ashfordeOU Wave Drag and the Whitcomb Area Rule (aerodynamics/high-speed/wave-drag-area-rule)
Use when the task is transonic wave drag: the Whitcomb area rule, cross-sectional area distributions, the Sears-Haack minimum-drag body, and drag divergence in high-speed configuration design.
Domain quick reference
- Whitcomb area rule (1952): at transonic speeds the zero-lift wave drag of a wing-body combination depends mainly on the streamwise distribution of the total cross-sectional area (fuselage plus wing and nacelle contributions), not on the details of the individual components. The rule follows from the equivalence between the aircraft and an equivalent body of revolution.
- Area-rule shaping: where the wing adds area, the fuselage is pinched so the total area distribution stays smooth; the coke-bottle waist. The pinch at a station is S_fuselage = S_target - S_wing, computed with area_rule_fuselage_area. area_rule_deviation gives the RMS distance of an actual distribution from its ideal smooth target; a rougher equivalent body costs more wave drag.
- Sears-Haack body: the minimum-wave-drag body of revolution for a given length and volume (Haack 1941, Sears 1947). Radius r(x) = r_max * (4 * (x / L) * (1 - x / L))^(3/4), zero at both ends and r_max at the midpoint. This is the shape the total area distribution should approach at transonic speeds.
- Volume: V = (3 * pi^2 / 16) * r_max^2 * L. A 15 m body with a 0.54 m maximum radius holds about 8.1 m^3.
- Zero-lift wave drag area: D/q = (9 * pi / 2) * (A_max / L)^2 with A_max = pi * r_max^2 (drag-area form; identical to the volume form D/q = 128 * V^2 / (pi * L^4)). Multiply by dynamic pressure q for the wave drag force. The wave drag coefficient based on A_max is C_Dw = (9 * pi / 2) * (A_max / L^2), about 0.11 for a fineness ratio of 10 and 0.44 for a fineness ratio of 5.
- Drag divergence: wave drag stays negligible below the critical Mach number and rises steeply past the drag-divergence Mach number M_DD, which sits roughly 0.05 to 0.08 above M_cr for typical sections; drag_divergence_mach applies that margin. The rise above M_DD is modeled as parabolic, Delta C_Dw = k * (M - M_DD)^2, with k an empirical configuration-dependent constant (wave_drag_rise_coef).
- Mach number effects: at a fixed Mach number, wave drag scales with the dynamic pressure and with the square of the body slenderness ratio A_max / L; sweep and supercritical sections push M_DD up, and this leaf's divergence estimate feeds the high-speed design loop.
- Range: the Sears-Haack and area-rule results are slender-body linearized results, valid in the transonic and low-supersonic regime for smooth, slender configurations; a drag-divergence Mach at or above 1 is out of domain.
- Validation anchor: NACA Report 824 (public domain) supplies the section data family the pack references; the area rule itself is public-domain US government work (NACA Report 1273) and is used here as summary only per standards-map.yaml.
Workflow
- Collect the body length L, maximum radius r_max (or the volume V), and the station-by-station total area distribution.
- Compute the Sears-Haack radius and area distributions with sears_haack_radius and sears_haack_area, and the volume with sears_haack_volume.
- Evaluate the zero-lift wave drag: the drag area with sears_haack_wave_drag_area, the coefficient with sears_haack_wave_drag_coef, and the force with wave_drag_force at the cruise dynamic pressure.
- Apply the area rule: at each station where the wing contributes area, size the fuselage pinch with area_rule_fuselage_area so the total stays on the smooth target; check the whole distribution with area_rule_deviation.
- Estimate M_DD with drag_divergence_mach from the section critical Mach, then the wave drag rise at the cruise Mach with wave_drag_rise_coef.
- Report the Sears-Haack values next to the actual configuration so the wave drag penalty of the real area distribution is visible.
Pitfalls
- Reading Raymer's drag-area form as a coefficient: D/q has units of area and must be multiplied by q; the coefficient C_Dw divides by A_max.
- Squaring A_max / L^2 instead of A_max / L in the drag area: the drag area is (9 * pi / 2) * (A_max / L)^2.
- Area ruling the fuselage alone: the rule applies to the total area distribution, wing and nacelle contributions included.
- Pinching the fuselage past zero area at a station: the wing contribution must stay below the target total.
- Expecting zero wave drag below M_cr: the area rule reduces the drag rise; it does not remove wave drag entirely.
- Applying slender-body results to short, blunt bodies: the Sears-Haack and equivalent-body results are linearized slender-body theory.
- Confusing critical Mach with drag-divergence Mach: M_DD is higher by about 0.05 to 0.08, and the wave drag rise is driven by M_DD.
- Treating the parabolic rise constant k as a physical constant: it is empirical and configuration dependent.
- Using the divergence estimate past M = 1: the parabolic rise model is transonic; a supersonic result is out of domain.
Behavior contract (gate 3)
The wave drag and area rule logic is exercised by the gate 3 contract test: scripts/test_wave_drag_area_rule.py against scripts/wave_drag_area_rule_logic.py (stdlib unittest, offline). Run: python3 scripts/test_wave_drag_area_rule.py
Compliance
- The Sears-Haack body, the area rule, and the drag-divergence relations are standard transonic aerodynamics content (public-domain textbook and report material, e.g. Raymer, Aircraft Design; Anderson, Fundamentals of Aerodynamics; Whitcomb, NACA Report 1273). Paraphrase and computed values only, no verbatim excerpts.
- Standards reference: NACA TR 824 (section data family, reference-only) per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.