Physics:Zero-lift drag coefficient
In aerodynamics, the zero-lift drag coefficient [math]\displaystyle{ C_{D,0} }[/math] 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 [math]\displaystyle{ C_{D,0} = C_D - C_{D,i} }[/math], where [math]\displaystyle{ C_D }[/math] is the total drag coefficient for a given power, speed, and altitude, and [math]\displaystyle{ C_{D,i} }[/math] 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 [math]\displaystyle{ C_{D,0} }[/math] 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 ([math]\displaystyle{ f }[/math]) which is simply the product of zero-lift drag coefficient and aircraft's wing area ([math]\displaystyle{ C_{D,0} \times S }[/math] where [math]\displaystyle{ S }[/math] 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:
- [math]\displaystyle{ V_{max}\ \propto\ \sqrt[3]{power/f} }[/math].[1]
Estimating zero-lift drag[1]
As noted earlier, [math]\displaystyle{ C_{D,0} = C_D - C_{D,i} }[/math].
The total drag coefficient can be estimated as:
- [math]\displaystyle{ C_D = \frac{550 \eta P}{\frac{1}{2} \rho_0 [\sigma S (1.47V)^3]} }[/math],
where [math]\displaystyle{ \eta }[/math] is the propulsive efficiency, P is engine power in horsepower, [math]\displaystyle{ \rho_0 }[/math] sea-level air density in slugs/cubic foot, [math]\displaystyle{ \sigma }[/math] 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 [math]\displaystyle{ \rho_0 }[/math], the equation is simplified to:
- [math]\displaystyle{ C_D = 1.456 \times 10^5 (\frac{\eta P}{\sigma S V^3}) }[/math].
The induced drag coefficient can be estimated as:
- [math]\displaystyle{ C_{D,i} = \frac{C_L^2}{\pi A\!\!\text{R} \epsilon} }[/math],
where [math]\displaystyle{ C_L }[/math] is the lift coefficient, AR is the aspect ratio, and [math]\displaystyle{ \epsilon }[/math] is the aircraft's efficiency factor.
Substituting for [math]\displaystyle{ C_L }[/math] gives:
- [math]\displaystyle{ C_{D,i}=\frac{4.822 \times 10^4}{A\!\!\text{R} \epsilon \sigma^2 V^4} (W/S)^2 }[/math],
where W/S is the wing loading in lb/ft2.
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