Talk:Lift-induced drag

Untitled
According to what is written it could be understood that there is two sources of induced drag: Due to lift generation( could be understood that it appears in 2-Dimentional flow) Due to tip effect (could be understood as 3-Dimentional since tip vortices are 3-d) the question is:
 * what is the equation of induced drag in 2-D flow?
 * what is the equation of induced drag in 3-D flow?

IF ANY ONE CAN HELP ME please mail me at mohamadcharif@yahoo.com —Preceding unsigned comment added by 193.227.168.132 (talk) 08:28, 14 April 2004

The 2-D flow means the same as a wing od infinite span to chord ratio - and in this case the lift induced drag is zero.

In real 3-D case induced drag is the function Ci=(Cy)^2/Pi*A and is caused by the tip vortexes (mainly) and other trailing vortexes.

andrzejmat 16:13, 7 April 2006 (UTC)

Simplification
I'm thinking of adding this, early in the article: "Lift induced drag is caused by the lift component of the wing being rotated backwards relative to the aircraft's motion, resulting in a drag component." Seem reasonable? Mat-C 16:31, 24 August 2005 (UTC)


 * This is how the article originally defined Lift-Induced drag, but since then others have taken issue with this approach saying that it's an over-simplistic picture and not reflective of what's truly going on. If you check through the history you'll see this, inclusing a diagram (which I drew) which is no longer used in the article. Graham 00:11, 25 August 2005 (UTC)

I think a better simplification is: "Lift induced drag is caused by the lift component of the wing moving the air down behind the wing, resulting in a drag component." manfred.ullrich@arcor.de —Preceding unsigned comment added by 217.233.126.65 (talk) 21:56, 20 October 2008 (UTC)

This is all technically incorrect and misleading. By DEFINITION lift is the component of the total force on the wing in the direction perpendicular to flight path. So, by definition, lift is always pointing upward. The more correct statement is to say that the total foce acting on the wing is slightly more tilted in the backward direction relative to the aircraft motion due to 3D effects, which results in greater drag. — Preceding unsigned comment added by 67.252.29.252 (talk) 04:05, 1 February 2014 (UTC)

Error
"k is the factor by which the induced drag exceeds that of a wing of infinite span typically 1.05 to 1.15" is not true of course. Not" a wing of infinite span", but should be there "a wing of otimum (eliptical) planform."

andrzejmat 16:13, 7 April 2006 (UTC)

Agreed.--FHBridges 17:12, 13 June 2006 (UTC)

"The wingtip vortices do not directly cause induced drag". Edit of 05.10, 9 October 2007
In this I chose to Undo the immediately prior edit on the grounds: 1) The wingtip vortex is not the Source of induced drag, it is a manifestation of the spanwise flow which is the Source. 2) The later section of the Undone text essentially repeated what was previously said in the Section. While it is true that the energy content of the tip vortex is equal to the energy loss due to induced drag, and of mathematical value to the aerodynamicist, it is confusing to a lay reader to see two apparently conficting statements as to the source in the section 'Source'. Geoffrey Wickham 05:32, 9 October 2007 (UTC) Ref: 'Theory Of Flight', Richard Von Mises (Dover Books) sections VIII & IX. Geoffrey Wickham 05:51, 9 October 2007 (UTC)

Exactly, here is one citation about it: "Wingtip vortices are sometimes described as a component of induced drag, however this is incorrect. Though wingtip vortices do cause some drag, this drag is parasitic in nature. However, wingtip vortices also have the effect of destroying much of a wing's lift. Thus, in order to compensate for them, the wing must fly at a higher angle of attack, thereby causing more induced drag. The wingtip vortices do not directly cause induced drag though."--Dajsinjo (talk) 14:22, 6 February 2009 (UTC)

Winglets and vortex formation. Possible errors?
Are these statements in error?

(1) It is implied that it is possible to reduce induced drag without thereby reducing circulation, and thus lift, in this way: "Provide a physical barrier to vortex formation." It is my understanding (as a layman) that this is physically impossible--that preventing vortices would necessarily reduce circulation, and that since lift is proportional to circulation, this would reduce lift, defeating the purpose of a wing.

(2) It is suggested in the following that winglets, which have become commonplace, work by providing a physical barrier to vortex formaton: "More recent aircraft have wingtip mounted winglets to oppose the formation of vortices." It is my understanding that this is completely incorrect, and that winglets actually work by increasing effective aspect ratio (effective length of the wing) without increasing wingspan. Not by "creating a barrier to vortex formation", which, as I suggested above, would be self-defeating.

Could someone with knowledge of aerodynamics comment? —Preceding unsigned comment added by Mark.camp (talk • contribs) 03:13, 11 August 2008 (UTC)


 * Hi Mark. No, the statements you have quoted are not in error.


 * (1) Circulation, and the Kutta-Joukowski theorem in particular, are concepts used in two-dimensional flow, whereas lift-induced drag is a concept used in three-dimensional flow.  I find it difficult to talk about the two together.  The angle of attack on an airfoil can always be adjusted (using the horizontal stabilizer) to provide exactly the amount of lift required for the aircraft to fly exactly as the pilot requires - either to fly straight and level, or to maneuver the aircraft.  For example, in straight and level flight the lift on an aircraft is equal to the weight.  Whether the aircraft has winglets or not, and regardless of the aspect ratio of the wing, lift is equal to the weight.  The angle of attack will be a little different with and without winglets, but lift has not been reduced.


 * (2) The function of winglets is to reduce lift-induced drag by reducing the intensity of trailing vortices.  The winglet does indeed provide a barrier to formation of trailing vortices, in the same way as an airfoil in a wind tunnel displays little or no induced drag if the airfoil spans the width of the wind tunnel from one wall to the other.  Whether the function of a winglet is described as "increasing the effective aspect ratio" or "providing a barrier to formation of trailing vortices" is a matter of choice - neither is incorrect.  Dolphin51 (talk) 05:21, 11 August 2008 (UTC)


 * Thanks for the corrections. I had gotten the impression from browsing the web that to minimize induced drag on a real 3D wing with finite span (not terminated at the wall of a wind tunnel), one must create an elliptical distribution of circulation along the span, which would minimize, rather than eliminate, tip vortices. From other articles, I had gotten the impression that adding a winglet does not create a barrier to tip vortices, that on the contrary, tip vortices still form, but that they are simply moved to the new extended wingtip (which is now the winglet).  I was told by someone in Boeing engineering that the reason for extending the wing up (into a winglet), rather than out, was that (a) the airplane could fit in smaller spaces at airports, and (b) the moment of inertia is less for a winglet than for a longer wing, which results in a lighter structure.  —Preceding unsigned comment added by Mark.camp (talk • contribs) 23:37, 12 August 2008 (UTC)

Erroneous drawing
Is there where I should place this comment? Lift, by definition is perpendicular to the freestream velocity vector. The little introductory drawing and the second drawing should rather have the verticle arrow labeled lift. Lerabee (talk) 03:42, 17 May 2009 (UTC)


 * Yes, Lerabee, this is the ideal place to ask questions and make suggestions about the article Lift-induced drag. I agree with your comment.  The diagram might be more accurate if the vectors presently labelled Lift were labelled Aerodynamic force, and the vertical vectors were labelled Lift.  Dolphin51 (talk) 06:42, 17 May 2009 (UTC)


 * I concur. I never liked the "tilting of the lift vector" explanation for this reason.  I think the intent (which is usually not made clear) is to suggest that there is a new reference freestream flow direction created locally nearer to and around the body by the net downwash.  The net downwash is somehow propagated by induction to rotate the more-local freestream flow so as to imply that the whole body sees it and is flying in it.  Even if that was explained, it's still nonsense for a number of reasons: 1) "Lift" is an arbitrarily-defined component of the net Aerodynamic force vector (as Dolphin said) so nothing about it gets "tilted", 2) It defies lift's definition as perpendicular to freestream, 3) It muddles the (wrong) idea that "lift is tilting" with the fact that AOA for the same Cl would increase as AR decreases and so it obfuscates understanding of this other more-correct idea, 4) It pulls attention away from the fundamental source of "sub-infinite aspect ratio inefficiencies" which is the extra energy put into vortices and into the higher-speed/smaller area downwash/forwardwash field, 5) etc.
 * Gummer85 (talk) 20:55, 18 May 2009 (UTC)


 * I'm taking the diagram out. It has been noted here for four years that it's wrong. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

Inversely proportional to airspeed
The article says "Unlike parasitic drag, induced drag is inversely proportional to the square of the airspeed." This is not strictly true. It has the unstated assumption that the lift coefficient is inversely proportional to the square of the airspeed. This is certainly true (only because the pilot actively makes it so) for an aircraft that is flying straight and level. It is not true for a wing at constant angle of attack with constant CL. I think the statement needs to be qualified (as well as the equations given below it about CL), or revised.

Perhaps this represents the difference between the mindsets of pilots and engineers =)MarcusMaximus (talk) 10:05, 16 September 2008 (UTC)


 * I attempted to fix that problem. I'm fairly confident my fix isn't wrong. Fresheneesz (talk) 23:32, 23 September 2008 (UTC)

Angle of attack + drag equation
Is there a way to relate the angle of attack to the variables in the drag equation? Obviously if you increase the angle of attack, the drag goes up - but where is that reflected in the equation? Fresheneesz (talk) 23:35, 23 September 2008 (UTC)


 * Your question is a good one, regularly asked by people getting an understanding of the drag equation and lift equation. The answer is that these two equations apply to a body or an airfoil in a particular orientation to the oncoming fluid.  (In the case of an airfoil we describe this as a particular angle of attack.)  If that orientation is changed the drag coefficient or lift coefficient also changes.


