Explaing lift tilt back in induce drag pictures?

Question

This is usual induced drag diagram.

What is black vector in diagram below?

If it is lift you cant decompose lift in lift(L in diagram) and induced drag (Di), this you can do only if this is resultant force(aerodynamic force) BUT If this is resultant force why is always drawn perpendicular to effective airflow(which is impossible in real fluid), resultant force is always slightly back from line that is perpendicular to eff. airflow?

Can someone explain this and draw vectors of lift and drag, without effective airflow and with eff. airflow? (keep in mind change in magnitudes of each vectors)

Answer 1

We start from scratch.

I think this is the best way to truly understand the concept and get rid of all these pseudo-explenations which are as correct as the "equal transit time" theory.

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So, we consider a wing as boring as possible i.e. a wing with rectangular planform, constant lift, no sweep, no taper, no twist, ... and obviously no tip so that there's no tip vortex either. In short we consider an "infinite rectangular wing" (aka "airfoil"). And to be even more boring, we consider the airflow as being inviscid (no boundary layer) and stationary (no manoeuvres) as well. In this way we have removed from our analysis all the possible aerodynamic forces except the ones that we need: lift and induced drag.

The airflow around such an airfoil is well known and looks like this:

(Source)

Schematically, the airflow goes up in front of the airfoil, accelerates on its upper part, goes down behind it and decelerates on the lower part. If the freestream airflow's speed could be subtracted, a "circulation" of air around the airfoil would be visible:

(Source: Daniel P. Raymer, Aircraft Design: A Conceptual Approach)

This circulation of air is what creates lift on the airfoil. What about drag? There's none. Why that? Because that circulation of air creates an aerodynamic force which can be only perpendicular to the freestream velocity. And this is lift by definition (at least in this case). Mathematically all this is translated via the famous Kutta–Joukowski theorem:

$L=\rho V \Gamma$

where $\Gamma$ (gamma) is the mathematical symbol used to represent the circulation.

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So, we got lift how do we get (induced) drag then? Easy! we give tips to our wing. Now lift cannot be constant anymore. Why that? Because now the wing ends and where it ends lift goes (more or less suddenly) to zero. One mm before the tip we have lift, one mm outside the tip we have zero lift. Why is this important? Because everywhere lift changes spanwise, another circulation originates around the wing but this time on the wing as seen from behind. This circulation (which I depicted with blue arrows in the next picture) is then transported back by the freestream:

(Source: this answer.)

Lift can change spanwise for several reasons: wing ends, airfoil geometry changes, chord (taper) changes, AoA (twist) changes, flap ends (like in the picture), ... Now, for the same reason for which the previous circulation around the airfoil could create a force exclusively perpendicular to the freestream (a lift), now, for exactly the same reason, this circulation released backward can only create a force parallel to the freestream. And this is drag by definition. Since this drag is induced by the spanwise variation of lift, it is called induced drag.

And if we expand a bit the mathematics of the Kutta–Joukowski theorem taking into account this second vortex as well, we get the famous relationship between lift and induced drag:

$C_{D_i} = \frac{C²_L}{\pi AR e}$

That's all actually. But...

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If one writes that the Kutta–Joukowski theorem explains that the aerodynamic force is proportional to the circulation around the airfoil and perpendicular to the freestream, everybody believes in and is happy with that. But somehow if one says the same thing to explain induced drag, this is not enough. Somehow something more tangible is needed for the induced drag to be understood. And here is where the confusing plethora of explanations about induced drag kicks in.

So, if you really needs a picture, I'd suggest the following one:

(Source: chapter 12 of this lectures by Brian J. Cantwell of Stanford university)

The upper part of picture shows again the circulation around the airfoil responsible for the lift. The bottom picture shows instead which is the equivalent effect on the airfoil of the second vortex (the one due to the end of lift): it bends the airflow locally downward by $U_z$. Is this equivalent effect physically correct? I don't know, but what I do know is that Kutta–Joukowski is still valid: an aerodynamic force is generated due to the circulation and its orientation is perpendicular to the airflow, just like before but! now the perpendicular orientation is the one titled backward and that's why the aerodynamic force is also drawn tilted backward:

(Source as before)

Now there's nothing left to do but decompose this "normal" aerodynamic force in a component perpendicular to the freestream (lift) and a component parallel to it (induced drag).

Answer 2

Context and intent are critical. Like so many other aerodynamic concepts, Lift and Drag are not real things, they are engineering, or mathematical constructs defined, designed and created to serve an engineering or calculation-based purpose. The tilt back you see in the diagram is probably because the author was trying to explain the Lift on the wing airfoil, not the airframe it might be attached to. Then, the Lift would be defined as the component of total aerodynamic force on the wing airfoil normal (perpendicular) to the airflow across the airfoil, not the airflow across the airframe.

