Wing Planform Design

Wing Planform Design (aerodynamics/wing-design/wing-planform-design)

Use when the task is the planform geometry and spanwise loading of a straight-tapered wing: the reference chords (root, tip, mean geometric, mean aerodynamic), the MAC spanwise station, the sweep reference-line conversion, the Schrenk spanwise load approximation, and the washout sizing for a root-first stall sequence.

Domain quick reference

• A straight-tapered (trapezoidal) wing is defined by the span b, the reference area S, and the taper ratio lambda = ct/cr in (0, 1]. The derived reference geometry is the aspect ratio AR = b^2/S, the root chord cr = 2S/(b(1+lambda)), the tip chord ct = lambda*cr, and the mean geometric chord cbar = S/b. Worked example, b = 35 m, S = 125 m^2, lambda = 0.3: AR = 9.8, cr = 5.4945 m, ct = 1.6484 m, cbar = 3.5714 m.
• Mean aerodynamic chord: MAC = (2/3)cr(1+lambda+lambda^2)/(1+lambda). For the example: MAC = 3.9166 m. The MAC is the chord of the equivalent rectangular wing that preserves the pitch moment behavior; it is the reference for the center of gravity limits and the neutral point, not the mean geometric chord.
• MAC spanwise station from the root: y_MAC = (b/6)(1+2lambda)/(1+lambda). For the example: y_MAC = 7.1795 m from the root centerline. Checks: rectangular wing (lambda = 1) gives y_MAC = b/4, a triangle (lambda toward 0) gives y_MAC = b/6.
• Local chord at span station eta = 2y/b in [0, 1]: c(eta) = cr(1 - (1-lambda)eta), eta = 0 at the root and eta = 1 at the tip. For the example at eta = 0.25: c = 4.5330 m.
• Sweep reference conversion: the chord-fraction line m (0 = leading edge, 0.25 = quarter chord, 0.5 = mid chord, 1 = trailing edge) obeys tan(Lambda_m) = tan(Lambda_LE) - 4m(1-lambda)/(AR(1+lambda)), so tan(Lambda_to) = tan(Lambda_from) + 4(from_ref - to_ref)(1-lambda)/(AR(1+lambda)). For the example with Lambda_LE = 30 deg: Lambda_c/4 = 27.5828 deg, Lambda_c/2 = 25.0542 deg, Lambda_TE = 19.6755 deg. A straight trailing edge needs tan(Lambda_LE) = 4(1-lambda)/(AR(1+lambda)), 12.3954 deg for the example.
• Schrenk spanwise loading approximation: the loading is the average of the elliptical distribution and the chord-proportional distribution, l(eta) = CL[2S/(pi b)sqrt(1-eta^2) + c(eta)/2], with the local lift coefficient cl(eta) = l(eta)/c(eta). The loading integrates to CL*S, so the approximation carries the exact total lift. For the example at CL = 0.5: cl_root = 0.4569 (eta = 0), cl = 0.4928 at eta = 0.25, cl = 0.5257 at eta = 0.5, and cl_tip = 0.25 (eta = 1). A rectangular wing at CL = 0.5 gives cl_root = 0.5683 and cl_tip = 0.25; the tip is always lightly loaded because the chord-proportional term halves at the tip.
• Washout: linear twist twists the tip nose-down relative to the root, alpha_eff(eta) = alpha_root - washout_tip*eta. The washout needed so the tip reaches its clmax no earlier than the root is epsilon = (tip_local_cl - tip_clmax)/a with the section slope a per radian, clamped at zero. Worked example: tip_local_cl = 1.15, tip_clmax = 1.0, a = 5.0265 per radian (thin wing, AR = 8) gives epsilon = 1.7098 deg. Root-first stall gives a benign stall with lateral control retained at the tip.

