Lift-to-drag ratio

The rates of change of lift and drag with angle of attack (AoA) are called respectively the lift and drag coefficients C_L and C_D. The varying ratio of lift to drag with AoA is often plotted in terms of these coefficients.

For any given value of lift, the AoA varies with speed. Graphs of C_L and C_D vs. speed are referred to as drag curves. Speed is shown increasing from left to right. The lift/drag ratio is given by the slope from the origin to some point on the curve and so the maximum L/D ratio does not occur at the point of least drag coefficient, the leftmost point. Instead, it occurs at a slightly greater speed. Designers will typically select a wing design which produces an L/D peak at the chosen cruising speed for a powered fixed-wing aircraft, thereby maximizing economy. Like all things in aeronautical engineering, the lift-to-drag ratio is not the only consideration for wing design. Performance at a high angle of attack and a gentle stall are also important.

Mathematically, the maximum lift-to-drag ratio can be estimated as:

${\displaystyle (L/D)_{\max }={\frac {1}{2}}{\sqrt {\frac {\pi \varepsilon AR}{C_{D,0}}}},}$[6]

where AR is the aspect ratio, ${\displaystyle \varepsilon }$ the span efficiency factor, a number less than but close to unity for long, straight edged wings, and ${\displaystyle C_{D,0}}$ the zero-lift drag coefficient.

Most importantly, the maximum lift-to-drag ratio is independent of the weight of the aircraft, the area of the wing, or the wing loading.

It can be shown that two main drivers of maximum lift-to-drag ratio for a fixed wing aircraft are wingspan and total wetted area. One method for estimating the zero-lift drag coefficient of an aircraft is the equivalent skin-friction method. For a well designed aircraft, zero-lift drag (or parasite drag) is mostly made up of skin friction drag plus a small percentage of pressure drag caused by flow separation. The method uses the equation:

${\displaystyle C_{D,0}=C_{\text{fe}}{\frac {S_{\text{wet}}}{S_{\text{ref}}}},}$[7]

where ${\displaystyle C_{\text{fe}}}$ is the equivalent skin friction coefficient, ${\displaystyle S_{\text{wet}}}$ is the wetted area and ${\displaystyle S_{\text{ref}}}$ is the wing reference area. The equivalent skin friction coefficient accounts for both separation drag and skin friction drag and is a fairly consistent value for aircraft types of the same class. Substituting this into the equation for maximum lift-to-drag ratio, along with the equation for aspect ratio (${\displaystyle b^{2}/S_{\text{ref}}}$), yields the equation:

${\displaystyle (L/D)_{\max }={\frac {1}{2}}{\sqrt {{\frac {\pi \varepsilon }{C_{\text{fe}}}}{\frac {b^{2}}{S_{\text{wet}}}}}}}$

where b is wingspan. The term ${\displaystyle b^{2}/S_{\text{wet}}}$ is known as the wetted aspect ratio. The equation demonstrates the importance of wetted aspect ratio in achieving an aerodynamically efficient design.

At very great speeds, lift to drag ratios tend to be lower. Concorde had a lift/drag ratio of about 7 at Mach 2, whereas a 747 is about 17 at about mach 0.85.

Dietrich Küchemann developed an empirical relationship for predicting L/D ratio for high Mach:[8]

${\displaystyle L/D_{\max }={\frac {4(M+3)}{M}}}$

where M is the Mach number. Windtunnel tests have shown this to be approximately accurate.