Oswald efficiency number

The Oswald efficiency, similar to the span efficiency, is a correction factor that represents the change in drag with lift of a three-dimensional wing or airplane, as compared with an ideal wing having the same aspect ratio and an elliptical lift distribution.[1]

Definition

The Oswald efficiency is defined for the cases where the overall coefficient of drag of the wing or airplane has a constant+quadratic dependence on the aircraft lift coefficient

${\displaystyle C_{D}=C_{D_{0}}+{\frac {(C_{L})^{2}}{\pi e_{0}AR}}}$

where

${\displaystyle C_{D}\;}$	is the overall drag coefficient,
${\displaystyle C_{D_{0}}\;}$	is the zero-lift drag coefficient,
${\displaystyle C_{L}\;}$	is the aircraft lift coefficient,
${\displaystyle \pi \;}$	is the circumference-to-diameter ratio of a circle,
${\displaystyle e_{0}\;}$	is the Oswald efficiency number
${\displaystyle AR}$	is the aspect ratio

For conventional fixed-wing aircraft with moderate aspect ratio and sweep, Oswald efficiency number with wing flaps retracted is typically between 0.7 and 0.85. At supersonic speeds, Oswald efficiency number decreases substantially. For example, at Mach 1.2 Oswald efficiency number is likely to be between 0.3 and 0.5.[1]

Comparison with span efficiency factor

It is frequently assumed that Oswald efficiency number is the same as the span efficiency factor which appears in Lifting-line theory, and in fact the same symbol e is typically used for both. But this assumes that the profile drag coefficient is independent of ${\displaystyle C_{L}}$, which is certainly not true in general. Assuming that the profile drag itself has a constant+quadratic dependence on ${\displaystyle C_{L}}$, an alternative drag coefficient breakdown can be given by

${\displaystyle C_{D}=c_{d_{0}}+c_{d_{2}}(C_{L})^{2}+{\frac {(C_{L})^{2}}{\pi eAR}}}$

where

${\displaystyle c_{d_{0}}\;}$	is the constant part of the profile drag coefficient,
${\displaystyle c_{d_{2}}\;}$	is the quadratic part of the profile drag coefficient,
${\displaystyle e\;}$	is the span efficiency factor from inviscid theory, such as Lifting-line theory

Equating the two ${\displaystyle C_{D}}$ expressions gives the relation between the Oswald efficiency number e₀ and the lifting-line span efficiency e.

${\displaystyle C_{D_{0}}=c_{d_{0}}}$

${\displaystyle {\frac {1}{e_{0}}}={\frac {1}{e}}+\pi ARc_{d_{2}}}$

For the typical situation ${\displaystyle c_{d_{2}}>0}$, we have ${\displaystyle e_{0}<e}$.

See also

• Lift-induced drag
• Lifting-line theory

Notes

1. Raymer, Daniel P., Aircraft Design: A Conceptual Approach, Section 12.6 (Fourth edition)

References

• Raymer, Daniel P. (2006). Aircraft Design: A Conceptual Approach, Fourth edition. AIAA Education Series. ISBN 1-56347-829-3
• Anderson, John D. (2008). Introduction to Flight, Sixth edition. McGrawHill. ISBN 0-07-126318-7
• PhD. William Bailey Oswald, http://calteches.library.caltech.edu/3961/1/Obituaries.pdf