 * The laws of physics explain why there is a linear relationship between aerodynamic force and fluid density, and a parabolic relationship between aerodynamic force and velocity. These relationships are easily represented by simple mathematical formulae, and hence the drag equation and lift equation.  However, the relationship between aerodynamic force and angle of attack is not accurately represented by any of the basic mathematical formulae so the drag and lift for any particular angle of attack must be determined experimentally.  This can still be accommodated in the drag and lift equations by using a coefficient.  That coefficient varies with angle of attack, but not in a way that can be represented easily by a basic mathematical formula.  It is common to use experimental methods to determine the relationship between the angle of attack and the force coefficients, and then use a graph or table to display that relationship.  The accompanying graph illustrates the point.  (It comes from Airfoil).  It would be possible to use mathematical formulae to represent the relationship between angle of attack and the force coefficients.  For example, one could use a Fourier series.  However, graphs and tables are much simpler and appear to be entirely adequate for most practical purposes.  Dolphin51 (talk) 02:17, 24 September 2008 (UTC)


 * A spline would be better. Nice graph – I'm puttin' it in the article. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

Oswald Efficiency Factor
-- Correction: " Ci=(Cy)^2/Pi*A " should be " Ci=(Cy)^2/Pi*A*e ", where "e" is the Oswald Efficiency Factor. For an elliptically loaded wing, e = 1, corresponding to the wing loading for minimum induced drag; for all other wing loadings, 0<e<1.

In the language of the article, "Cdi=(CL)^2*k/Pi*AR", k=1/e, and is a term corresponding to the efficiency of the wing. The difference is semantic, but in practice, most aerospace engineers tend to use the Oswald efficiency factor, and it is used in most aerodynamic textbooks (in the US), including the always popular aerodynamic text by Anderson.

Airplanenerd (talk) 05:18, 6 April 2009 (UTC)
 * See next section. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

Definiton of Cdi
Cdi is given as K*Cl^2/Pi*AR. This is incorrect. The correct form is K*Cl^2 OR Cl^2/Pi*e*AR. This is because K = 1/Pi*e*AR. To clarify, Cdi= Induced drag Coefficient, K = drag polar constant, Cl = Coefficient of Lift, e = Oswald efficiency factor, and AR = aspect ratio. Mathiusdragoon (talk) 14:40, 22 May 2009 (UTC)

I call "flakey" on this article.
I saw 206.55.186.247's changes and I liked them. I had been thinking about that change myself. Kinetic energy imparted into the downwash/vortex field is the fundamental source of induced drag. There's nothing magical, it's just an energy/power balance. When AR is infinite, there is no energy in the downwash/vortex field, and there is no induced drag.

Also, the "tilting lift vector" explanation is time-tested, but it's crap. It is akin to the "equal transit time" model. That is, it's right in some ways giving it the illusion of validity, but it's wrong in fundamental ways and it's incomplete. The tilted-lift vector begs more questions than it (supposedly) answers.


 * 1) The explanation should start with the fundamental source of induced drag which is:


 * For a constant downward rate of change of momentum (constant lift in opposition to constant weight), more kinetic energy is put into the downwash (+vortex) field as the cross section area of the downwash field becomes less. Both reduced AR and slower speeds reduce the cross section area of the downwash field.  Reduced AR reduces the cross section area of the downwash field because for a constant wing area and planform shape, reduced AR means reduced wingspan.  The crossflow dimension of the downwash field is closely equal to span.  The cross section area of the downwash field is also reduced at slower speeds (higher Cls).  In this case, the cross section area of the downwash field is reduced along the direction of flight.  That is, for a given amount of momentum (impulse) imparted downward to the air, the plane travels a shorter distance.


 * The fraction of kinetic energy put into the vortices compared to the downwash field (and its increase with reduced AR + slower speed) may or may not have much to do with spanwise flow (I'm skeptical in this regard, although I suspect it has something to do with sub-optimal (non-elliptical) lift distribution and Oswald efficiency). I'm still muddled with regard to the amount of K.E. in the vortices vs. the actual downwash field, but in this model, K.E. in both the downwash field and vortices becomes zero when AR becomes infinite.  When AR is infinite, downwash speed is zero (no K.E. lost there), and there are no vortices (no K.E. lost there either), and induced drag is, well, also zero(!).


 * 2) After the fundamental source of loss of energy (and power) is discussed, the mechanism by which this alters the total aerodynamic force (and it's component drag) should be discussed. That is, exactly how is more drag induced as more kinetic energy (power) gets put into the downwash/vortex field?  What are the complete and consistent mechanics of it?  This is an area I am unfamiliar with and could use some help on.  The tilted lift vector is inadequate here.  It doesn't give the amount of change to local AOA which would be needed to support the "inversely proportional to speed" wording.  Somewhere out there in the reliable literature someone must have thought about this well and explained it well.  (I emphasize "reliable" because much of our literature merely promulgates galloping error.  Just because it's at a NASA web site doesn't make it reliable.  We oughta know better.)


 * 3) Last, a discussion of how "everything effects everything else" until an equilibrium is settled into. This is something missing in a lot of our "aero" explanations.  Sure, "vortices produce downwash" but downwash also produces vortices and so on and so on, for a simple example.

I'm still working this "correct & complete" understanding out in my own head. I've still got holes in my own understanding here (not in my head :-) ), The more we know, the more we know about what we don't know.  I do know however, that much of this article as it stands is diversionary not explanatory.  People can easily come away thinking they understand when they actually don't.

--Gummer85 (talk) 19:41, 29 May 2009 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: Move. Jafeluv (talk) 17:25, 18 May 2010 (UTC)

Lift–induced drag → — This article was improperly moved from "Lift-induced drag" (with hyphen) to "Lift–induced drag" (with en dash), and the move needs to be reverted by an admin. "Lift-induced" here is simply a compound modifier, and there is no disjunction between "lift" and "induced"; in fact, there can't be because they're different parts of speech. Stephan Leeds (talk) 01:12, 10 May 2010 (UTC)


 * I have no objection to the change proposed by Stephan Leeds. Lift-induced drag is spelled with a hyphen but the article title is no longer spelled with a true hyphen.  Dolphin  ( t ) 01:23, 10 May 2010 (UTC)


 * Support, since the ndash is being improperly used here. — Twas Now ( talk • contribs • e-mail ) 16:40, 10 May 2010 (UTC)


 * Support. That should be a hyphen, not an en dash. TheFeds 01:18, 18 May 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Role of vortexes
This article went wrong at this edit http://en.wikipedia.org/w/index.php?title=Lift-induced_drag&oldid=1383804 when the paragraph about wing tip vortexes was added. Wingtip vortexes do not cause drag, and span wise flow resulting in flow around the wing tip is a minor contribution to the twin vortexes which form behind the aircraft. The major contribution is lift itself. Lift is the result of the aircraft wing thrusting a large volume of air downward. (The manner in which this air is thrust downward by the wing is that the wing acts as an inclined plane. And this is the proper explanation of induced drag. The rest is distracting details about how it acts as an inclined plane.) The large volume of downward moving air in the midst of still air results in the vortexes at the boundaries between the moving and still air beginning at each wing tip and extending behind the aircraft. These vortexes would exist even in the absence of span wise flow around the wingtips, and they are responsible for the dissipation of ALL the power deposited into the air as a result of induced drag. They are not the cause of induced drag only the result of it. Any attempt to reduce these vortexes for a wing of given aspect ratio will either be achieved at the expense of lift (by reducing it) or will result in counter productive added weight and/or drag.

There is energy dissipation which is the result of span wise flow on the wing. Air subject to greater than ambient pressure under the wing will tend to flow out from under the wing in all directions including span wise toward the tip. Air subject to less than ambient pressure on top of the wing will tend to flow inward from all directions causing span wise flow from the wing tip. It can be seen from the geometry of this flow compared to the wing that the span wise flow reduces the effective angle of attack by increasing the effective chord of the wing relative to air flowing over the wing. Another way of looking at losses due to span wise flow is that it is momentum, and thus power imparted to the airflow which does not contribute to lift but decreases it instead. This is all that is needed to explain the loss of efficiency of a wing due to span wise flow over the wing surfaces. This wasted power is also dissipated in the vortexes created by the wing both at the wing tip as described and as vorticity created at the trailing edge of the wing as the span wise flow of the top of the wing meets the span wise flow of the bottom of the wing traveling in the opposite direction. Now here is where a wing tip treatment can do some good. Anything which will efficiently limit span wise flow or recover the momentum from it will improve the efficiency of a wing. This must be what effective wing tip treatments such as winglets are accomplishing. They're reducing the vortexes, but that is incidental to improving the wing performance by dealing with span wise flow.

In short, the large vortexes generated behind an aircraft wing are not the cause of drag on the wing, they are a result of it and they are the dissipation, or mixing, of the power deposited into the air near the wings with the bulk of the air nearby.

The effect of aspect ratio on wing efficiency can also be explained without reference to vortexes, and indeed vortexes have nothing to do with it:

f = ma where f is the component of the total force which acts perpendicular to the motion of the wing and, m is the mass of the air thrust downward by the wing, and a is the acceleration of the air thrust downward by the wing. The weight of the aircraft is equal to f.

v = at or a = v/t where t is the time the air thrust downward by the wing is in contact with the wing.

f = mv/t by substitution and thus we obtain

ft = mv or the relationship between impulse and momentum.

