Pitchfork bifurcation

2013-10-08T21:47:23Z

130.88.166.46: /* References */

In [[aerodynamics]], the '''zero-lift drag coefficient''' <math>C_{D,0}</math> is a dimensionless parameter which relates an aircraft's zero-lift [[drag (physics)|drag]] [[force]] to its size, speed, and flying altitude.

Mathematically, zero-lift [[drag coefficient]] is defined as <math>C_{D,0} = C_D - C_{D,i}</math>, where <math>C_D</math> is the total drag coefficient for a given power, speed, and altitude, and <math>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>C_{D,0}</math> value of 0.0161 for the streamlined [[P-51 Mustang]] of [[World War II]]<ref name="Loftin">{{cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468|url=http://www.hq.nasa.gov/pao/History/SP-468/cover.htm|accessdate=2006-04-22}}</ref> which compares very favorably even with the best modern aircraft.

The zero-lift drag coefficient can be more easily conceptualized as the '''drag area''' ('''<math>f</math>''') which is simply the product of zero-lift drag coefficient and aircraft's wing area (<math>C_{D,0} \times S</math> where <math>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 {{convert|8.73|sqft|m2|abbr=on}}, compared to {{convert|3.80|sqft|m2|abbr=on}} for the P-51. Both aircraft have a similar wing area, again reflecting the Mustang's superior aerodynamics in spite of much larger size.<ref name="Loftin" /> 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>V_{max}\ \propto\ \sqrt[3]{power/f}</math>.<ref name="Loftin"/>

==Estimating zero-lift drag<ref name="Loftin"/>==
As noted earlier, <math>C_{D,0} = C_D - C_{D,i}</math>.

The total drag coefficient can be estimated as:

:<math>C_D = \frac{550 \eta P}{\frac{1}{2} \rho_0 [\sigma S (1.47V)^3]}</math>,

where <math>\eta</math> is the [[propulsive efficiency]], P is engine power in [[horsepower]], <math>\rho_0</math> sea-level air density in [[slug (mass)|slugs]]/cubic foot, <math>\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>\rho_0</math>, the equation is simplified to:

:<math>C_D = 1.456 \times 10^5 (\frac{\eta P}{\sigma S V^3})</math>.

The induced drag coefficient can be estimated as:

:<math>C_{D,i} = \frac{C_L^2}{\pi A \epsilon}</math>,

where <math>C_L</math> is the [[lift coefficient]], ''A'' is the [[aspect ratio (wing)|aspect ratio]], and <math>\epsilon</math> is the aircraft's efficiency factor.

Substituting for <math>C_L</math> gives:

:<math>C_{D,i}=\frac{4.822 \times 10^4}{A \epsilon \sigma^2 V^4} (W/S)^2</math>,

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

==References==

{{reflist}}

[[Category:Aerodynamics]]

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Pitchfork bifurcation