Strominger's equations: Difference between revisions

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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 (wing)|aspect ratio]] and an elliptical lift distribution.<ref name=DPR12.6>Raymer, Daniel P., ''Aircraft Design: A Conceptual Approach'', Section 12.6  (Fourth edition)</ref>


==Definition==
The Oswald efficiency is defined for the cases where the overall [[Drag coefficient|coefficient of drag]] of the wing or airplane has a constant+quadratic dependence on the [[Lift coefficient#Aircraft lift coefficient|aircraft lift coefficient]]


:<math>C_D = C_{D_0} + \frac{(C_L)^2}{\pi e_0 AR}</math>
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where
:{| border="0"
|-
| <math>C_D\;</math>           || is the overall [[drag coefficient]],
|-
| <math>C_{D_0}\;</math>      || is the [[zero-lift drag coefficient]],
|-
| <math>C_L\;</math>          || is the [[Lift coefficient#Aircraft lift coefficient|aircraft lift coefficient]],
|-
| <math>\pi\;</math>           || is the [[circumference-to-diameter ratio]] of a circle,
|-
| <math>e_0\;</math>            || is the Oswald efficiency number
|-
| <math>AR</math>      || is the [[Aspect ratio (wing)|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.<ref name=DPR12.6/>
 
==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 bothBut this assumes that the profile drag coefficient is independent of <math>C_L</math>, which is certainly not true in general. Assuming that the profile drag itself has a constant+quadratic dependence on <math>C_L</math>,
an alternative drag coefficient breakdown can be given by{{citation needed|date=May 2010}}
 
:<math>C_D = c_{d_0} + c_{d_2} (C_L)^2  + \frac{(C_L)^2}{\pi e AR}</math>
 
where
:{| border="0"
|-
| <math>c_{d_0}\;</math>       || is the constant part of the profile drag coefficient,
|-
| <math>c_{d_2}\;</math>       || is the quadratic part of the profile drag coefficient,
|-
| <math>e\;</math>        || is the wing span efficiency factor from inviscid theory, such as [[Lifting-line theory]]
|}
 
Equating the two <math>C_D</math> expressions gives the relation between the Oswald efficiency number ''e<sub>0</sub>'' and the lifting-line span efficiency ''e''.
 
:<math> C_{D_0} = c_{d_0}</math>
:<math>\frac{1}{e_0} = \frac{1}{e} + \pi AR c_{d_2}</math>
 
For the typical situation <math>c_{d_2}>0 </math>, we have <math>e_0 < e</math>.
 
== See also ==
* [[Lift-induced drag]]
* [[Lifting-line theory]]
 
==Notes==
{{Reflist}}
 
==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
 
{{DEFAULTSORT:Oswald Efficiency Number}}
[[Category:Aerospace engineering]]
[[Category:Aircraft wing design]]
 
 
{{engineering-stub}}

Latest revision as of 03:46, 8 December 2014


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