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I know simmilar questions are all over the internet (and this website too), but there's one thing that i just cannot comprehend and none of the articles i read could explain it properly.

According to wikipedia the coeficient of drag depends on the aspect ratio by the following way: $$ C_D=C_{D0}+ \frac{(C_L)^2}{\pi e AR} $$ But when i search why does it have this negative impact, i read things like "high AR creates stronger vortices at the tip of wings, which reduce lift".
But reducing lift reduces the numerator of the expression, so that should in fact REDUCE the drag. Does the $C_L$ somehow also depend on $AR$? If yes, is the dependancy also linear or less?

And another thing that blows my mind are statements, that wing which spans the whole width of a wind tunel produces no induced drag, because it has no wing tips. Are you telling me that the induced drag is only caused by wing tips? From the pictures at How complete is our understanding of lift? one would think that drag is natural part of lift and is caused by the airfoil even in 2D without considering wingtips.

So to sum up, can you please properly explain me:

  1. How does the aspect ratio influence induced drag, why is the AR in denominator
  2. How does it influence lift
  3. What really happens at the wing tips
  4. What happens in the wind tunel on a wing that spans its whole width
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  • $\begingroup$ low aspect would create stronger vortices for a given wing area. There is good and bad info here, but in time in sorts out. My understanding is induced drag is caused by raising angle of attack to create lift. This also increases drag, so induced drag is added to CDo. Hope info here helps clear it up. $\endgroup$ – Robert DiGiovanni Dec 27 '18 at 2:18
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Your suspicion is leading you in the right direction. A lot of what you read on the Internet about induced drag is at least misleading if not blatantly wrong. But it is repeated by sloppy authors who don't ask the right questions.

If you have read stuff like "high AR creates stronger vortices at the tip of wings, which reduce lift", you hit such a completely wrong statement. The misleading version of it is high AR creates weaker vortices. For a hopefully better explanation, please turn to this answer.

Now to your questions:

  1. How does the aspect ratio influence induced drag, why is the AR in denominator?

The surprising answer: It is the lift per unit of span and the inverse of the square of flight speed which determine induced drag. Another influence is the distribution of lift over span, but aspect ratio does not play a role. See this excellent answer by @DeltaLima for an explanation, or this one for a longer one.

  1. How does it influence lift?

Indirectly, by increasing the lift curve slope.

  1. What really happens at the wing tips?

On the top side, suction turns the airflow inwards so the air gets a rotational component and some of it flows around the tip from bottom to top. Due to the small wingtip curvature, this vortex has low pressure at its core (and lets air condense early), but is overall weak and rather insignificant. The real trailing vortex results from a rollup of the wake, but even this does not create drag.

  1. What happens in the wind tunnel on a wing that spans its whole width?

Please read the answers to this question.

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  • $\begingroup$ I have complementary question. You wrote "A higher wingspan allows to capture more air for lift creation". Which means at lower AR, more air is "leaking" at the tips and not participating in creating lift. But it still participates in creating drag, why? The linked answers just play with transformation of the equation to show the relations mathematically. But i'm searching more for an aswer WHY the equation looks as it does. Let's say i want to keep total wing area same. Why does the low AR negatively impact drag, why the leaked air doesn't create lift, but still creates induced drag? $\endgroup$ – Youda008 Dec 29 '18 at 12:52
  • $\begingroup$ Also, when they break up the equation and show that induced drag in fact depend on span, i wonder what happens if i increase the wing area chord-wise. Why does span-wise increase of area reduce induced drag, but chord-wise increase of area doesn't? $\endgroup$ – Youda008 Dec 29 '18 at 15:08
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    $\begingroup$ @Youda008: There is no air which creates only lift or drag. All air creates an aerodynamic force which is per definition split into lift and drag components.Lower AR means more deflection of less air for the same lift force, and more deflection means more backward tilt of the aero force. The backward-facing component of this force (without friction-related components) is induced drag. $\endgroup$ – Peter Kämpf Dec 29 '18 at 15:43
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    $\begingroup$ @Youda008: More chord only means that the same air can be influenced over a longer flow path. This means more lift, but also more backward tilt. More span can influence more air which needs less deflection to produce the same lift. Less deflection = less backward tilt = lower induced drag. If that does not help, I am out of my wits to explain it. $\endgroup$ – Peter Kämpf Dec 29 '18 at 15:47
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    $\begingroup$ @Youda008: I prefer to say "lengthening the wing chord-wise", but yes, it is roughly equivalent. $\endgroup$ – Peter Kämpf Dec 29 '18 at 23:16
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Well, let's pick those pieces of your mind up and get this sorted out.

CDtotal = CDo + Clsquared/pi x e x AR

Reading the NASA website the following is deduced.

  1. CDo is the zero lift drag, essentially everything except what produces lift. For simplicity they only consider induced drag from the wing, even though fuselage and tail drag will change with AoA.

  2. The rest is induced drag from the wing. Drag increases with increase in Cl squared, more drag is a product of more lift. Drag decreases with increased AR. Look at a sailplane.

Why does increased Aspect ratio make a wing more efficient (more lift per unit of thrust) for the same area? Simply because there is a pressure "leak" at the end of the wing, making it produce less lift per area than the rest of the wing. The smaller the wingtip chord is relative to the area of the wing, the smaller the percentage of this lift loss is relative to total lift. There for, a lower aspect wing of equal area needs a higher AOA to produce the same lift, producing more drag (and more energetic vorticies). The plane must add more power to maintain velocity, hence the higher aspect wing is more efficient. This is how aspect ratio is related to drag, and why it's value is in the denominator.

Remember, vortices are the RESULT of lift creation. They are literally an expression of how much energy the passing wing has imparted to the air. Lift is how efficiently the air moving energy was used. Drag = Thrust!

Finally, a wing that spans the entire wind tunnel would be useful to focus on the airfoil shape performance at various AoA and to provide an "infinite" AR data point to compare with a range of AR to be tested.

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