Supercritical Airfoil

Supercritical Airfoil Design and Analysis (aerodynamics/high-speed/supercritical-airfoil)

Use when the task is supercritical airfoil design and analysis: the flat upper surface, aft loading, drag-divergence Mach, wave-drag reduction, terminating shock strength, and the cruise lift coefficient of the section.

Domain quick reference

• Supercritical airfoil: a section whose nearly flat upper surface keeps the local Mach number barely supersonic over a long region. The weak supersonic pocket replaces the strong suction peak of a conventional section, so the terminating shock at its rear is weaker and wave drag at high subsonic Mach is reduced.
• Terminating shock strength: the static pressure ratio p2/p1 across the normal shock that closes the supersonic pocket, p2/p1 = 1 + 2gamma/(gamma+1)(M^2 - 1), evaluated at the local Mach just ahead of the shock (gamma = 1.4). A conventional section at M 0.8 accelerates the upper surface to about M 1.3, ratio about 1.81; the flat top of a supercritical section holds it near M 1.15, ratio about 1.38. Weaker shock means less wave drag and less shock/boundary-layer separation risk.
• Wave-drag penalty: above drag divergence, wave drag grows roughly with the cube of (M - M_DD). The penalty index (M - M_DD)^3 is zero at or below M_DD and rises steeply above it.
• Drag-divergence Mach (Korn rule of thumb): M_DD = 0.95 - t/c - C_L/10 for a supercritical section and M_DD = 0.90 - t/c - C_L/10 for a conventional section, with t/c the thickness ratio and C_L the cruise lift coefficient. The 0.05 offset is the wave-drag-reduction benefit. Example: t/c 0.12 at C_L 0.5 gives M_DD 0.78 supercritical versus 0.73 conventional.
• Thickness and lift at fixed Mach (inverse Korn): at M 0.8 and C_L 0.5 the supercritical section still carries t/c 0.10 while the conventional section is limited to about 0.05; equivalently, at M 0.8 with t/c 0.10 the supercritical section carries C_L 0.5 while the conventional section has no cruise lift left. This is the classic "same Mach, roughly twice the thickness" trade.
• Aft loading: camber and loading concentrated near the trailing edge recover the lift lost by the flat upper surface, at the price of a more negative pitching moment about the aerodynamic center: about -0.12 for a typical supercritical section versus -0.06 for a conventional transport section. The trim drag this creates is part of the trade against the wave-drag saving.
• Range: the Korn rule is a high-subsonic design rule of thumb, valid for t/c in (0.02, 0.30), C_L in [0, 1.5), and flight Mach below 1; supersonic cruise is out of domain.
• Validation anchor: NACA Report 824 (public domain) supplies the classic section data that the design trade is compared against.

Workflow

1. Fix the design point: cruise Mach M, target cruise lift coefficient C_L, and thickness ratio t/c.
2. Compute the drag-divergence Mach with drag_divergence_mach for both section types; keep the cruise Mach below M_DD.
3. Size the section: max_thickness_ratio at fixed Mach and C_L, or max_cruise_lift_coefficient at fixed Mach and t/c, for the chosen type.
4. When M exceeds M_DD, estimate the wave-drag penalty with wave_drag_penalty.
5. Assess the terminating shock strength with terminating_shock_strength at the local Mach ahead of the shock on the upper surface.
6. Note the aft-loading pitching moment with aft_loading_moment and budget the trim drag in the tail sizing.

Pitfalls

• Reading a supercritical claim at the free-stream Mach as if the local upper-surface Mach were the same; the flat top keeps the local Mach barely supersonic, which is the whole mechanism.
• Using the conventional base (0.90) for a supercritical section or the reverse; the 0.05 offset is the benefit being quantified.
• Treating the Korn rule as exact; it is a design rule of thumb calibrated against wind-tunnel data, not a theory.
• Ignoring the aft-loading moment: the more negative C_m,ac needs a tail download and costs trim drag that eats part of the wave-drag saving.
• Expecting the flat upper surface to keep the lift unchanged; the flat top loses front loading and aft loading is what recovers the cruise lift coefficient.
• Applying the rules at or above M 1; the supercritical benefit is a high-subsonic effect and supersonic cruise is out of domain.
• Assuming the terminating shock disappears; it is weakened, not eliminated, and the shock/boundary-layer interaction at its foot still needs care.
• Confusing drag-divergence Mach with the first-sonic-point Mach; M_DD sits well above it, and first-sonic-point estimation belongs to the transonic-similarity leaf.

Behavior contract (gate 3)

The design logic is exercised by the gate 3 contract test: scripts/test_supercritical_airfoil.py against scripts/supercritical_airfoil_logic.py (stdlib unittest, offline). Run: python3 scripts/test_supercritical_airfoil.py

Compliance

• The Korn rule and the flat-upper-surface/aft-loading mechanism are standard public-domain textbook content (Anderson, Fundamentals of Aerodynamics; Mason, Configuration Aerodynamics; Whitcomb's NASA work on supercritical airfoils is US Government public-domain); paraphrase and computed values only, no verbatim excerpts of any standard.
• Standards reference: NACA TR 824 (classic airfoil section data, reference-only) per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.