## The quest for e

Estimating e seems to not be so trivial and causes lots of thinking – it does not seem to be directly applicable by the book:

Daniel Raymer says in his book that e (Oswald’s efficiency factor) is normally between 0.7 and 0.85 (the e that is below 1.0 comes from the deviation from the perfect elliptic lift distribution). Jon Anderson Jr. says on his book Aircraft Performance & Design that on general aviation aircraft, the e is normally 0.6. And in one example aircraft design in the book Anderson then goes and uses e that is 0.9. It has quite large impact on the estimation results, so it would be better to estimate it right.

Then there are multiple equations for estimating e, in Raymer’s and Anderson’s books. All produce different results, and as a result, the K will be different. And the K has effect on L/Dmax. Interesting enough – the L/Dmax, if the e is estimated with any of the equations provided or assumed as 0.6 as Anderson recommends, the Diamond DA42 Twin Star should have L/D ratio around 12 instead of 18 it in reality has. It has been said that these estimation equations apply only to “normal” aspect ratios. It would be interesting to know what is “normal” aspect ratio – DA42 has AR=12 and the LH10 has AR=14. Maybe that is “higher than normal” then and maybe I have had the privilege to fly “not so normal airplane” when flying the Twin Star. Normal or abnormal, it is an excellent aircraft which is very much fun to fly.

So if I am estimating the L/Dmax of aircraft that has AR=14, and has tapered wing, it seems that quite high e value needs to be picked up. The estimation equation is a heavy generalization though, it does not take into account that on which CL the low drag laminar bucked is (it rather seems that the equations assume turbulent flow).

Just in case – it´s all unchanged – so far..

http://selair.selkirk.bc.ca/aerodynamics1/Performance/Page3.html

Just in case – it´s all unchanged – so far..

http://selair.selkirk.bc.ca/aerodynamics1/Performance/Page3.html

There are many different “e”s, and it is VERY confusing keeping track of all of them.

1. The “e” you are interested in for overall induced (due-to-lift) drag is the Oswald efficiency factor, simply called “e” or “e_viscous”

This accounts for both

(a) the inviscid “vortex” drag, which is purely the deviation of the lift distribution from elliptical. This is often called “u” or “e_inviscid”, and can be calculated using a lifting line (unswept wings) or lifting surface method. Note that e_inv sometimes includes an additional knockdown factor for the modification of the wing lift distribution by the presence of the fuselage. This factor is usually called “s”. ie. u*s=e_inv

(b) the viscous drag-due-to-lift on the 2-D airfoil shape (over and above the zero-lift drag on the airfoil)

Ilan Kroo (http://adg.stanford.edu/aa241/drag/induceddrag.html) gives the (viscous) Oswald efficiency factor as follows:

e = 1/((1/(u*s))+(pi*AR*K*CD_p))

Where u*s=e_inv and where K is a factor you can obtain from wind tunnel data (ie. a viscous test of the airfoil shape)

2. There is another, different e factor used in computing the 3-D lift slope. This factor is usually called “e_1”. I’ve never actually computed it (apparently, it’s really tricky to do), and most people just assume e_1 = e_oswald, but apparently it makes a small but significant difference in computation of lift lift slopes. I’ve been looking to this myself lately, and the best advice I’ve gotten is to use the lifting line or lifting surface method at a couple of different AOAs and then interpolate between them for the lift slope.

There are many different “e”s, and it is VERY confusing keeping track of all of them.

1. The “e” you are interested in for overall induced (due-to-lift) drag is the Oswald efficiency factor, simply called “e” or “e_viscous”

This accounts for both

(a) the inviscid “vortex” drag, which is purely the deviation of the lift distribution from elliptical. This is often called “u” or “e_inviscid”, and can be calculated using a lifting line (unswept wings) or lifting surface method. Note that e_inv sometimes includes an additional knockdown factor for the modification of the wing lift distribution by the presence of the fuselage. This factor is usually called “s”. ie. u*s=e_inv

(b) the viscous drag-due-to-lift on the 2-D airfoil shape (over and above the zero-lift drag on the airfoil)

Ilan Kroo (http://adg.stanford.edu/aa241/drag/induceddrag.html) gives the (viscous) Oswald efficiency factor as follows:

e = 1/((1/(u*s))+(pi*AR*K*CD_p))

Where u*s=e_inv and where K is a factor you can obtain from wind tunnel data (ie. a viscous test of the airfoil shape)

2. There is another, different e factor used in computing the 3-D lift slope. This factor is usually called “e_1”. I’ve never actually computed it (apparently, it’s really tricky to do), and most people just assume e_1 = e_oswald, but apparently it makes a small but significant difference in computation of lift lift slopes. I’ve been looking to this myself lately, and the best advice I’ve gotten is to use the lifting line or lifting surface method at a couple of different AOAs and then interpolate between them for the lift slope.

Thanks for your comments!

Thanks for your comments!