Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Why does induced drag decrease as airspeed increases?

Hello,

induced drag is a result of the wing vortices inducing downwash - a velocity component w perpendicular to the oncoming flow direction. My first question is: is this really the full truth? Is there really no velocity component induced in the direction of the flow? Why should there only be a component perpendicular to the flow? So that means no matter what AOA, the wingtip vortices only induce a velocity component perpendicular to the oncoming flow although they result from a spanwise flow of air and therefore should rotate in a plane perpendicular to the airfoil chord and not the oncoming flow direction.

My actual question is: let's assume a steady-state horizontal flight, i.e. L = W and F = D. Now I increase the airspeed - to maintain the current state I decrease the AOA because as an increase of airspeed leads to a higher lift L, to hold it constant I therefore decrease the AOA. So at the end there's no change in lift L, but still, the induced drag changes. I thought the induced drag is a direct result of the pressure difference between the upper surface and the lower surface of the wing. So as there's no change in lift there's no change in pressure difference and hence there should be no change regarding the vortices and the induced drag. But apparently that's not the case. So what's the explanation for that?

I hope someone can clear this problem up for me. Thanks!

Regards,

B737200
Posts: 211
Joined: Sun Feb 20, 2005 5:30 pm

### Re: Why does induced drag decrease as airspeed increases?

Disclaimer: It has been a while since I studied this and don't use it day to day so others may be able to give you a more accurate reply.

Firstly let us look at the induced drag coefficient and drag force:
Let your induced drag force be D_i and your Induced Drag Coefficient be C_DI:
==> D_I = C_DI*(1/2*rho*v^2)*S

Where,
rho is the air density,
v is the freestream velocity
S is your wing surface area.

Therefore, for a given C_DI the induced drag force will increase the faster the go.

Now let us look at C_DI:

Lifting line theory provides the following result:
C_DI=[([C_L]^2)/(Pi*AR)]*(1+d)

Where,
C_DI is the induced drag coefficient.
C_L is the lift coefficient
Pi is the number pi.
AR is the wing aspect ratio.
d is a number made up of the fourier series coefficients A3, A5, A7.... these are derived from the monoplane equation which I will not go into.

Therefore we can see that C_DI increases as C_L increases and vice versa.

So, looking at both equations:
1) It is true that the faster you go the more drag you will be producing, however this only holds for a specific C_DI.
2) When you change the angle of attack you will also be changing the AOA and changing the CL.
3) Whilst there is a direct increase in induced drag as velocity increases (for a specific C_DI), for level flight, if we reduce the AOA to reduce the CL we will have a drop a corresponding drop in C_DI.

B737200
Posts: 211
Joined: Sun Feb 20, 2005 5:30 pm

### Re: Why does induced drag decrease as airspeed increases?

Combining the two equations:

D_I = C_DI*(1/2*rho*v^2)*S
D_I = (([C_L]^2)/(Pi*AR)]*(1+d))*(1/2*rho*v^2)*S
D_I = {([C_L]^2)*(1+d)*(rho*V^2)*S}/(2*Pi*AR)

But looking at the lift equation:

L=C_L*1/2*rho*v^2*S
C_L=2L/(rho*v^2*S)

Substituting into the above copied:

D_I = {([2L/(rho*v^2*S)]^2)*(1+d)*(rho*V^2)*S}/(2*Pi*AR)
D_I = {[(4L^2)/(rho^2*V^4*S^2]*(1+d)*(rho*v^2)*S}/(2*Pi*AR)
D_I = (4L^2)*(1+d)*[((rho*v^2)*S)/(rho^2*v^4*S^2)]/(2*Pi*AR)
D_I = (4L^2)*(1+d)*(1/(rho*v^2*S))/(2*Pi*AR)
D_I = [(4L^2)*(1+d)] / (2*Pi*AR*rho*v^2*S)

Therefore, assuming I didn't make a mistake in my maths, the induced drag D_I will be proportionate to 1/v^2 for a given weight if we assume that L=W, i.e. level flight. The higher the speed the lower the D_I.

We are gaining more drag directly because of the faster speed but loosing even more from the lower C_L caused by the faster speed. Kind of like we gain +5 drag units but lose -10 drag units so the sum is +5-10 = -10 units.

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

Thank you for your response! But that doesn't quite answer my question.
No change in lift, airspeed increase, AOA decrease yield an decrease of induced drag. What's the (physical) cause for that? Because as I've already said, it is my understanding that induced drag only changes as lift changes.

