Thin Airfoil Section Theory

Thin Airfoil Section Theory (aerodynamics/airfoil/thin-airfoil-section-theory)

Use when you must turn a thin airfoil's camber line into its analytic section aerodynamics: the Glauert sine-series coefficients of the camber slope, the zero-lift angle, the section lift curve and the quarter-chord pitching moment. This leaf computes the closed-form thin-airfoil solution in pure stdlib Python, with trapezoid quadrature used only for the Fourier integrals over the theta transform. It pairs with aerodynamics/airfoil/airfoil-geometry, which produces the mean camber line ordinates and slope consumed here (both leaves can be cross-checked on the same NACA 2412 camber line), and it supplies the analytic anchor that the numerical and viscous section layers beneath it verify against. Section coefficients are derived here from the camber line; the finite-wing slope leaf aerodynamics/drag-polars/lift-curve-slope instead consumes alpha_zero and the section slope as given inputs for its aspect-ratio and sweep corrections, so it is the complement, not the duplicate.

Domain quick reference

• Glauert theta transform: x = (1 - cos(theta))/2 maps the chord x in [0, 1] onto theta in [0, pi], clustering stations at the leading edge where the camber slope varies fastest. All integrals run over this theta grid.
• Camber slope input routes: NACA 4-digit mean line (m, p) with dz/dx = (2m/p^2)(p - x) for x <= p and dz/dx = (2m/(1-p)^2)(p - x) for x >= p (zero at x = p, matching the airfoil-geometry formulas); polynomial mean line z = sum(c_i x^i) with dz/dx = sum(i c_i x^(i-1)); or sampled (x, z) stations read as piecewise-linear camber with constant slope per segment.
• Glauert sine-series coefficients (gamma(theta) = 2U(A0 (1+cos)/sin + A1 sin(theta) + A2 sin(2 theta))): A0 = alpha - (1/pi) int dz/dx dtheta, A1 = (2/pi) int dz/dx cos(theta) dtheta, A2 = (2/pi) int dz/dx cos(2 theta) dtheta.
• Zero-lift angle: alpha_L0 = (1/pi) int dz/dx (1 - cos(theta)) dtheta, identically -A0(alpha = 0) - A1/2; the identity A0(alpha_L0) = -A1/2 holds exactly, so cl vanishes at the zero-lift angle.
• Section lift: cl = 2 pi A0 + pi A1 = 2 pi (alpha - alpha_L0).
• Quarter-chord pitching moment (aerodynamic center at c/4): cm_c4 = (pi/4)(A2 - A1), equivalently cm_le + cl/4 with cm_le = -(pi/2)(A0 + A1) + (pi/4) A2. The mistranscribed reading -(pi/4)(A1 + A2) is off by (pi/2) A2 and fails the identity; it is rejected (it coincides with the classical form only at A2 = 0).
• Center of pressure: x_cp/c = 1/4 - cm_c4/cl, defined only at nonzero lift.
• KNOWN-EXACT closed forms: the parabolic mean line z = 4m x (1 - x) (the NACA 4-digit mean line at p = 0.5) gives alpha_L0 = -2m, A1 = 4m, A2 = 0, cm_c4 = -pi m exactly; a flat mean line gives A0 = alpha and zero everything else. Incompressible potential flow only: no Mach effects, no viscosity, no stall.

Workflow

1. Choose the camber-line input route: NACA 4-digit mean-line parameters (m, p) with glauert_coefficients_naca4, polynomial power-series coefficients with glauert_coefficients_poly, or sampled (x, z) camber stations spanning [0, 1] with glauert_coefficients_points. All three take alpha_rad first and default to n_theta = 40000 quadrature intervals.
2. Run the Glauert sine-series decomposition: each route integrates the camber slope dz/dx by trapezoid quadrature on the uniform theta grid (endpoints included) and returns A0, A1, A2 and the zero-lift angle alpha_L0 as a 4-tuple.
3. Recover the section lift coefficient with lift_coefficient(alpha_rad, alpha_L0_rad) = 2 pi (alpha - alpha_L0); cross-check the identity cl = 2 pi A0 + pi A1 on the same coefficients.
4. Read the quarter-chord pitching-moment coefficient with quarter_chord_moment(A1, A2) = (pi/4)(A2 - A1); verify via the independent route cm_le + cl/4 when the full series is available.
5. Locate the center of pressure with center_of_pressure_over_chord(cl, cm_c4) = 1/4 - cm_c4/cl; the function raises ValueError at |cl| < 1e-12, where the cp is not defined.
6. Confirm the deterministic checks with the contract test scripts/test_thin_airfoil_section_theory.py.