Now lets choose two aircraft of the same weight, with the same wing area traveling forward at the same speed. Let aircraft two have four times the wing aspect ratio of aircraft one which means that the wing chord of aircraft two will be one half that of aircraft one. Let's use capital letters for aircraft two since I don't want to bother with superscripts or subscripts. So;

F = f

T = t/2 since the speed is the same but the chord is 1/2 so MV = mv/2 or the momentum of the air thrust downward by aircraft two is 1/2 that of aircraft one. But how to apportion this difference among M and V? Even though it might seem that M = 2m because the span of aircraft two is twice that of aircraft one it seems more likely that M = m because V is likely = v/2 and so the depth of the air moved by aircraft two is 1/2 that of aircraft one - approximately.

So let's look at it from the perspective of energy.

e = mvv/2

E = MVV/2

if V = v/2 then

E = Mvv/8 = e/4

Energy over time is power p so P = p/4, but power is also force times velocity and the power forcing the air downward is the same power expended against induced drag so the induced drag of aircraft two is 1/4 that of aircraft one and by the small angle approximation the angle of attack of aircraft two is about 1/4 that of aircraft one. There you have the best explanation of why a higher aspect ratio wing is more efficient. You might also notice that 1/4 is the inverse of the difference in aspect ratio which agrees with the commonly accepted equation for induced drag. This has nothing to do with vortexes.

Now let's consider span wise flow. It seems clear to me that the same impulse - momentum - power argument can be made for the span wise flow based on the lesser chord of the wing on aircraft two so the higher aspect ratio wing will have less span wise flow losses as well and by a very similar proportion.

Notice that this discussion precludes vertical winglets on the tips of wings from acting to increase the aspect ratio of a wing. They can only have effect by reducing the inward span wise flow on top of the wing.

I did not add all this discussion to the article because I don't feel it belongs here. Parts of it probably belong in other articles. I put it here to make my point that this article should be reverted to the version just prior to http://en.wikipedia.org/w/index.php?title=Lift-induced_drag&oldid=1383804 since that version is essentially correct, simple and to the point.

BTW, I got into this because I found that the "FAA Pilots Handbook of Aeronautical Information" also provides the bogus vortexes cause drag explanation, and I find that particularly embarrassing. —Preceding unsigned comment added by 71.216.55.126 (talk) 04:58, 13 July 2010 (UTC)

Preceding added by me, 71.216.55.126, when I was not logged in.Rickanwp (talk) 05:43, 13 July 2010 (UTC)


 * I think you are correctly describing the essence of the 3d flow, where drag losses come from the vertical flow field's momentum in the wake, which is lost at the rate you describe. The reason vortices are cited is that they provide the mechanism for that energy to be lost, in returning the downwash around and back into airmass above the wake, a mechanism for dissapative loss is introduced. Meanwhile, in 2-D...


 * In an infinite wing (with incompressible (M<<1), inviscid (Re>>1e6) flow, modeled by a section spanning the entire tunnel, that can't happen. The air has to spring back into place and occupy its original streamline eventually. Since we're subsonic, the wing can "feel" that. It's like running on springy rubber track instead of a squishy beach that absorbs your every footfall.  The implication is that, without spanwise flow & the vortices, the downstream flow would return to horizontal streamlines, and there'd be no drag. So, while "blaming it on the vortices" does seem a poor description, I think it's basically right.  98.245.118.57 (talk) 01:49, 1 November 2010 (UTC) mark

"Lift" in First Diagram is Wrong
"Lift" is the component of the pressure-gradient force that is perpendicular to the relative wind; "Induced Drag" is the component that is parallel. The vector diagram at the beginning of the article labels the entire pressure vector as "Lift", which is just wrong. — Preceding unsigned comment added by 69.138.184.248 (talk) 19:43, 13 August 2011 (UTC)


 * It has been removed. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

"Lift" in Second Diagram is Also Wrong
Is anyone actually reading this discussion? Helloooo? — Preceding unsigned comment added by 69.138.184.142 (talk) 23:25, 24 December 2011 (UTC)


 * Also removed. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

Diagrams
On 26 February User:GliderMaven edited this article, and also Drag (physics), by removing diagrams illustrating the origin of lift-induced drag. In the edit summary, GliderMavern wrote ''Removed diagrams are incorrect. Lift is perpendicular to flow, not to wing. Lift-induced drag is not what is shown in diagram, it’s a wingtip effect.''

I have restored the diagrams and started this discussion so that interested Users can debate the accuracy or otherwise of the diagrams. Dolphin ( t ) 03:00, 27 February 2012 (UTC)


 * My view is that these diagrams are typical of the simple diagrams presented in aerodynamics textbooks to illustrate the background to lift-induced drag. I'm sure it is possible to produce diagrams that are superior to these, but we don't have them and until someone produces them and makes them available to Wikipedia we should continue to use the present ones.


 * I can make the following comments about GliderMaven's edit summary:
 * Removed diagrams are incorrect
 * No they aren't. They might not be perfect, but they can't be described as incorrect.
 * They are incorrect.GliderMaven (talk) 03:28, 27 February 2012 (UTC)


 * Lift is perpendicular to flow, not to wing.
 * Correct. Lift is defined to be perpendicular to the vector representing the relative velocity between the airfoil and the freestream.
 * And in the diagram the lift force is perpendicular to the wing, making them WRONG. Which bit don't you understand?GliderMaven (talk) 03:28, 27 February 2012 (UTC)


 * Lift-induced drag is not what is shown in diagram
 * This diagram is typical of the simple diagrams used in many textbooks on aerodynamics and other books written for student pilots.
 * Since it's not referenced to a reliable source I don't care. The only reference I found when I checked did not have this sort of diagram and did not describe it like that diagram either.GliderMaven (talk) 03:28, 27 February 2012 (UTC)


 * It’s a wingtip effect.
 * Partly correct. The strongest trailing vortices leave the wing at the wingtip, but there are trailing vortices leaving from other parts of the wing, such as the outboard end of the flaps when flaps are extended.  There is also trailing vorticity leaving the wing at any change in chord, change in airfoil section, and change in incidence.  Induced drag exists generally on any three-dimensional wing.  It is not purely a wing-tip effect.  Dolphin  ( t ) 03:13, 27 February 2012 (UTC)


 * Yes, and the freaking incorrect diagram is 2 dimensional, and wrong in every other respect as well, and does not even match the text.GliderMaven (talk) 03:41, 27 February 2012 (UTC)


 * Since, by definition, the lift force is always at 90 degrees to the airflow. In the diagram it's 90 degrees to the wing, which is not parallel to the flow. These completely unreferenced diagrams are therefore unequivocally wrong!!!GliderMaven (talk) 03:28, 27 February 2012 (UTC)

Thanks for joining this discussion. I will add a reference or two for these diagrams in a few hours.

This diagram show two lift vectors – one is red and labelled Lift; and the other appears to be grey in color but it isn't labelled. It also shows two velocity vectors – one is red and labelled Effective Relative Airflow; and the other is blue and labelled Relative Airflow (Free Stream).

The red lift vector is perpendicular to the red velocity vector (Effective Relative Airflow). The grey lift vector is perpendicular to the blue velocity vector (Relative Airflow (Free Stream)). Neither lift vector is perpendicular to the chord line of the airfoil.

The angle between these pairs of vectors is correctly labelled the induced downwash angle ε.

You might recall the induced downwash angle ε is given by:
 * $$\epsilon = \frac{C_L}{ \pi \lambda} $$
 * where CL is wing lift coefficient and λ is wing aspect ratio.

The coefficient of induced drag is given by:
 * $$C_{Di} = \frac{C_L^2}{ \pi \lambda} $$

Notice the similarity? The reason for the similarity is that the coefficient of induced drag is equal to the induced downwash angle multiplied by the wing lift coefficient. If the induced downwash angle disappears then induced drag also disappears. As the induced downwash angle increases, induced drag also increases. That is what diagrams of this kind are attempting to illustrate. Dolphin ( t ) 04:05, 27 February 2012 (UTC)


 * The lift vector is not defined relative to the downwash angle, it's referenced to the free stream, so at best this diagram is not using standard descriptions.GliderMaven (talk) 04:29, 27 February 2012 (UTC)


 * According to the article "Theoretically a wing of infinite span and constant airfoil section would produce no induced drag." Those diagrams are 2 dimensional and hence have effectively infinite span. Also the NASA website says: "Induced drag is a three dimensional effect related to the wing tips; induced drag is a wing tip effect. So if the wing tip represents only a small fraction of the total wing area, the induced drag will be small. Again, long thin wings have low induced drag." which is essentially the same thing. So as far as I can tell, these diagrams are unreferenced OR and seem to be wrong.GliderMaven (talk) 04:29, 27 February 2012 (UTC)


 * So I'm not able to tie up at all what you're saying with the references.GliderMaven (talk) 04:29, 27 February 2012 (UTC)


 * As promised, I have been able to locate some reliable published sources - one American and one British. The diagram shown at the top of this discussion thread is almost an exact copy from Figure 1.30 out of the NAVWEPS publication Aerodynamics for Naval Aviators, now available publicly under the author name Hugh Hurt Jr.  I have added this as an in-line citation and restored the diagram.  In Laurie Clancy's book Aerodynamics, Figure 5.24 is almost the same.  I have added this as an in-line citation to support the other diagram.


