The Drag Polar

The drag polar (also called "Lilienthal polar") is a graphical plot of CL vs CD for a given aircraft. It shows the aerodynamic behaviour across the entire angle-of-attack range.

Layout

code

CL
|       *  ←  CL_max (stall)
|     *
|    *
|   *
|  *
| *      ←  tangent from origin = (L/D)_max
|*
|
+-------------- CD
0      CD_min   CD_max

Y-axis: CL.
X-axis: CD.
Curve: (CD, CL) pairs at various α.

Key points on the polar

1. CD_min (min drag)

Point of minimum CD. Corresponds to a CL near zero (or 0 for symmetric airfoil).

V_max (max speed) is achieved here (max v at min drag at given power).

2. (L/D)_max — tangent from origin

The point at which CL/CD is maximum. On the polar: tangent from origin to the curve.

V_(L/D)_max = Vbg (best glide speed).
Best glide ratio = L/D there.

3. CL_max — top of polar

The highest attainable CL. Above α_stall, CL drops → polar curves back.

Vs (stall speed) at this CL.

4. Sink rate minimum

Point at which CL³/CD² is maximum (for power-off descent). Typically between (L/D)_max and CL_max.

V_min_sink = best endurance at power off.

Worked example — Cessna 172 (typical values)

Point	CL	CD	CL/CD	Speed
Cruise	0.3	0.04	7.5	about 105 KTAS
(L/D)_max	0.8	0.09	8.9	about 65 KIAS
Full flaps, CL_max	1.8	0.30	6.0	Vs0 ≈ 51 KIAS

(Values are approximations; POH is binding.)

Different polars

Flap configuration

Flaps up (cruise): narrow polar, high (L/D)_max, low CL_max.
Flaps 10° (T/O): similar, slightly shifted.
Full flaps (landing): wider polar, higher CL_max, lower (L/D)_max.

Landing gear

Gear up: less drag, higher (L/D)_max.
Gear down: more drag.

Icing

Icing: polar distorted — CL_max drops, CD rises — overall performance degraded.

Operational application

Glide: Vbg = V at (L/D)_max → max range from engine failure.
Max range cruise (jet): at (L/D)_max for max range (jet: constant specific fuel consumption).
Max range cruise (prop): at slightly higher speed (about V_(L/D)_max × 1.15).
Max endurance: at V_min_power (propeller) or V_(L/D)_max (jet).

Lilienthal — historical

Otto Lilienthal (1848–1896, Germany) first systematised the measurement of polars through his glider experiments. His polars formed the basis for the Wright Brothers' designs.