadrature. Harmonic refinement uses at least eight source points per selected harmonic. Warnings appear when the largest wing-drag change, relative to the largest refined wing drag, exceeds 0.5% for harmonics or 0.2% for wake quadrature. At the defaults these changes are below 0.1%. Stronger overlap can require considerably more harmonics.

The sweep solves 51 lateral positions with 32 harmonics by default, then uses shape-preserving cubic interpolation for the chart. Screening every point and checking selected-resolution convergence made 61 samples exceed the approximate 3 s budget; 51 retains the same lifting-line resolution with a 0.02-span lateral grid. The sampled optimum can therefore lie about 0.01 span from a continuous optimum. To keep large formations responsive, the sweep caps harmonics at max(12, floor(96/N)), wake quadrature at max(96, 8 × sweep harmonics), and positions at max(13, floor(459/N²)); requested values remain upper limits. This costs lateral optimum precision and can smooth narrow wake features; read the actual counts in the calculation and refine the selected geometry independently. The main drag and endurance results always use the requested harmonics and wake quadrature. The search excludes samples exceeding the entered section lift limit, the 15° small-angle incidence envelope, or 8° tip-to-tip control twist. Curves have gaps next to excluded samples. The lowest-drag remaining sample is checked again at the full selected resolution; no recommendation is displayed unless this confirmation passes. The best spacing is sampled within the stated lateral window at the entered stagger/height.

Actual control drag and safe clearance may remove part or all of the ideal benefit. The top-view drawing shows bound wings and representative wake edges; vertical offsets are in the position table. Battery endurance is only for fixed-wing electric propulsion at constant speed; no reserve beyond usable energy, wind, climb, mission overhead, battery derating or rendezvous cost is included. NASA and Airbus fuel-saving numbers are external flight evidence, not validation of this model for their aircraft.