 * Aerodynamics for Naval Aviators, p.66, says: "Another important influence of the induced flow is the orientation of the actual lift on a wing. Figure 1.30 illustrates the fact that the lift produced by the wing sections is perpendicular to the average relative wind.  Since the average relative wind is inclined downward, the section lift is inclined aft by the same amount – the induced angle of attack αi.  The lift and drag of a wing must continue to be referred perpendicular and parallel to the remote free stream ahead of the wing.  In this respect, the lift on a wing has a component of force parallel to the remote free stream.  This component of lift in the drag direction is the undesirable – but unavoidable – consequence of developing lift with a finite wing and is termed INDUCED DRAG, Di.  Induced drag is separate from the drag due to form and friction and is due simply to the development of lift."


 * Clancy (p.80) says: "The downwash angle at the wing is given by $$\epsilon = \frac{C_L}{ \pi \lambda}.$$ This implies that the effective incidence of the wing is reduced by ε as shown in Fig 5.24. Since the effective flow direction is rotated through an angle ε, the effective lift force, Leff, is tilted backwards through the same angle, since it has to be normal to the effective airflow.  This force now has a component in the free stream direction, and this is the induced drag, Di.  It is additional to the profile drag, which is not significantly affected by turning through the small angle ε."


 * Dolphin ( t ) 07:25, 27 February 2012 (UTC)


 * These diagrams are wrong. They disagree with every source I have located, they disagree with Wikipedia's lift article, they disagree with the text of this article, they disagree with every single thing I have found everywhere on the web; without exception, and without exaggeration.GliderMaven (talk) 13:36, 27 February 2012 (UTC)


 * In particular they disagree with this: http://www.pilotsweb.com/principle/liftdrag.htm where the diagrams are superficially similar, but the lift force is shown in a different direction.


 * How many ways can I say this? These diagrams, as they currently stand, are wrong.GliderMaven (talk) 13:36, 27 February 2012 (UTC)


 * The idea of Leff is NOT the same as Lift, not ever, never.GliderMaven (talk) 13:38, 27 February 2012 (UTC)


 * You keep writing that they are wrong; and that they disagree with other things, but you seem reluctant to explain why you believe they are wrong, and what other things they disagree with. Talk pages are intended to be used to persuade and explain, not to assert.  I can be persuaded by objective and rational discussion but I am generally unmoved by displays of anger or annoyance or assertion.
 * You also seem to be working under the impression that any physical phenomenon can only have one correct explanation. There is no reason to believe that.  Most physical phenomena can be explained in more than one way.  It is unreasonable to say that if there are two different explanations for a physical phenomenon, one of them must be wrong.  For example, in explaining the lift on an airfoil, use can be made of Bernoulli's principle, conservation of momentum, lifting-line theory, thin-airfoil theory, Kutta-Joukowski theorem, computational fluid dynamics, and others.  All have their strengths and weaknesses, but none of them is wrong.  Dolphin  ( t ) 22:01, 27 February 2012 (UTC)


 * In response to the assertion that the diagram at http://www.pilotsweb.com/principle/liftdrag.htm is different to the one above, as far as I can tell both diagrams are showing exactly the same thing. Sure the Lift vector and Effective Lift vector are labelled, whereas this is only implied in the image above, but that is surely clarified in the text. In any event, "pilotsweb.com" is hardly a reliable source. Dolphin has provided some adequate references and a good explanation of why this diagram is taught as part of aerodynamics and student pilot studies. I will try and locate some additional references.
 * The lift force vector in the diagram is perpendicular to the Relative (grey vector) and Effective Relative (red vector) Air Flows, not the chord of the wing. The diagram isn't perfect, but in terms of explaining induced downwash (and therefore drag) in a 2D diagram, I think it does a reasonable job. GliderMaven, perhaps you'd like to create a revised version of the diagram, or explain the changes that you would make so that we can discuss the merits of those changes? If it is just because you think "Lift" should read "Effective Lift" I don't think that warrants removal of the image altogether.    Jo h n wa lt on   22:55, 27 February 2012 (UTC)
 * I'm removing it until it's been corrected; it's simply wrong!GliderMaven (talk) 20:47, 28 February 2012 (UTC)
 * The 'lift' described in the article here, is NOT the same lift as shown in that diagram. I do not even believe dolphin when he says that that's what the source really says. Either he's wrong, or the source is, or both.GliderMaven (talk) 20:53, 28 February 2012 (UTC)

@GliderMaven: JohnWalton created this diagram. In his post above he has invited you to explain the changes that you would make so that we can discuss the merits of those changes? Do you intend to explain your point of view so that JohnWalton can correct it? Dolphin ( t ) 21:26, 28 February 2012 (UTC)
 * Look, I've pointed out that it's wrong, I've compared it with many other sources, that all say it's wrong. Clearly, the only way it's ever going to get fixed (IF it's ever going to get fixed) is to remove it until it's fixed. Nobody has done a single thing other than revert war back this incorrect, and deceptive material, repeatedly. This is not the way it's supposed to work, if somebody points out something is wrong, it should be IMMEDIATELY removed and kept out until it can be fixed. You guys are just completely out of line here.GliderMaven (talk) 19:20, 2 March 2012 (UTC)

Hi guys. I have to agree with GliderMaven. I first saw that so-called "explanation" diagram in 1981 in Freshman "Intro to Aero" class. It was dubious then as it is now for many of the reasons given by GliderMaven. The problem is that it merely "feels" like an explanation while it actually connects nothing to nothing. It redefines "lift" as now somehow being not perpendicular to flow, and so on, and so on.

Induced drag comes from the adding of kinetic energy into the wake (conceptually, in speed perpendicular to the flow ("downward"), but also in the vortices in a way I don't fully get). This imparting of KE into the "downward" flow and vortices comes in association ("somehow" :-) ) with redistribution of pressure on the body. The change in pressure distribution results in forces in the aft flow direction which is the actual drag induced by the lift.  The awful diagram purports (and has done so for decades) to give a physical connection between the production of lift and the subsequent induction of drag, but it does nothing except "induce" non critical thinkers into merely thinking they understand when they don't.

This diagram belongs in the trash heap along with that old "lift explanation" that says a molecule has to race around the top of an airfoil to meet up with its partner it split from at the leading edge. You know what I'm talking about.

As far as "reliable sources" go. If the diagram is from an otherwise reliable reference (and this diagram usually does appear in references that are otherwise reliable), that is not cause for inclusion. Reliability is determined by judgement of knowledgeable editors, there is just no other way to make that assessment. "Reliable sources" of the 1300's would have the sun going around the Earth, but we know better than to include it because we judge parts of those sources to be unreliable in light of what we know as editors. We would never include a diagram of the sun going around the Earth as an explanation of why day and night follow each other -- even if that 1300's source was otherwise mostly reliable in the light of modern days.

If we really want to make the article better, we need to find a source that eschews that crappy diagram for a more critical and modern explanation. I'm sure there is at least one out there.

Skyway (talk) 20:49, 25 May 2012 (UTC)
 * Hi Skyway. Thanks for taking an interest in this subject.  When GliderMaven first erased the diagram it was not explicitly supported by an in-line citation so there was some legitimacy in his actions. (It would have been better if he had begun by going to User talk:Johnwalton, the creator of the diagram, and discussing his concerns directly with him.)  Since then I have added three reliable published sources to support the diagram and I think Johnwalton also made an adjustment to the diagram itself.  I understand GliderMaven is now reasonably happy with the diagram.  On this discussion thread, on 27 February, Johnwalton asked GliderMaven to advise what problems he had with the diagram so they could discuss it.  To the best of my knowledge, GliderMaven has never responded to Johnwalton's request.
 * Throughout this discussion thread GliderMaven displayed one problem that is very common among newbies - he devotes a lot of words to displaying his assertiveness, and denigrating the object of his displeasure (in this case a diagram). He is obviously quite skilled at this aspect of debate.  Unfortunately, he never really got started into providing an objective, dispassionate explanation that might convince other Users that there was a problem, and that his suggested solution might be better than the status quo.  He is certainly not alone in displaying this problem.  For every User on Wikipedia who is skilled at writing words that get to the heart of an issue without inadvertantly putting out a smokescreen of anger and intimidation, there is a hundred whose skill doesn't yet go beyond writing words that convey their anger.  (For example, making a strong and assertive claim that something is wrong is not the same as objectively explaining why one believes something is wrong.  In my view, most experienced Users on Wikipedia are totally unmoved by claims, even claims made assertively, but they are open to persuasion by objective reasoning.)
 * Your edit above is a little like those of GliderMaven. Without question, you have succeeded in telling all Users who are interested in this article that you don't like the diagram in question; it reminds you of a diagram you first saw in a Freshman class; it irritates you and you don't like it; you claim it belongs on the trash heap.  You have attempted to denigrate the diagram by facetiously comparing it with diagrams from pre-Copernican times that showed the sun orbiting the Earth.  As a vehicle for conveying objective reasoning your edit didn't get out of first gear.
 * You have written that Induced drag comes from the adding of kinetic energy into the wake. I agree that is one way of explaining drag in general but if you are suggesting it is the only way of explaining induced-drag, I would have to disagree with that.  I'm sure you are aware that in science and math there are usually two or more ways of satisfactorily explaining any phenomenon.  Just because you have found an explanation that is satisfactory it doesn't mean that all alternative explanations must be unsatisfactory.  If you can produce a diagram that shows how the induced-drag force is related to the addition of kinetic energy to the wake, please go ahead and add it to the article.
 * You have written that reliability is determined by judgement of knowledgeable editors. There may be some truth in that.  Are you able to quote some part of WP:Verifiability that clarifies your point?
 * User talk:Johnwalton has invited anyone having an issue with his diagram to discuss it with him. I'm sure he would be delighted to hear from you.
 * Finally, you have written If we really want to make the article better, we need to find a source that eschews that crappy diagram for a more critical and modern explanation. I agree that is the Wikipedia method of tackling a problem.  Please feel free to go for it!  Dolphin  ( t ) 05:57, 26 May 2012 (UTC)

