Lift Curve Slope Estimation (aerodynamics/drag-polars/lift-curve-slope)
Use when the task is wing lift curve slope estimation from section
data: thin-airfoil theory for the section slope, lifting-line theory
for the finite wing, simple sweep theory, the Prandtl-Glauert Mach
correction, and lift coefficient from angle of attack.
Domain quick reference
- Section slope: thin-airfoil theory gives a0 = 2 * pi per radian for
a flat plate or thin uncambered section; use a measured or computed
value when one is available. Camber shifts the zero-lift angle, it
does not change the thin-airfoil slope.
- Finite wing (lifting-line theory, span efficiency e):
a = a0 / (1 + a0 / (pi * e * AR)). With a0 = 2 * pi and e = 1
(elliptic loading) this reduces to a = a0 * AR / (AR + 2).
- Sweep (simple sweep theory): a_swept = a * cos(sweep), sweep in
degrees, documented range 0 <= sweep < 90 where cos(sweep) > 0.
- Mach (Prandtl-Glauert): a_mach = a / sqrt(1 - M^2), documented
range 0 <= M < 0.7. The correction is a subsonic small-disturbance
result and becomes invalid as transonic effects appear above 0.7.
- Combined order: section slope, then finite wing, then sweep, then
Mach. Each step consumes the slope produced by the previous step.
- Lift coefficient: C_L = a * (alpha - alpha_zero), alpha and
alpha_zero in degrees, a per radian. The optional stall guard flags
angles whose linear prediction exceeds the stall limit.
- Validation anchor: NACA Report 824 (public domain) supplies section
lift data; the thin-airfoil slope 2 * pi is the theoretical limit
that measured thin sections approach.
Workflow
- Choose the section slope with airfoil_slope(a0), or the default
thin-airfoil value 2 * pi.
- Correct for the finite wing with finite_wing_slope(a0, ar, e).
- Apply the sweep correction with sweep_correction(a, sweep_deg).
- Apply the Mach correction with mach_correction(a, mach).
- Or chain all three corrections in one call:
wing_lift_curve_slope(ar, a0, sweep_deg, mach, e).
- Predict the lift coefficient at the operating angle with
lift_coefficient(a, alpha_deg, alpha_zero_deg, stall_cl).
Pitfalls
- Using the section slope as the wing slope: a finite aspect ratio
always reduces the slope below 2 * pi.
- Accepting sweep at or beyond 90 degrees where cos(sweep) <= 0.
- Accepting Mach at or above 0.7 where Prandtl-Glauert breaks down.
- Mixing degrees and radians: a is per radian, alpha is in degrees.
- Ignoring alpha_zero for cambered sections: C_L = 0 at the
zero-lift angle, not at alpha = 0.
- Reordering the correction chain when the intermediate values are
reported for design use.
Behavior contract (gate 3)
The slope and lift coefficient logic is exercised by the gate 3
contract test: scripts/test_lift_curve.py against
scripts/lift_curve_logic.py (stdlib unittest, offline). Run:
python3 scripts/test_lift_curve.py
Compliance
- NACA Report 824 is US government work (public domain); summary and
physics values only, per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.
1---2name: lift-curve-slope3description: Use when you must estimate the lift curve slope of a wing from section data: compute the thin-airfoil section slope a0 = 2*pi per radian, correct it for finite aspect ratio with the lifting-line formula a = a0 / (1 + a0 / (pi * e * AR)), apply the simple sweep theory cosine correction, apply the Prandtl-Glauert Mach correction a / sqrt(1 - M^2) with a documented M < 0.7 limit, and predict lift coefficient from angle of attack with C_L = a * (alpha - alpha_zero), including an optional stall guard. Produces the corrected wing slope and lift coefficient for a given angle of attack that feed wing sizing and performance estimates. Trigger: lift curve slope, thin-airfoil theory, finite wing correction, aspect ratio, sweep correction, Prandtl-Glauert, lift coefficient.4license: Apache-2.05---67# Lift Curve Slope Estimation (aerodynamics/drag-polars/lift-curve-slope)89Use when the task is wing lift curve slope estimation from section10data: thin-airfoil theory for the section slope, lifting-line theory11for the finite wing, simple sweep theory, the Prandtl-Glauert Mach12correction, and lift coefficient from angle of attack.1314## Domain quick reference1516- Section slope: thin-airfoil theory gives a0 = 2 * pi per radian for17 a flat plate or thin uncambered section; use a measured or computed18 value when one is available. Camber shifts the zero-lift angle, it19 does not change the thin-airfoil slope.20- Finite wing (lifting-line theory, span efficiency e):21 a = a0 / (1 + a0 / (pi * e * AR)). With a0 = 2 * pi and e = 122 (elliptic loading) this reduces to a = a0 * AR / (AR + 2).23- Sweep (simple sweep theory): a_swept = a * cos(sweep), sweep in24 degrees, documented range 0 <= sweep < 90 where cos(sweep) > 0.25- Mach (Prandtl-Glauert): a_mach = a / sqrt(1 - M^2), documented26 range 0 <= M < 0.7. The correction is a subsonic small-disturbance27 result and becomes invalid as transonic effects appear above 0.7.28- Combined order: section slope, then finite wing, then sweep, then29 Mach. Each step consumes the slope produced by the previous step.30- Lift coefficient: C_L = a * (alpha - alpha_zero), alpha and31 alpha_zero in degrees, a per radian. The optional stall guard flags32 angles whose linear prediction exceeds the stall limit.33- Validation anchor: NACA Report 824 (public domain) supplies section34 lift data; the thin-airfoil slope 2 * pi is the theoretical limit35 that measured thin sections approach.3637## Workflow38391. Choose the section slope with airfoil_slope(a0), or the default40 thin-airfoil value 2 * pi.412. Correct for the finite wing with finite_wing_slope(a0, ar, e).423. Apply the sweep correction with sweep_correction(a, sweep_deg).434. Apply the Mach correction with mach_correction(a, mach).445. Or chain all three corrections in one call:45 wing_lift_curve_slope(ar, a0, sweep_deg, mach, e).466. Predict the lift coefficient at the operating angle with47 lift_coefficient(a, alpha_deg, alpha_zero_deg, stall_cl).4849## Pitfalls5051- Using the section slope as the wing slope: a finite aspect ratio52 always reduces the slope below 2 * pi.53- Accepting sweep at or beyond 90 degrees where cos(sweep) <= 0.54- Accepting Mach at or above 0.7 where Prandtl-Glauert breaks down.55- Mixing degrees and radians: a is per radian, alpha is in degrees.56- Ignoring alpha_zero for cambered sections: C_L = 0 at the57 zero-lift angle, not at alpha = 0.58- Reordering the correction chain when the intermediate values are59 reported for design use.6061## Behavior contract (gate 3)6263The slope and lift coefficient logic is exercised by the gate 364contract test: scripts/test_lift_curve.py against65scripts/lift_curve_logic.py (stdlib unittest, offline). Run:66python3 scripts/test_lift_curve.py6768## Compliance6970- NACA Report 824 is US government work (public domain); summary and71 physics values only, per standards-map.yaml.72- compliance: STANDARDS-REF, gated: false.
ashfordeOU