Worked example

NACA 2412 mean line (m = 0.02, p = 0.4) at alpha = 4 deg = 0.069813170080 rad, the section whose camber line airfoil-geometry also handles. Real module outputs (trapezoid quadrature, n_theta = 40000):

• glauert_coefficients_naca4(0.069813170080, 0.02, 0.4): A0 = 0.065320283700, A1 = 0.081495141601, A2 = 0.013861276465, alpha_L0 = -0.036254684420 rad = -2.077240404870 deg, within a few hundredths of a degree of the measured 2412 zero-lift angle near -2 deg.
• Zero-lift identity: alpha_L0 = -A0(0) - A1/2 agrees with the direct integral to 1.3e-16; A0(alpha_L0) = -0.040747570801 equals -A1/2.
• Section lift: cl = 2 pi (0.069813170080 + 0.036254684420) = 0.666443984960, matching cl = 2 pi A0 + pi A1 to 7.8e-16; at alpha = 0 the camber alone lifts at cl = 0.227794900467.
• Quarter-chord moment: cm_c4 = (pi/4)(A2 - A1) = -0.053119513461, near the familiar measured 2412 cm0.25 of about -0.05; the route cm_le + cl/4 gives the same value to 1.9e-16, while the mistranscribed -(pi/4)(A1 + A2) = -0.074892755617 is off by (pi/2) A2 = +0.021773242156.
• Center of pressure: x_cp/c = 1/4 - cm_c4/cl = 0.329705893759: the cp sits at 0.330 chord at 4 deg and moves back toward the quarter chord as cl grows.
• Quadrature convergence: A1 at n_theta = 40000 agrees with n_theta = 1000000 to 3.0e-13, so the default resolution is converged at the kinked 2412 slope to all printed digits.

KNOWN-EXACT validation on the parabolic mean line z = 4m x (1 - x), m = 0.04 (polynomial route, coeffs [0.0, 0.16, -0.16]) at 4 deg: exact closed forms are alpha_L0 = -2m = -0.08 rad, A1 = 4m = 0.16, A2 = 0, cm_c4 = -pi m = -0.125663706144, cl(4 deg) = 0.941303909067, cl(0) = 4 pi m = 0.502654824574, x_cp/c(4 deg) = 0.383499611478. Quadrature recovers them to 1e-15, and the NACA 4-digit route at p = 0.5 reproduces the parabola exactly (A1 = 0.160000000000, alpha_L0 = -0.080000000000, cm_c4 = -0.125663706144). A flat mean line (coeffs [0.0, 0.0]) gives A0 = 0.069813170080 = alpha with A1 = A2 = alpha_L0 = cm_c4 = 0.

Sampled camber stations: a 4001-station sample of the parabola gives alpha_L0 = -0.079999658257 (3.4e-7 from -2m) and A1 = 0.159999318515 (6.8e-7 from 4m); a 4001-station sample of the 2412 mean line gives alpha_L0 = -0.036254564543 rad and cm_c4 = -0.053119329442, within 1.2e-7 and 1.8e-7 of the analytic route respectively.