By your leave, Dolphin. The talk page allows for some passion in argument. Witness: Your 4016-character investment in a response. Even then, it should have been obvious I was talking specifically about the diagram and its poor explanation, and that there was nothing personal about anyone or anyone's style. Indeed I had noted GliderMaven's "style" and ignored it in favor of attending to his actual point, which I thought was valid. Sorry I offended your sensibilities and that my Copernican example didn't sit well with you. Skyway (talk) 19:52, 27 May 2012 (UTC)

I agree that the diagram is wrong, and I have removed it. There are several things wrong with it. If the vector labeled "L" is supposed to be the lift, then it's too long compared to "Leff", which is supposed to be the total force. And if the angle of attack (α) is zero, the downwash angle will still be non-zero (because the lift is positive), whereas this diagram makes it look as thought ε is always smaller than α. Also, the total force is not necessarily exactly perpendicular to the downwash direction, as implied by the diagram. I think a better diagram should be used, or else the caption should explain what is wrong with this one. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)
 * @Eric Kvaalen: The diagram was created by User:Johnwalton. In this edit on this thread on 27 Feb 2012, John invites anyone with a problem or a suggestion for his diagram to contact him and discuss it. (That is the way Wikipedia works best.) I notice you haven't contacted John to discuss your suggestions. Please do so.
 * In your explanation of why you think the diagram is "wrong" you have written if the angle of attack (α) is zero, the downwash angle will still be non-zero. I don't know what you are aiming at here. This is a diagram in which the angle of attack is not zero. Please clarify what you are trying to say.
 * In your explanation you talk about total force. The diagram does not contain a vector labeled total force. Wikipedia doesn't have an article on total force. Do you mean aerodynamic force?
 * Until you engage Johnwalton in discussion about his diagram, or until you produce a better one, the diagram should remain in place. It is supported by adequate references to reliable published sources. It may not be perfect but Wikipedia does not have a policy that says anything like "delete everything that is not perfect." I will restore it. Dolphin  ( t ) 12:37, 16 December 2013 (UTC)


 * I've just sent John Walton an email.


 * The diagram is meant to apply to any angle of attack, including zero. But it's a minor point. Actually, I don't think that the "downwash angle" is well defined. The air just aft of the wing may be going in a certain direction, but if you go a bit further back, it will be going more horizontal. Otherwise it would push all the air at lower altitudes down and pretty soon you'd have the whole atmosphere moving down at that angle!


 * True, there's no vector labeled "total force", but that's what the Leff is – or I should say total aerodynamic force (it doesn't include gravity or the force exerted by the engines or fuselage).


 * I think the diagram is misleading (as do several other people here) and shouldn't be used as it is.


 * Eric Kvaalen (talk) 10:37, 18 December 2013 (UTC)
 * @Eric Kvaalen: Thanks for writing to John Walton. Despite numerous others not being comfortable with the diagram, I believe you are the first to actually do something as constructive as contacting John in response to his invitation to do so. We will wait and see his response. Hopefully he will amend the diagram in response to your comments.
 * The diagram is presently supported by three reliable published sources - Hurt, Clancy and Kermode. John Walton's diagram and that by Clancy both apply the label effective lift to the rear lift vector, the one tilted backwards slightly. The diagram by Hurt applies the label effective lift to the front lift vector, the one that isn't tilted backwards! (So even reliable published authors don't agree always agree on the best way to present and label their diagrams.) Kermode doesn't use the terminology effective lift - his diagram states Direction of lift tilted backwards owing to deflection of airflow. Dolphin  ( t ) 06:30, 19 December 2013 (UTC)
 * @Eric Kvaalen. I see that John Walton hasn't responded to your email a few days ago. We may have to wait a while before he comes along.
 * I have made some changes to the work you did on the caption. Let me know what you think of them. (See my diff.)
 * You wrote This diagram shows the direction of the total force to be perpendicular to the direction of the air leaving the wing, which is not accurate. The diagram doesn't show the direction of the air leaving the wing. It will leave the wing parallel to the trailing edge of the wing - see Kutta condition. Practical airfoils don't have a sharp trailing edge and in practice the air leaves the trailing edge parallel to the underside of the wing. (The topside of the wing sees separated flow or even reverse flow.) Imagine a typical lifting wing operating at an angle of attack 10 degrees - the air leaves the wing heading downwards at about 10 degrees below the horizontal. But the air arrives at the leading edge heading upwards at about 9 degrees. The difference in the two angles is 1 degree and this is the downwash. Our diagram explicitly shows the downwash angle (ie about 1 degree) whereas the air leaves the trailing edge of the wing at about 10 degrees. In 2-dimensional flow there is no downwash so if the air leaves the trailing edge heading 10 degrees downwards relative to the horizon, the air arrives at the leading edge heading upwards at 10 degrees relative to the horizon. Dolphin  ( t ) 06:27, 22 December 2013 (UTC)

Well, I see several things to remark on. First of all, it's true I didn't understand downwash to mean what you say. But what you have just written doesn't really correspond to how the Downwash article defines it (which I hadn't read). That article says it's the change in direction, which in your example would be 19°, not 1°.

I don't see why the force on the wing should be oriented 1° backwards if the air hits the wing going upwards at 9° and leaves going downwards at 10°. It seems to me the force should be oriented 0.5° backwards from vertical in that case (which would give a lift-to-drag ratio of about 115!).

Aren't you contradicting yourself when you say that the Kutta condition applies and then you say that the topside of the wing sees separated or even reverse flow? (The article Kutta condition has a couple sentences I don't understand: "If the trailing edge has a non-zero angle, the flow velocity there must be zero. At a cusped trailing edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the trailing edge.")

Why do you say that in 2D flow there is no downwash? Surely if you have a wing section of constant cross section spanning the width of a wind tunnel (so it's like 2D), it will still generate lift. But as I said on the 18th, a wing cannot cause a change in direction that is maintained forever as the air continues away from the wing.

Now, as to your version of the caption, you say the red vector is perpendicular to the airflow in the vicinity of the wing – but what is that? It changes from place to place!

You say it represents the lift on the airfoil section in two-dimensional flow at the same angle of attack. Why say 2D? And it's not the "lift", it's the total force.

Finally, you say the component of "Leff" parallel to the free stream is the induced drag on the wing. Well, actually it's the total drag isn't it? The induced drag is the total minus the drag you get when there's no lift.

By the way, I looked at your user page and saw a little box saying something silly about scientists, but that's off the subject!

Eric Kvaalen (talk) 19:29, 23 December 2013 (UTC)


 * @Eric Kvaalen: You have asked some good questions. I will do my best to provide equally good answers. Part of the problem is that in aviation the word “downwash” is applied in two different ways. I will try to explain the two different applications.


 * Firstly, in the case of a wing of finite span, the trailing vortices induce a small downwash in the flow around the wing and in the wake behind the wing. This downwash causes the flowfield to be rotated through a small angle epsilon ($$\epsilon$$) and the lift vector to be tilted backwards through the same small angle $$\epsilon$$. Most textbooks on aerodynamics show that, for an elliptical wing, this small angle $$\epsilon$$ is constant across the entire wing span and is given by:


 * $$\epsilon = \frac{C_L}{ \pi \lambda}$$


 * where CL is the lift coefficient of the wing and $$\lambda$$ is the aspect ratio of the wing.


 * Some authors, such as John D Anderson, call $$\epsilon$$ the induced angle of attack. (If the geometric angle of attack is 10 degrees and the induced angle of attack $$\epsilon$$ is 1 degree, the effective angle of attack is 9 degrees.)


 * Are you saying that the air way out in front is moving downwards with a slope of 1°, so the effective angle of attack is 9° instead of 10°? Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)
 * Secondly, in the case of any wing generating lift, including a wing of infinite span, the air approaching the wing is deflected upwards, and the air leaving the wing is deflected downward. These motions are often described as “upwash” and “downwash” respectively. Both the upwash and the downwash are clearly visible in the attached diagram of the streamlines around an airfoil. In 2-dimensional flow (such as a wing of rectangular planform that completely spans a wind tunnel) the angle of upwash is equal in magnitude but opposite in sign to the downwash; and $$\epsilon$$ is zero because the aspect ratio of the wing is infinite. (You can see the advantage in calling epsilon the induced angle of attack rather than the downwash because it avoids the problem of having two different kinds of downwash.) The Wikipedia article Downwash talks about this second kind of downwash, not $$\epsilon$$, not the kind that Anderson and some others call the induced angle of attack.


 * Isn't it true that in a wind tunnel with a wing completely spanning it there will be some induced drag? Which would mean that tan(ε) (=0) would not be equal to induced drag over lift. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * In the flow around a wing of finite span, the effect of $$\epsilon$$ is to slightly reduce the upwash angle, and slightly increase the downwash angle. This rotation of the flow field around the wing through the angle $$\epsilon$$ causes the lift vector to also be rotated to the rear through the same angle $$\epsilon$$, and that is the origin of lift-induced drag.


 * You have also asked some other questions. I will try to answer them shortly. Dolphin  ( t ) 11:25, 24 December 2013 (UTC)

@Eric Kvaalen: Here are my suggested answers to some of your questions:


 * EK: It seems to me the force should be oriented 0.5° backwards from vertical in that case (which would give a lift-to-drag ratio of about 115!).
 * D51: You are right – the flowfield has been rotated through 0.5°. The ratio of lift to induced drag is about 115. However, in addition to the induced drag there is also the parasite drag. The two combine to give the total drag. The ratio of lift-to-drag refers to total drag and will be significantly lower than 115.


