Zero-lift drag coefficient

In aerodynamics, the zero-lift drag coefficient ${\displaystyle C_{D,0}}$ is a dimensionless parameter which relates an aircraft's zero-lift drag force to its size, speed, and flying altitude.

Mathematically, zero-lift drag coefficient is defined as ${\displaystyle C_{D,0}=C_{D}-C_{D,i}}$, where ${\displaystyle C_{D}}$ is the total drag coefficient for a given power, speed, and altitude, and ${\displaystyle C_{D,i}}$ is the lift-induced drag coefficient at the same conditions. Thus, zero-lift drag coefficient is reflective of parasitic drag which makes it very useful in understanding how "clean" or streamlined an aircraft's aerodynamics are. For example, a Sopwith Camel biplane of World War I which had many wires and bracing struts as well as fixed landing gear, had a zero-lift drag coefficient of approximately 0.0378. Compare a ${\displaystyle C_{D,0}}$ value of 0.0161 for the streamlined P-51 Mustang of World War II^[1] which compares very favorably even with the best modern aircraft.

The drag at zero-lift can be more easily conceptualized as the drag area (${\displaystyle f}$) which is simply the product of zero-lift drag coefficient and aircraft's wing area (${\displaystyle C_{D,0}\times S}$ where ${\displaystyle S}$ is the wing area). Parasitic drag experienced by an aircraft with a given drag area is approximately equal to the drag of a flat square disk with the same area which is held perpendicular to the direction of flight. The Sopwith Camel has a drag area of 8.73 sq ft (0.811 m²), compared to 3.80 sq ft (0.353 m²) for the P-51 Mustang. Both aircraft have a similar wing area, again reflecting the Mustang's superior aerodynamics in spite of much larger size.^[1] In another comparison with the Camel, a very large but streamlined aircraft such as the Lockheed Constellation has a considerably smaller zero-lift drag coefficient (0.0211 vs. 0.0378) in spite of having a much larger drag area (34.82 ft² vs. 8.73 ft²).

Furthermore, an aircraft's maximum speed is proportional to the cube root of the ratio of power to drag area, that is:

${\displaystyle V_{max}\ \propto \ {\sqrt[{3}]{power/f}}}$.^[1]

Estimating zero-lift drag^[1]

As noted earlier, ${\displaystyle C_{D,0}=C_{D}-C_{D,i}}$ .

The total drag coefficient can be estimated as:

${\displaystyle C_{D}={\frac {550\eta P}{{\frac {1}{2}}\rho _{0}[\sigma S(1.47V)^{3}]}}}$ ,

where ${\displaystyle \eta }$  is the propulsive efficiency, P is engine power in horsepower, ${\displaystyle \rho _{0}}$  sea-level air density in slugs/cubic foot, ${\displaystyle \sigma }$  is the atmospheric density ratio for an altitude other than sea level, S is the aircraft's wing area in square feet, and V is the aircraft's speed in miles per hour. Substituting 0.002378 for ${\displaystyle \rho _{0}}$ , the equation is simplified to:

${\displaystyle C_{D}=1.456\times 10^{5}({\frac {\eta P}{\sigma SV^{3}}})}$ .

The induced drag coefficient can be estimated as:

${\displaystyle C_{D,i}={\frac {C_{L}^{2}}{\pi A\!\!{\text{R}}\epsilon }}}$ ,

where ${\displaystyle C_{L}}$  is the lift coefficient, AR is the aspect ratio, and ${\displaystyle \epsilon }$  is the aircraft's efficiency factor.

Substituting for ${\displaystyle C_{L}}$  gives:

${\displaystyle C_{D,i}={\frac {4.822\times 10^{4}}{A\!\!{\text{R}}\epsilon \sigma ^{2}V^{4}}}(W/S)^{2}}$ ,

where W/S is the wing loading in lb/ft².

References

1. Loftin, LK Jr. "Quest for performance: The evolution of modern aircraft. NASA SP-468". Retrieved 2006-04-22.