Aviation Calcs

Drag polar / L/D max

Drag Polar and L/D Max Calculator

Enter parasite drag CD0 and either an Oswald efficiency and aspect ratio or a direct induced-drag factor k. You get the quadratic polar, (L/D)max, and the lift coefficients that maximize propeller and jet range and endurance. Optionally enter a CL to read CD and L/D at that point.

Induced-drag factor from

k = 1 / (π e AR). A clean subsonic airplane often has e around 0.7–0.85.

Maximum L/D

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The quadratic polar

CD = CD0 + k CL²   with   k = 1 / (π e AR)

At (L/D)max, induced drag equals parasite drag:

CL* = √(CD0 / k)   CD* = 2 CD0   (L/D)max = 1 / (2 √(CD0 k))

For CD0 = 0.02, e = 0.8, and AR = 8, k is about 0.0497 and (L/D)max is about 15.85, at CL* ≈ 0.634.

Range and endurance lift coefficients

On this polar, the same CL* is the condition for propeller maximum range and for jet maximum endurance (maximum L/D). Propeller maximum endurance is at max CL3/2/CD, or CL = √(3 CD0/k) = √3 CL*. Jet maximum range is at max CL1/2/CD, or CL = CL* / √3. Feed L/D and speed into the Breguet range and endurance calculator for a distance or time.

Sources

Anderson, Introduction to Flight and Aircraft Performance and Design; Raymer, Aircraft Design: A Conceptual Approach; the induced-drag factor k = 1/(π e AR) as used in NACA / NASA and standard design texts; FAA PHAK (L/D, induced drag). For CL and CD from a measured force, see the lift and drag coefficient calculator. For wing AR, see wing loading and aspect ratio.

Frequently asked questions

What is a typical Oswald efficiency e?

For a clean, moderate-aspect-ratio subsonic airplane, 0.7 to 0.85 is a common planning range. A well-designed sailplane can be higher; a dirty or highly swept configuration can be lower. The page will not accept a non-physical e.

Is (L/D)max the jet best-range condition?

Not on the usual Breguet jet-range argument. Jet maximum range is at max CL1/2/CD, a slightly lower lift coefficient than (L/D)max. Jet maximum endurance, and propeller maximum range, are at (L/D)max.

Can I use this as a flight-test polar?

No. A real polar includes camber, trim drag, Reynolds-number and Mach effects, and configuration changes. This is the textbook parabola used in conceptual design and performance class.