 * EK: Aren't you contradicting yourself when you say that the Kutta condition applies and then you say that the topside of the wing sees separated or even reverse flow?
 * D51: The Kutta condition implies that where a flow leaves a sharp trailing edge, the streamlines are parallel to the sharp trailing edge. The trailing edge of a wing is not a sharp edge (not a cusp) – the upper and lower surfaces meet at an angle that is greater than 0°. This invites the question as to whether the flow departs parallel to the upper surface at the trailing edge, or the lower surface at the trailing edge. The answer is the lower trailing edge.


 * I don't think that's correct. Surely they compromise! The border between the two streams would go off at an angle that is between the slope of the upper surface and the slope of the lower surface. (This implies that there is a high-pressure region right at the edge, causing a deviation in each of the two streams.) Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * EK: The article Kutta condition has a couple sentences I don't understand.
 * D51: These aren’t good sentences. I don’t understand them either.


 * I think I understand them now. "If the trailing edge has a non-zero angle, the flow velocity there must be zero. At a cusped trailing edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the trailing edge."
 * That means, if at the trailing edge the upper and lower surfaces meet at some positive angle (as is the case with real wings), then (as I just said above) there has to be a region of high pressure there and the directions of flow change abruptly (by a very small angle of course). That implies (assuming Euler's equations of inviscid motion) that the velocity right at the edge is zero! (The velocity comes from the derivative of the function $$z^{180/(180-\epsilon/2)}$$, and that derivative goes to zero at z=0. See Potential flow.) But if the trailing edge is a cusp, meaning that the surfaces are curved and meet at an angle of zero, then the velocity will not be zero. If the velocity above and below are not equal, then by the Bernoulli principle the pressures will not be equal. But I'm not sure that that is impossible. It would cause the flow to curve toward the faster stream. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * EK: Why do you say that in 2D flow there is no downwash?
 * D51: For the reasons I tried to explain in my 24 December edit, I will change my terminology and say that in 2D flow there is no induced angle of attack ($$\epsilon$$ is zero). In 2D flow there is parasite drag but no induced drag. There is upwash as the flow approaches the  airfoil and downwash as the flow departs the airfoil, but no spanwise flow and no induced angle of attack $$\epsilon$$, and therefore no induced drag.


 * Well, I think you're redefining induced drag in that case. Induced drag should mean (in my opinion) the excess drag that is produced when there is lift. There is a certain (negative) angle of attack at which there is no lift, and the drag is very low. When the angle of attack is increased, both lift and drag increase (I think – maybe the drag decreases initially). The difference between the new drag and the drag at zero lift is the induced drag. And as I said earlier, I think it would be non-zero even in 2D flow. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * EK: Surely if you have a wing section of constant cross section spanning the width of a wind tunnel (so it's like 2D), it will still generate lift.
 * D51: Yes, it will generate lift. If the flow rises at 10° to meet the airfoil, and then descends at 10° as it departs the airfoil, the airfoil has exerted a force on the fluid to turn the flow through 20°. There is another force that is equal in magnitude but opposite in direction – this is the upward force on the airfoil that is called lift. However, as this airfoil spans the width of the wind tunnel there is no spanwise flow, no induced angle of attack $$\epsilon$$ and no induced drag.
 * See my comment above. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * EK: you say the red vector is perpendicular to the airflow in the vicinity of the wing – but what is that? It changes from place to place!
 * D51: The diagram shows an airfoil section operating at a particular value of effective angle of attack (unlabeled.) The red vector Leff is the lift on this airfoil section in 2D flow; the grey line labelled L shows the orientation of the lift on this airfoil section in 3D flow at the same effective angle of attack. The geometric angle of attack on the 3D airfoil is equal to the effective angle of attack plus the induced angle of attack $$\epsilon$$. The effect of the trailing vortices is to rotate the flow field (in the vicinity of the wing) clockwise through an angle $$\epsilon$$. In 2D flow there is no spanwise flow, no trailing vortices, no induced angle of attack $$\epsilon$$ and no induced drag.


 * Well, if that's what L is, then the diagram is not clear at all. I thought it was the lift when the angle of attack is zero or slightly negative. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)


 * EK: And it's not the "lift", it's the total force.
 * D51: Aerodynamic force, or total force if you prefer, contains parasite drag. Parasite drag is omitted from this diagram. This diagram is intended to show only lift and induced drag.


 * In that case, it's the sum of the lift and the induced drag. Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)

It can be a complex subject. I hope this is helpful. Dolphin ( t ) 11:49, 25 December 2013 (UTC)
 * @Eric Kvaalen: In my explanations above, I have relied on the concepts of geometric angle of attack, effective angle of attack, and induced angle of attack $$\epsilon$$. The aerodynamic force on an airfoil (2D flow) or a wing (3D flow) is determined by the effective angle of attack. (In Aerodynamics for Naval Aviators by H.H. Hurt it is called the "section angle of attack" - p.66 in my copy.) The lift on an airfoil or wing is defined to be the component of aerodynamic force perpendicular to the relative direction of the remote free stream and the airfoil or wing. The angle between the remote free stream and the chord line is called the geometric angle of attack. (In Aerodynamics for Naval Aviators it is called the "wing angle of attack" - p.66 in my copy.)


 * The difference between the geometric angle of attack and the effective angle of attack is the induced angle of attack $$\epsilon$$. (In Aerodynamics for Naval Aviators it is called the "induced angle of attack" - p.66 in my copy.) In 2D flow there are no trailing vortices, induced angle of attack $$\epsilon$$ is zero and the geometric angle of attack is equal to the effective angle of attack; the wing does not experience induced drag. In 3D flow there are trailing vortices, induced angle of attack $$\epsilon$$ is greater than zero and is given by the formula given above, the geometric angle of attack is greater than the effective angle of attack, and the wing experiences induced drag.


 * In 2D flow, $$\epsilon$$ is zero so the lift vector is not tilted backwards so there is no induced drag. Dolphin  ( t ) 05:41, 27 December 2013 (UTC)

Influence of aspect ratio
In the “Reducing induced drag” section, the statement “a high aspect ratio wing will produce less induced drag than a wing of low aspect ratio” is a common and persistent mistake in aeronautical circles. It is simply wrong. It can actually be deduced quite easily from the equation written in the very next section: “Calculation of induced drag”. There it can be seen that Di is apparently inversely proportional to S and to AR. But AR=b²/S, so in fact Di is inversely proportional to b² alone, and does not depend on S. So comparing two identical airplanes with: same weight, same speed, and same lift distribution (i.e. same e), the lower Di will be that of the plane with the longest wing span, regardless of having maybe a smaller AR, due to a sufficiently bigger S (bigger mean chord). Enrico Lucarelli (talk) 10:38, 5 December 2012 (UTC)
 * There is much truth in what Enrico has written. However, statements that "A is directly proportional to B" or "C is inversely proportional to D" are often true only when linked to a particular equation. If a different equation is chosen, the statement may be different. I can illustrate this by considering the power P in a resistor R caused by a direct current flowing through the resistor. If I choose the equation:


 * $$P = I^2 R$$ I can say the power is directly proportional to the resistance R, but if I choose the equation:


 * $$P = \frac{V^2}{R}$$ I can say the power is inversely proportional to the resistance R.


 * This is a paradox and both statements are correct. Power is directly proportional to R if the current is fixed and the voltage varies or is unknown; and power is inversely proportional to R if the voltage is fixed and the current varies or is unknown.


 * If the area S of a wing is fixed and the span varies or is unknown, the induced drag is inversely proportional to the aspect ratio. But if the span is fixed and the area varies or is unknown, we can use the following form of the equation:
 * $$D_i = \frac{L^2}{\frac{1}{2} \rho_0 V_e^2 \pi e b^2} $$


 * Using this form of the equation, the induced drag is inversely proportional to the square of the span and is independent of aspect ratio, just as Enrico has said. Dolphin  ( t ) 11:59, 5 December 2012 (UTC)


 * Interesting example there with P = I*V = I²*R = V²/R, Dolphin. You are right in stating that one has to be careful when analyzing how something depends on something else. In your example the apparent paradox raises from the fact that I, V, and R are not independent variables. Because of ohms law one variable cannot be changed without changing at least one of the other two.
 * In a similar way AR, S, and b² depend on each other according to the formula AR=b²/S. The fact is that induced drag depends only inversely on span squared. But AR and S have been “cheated “into Di´s formula; by dividing and multiplying by S. b² suddenly disappears and it looks like Di depends instead inversely on S and AR. Does now Di actually depend on S? Of course not! If this trick would hold, I could make Di depend on the diameter of the moon! All I´d have to do is introduce a new invented parameter Z=b²/diameter of the moon.
 * So allow me to put clear what I belive to be the right statement. AS FAR SURFACE, SPAN AND ASPECT RATIO ARE CONCERNED, i.e. MAINTAINING ALL THE OTHER PARAMETERS CONSTANT:
 * 1.- Induced drag depends only on the span. It decreases if span increases. It does not depend on the surface area and, in consequence, it has no unique relation with the aspect ratio.
 * 2.- Parasitic drag depends only on the surface: Dp=CD0*q*S. It decreases if surface decreases. It does not depend on the span, and, in consequence, it has no unique relation with the aspect ratio.
 * 3.- On the other hand, both increasing the span as decreasing the surface will increase the aspect ratio. As a consequence, reducing both induced drag and parasitic drag, and hence reducing the total drag, will yield a high aspect ratio.
 * Interestingly, not even the total drag has a unique relation with the aspect ratio: playing with numbers, you can find values of span and surface for A and for B that make total drag of A smaller than total drag of B and yet AR of A smaller than AR of B.Enrico Lucarelli (talk) 09:17, 6 December 2012 (UTC)


