Winglet Design

Winglet Design (aerodynamics/wing-design/winglet-design)

Use when the task is the wingtip device trade for induced-drag reduction: sizing the winglet height and cant from the reference wing geometry and a target drag reduction, and weighing the drag gain against the root bending moment penalty. This leaf implements the effective-span extension model in pure Python, stdlib only. It pairs with aerodynamics/wing-design/wing-planform-design for the reference planform, and with vehicle-design/sizing/wing-planform-sizing and the drag-polar leaves for the system context. The improved span efficiency and bending penalty models are documented conceptual approximations for a preliminary trade; a real winglet design needs a VLM/CFD pass and a structural FEM pass.

Domain quick reference

• Effective span extension: extension = K_HEIGHT * height_frac, with K_HEIGHT = 0.8 the documented fraction of the winglet height that acts as span extension. height_frac is the winglet height over the local semi-span.
• Cant weighting: cant_factor = cos(cant_deg). A vertical winglet (cant 0) keeps the full effect; a flat tip (cant 90) loses it.
• Extended span and effective aspect ratio: b_eff = b * (1 + 2 * cant_factor * K_HEIGHT * height_frac) adds both tips, and AR_eff = b_eff^2 / area.
• Improved span efficiency: e_eff = 1 - (1 - e_base) / (AR_eff / AR) with AR = span^2 / area. Documented approximation: the drag factor k = 1 / (pi * e * AR) shrinks with the effective-AR gain.
• Induced drag: k = 1 / (pi * e * AR) and cd_i = cl^2 * k at the reference lift coefficient.
• Drag reduction: reduction_pct = 100 * (1 - cd_i_wl / cd_i_base).
• Root bending penalty: penalty_pct = cant_factor * K_HEIGHT * height_frac * 100 * (1 + 0.5 * height_frac). Approximate scaling, the winglet load acts near the tip so the added root moment grows roughly with the height fraction.
• Sizing: size_winglet bisects height_frac in [0.01, 0.5] to a 0.1 pct reduction tolerance and returns the physical height height_m = height_frac * span / 2 (semi-span local reference).
• Units are SI: m, m^2, degrees.

Workflow

1. Take the reference wing geometry from the planform design: span, area, base span efficiency e_base, and the lift coefficient cl_ref for the drag check.
2. Read off the winglet height fraction and cant, or the target reduction percent when sizing.
3. Compute the span extension with effective_span_extension and the cant weighting with cant_factor.
4. Get the effective aspect ratio with ar_eff and the improved span efficiency with e_winglet.
5. Get the induced-drag factors with induced_drag_factor, then the drag coefficients with cd_i, and the gain with drag_reduction_pct.
6. Check the structural side with root_bending_penalty_pct.
7. To size the device, run size_winglet with the target reduction and inspect the returned height fraction, height, ar_eff, e_eff, cd_i, reduction_pct and bending_penalty_pct.
8. Confirm the deterministic checks with the contract test scripts/test_winglet_design.py.

Worked example

Wing: span 30 m, area 100 m^2 (AR 9), e_base 0.80, cl_ref 0.5. Direct case height_frac 0.12, cant 0 deg, taper 0.35.

• Extension: 0.8 * 0.12 = 0.096; cant factor 1.0.
• Extended span: 30 * (1 + 2 * 1.0 * 0.096) = 35.76 m; AR_eff = 35.76^2 / 100 = 12.788.
• Span efficiency: e_eff = 1 - 0.2 / (12.788 / 9) = 0.85924.
• Drag factors: k_base = 1 / (pi * 0.8 * 9) = 0.044210; k_wl = 1 / (pi * 0.85924 * 12.788) = 0.028969.
• Drag coefficients: cd_i base = 0.25 * 0.044210 = 0.011052; cd_i wl = 0.25 * 0.028969 = 0.007242.
• Reduction: 100 * (1 - 0.007242 / 0.011052) = 34.47 pct.
• Bending penalty: 1.0 * 0.8 * 0.12 * 100 * (1 + 0.06) = 10.18 pct.
• Sizing case: size_winglet(30, 100, 0.8, 25, 0.5) returns height_frac 0.0784, height_m 1.176 m, ar_eff 11.40, e_eff 0.8421, reduction_pct 25.00 within the 0.1 pct tolerance, bending_penalty_pct 6.52.

