Library / Aerospace / AER-22
Aerospace
Finite-wing lift-curve slope
Slope of lift coefficient against angle of attack, per radian, for a finite wing, from the two-dimensional slope a0 and the aspect ratio.
- Ref
- AER-22
- Rev
- v1
- Field
- Aerospace
- Result
- a [1/rad]
A finite wing generates less lift per radian than the 2D section because of the downwash from its trailing vortices. Lifting-line theory gives a = a₀/[1 + a₀/(πe AR)]. At high aspect ratio it approaches a₀, and at low aspect ratio it falls to a small fraction of it. With a₀ = 2π and e = 1, the slope is 2π/(1 + 2/AR).
Slope of lift coefficient against angle of attack, per radian, for a finite wing, from the two-dimensional slope a0 and the aspect ratio. e is a lift-slope or span-efficiency correction suited to the model, equal to 1 for the ideal elliptic case. Unswept, attached, subsonic lifting-line regime.
Calculator
- Lift-curve slope per degree0.08773 1/°
- Slope as a fraction of the 2D value, a/a₀0.8
- Lift coefficient at 5° angle of attack0.4386
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The wing slope is 5.0265 per radian (0.08773 per degree), 80 % of the 2D value.
- Each degree of angle of attack adds 0.0877 to the lift coefficient, so 5° gives CL ≈ 0.439 before stall.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| AR Aspect ratio | 2 to 30 | 3.142 | 5.89 | 2.749 | +10% | +1.85 % |
| a0 Two-dimensional lift slope | 4 to 7 1/rad | 3.451 | 5.475 | 2.024 | +10% | +7.84 % |
| e Span efficiency correction | 0.7 to 1 | 4.63 | 5.027 | 0.3968 | +10% | +1.85 % |
Sweep of Aspect ratio
| Aspect ratio | Lift-curve slope (1/rad) |
|---|---|
| 2 | 3.142 |
| 5.5 | 4.608 |
| 9 | 5.141 |
| 12.5 | 5.417 |
| 16 | 5.585 |
| 19.5 | 5.699 |
| 23 | 5.781 |
| 26.5 | 5.842 |
| 30 | 5.89 |
Sweep one variable
Variables and how each one contributes
Lift-curve slope
What the equation gives you. Everything else feeds it.
Two-dimensional lift slope
The two-dimensional lift-curve slope of the section, per radian (2π ideal). Raises the wing slope, but less than one-for-one since the downwash reduces it. Raising it by 10 % (to 6.912) moves the result by +7.84 %, from 5.027 to 5.421 1/rad.
Span efficiency correction
The span-efficiency or lift-slope correction factor (1 for the ideal elliptic case). A lower factor means more downwash and a lower slope. Raising it by 10 % (to 1.1) moves the result by +1.85 %, from 5.027 to 5.12 1/rad.
Aspect ratio
The wing aspect ratio. A higher aspect ratio raises the slope toward the 2D value. Raising it by 10 % (to 8.8) moves the result by +1.85 %, from 5.027 to 5.12 1/rad.
Worked example with the values above
- Downwash term: a₀/(πe AR) = 6.283 / (π × 1 × 8) = 0.25.
- Divisor: 1 + 0.25 = 1.25.
- a = 6.283 / 1.25 = 5.02655 1/rad.
Assumptions and limits
- A symmetric, straight, linearly tapered wing with a planform area measured as a projected area.
- For the lift slope: attached subsonic flow, small angles and a lifting-line model.
Common mistakes
- Using the root chord instead of the mean chord to find aspect ratio.
- Mixing up semi-span and full span.
- Using the mean geometric chord to normalise pitching moment instead of the mean aerodynamic chord.
Statistics and uncertainty
Planform geometry is exact, and the aerodynamic estimate is not: lifting-line theory predicts lift-curve slope to about 5 % for unswept wings of aspect ratio above 4, and it overpredicts for low aspect ratio, swept and highly tapered wings.
Where it is used
- Setting reference geometry for coefficients, and predicting how lift grows with angle of attack.
- Locating the aerodynamic centre and the centre of gravity limits.
Maths you need
- Taper ratio: λ = tip chord divided by root chord
- Aspect ratio: AR = b²/S
- Lift-curve slope: the rate of change of lift coefficient with angle of attack