Workflow

1. Collect the planform inputs: span, reference area, taper ratio, and the sweep angle on one reference line (usually the quarter chord).
2. Compute the derived geometry: aspect_ratio(span, area) and trapezoidal_chords(span, area, taper) for the root chord, tip chord, and mean geometric chord; confirm the taper with taper_ratio(tip, root) when only the chords are known.
3. Compute the mean aerodynamic chord with mean_aerodynamic_chord(root, taper) and its spanwise station with mac_span_station(span, taper); report the MAC station for the CG limits and the neutral point reference.
4. Convert the sweep line with sweep_convert(sweep_deg, from_ref, to_ref, span, area, taper) when the leading edge or the mid chord is the known line; the quarter chord is the standard reference for aerodynamic analysis.
5. Compute the spanwise loading with schrenk_loading(span, area, taper, cl_wing, eta) at the stations of interest (root, the MAC station, the planform break, the tip) and check that the local cl at the tip stays below the tip clmax; the loading integrates to CL*S by construction.
6. Size the washout with washout_required(root_clmax, tip_clmax, root_local_cl, tip_local_cl, section_slope), using the tip clmax for the thinner tip section and the local cl values from step 5 at the design CL; apply the twist with linear_washout_angle(alpha_root_deg, washout_tip_deg, eta).
7. Record the reference chords, the sweep line, and the washout in the wing geometry definition; the lift curve slope and the detailed loading belong to the lift-curve-slope and vortex-lattice-method leaves.

Pitfalls

• Routing geometry here: deriving the reference geometry (chords, MAC, sweep line, taper) and the Schrenk loading from a given planform is this leaf. Sizing the planform from the takeoff gross weight and the wing loading belongs to vehicle-design/sizing/wing-planform-sizing; that leaf produces the planform, this leaf analyzes it.
• Routing detailed loading there: a numerical panel solution with horseshoe vortices, influence coefficients, downwash, and induced drag belongs to vortex-lattice-method. Schrenk is the hand approximation for conceptual checks, not a vortex solution.
• Routing sweep effects there: the effect of sweep on the section Mach number and the critical Mach (simple sweep theory) belongs to swept-wing-aerodynamics; this leaf only converts the sweep reference line as geometry.
• Routing cl-alpha there: the lift curve slope corrections (aspect ratio, sweep, Mach) belong to lift-curve-slope; this leaf provides the aspect ratio and sweep line it consumes, not the slope itself.
• Routing sections there: 2D section shape, camber, and NACA naming belong to airfoil-geometry; this leaf treats the wing planform, not the section contour.
• Routing clmax increments there: flap and slat clmax increments and the stall speed belong to high-lift-systems; this leaf sizes the geometric twist for stall sequencing, it does not compute clmax.
• Confusing MAC with the mean geometric chord: the mean geometric chord is S/b and locates the planform area; the MAC is the moment-preserving reference chord and its spanwise station is y_MAC, not the centroid station. Using cbar in place of MAC shifts the CG limit reference.
• Forgetting the MAC station: the MAC value alone is not enough; the CG and neutral point limits are measured at y_MAC, which depends on the taper ratio through (1+2lambda)/(1+lambda).
• Sign conventions in the sweep conversion: the conversion adds 4(from_ref - to_ref)(1-lambda)/(AR(1+lambda)); converting from the leading edge (0) to the quarter chord (0.25) subtracts, so the quarter-chord sweep is always less than the leading-edge sweep for a tapered wing. Reversing the sign gives a swept-forward result.
• Clamping the washout: washout_required never returns a negative value; a negative excess means the tip is already the lighter loaded station and adding wash-in would need an explicit design decision, not the default.

Behavior contract (gate 3)

The reference geometry, sweep conversion, Schrenk loading, and washout sizing are exercised by the gate 3 contract test: scripts/test_wing_planform.py against scripts/wing_planform_logic.py (stdlib unittest, offline). Run: python3 scripts/test_wing_planform.py

Compliance

• Standards referenced, not reproduced: NACA Report 824 (public domain) supplies the section lift data anchor for the clmax values used in the washout sizing; the planform formulas, the Schrenk approximation, and the twist sizing are common aerodynamic design knowledge, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.