The overall aerodynamic force acting on a wing that is creating lift is not straight upward but is inclined backward. We usually consider the vertical and horizontal axes of the force separately. Lift is defined as the portion of the resultant force that is perpendicular to the relative wind, and induced drag is the portion parallel with it. The slope of the angle of the resultant force is the L/D ratio. This ratio changes with angle of attack as shown in the following diagram. As the AoA increases this vector leans further back creating a different L/D ratio (more drag for the same amount of lift).

enter image description here

Then looking from the wing's point of view, the portion of the RF perpendicular to the chord is the normal force (N) and the part parallel to it is the axial force (A). We can see that the the angle of the RF with respect to the wing chord is less affected by changes in AoA.

enter image description here

I've never seen a chart that graphs out the N/A relationship. Does this ratio change at all with AoA or is it a fixed property of the wing design? If it is not fixed, what dynamic factors (speed, AoA, etc.) affect the direction of the resultant force?

  • $\begingroup$ R = L + D, should be basic vector math. $\endgroup$
    – Ron Beyer
    Jun 5, 2016 at 1:46
  • $\begingroup$ @RonBeyer ?? R squared = L squared + D squared. I'm not asking about the trigonometry. I'm asking about the physics of induced drag. $\endgroup$
    – TomMcW
    Jun 5, 2016 at 4:06
  • $\begingroup$ @RonBeyer I re-worded it. Maybe it makes more sense what I'm asking. $\endgroup$
    – TomMcW
    Jun 5, 2016 at 5:09
  • $\begingroup$ @TomMcW your drawings are a bit misleading. In the first two diagrams, you set your R force perpendicular to the wing chord. In the last diagram, you set it at an angle. These diagrams do not represent an equivalent situation, is this intentional? $\endgroup$
    – Sanchises
    Jun 5, 2016 at 9:19
  • 1
    $\begingroup$ @TomMcW The change in resultant in N/A will be the same as that of the resultant vector in L/D case- they're the same. You're seeing difference because in case of the L/D case, the angle of attack gets added to the change. The only difference is that the co-ordinate system is changed. For L/D case, the angle of attack, alpha, gets added progressively to the resultant vector. $\endgroup$
    – aeroalias
    Jun 5, 2016 at 12:32

2 Answers 2


Yes, the resulting force is not perpendicular to the wing's chord but can point either way, slightly forward and backward. The very first polar charts which were published show this nicely, because those plots were true polar plots and used the wing chord as the reference orientation for 0°.

Illustration on page 196 of Otto Lilienthal's "Der Vogelflug als Grundlage der Fliegekunst"

Illustration on page 196 of Otto Lilienthal's "Der Vogelflug als Grundlage der Fliegekunst" showing measurements on model wings as polar plots (source). Note the left part: Here the forces are plotted in the wing section's coordinate system.

The N/A relationship was covered by the work of Edward C. Polhamus, who did a lot of research on this at NACA Langley and published several papers with equations for calculating the leading edge thrust, the force which helps to tilt the resultant force vector forward.

  • $\begingroup$ While I don't have the required background to develop, I feel the induced angle of attack, resulting from the downwash, which creates the induced drag (which in turn makes the difference between the lift component and the whole aerodynamic force) is a main point of the answer. ai(y0)=atan{-w(y0)/V∞} seems the equation that defines this difference, as published here. See also. $\endgroup$
    – mins
    Jun 5, 2016 at 14:33
  • $\begingroup$ @mins Sheesh, I'll have to delve into peter's and your links when I've had more sleep! When we say it's the downwash that creates the induced drag I always feel that the term is a little confusing. If you look at it in Newtonian terms the drag portion of the angle must be caused by "forward-wash." In other words the wing imparts momentum to the air not only downward but slightly forward. It's the same movement of air, like you said in your other comment, but the word downwash was a point of confusion for me when I was trying to understand induced drag. $\endgroup$
    – TomMcW
    Jun 5, 2016 at 17:10
  • $\begingroup$ @Peter I don't speak German so I used Google Translate on the labels. Did the best I could with the results it gave me. Let me know if something could be worded better. I couldn't figure out where "normal" was supposed to go in that one label. The sentence Google gave me didn't make sense. Oddly enough, that's the only word on the page that's the same in both languages but I couldn't figure out the grammar. $\endgroup$
    – TomMcW
    Jun 5, 2016 at 18:35
  • $\begingroup$ @TomMcW: Thanks a lot for the edit! Normal means perpendicular, this is for the 90° vector only. What probably also needs explaining is the "rotierend gemessen" part. Lilienthal used a special machine which swung small wings around on a horizontal arm as a poor man's wind tunnel, and that is where the results are from. $\endgroup$ Jun 5, 2016 at 19:14
  • $\begingroup$ @TomMcW: The downwash starts by the tip vortexes, the air going from the bottom side to the up side of the wing. So the downwash isn't anyway a simple move downwards. You'll find such elements in the links provided. $\endgroup$
    – mins
    Jun 5, 2016 at 21:27

The angle of attack changes the L/D ratio, yes. Image the angle increased to 90 degrees (coming straight from the bottom). I suppose you could call that 100% lift and 0% drag.

Not sure how useful that is... From another point of view, that's 100% drag and 0% lift. Pugachev's Cobra.

  • $\begingroup$ This answers the first question, "Does this ratio change at all with AoA or is it a fixed property of the wing design?" This doesn't anwer the second part of the question, "If it is not fixed, what dynamic factors (speed, AoA, etc.) affect the direction of the resultant force?". $\endgroup$
    – Steve
    Jun 5, 2016 at 9:52

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