 * Enrico, I don't think it's reasonable to talk about what happens when you vary S, b, and AR only, keeping everything else fixed – because "everything else" includes the lift (L), and normally when you vary S you change L. Of course, for a given weight of plane, flying with 1 g (for example, a straight horizontal path), the lift will be equal to that given weight. But the equation in question can be applied to many other comparisons. Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)

I'm with Enrico here. Aspect ratio only affects drag if the wing area is held constant, is the span varies. If aspect ratio is varied, but the span is held constant, the induced drag WILL STAY THE SAME. That is far from obvious in this article. I've had to explain this to numerous people who look at this article and think they can reduce Di by reducing the chord... Pictsidhe (talk) 13:44, 17 July 2016 (UTC)


 * Let me clarify what I meant in my comment just above (16 Dec. 2013). What Enrico said is only true if the lift and the speed are held constant. Let's take a different case — a constant angle of attack and constant speed. Then the lift equals some coefficient of lift (which depends on the angle of attack) times the area, the density, and the square of the speed:


 * $$L=C_LS\rho_0 V_e^2$$


 * So then the lift-induced drag


 * $$D_i = \frac{L^2}{\frac{1}{2} \rho_0 V_e^2 \pi e b^2} $$


 * becomes:


 * $$D_i = \frac{C_L^2S\rho_0V_e^2}{\frac{1}{2}\pi e AR} $$


 * or


 * $$D_i = \frac{C_L^2b^2\rho_0V_e^2}{\frac{1}{2}\pi e AR^2} $$


 * So now, instead of depending only on the span, it depends on the aspect ratio. Since $$S/AR$$ is the chord squared, you could say it depends only on the chord.


 * By the way, Dolphin, I made several comments on this page after your last comment of 2013. I didn't Ping you so I don't know whether you saw them.


 * Eric Kvaalen (talk) 08:28, 25 September 2019 (UTC)

This thread was started more than 8 years ago but no significant change appears to have been made in response to the criticism by u|Enrico Lucarelli. Better late than never, I made some changes to reduce the emphasis on aspect ratio, and to remove some unsourced, non-encyclopaedic explanations of why induced drag occurs. See my diff. Dolphin ( t ) 12:29, 19 May 2021 (UTC)

Am I summarising this correctly: Induced drag does not depend on aspect ratio. It does depend on span. So we should say it depends on span, with references. cagliost (talk) 08:39, 27 May 2021 (UTC)


 * Apologies for the 13-month delay in getting you a response. It isn’t reasonable to be emphatic and say induced drag does depend on one parameter but does not depend on another. Comments about induced drag must clarify which parameters are considered fixed and which are allowed to vary. For example we can say “For a given (fixed) wing area, weight and airspeed, induced drag is inversely proportional to the square of the wingspan”; alternatively we can say “For a given (fixed) wing span, weight and airspeed, induced drag is also fixed but profile drag varies with wing area.” See the first few posts in this discussion thread. Dolphin ( t ) 05:38, 16 July 2022 (UTC)

Horizontal winglet edit
The edit made by 157.127.124.15 on 23 July 2012 states: "Of course the best way to reduce flow around the wingtip would be to have the winglet horizontal to provide lift from the extended wing, but this approach restricts the number of hangers that the aircraft may use."

The addition of this statement seems erratic in the context of the paragraph and conceptually wrong. Getting lift from this "horizontal winglet" would also generate induced drag. I might be able to accept a nonlifting airfoil as the horizontal winglet but this seems impractical, and would degredate lift on wing surfaces below the 'winglet'. I have been unable to find an example of this. Perhaps i'm interpretting this wrong.

lastly: "this approach restricts the number of hangers that the aircraft may use" - this is silly.

Thoughts? I'm going to remove this.

Viola00 (talk) 20:16, 26 February 2013 (UTC)
 * I agree that the edit did not improve the quality of the article, especially as it was unsourced. Thanks for erasing it. Dolphin  ( t ) 23:32, 26 February 2013 (UTC)

Explanation of my edit
I have done an edit comprising the following:


 * 1) Removed the diagram at the beginning because it is inaccurate. See what I wrote under for more explanation.
 * 2) Modified the sentence saying that a wing of infinite span would produce no induced drag. First of all, it's not true because the surface area would also be infinite. But also, the idea expressed in the equation saying that induced drag is inversely proportional to aspect ratio is only approximate. Induced drag is not only due to wing-tip vortices. If you have a section of wing spanning the width of a wind tunnel, and vary its angle of attack, the drag will change. (Surely the drag at 15° will be greater than at 0° or −5°.) The increase above the minimum is induced drag.
 * 3) Removed the diagram at the end (see right).[[Image:drag.jpg|right|thumb|Curves showing induced, parasitic, and combined drag vs airspeed]] It is also wrong. Maximum range occurs at the same airspeed as minimum drag (since energy equals force times distance). The minimum fuel flow rate should be at an airspeed lower than that of minimum drag, namely where drag times speed is minimized. This would be a good diagram to include if it is corrected.
 * 4) Added a diagram of lift and drag (mentioned in this Discussion page a few years ago by user:Dolphin51).
 * 5) Rearranged the last section, since the speed of minimum drag is also the speed of greatest range (though these are not constant, since the weight of the plane decreases with time). Added a parenthesis on other considerations involved in choosing the best speed and altitude.

Eric Kvaalen (talk) 08:52, 16 December 2013 (UTC)
 * Dolphin, do you have any comment on what I wrote about choosing the best speed and altitude? And do you understand the phrase about "the speed at which the best gradient of climb ... is achieved"? Eric Kvaalen (talk) 09:58, 8 January 2014 (UTC)

1) I'm sorry, but you have your very own definition for induced drag, that is not the same it means in aerodynamics. Increase of drag above minimum for an wing spanning the full width of the windtunnel has nothing to do with induced drag, which is zero in that case.

2) Parasite drag is not constant, but depends on several factors, including angle of attack, but not amount of lift. Please see the second last paragraph onn this link: https://www.grc.nasa.gov/www/k-12/airplane/inclind.html link "Since the amount of drag generated at zero angle and the location of the stall point must usually be determined experimentally, aerodynamicists include the effects of inclination in the drag coefficient. But this presents an additional problem. There is another factor which affects the amount of drag produced by a finite wing. The effect is called induced drag or drag due to lift. The flow around the wing tips of a finite wing create an "induced" angle of attack on the wing near the tips. As the angle increases, the lift coefficient increases and this changes the amount of the induced drag. To separate the effects of angle of attack on drag, and drag due to lift, aerodynamicists often use two wing models. The wing model to determine angle of attack effects is long and thin, and may span the entire tunnel to produce a "two-dimensional" airfoil. Another model is used to determine the effects of the wing tips on the drag."

3) Your statement :"Maximum range occurs at the same airspeed as minimum drag (since energy equals force times distance)" would only be correct, if efficiency would remain the same, and that is a wrong assumption in general, and very far from the truth in case on jet aircrafts, propeller planes might be close enough as a rough approximation. In jets, the simplest approximation is assuming the thrust to be linearly related to fuel bur/hour, not the power. As a result, for jets the speed of minimum drag is significantly less than the speed of greatest range. 86.50.116.35 (talk) 02:54, 3 July 2016 (UTC)


 * You didn't Ping me, so I didn't see your comment. I guess you're right that my understanding of the term "lift-induced drag" was not the way it is normally defined. It seems to me that the way it is normally defined (as given in your NASA link for school children - here's an archive link) is rather strange, maybe even useless. You can't know what it actually is except by subtracting out the part of the drag which is not due to the finite span and the vortices. To do that you have to do a wind tunnel test with an "infinite wing", that is, a wing that spans the whole wind tunnel. And they talk about an induced angle of attack, but how do you define the direction of the wind on the wing. It all depends where you look!


 * On your third point, you're right, I forgot about the fact that efficiency is not constant. But I don't think the approximation that thrust is proportional to fuel burn per hour, independent of speed, is good. The thrust-specific fuel consumption must go up with speed, eventually becoming more or less proportional to speed. That means that the rate of fuel consumption will have its minimum (as a function of speed) at a lower speed than where the drag is minimized. But (as you say) the speed for maximum range will be higher (at least for jets) than the speed where the drag is minimized.


 * Eric Kvaalen (talk) 08:28, 25 September 2019 (UTC)

Wing Fences
I removed the unreferenced mention of wing fences from the "Reducing induced drag" section. http://www.aerospaceweb.org/question/aerodynamics/q0228.shtml, referenced at wing fence, says fences create vortices to delay wing stall, but vortices also create drag. Can fences reduce induced drag? Burninthruthesky (talk) 19:37, 2 May 2015 (UTC)
 * Anything that alters the spanwise lift distribution so that it more closely resembles an elliptical shape will reduce the lift-induced drag. I can imagine wing fences causing jump discontinuities in the spanwise distribution but I can't imagine them promoting an elliptical distribution. Therefore I think we have to be sceptical about any claim that they reduce lift-induced drag. Besides, lift-induced drag is more of a problem in low-speed aircraft whereas fences are mostly a device for dealing with problems in transonic and supersonic aircraft. Dolphin  ( t ) 07:14, 3 May 2015 (UTC)

Thesis material crammed in as section "Minimum induced drag for generic non-planar systems"
Hi, I'm trying to help improve this article, and I had a lot of changes reversed, which I understand because I made a lot of them.