Answer 3

A similar image can be found in the Wikipedia entry on "Induced drag"-- reproduced below:¹

The accompanying text reads (bolding and italics added):

"Induced drag is related to the amount of induced downwash in the vicinity of the wing. The grey vertical line labeled "L" is perpendicular to the free stream and indicates the orientation of the lift on the wing. The red vector labeled "L_eff" is perpendicular to the actual airflow in the vicinity of the wing; it represents the lift on the airfoil section in two–dimensional flow at the same angle of attack. The lift generated by the wing has been tilted rearwards through an angle equal to the angle of the downwash in three-dimensional flow. The component of "L_eff" parallel to the free stream is the induced drag on the wing."

Sources cited for the illustration and accompanying text include:

Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30, NAVWEPS 00-80T-80

Clancy, L.J. (1975) Aerodynamics. Pitman Publishing Limited, London. ISBN 0-273-01120-0

Kermode, A.C. (1972). Mechanics of Flight, Figure 3.29, Ninth edition. Longman Scientific & Technical, England. ISBN 0-582-42254-X

McLean, Doug (2005). Wingtip Devices: What They Do and How They Do It (PDF). 2005 Boeing Performance and Flight Operations Engineering Conference.

Note that only the vector that is perpendicular to the free-stream relative wind-- i.e. the grey arrow marked "L" -- truly meets the strict definition of a Lift vector. The free-stream relative wind is the direction of the relative wind far in front of the aircraft, beyond any influence of the aircraft on the direction of the local relative wind.

Note also that the concept of "effective relative airflow" is only a theoretical one-- at nearly every point near the airfoil, the direction of the actual relative airflow is different than shown by the dashed red line. The actual relative airflow curves upward as it approaches the airfoil, curves downward behind the airfoil, etc in a complex pattern.

The explanation is apparently suggesting that this net change in the overall direction of the "effective relative airflow" would not exist in 2-D flow, e.g. if the airfoil fully spanned the test section of the a wind tunnel, meeting the wall on each side. It's far from obvious why this should be so. Certainly in such a case, there would still be some drag associated with the production of lift, but perhaps by some definitions this drag would not include any component called "induced drag".

This ASE answer is not intended to represent that this explanation is, or is not, a fully valid explanation of the origin of induced drag!

At any rate, the question seems to be trying to understand which of the vectors in the diagram can be seen as the "decomposed" constituents of other vectors. In the diagram in the original question, the heavy black arrow-- which is the vector labelled "L_eff" in this answer-- is the vector sum of the "L" (Lift) vector and the "D_i" (Induced Drag) vector. Therefore Induced Drag can be viewed as resulting directly from the fact that the L_eff vector is tilted aft, due to the effective relative airflow being inclined downward due to downwash. In the diagram attached to this answer, it appears that the intention is the same, though a close look shows that the vectors as actually drawn do not quite add up this way.

For more on this concept, it may be very enlightening to read the "Induced Drag" section of the "Aerodynamics for Naval Aviators" text by H.H. Hurt Jr, revised 1965. A pdf is downloadable from the FAA website here, and the relevant content starts on page 66. Note that this source specifically addresses the upwash in front of the airfoil, as well as the downwash behind the airfoil. Differences between 2-D and 3-D flow, and the significance of the "bound vortex system" that develops in 3-D flow, are addressed in detail (see especially pages 61-66.) Here's a "teaser" from page 66 that may inspire some to explore further: "Hence, the sections of the wing operate in an average relative wind which is inclined downward one-half the final downwash angle."

If you are interested in understanding the flow of air around airfoils, including the concept of "circulation", you may also find it worthwhile to read the "Airfoils and Airflow" section of John S Denker's excellent "See How It Flies" website.

BUT If this is resultant force why is always drawn perpendicular to effective airflow (which is impossible in real fluid)

These diagrams are not attempting to make any representation that L_eff is the net resultant force arising from the interaction between the wing and the airflow. In real fluid, it goes without saying that there is an additional Drag component, separate from the Induced Drag component, which could be drawn perpendicular to the L_eff arrow, or perpendicular to the L arrow, according to whichever better suited our purposes. (Only the latter would truly meet the strict definition of a Drag component.) Therefore there is no valid objection that the situation depicted by these diagrams "is impossible in real fluid." The remaining Drag component has simply been omitted for clarity, just as the Weight vector has been. The intent of the diagrams is only to explain the origin of the Induced Drag component, not to show all the forces, or even all the aerodynamic forces, acting on the wing.

Footnotes:

1. Accessed 11/17/2022

Answer 4

I find answer in book from aerospace university.

Black vector is resultant force(Fr), but some authors like Doug Mclean(Understanding Aerodynamics: Arguing from the Real Physics) call it "apparent lift vector".

"If this is resultant force why is always drawn perpendicular to effective airflow?(which is impossible in real fluid)"

Because from Kutta–Joukowski theorem resultant force(Fr) per unit of span is perpendicular to effective airflow.

(Remember that this theory refer to inviscid flow.)

So if downwash is zero, resultant force is now perpendicular to relative airflow(at picture), so induced drag is zero.