Prandtl and Helmbold
Prandtl lifting-line theory gives a = a0 / (1 + a0 / (π e AR)). Helmbold’s formula, often the better first estimate at low aspect ratio, is a = a0 / (√(1 + (a0 / (π AR))²) + a0 / (π AR)). With a0 = 2π and AR = 6, e = 1, the Prandtl slope is exactly 3π/2 ≈ 4.712 per radian (about 0.0822 per degree), and Helmbold sits a little below that. At very large AR both formulas recover a0; at very small AR, Helmbold is the more conservative reduction.
These are incompressible, inviscid corrections. They are not a stall model and they are not a transonic lift slope. For the section slope itself see thin-airfoil lift; for AR from a span and area, see wing loading and aspect ratio; for induced drag from e and AR, see the drag polar.
Sources
- Prandtl lifting-line theory, finite-wing lift-curve slope.
- Helmbold’s finite-wing lift-slope formula (low aspect ratio).
- Anderson, Fundamentals of Aerodynamics; Phillips, Mechanics of Flight.
Frequently asked questions
Which number should I use, Prandtl or Helmbold?
Prandtl with a realistic Oswald e is the usual lifting-line result at moderate and high AR. Helmbold is often preferred for low-AR wings (fighters, delta planforms). Compare both.
What if I only have a slope per degree?
Multiply by 180/π to get a0 per radian, or leave a0 blank and start from the thin-airfoil 2π.
Does this include compressibility?
No. It is an incompressible correction. At higher Mach numbers a Prandtl–Glauert factor, not implemented here, is applied first.