So I'll just start with my questions, suggestions, changes one at a time to start the discussion.

I like the topic introduced in "Minimum induced drag for generic non-planar systems" section, and want to keep the final figure: "Nonplanar wings: results", with the text to introduce it. "Demasi Luciano et al discuss the efficiency of nonplanar wings. Here the efficiency is the ratio between its aerodynamic efficiency and the efficiency of a cantilevered wing with the same wing span and lift."

But the rest of the section seems to be a simple paste of a few paragraphs from a thesis/paper, and not at all in encyclopedic style or level. I propose the rest of this section be deleted. Any objections? Hess88 (talk) 02:08, 15 April 2016 (UTC)


 * I think the non planar section would better illustrated by this article: http://adg.stanford.edu/aa241/wingdesign/nonplanar.html Pictsidhe (talk) 14:00, 17 July 2016 (UTC)

Effect of induced drag
On 22 March 2022 inserted a new section titled Effect of induced drag. This new section begins “A 2000 study found that for commercial airliners, induced drag was the second-largest component of total drag, at 37%.” Two sources are cited but no mention is made of page number or sub-section number etc.

This new section is incompatible with the rest of the article which is based on a classical analytical explanation of lift-induced drag and the manner in which it varies with lift coefficient. The article contains a useful diagram showing the way induced drag and parasitic drag combine to constitute total drag. It clearly shows that, at high speeds, induced drag is a diminishing component of total drag; and at low speeds, approaching the stall, induced drag approaches 100% of total drag. Clearly, the statement that induced drag is 37% of total drag is talking about something very different to what the majority of the article is talking about. Perhaps the 2000 study is addressing the overall economic cost of induced drag.

I have perused the two cited sources but I found nothing to support the new section. Readers wishing to verify the content of the new section shouldn’t have to read an entire source document to find that verification; it should be quickly accessible by navigating to the cited page number, sub-section etc.

In its present state, this new section is unsatisfactory. It might be possible for it to be improved to the point that it becomes satisfactory. If not, it should be removed. Dolphin ( t ) 13:40, 23 March 2022 (UTC)


 * I've added page numbers, and updated the article to indicate the figures refer to "a typical civil transport aircraft".
 * This new section is useful because it gives the reader some idea of how significant induced drag is, compared to skin friction drag etc. Even though drag varies with speed and the type of aircraft, there are going to be typical ratios for typical flight regimes, such as cruise speed at cruise altitude.
 * Unfortunately the sources do not say whether the figures refer to the total fuel cost over the whole flight, or only the fuel cost during cruise flight, or something else. I don't want to make more specific claims that can't be justified by the source material, but the solution should not be to remove the whole section. Feel free to change the wording if you think necessary. cagliost (talk) 14:19, 23 March 2022 (UTC)


 * Do the sources describe the figures as referring to fuel cost? The new section would have more meaning if the concept of fuel cost were introduced. The statement that induced drag is 37% of total drag is incompatible with the rest of the article to the point that many readers will be able to point to it and say it is incorrect. Dolphin ( t ) 00:47, 24 March 2022 (UTC)


 * The context in source Marec 2000 is fuel efficiency, but it doesn't say that outright. It is taken as obvious, and I don't think it will confuse Wikipedia readers either. The full relevant quote from Marec is: "Figure 4 recalls the drag breakdown of a typical civil transport aircraft. It shows that skin friction drag and lift induced drag represent together more than 80% of the total drag and may offer the highest potential for drag reduction. The other components only represent about 20%, but cannot be neglected. // Fig. 4 - Drag breakdown of a typical transport aircraft."
 * I think you're making a mountain out of a molehill. Readers will understand that induced drag doesn't always account for 37% of total drag, but that it is an average figure in typical conditions. cagliost (talk) 11:14, 24 March 2022 (UTC)
 * I've folded the section into the "Reducing induced drag" section, and added a bit about cost reduction to provide context. cagliost (talk) 11:20, 24 March 2022 (UTC)


 * The article quotes two figures - 37% and almost 48%, but neither you nor I is confident that we know what those figures mean. That suggests the sources may not be sufficiently reliable as documents presenting information.


 * I suggest we either change the text to say something like “induced drag accounts for 37% of the fuel consumed in a typical flight”; or we remove the precise figures 37% and 48% and simply say induced drag is the second largest component of drag during a typical flight, and skin friction drag is the largest component.


 * It isn’t reasonable to insert some information that we don’t fully understand, on the grounds that we hope readers will be able to understand it. Dolphin ( t ) 14:58, 24 March 2022 (UTC)


 * I found "Control of Turbulent Flows for Skin Friction Drag Reduction" by Eric Coustols, in "Control of Flow Instabilities and Unsteady Flows" edited by G.E.A. Meier, G.H. Schnerr, pg 156]. "These two drags account for respectively 48% and 37% of the total drag of a modern subsonic transport aircraft."
 * Coustols cites: Robert J.P.: Drag reduction: an industrial challenge, in: Special Course on Skin Friction Drag Reduction (ed. J. Cousteix) AGARD Report 786, 1992, Paper 2. Pg 2-13, Figure 3. This simply says "% Total aircraft drag" of a "Typical wide-body twin in cruise".
 * "in cruise" is the crucial phrase, we now have a specific and meaningful claim. I'll update the article accordingly. cagliost (talk) 16:21, 24 March 2022 (UTC)

Diagram incorrect?
This famous diagram labels the backwards component of the aerodynamic reaction force "induced drag". However, isn't that wrong? I think this backwards component should be labelled "total drag". cagliost (talk) 12:30, 25 March 2022 (UTC)


 * The diagram is correct. See the discussion above under the sub-heading “Diagrams”. Dolphin ( t ) 13:41, 25 March 2022 (UTC)

Relevance of profile drag
HI Cagliost. The article presently contains the sentence “A two-dimensional wing can still reduce drag for a given lift, by travelling faster and reducing its angle of attack, therefore reducing profile drag.” It is tagged “citation needed”. On 22 April you responded by providing an in-line citation pointing to a NASA web site. See your diff. Thanks for providing that.

I have perused the text you cited in support of the sentence but I can find nothing to support the sentence quoted above. If you still believe the NASA website contains some words that are truly relevant to the sentence in question, please let me know which words you have in mind. You can do so by quoting the words exactly, like this: “To separate the effects of angle of attack on drag, and drag due to lift, aerodynamicists often use two wing models.”

Secondly, I would appreciate your view on why you think a sentence dedicated to reducing profile drag deserves mention in a sub-section titled “Reducing induced drag”. Thanks. Dolphin</i> ( t ) 13:34, 23 April 2022 (UTC)


 * Ten days have passed so I will remove the offending sentence. <i style="color: green;">Dolphin</i> ( t ) 11:34, 3 May 2022 (UTC)

Disagree. The source provided, titled "Inclination Effects on Drag", clearly states "as angle increases, drag increases". It's relevant because lift-induced drag is not the only cause of the effect where (at slow speeds) drag decreases as speed increases. It's helpful to the reader to understand that there are other causes of this effect, otherwise they might be left with the impression that lift-induced drag is the only cause. cagliost (talk) 11:40, 3 May 2022 (UTC)


 * Are you suggesting the five words “As angle increases, drag increases” serve as verification for the sentence, about 25 words, quoted at the start of this thread? <i style="color: green;">Dolphin</i> ( t ) 01:15, 4 May 2022 (UTC)


 * I am not sure on the status of this content but yes, reducing AoA and thereby reducing induced drag to increase speed for the same amount of lift, is what every plane does to get away from an approaching stall. There is of course a sweet spot, known as cruising speed, above which drag rises excessively again despite ever-reducing AoA, but that is down to form drag and not to induced drag. This is such basic stuff it should not need inline citation, though if somebody still insists, there must be dozens of suitable sources; I must have several on my shelves. &mdash; Cheers, Steelpillow (Talk) 09:08, 16 July 2022 (UTC)


 * I suspect your comments aren’t closely related to the subject of this discussion thread. There was a sentence, since deleted, that began “A two-dimensional wing ... ...” In contrast, you have written “... is what every plane does ...” As you know, every aeroplane has at least one wing and it is three-dimensional.
 * The offending sentence included “... by travelling faster ... therefore reducing profile drag.” You have correctly contradicted this suggestion by writing “... drag rises excessively... but that is down to form drag ...”
 * However, if one of the sources on your bookshelf supports the idea that, generally speaking, profile drag can be reduced by travelling faster, please let us know what that source says. It will make interesting reading. <i style="color: green;">Dolphin</i> ( t ) 10:42, 16 July 2022 (UTC)
 * Silly me. I assumed that "two-dimensional" refers to the analytical method. I also assumed that the material was discussing the article topic, i.e. induced drag, so I missed the fact that it was explicitly about profile drag. Also, I agree that sourcing "as angle increases, drag increases" does not qualify whether this is net drag or profile drag, nor whether it is for constant lift or constant speed; my assumption (ooer!) would be that it is net drag at a given speed, however for constant lift the plane would slow down at higher AoA (c.f. Concorde) and all bets are off. All in all, this is such an irrelevant mess that I have somewhat changed my mind and I agree that it is probably best avoided. &mdash; Cheers, Steelpillow (Talk) 11:31, 16 July 2022 (UTC)