Verification

• glauert_coefficients_naca4(0.069813170080, 0.02, 0.4) returns A0 = 0.065320283700, A1 = 0.081495141601, A2 = 0.013861276465, alpha_L0 = -0.036254684420 within 1e-9 of the spec anchors.
• lift_coefficient gives 0.666443984960 at 4 deg and 0.227794900467 at alpha = 0; quarter_chord_moment gives -0.053119513461; center_of_pressure_over_chord gives 0.329705893759, all within 1e-9.
• Closed forms hold: parabolic alpha_L0 = -2m, A1 = 4m, A2 = 0, cm_c4 = -pi m; flat mean line A0 = alpha with the rest zero; the identities alpha_L0 = -A0(0) - A1/2, A0(alpha_L0) = -A1/2, cl = 2 pi A0 + pi A1 = 2 pi (alpha - alpha_L0) and cm_c4 = cm_le + cl/4 all hold to 1e-12.
• Convergence: A1 at n_theta = 1000000 differs from n_theta = 40000 by 3.0e-13 on the 2412 mean line.
• ValueError rejection: m = -0.01 and m = 0.2 (4-digit camber range), p = 0.0 and p = 1.0, n_theta below 100 or non-int, empty or non-list polynomial coeffs, non-monotone sample stations, sample stations not spanning [0, 1], and center_of_pressure_over_chord at |cl| < 1e-12.
• Run the deterministic contract test offline: python3 scripts/test_thin_airfoil_section_theory.py (33 tests, under 2 s).

Related leaves

• aerodynamics/airfoil/airfoil-geometry: produces the mean camber line ordinates and slope consumed here; the NACA 2412 cross-check airfoil is shared by both leaves.
• aerodynamics/drag-polars/lift-curve-slope: the finite-wing slope leaf that consumes alpha_zero and the section slope as inputs; complementary scope, never a duplicate of the camber-line derivation.
• aerodynamics/airfoil/xfoil-analysis: numerical viscous section results that use the analytic section solution as their verification anchor.
• aerodynamics/cfd/panel-method: the numerical potential layer beneath the closed-form section solution.

Pitfalls

• Transcribing the quarter-chord moment with the wrong sign: the anchor rejects cm_c4 = -(pi/4)(A1 + A2), which is off by (pi/2) A2 and fails the identity cm_c4 = cm_le + cl/4. The classical relation is cm_c4 = (pi/4)(A2 - A1).
• Deriving section data that is input elsewhere: alpha_zero and the section slope are inputs to the lift-curve-slope leaf for its finite aspect-ratio and sweep work; this leaf derives them from the camber line instead. Do not reuse the lift-curve-slope routing tag for this leaf's camber-line claim.
• Reading the camber as lift: the flat mean line still lifts at cl = 2 pi alpha; only the zero-lift angle, not the slope 2 pi, is set by camber.
• Calling the center of pressure at zero lift: x_cp/c = 1/4 - cm_c4/cl diverges as cl approaches zero; the module raises ValueError at |cl| < 1e-12 rather than return a meaningless station.
• Feeding sample stations that are not strictly increasing or that do not span exactly [0, 1]: the piecewise-linear route rejects both, since a camber line must cover the full chord once.
• Under-resolving the quadrature: kinked 4-digit slopes converge slowly (A1 needs about 40000 intervals to reach 1e-12); the default n_theta = 40000 is converged, but n_theta below 100 is rejected.
• Extending beyond the model: no Mach correction, viscosity or stall lives in this leaf; it is the incompressible potential-flow section solution only.

Behavior contract (gate 3)

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

python3 scripts/test_thin_airfoil_section_theory.py

The 33 tests cover the NACA 2412 worked example at 4 deg (all four Glauert coefficients, the zero-lift angle in rad and deg, cl at 4 deg and at zero alpha, the quarter-chord moment, the center of pressure), the closed-form identities (zero-lift angle from -A0(0) - A1/2, A0(alpha_L0) = -A1/2, the two cl routes, cm_c4 = cm_le + cl/4, rejection of the A1 + A2 mistranscription), the KNOWN-EXACT parabolic mean line through the polynomial route and the NACA p = 0.5 crossover, the flat mean line, quadrature convergence at one million intervals, the 4001-station sampled camber routes, and ValueError rejection of every non-physical input listed in the spec. All numeric asserts are order-safe (assertAlmostEqual with delta or math.isclose); no exact float equality on computed sums.

Compliance

• NACA Report 824 (naca-tr-824) frames the thin-airfoil theory context as US government work in the public domain; the relations above are standard engineering methodology from Anderson and Katz and Plotkin, summary-only per standards-map.yaml, never reproduced verbatim.
• compliance: STANDARDS-REF, gated: false.