Verification

• Confirm ar_eff(30, 100, 0.12, 0.0) returns 12.788 within 0.01.
• Confirm e_winglet(0.8, 0.12, 0.0) returns 0.85924 within 1e-4.
• Confirm induced_drag_factor(0.8, 9.0) returns 0.044210 within 1e-6 and the winglet factor matches 1 / (pi * e_eff * AR_eff).
• Confirm cd_i base 0.011052 and the drag reduction 34.47 pct against the worked example band (34.43 within 0.05).
• Confirm root_bending_penalty_pct(0.12, 0.0) returns 10.18 within 0.05 and drops to zero for a flat tip at 90 deg cant.
• Confirm size_winglet with target 25 pct returns a height fraction in [0.05, 0.12] and a reduction within 0.1 pct of the target.
• Confirm every non-positive span, area, lift coefficient, span efficiency outside (0, 1], height fraction outside [0, 0.6], cant outside [-90, 90] degrees, taper outside (0, 1] and target reduction outside (0, 100) raises ValueError.
• Run the contract test offline: python3 scripts/test_winglet_design.py (31 tests, deterministic).

Related leaves

• aerodynamics/wing-design/wing-planform-design: the reference planform geometry and spanwise loading the winglet sizing starts from.
• vehicle-design/sizing/wing-planform-sizing: vehicle-level wing area sizing context.
• aerodynamics/drag-polars/drag-polar and aerodynamics/drag-polars/parasite-drag: the full polar the induced term feeds into.
• aerodynamics/cfd/vortex-lattice-method: the higher-fidelity follow-on for the spanwise loading with the winglet fitted.
• structures/fem/calculix-linear: the structural follow-on for the root bending check.

Pitfalls

• Sizing the winglet against the cruise drag polar without a reference lift coefficient: cd_i scales as cl^2, so the 34.47 pct reduction in the worked example is only valid at cl_ref = 0.5 - report the reduction with the lift coefficient it was computed at.
• Counting the full winglet height as span: only K_HEIGHT = 0.8 of the height acts as span extension, and only the cant-weighted part (cant_factor = cos(cant_deg)) contributes, so a canted or flat tip gains less effective span than its physical height suggests.
• Trading drag reduction without the bending check: the winglet that cuts induced drag 34.47 pct also adds 10.18 pct root bending moment in the worked example - the tip-device trade must weigh root_bending_penalty_pct against the drag gain.
• Treating the approximate models as a final design: the e_eff improvement and bending-penalty relations are documented conceptual approximations for a preliminary trade; a real winglet needs the VLM or CFD pass and the structural FEM pass named in the related leaves.
• Feeding non-physical geometry: span, area and lift coefficient must be positive, span efficiency in (0, 1], height fraction in [0, 0.6], cant in [-90, 90] deg, taper in (0, 1] and target reduction in (0, 100) - everything else raises ValueError.
• Confusing the sizing reference: size_winglet bisects the height fraction on the local semi-span reference and returns the physical height as height_frac * span / 2, so the returned height_m belongs to the semi-span local definition, not the full span.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_winglet_design.py

The test covers the worked example contract (AR_eff 12.788, e_eff 0.85924, k_base 0.044210, drag reduction 34.47 pct, bending penalty 10.18 pct), the cant weighting bounds, the effective-aspect-ratio identity, the sizing bisection contract and its monotonicity, the height from the semi-span reference, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is the classic induced-drag and airfoil-data basis and FAR 25 the structural and airworthiness context; both are cited by name only with the model relations stated as standard engineering methodology, summary-only per standards-map.yaml.
• The e_eff improvement and bending-penalty models are documented conceptual approximations for a preliminary trade.
• compliance: STANDARDS-REF, gated: false.