And regarding the rotation plane of the vortices, a delta wing taking off with high AOA is a more extreme example:

But I still can't really tell if the vortices are rotating in the plane perpendicular to the wing surface / airfoil chord (i.e. in the z-y-plane in that image) and hence also inducing a velocity component in the opposite direction of the oncoming flow or if they rotate in the plane perpendicular to the oncoming flow direction.

QuarkFly
Posts: 336
Joined: Mon Aug 15, 2016 4:20 pm

### Re: Why does induced drag decrease as airspeed increases?

Simple answer without getting into equations and vortices…

Induced drag is caused by the wing lift, the lift produces the downwash of air required by Newton’s third law, the equal-and-opposite of lift. When an aircraft goes faster, this downwash of air is spread out over a larger area during a given time, thus the wing has to work less hard to produce the downwash…so the related induced drag is also smaller.

In fact, induced drag drops in proportion to the aircraft’s velocity-squared for the same lift.
Always take the Red Eye if possible

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

I found these two statements (http://www.paraglidingforum.com/viewtopic.php?t=31131):

Josh Cohn wrote:
Lift is generated by deflecting air downwards (actually perpendicular to the glide path). At higher speeds the deflection is spread out over a longer distance per unit time. This gives the air less opportunity to flow spanwise around the wing, which would generate tip vortices.

You can feel and see this effect if you skim your hand over the surface of a pool slower and faster.

To generate lift, the wing accelerates air downwards, applying (inducing) a downwards velocity (v) to the mass (m) of air it encounters in unit time (t) so that in steady flight mv/t = total lift.

At higher speeds the wing encounters more mass of air. m is larger so v must be smaller if mv/t = total lift.

The kinetic energy lost to the wake is 1/2mv^2. So at higher speeds where m is larger and v is smaller the kinetic energy lost in the wake is lower. Therefore induced drag decreases as speed increases.

WIederling
Posts: 7333
Joined: Sun Sep 13, 2015 2:15 pm

### Re: Why does induced drag decrease as airspeed increases?

Actually the same mechanics you see in propulsive efficiency.
With I(mpulse) constant increase M and reduce delta V.
Murphy is an optimist

Florianopolis
Posts: 319
Joined: Sun Mar 29, 2015 2:54 pm

### Re: Why does induced drag decrease as airspeed increases?

Don't forget your lift vector also has a backwards component, which is more backwards at higher angles of attack.

This has a pretty good discussion of the topic, although it is a little technical. Starts on page 66, runs for eight pages.
www.faa.gov/regulations_policies/handbo ... 80T-80.pdf

Afstand
Posts: 11
Joined: Wed Jun 22, 2016 11:51 am

### Re: Why does induced drag decrease as airspeed increases?

It is very simple without complex formulas: flying slower you "through down" less air, and to maintain the same impulse you need to through it down faster. The energy needed is proportional to the mass and proportional to the square of speed. So flying two times faster you through down two times more airmass, but at a half of a velocity, thus using half as much energy.

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

Florianopolis wrote:
Don't forget your lift vector also has a backwards component, which is more backwards at higher angles of attack.

Yes, but then I ask why's that? There's no direct physical explanation why the backwards component (induced drag) is greater just because of a higher AOA. The backwards component is greater because the effective lift is higher when increasing the AOA while keeping the speed constant or because the speed has to be lower when increasing the AOA to keep the effective lift constant. Both lift (pressure difference) and speed (see explanations above) give the direct physical explanation for the resulting backwards component / induced drag - at least for me.

In other words: the lift generated by the wing doesn't tilt rearwards because of the wing tilting relative to the free stream (AOA) but because lift (pressure difference) itself results in induced drag and airspeed has an impact on the induced drag as described in the previous posts.

Andre3K
Posts: 365
Joined: Tue May 30, 2017 10:11 pm

### Re: Why does induced drag decrease as airspeed increases?

Think of it like an aircraft taking off on floats, or a speed boat. The water (air) has more time to contact the surface and cause an interaction at slower speeds, and it takes a greater angle of attack to displace a certain amount of fluid(water or air). As the speed increases the angle of attack and thus also the amount of time (or the amount of "stuff) the fluid comes into contact with the wing or hull decreases.

Just a random thought that's probably way off. I'd go with the super technical explanations above.

mxaxai
Posts: 732
Joined: Sat Jun 18, 2016 7:29 am

### Re: Why does induced drag decrease as airspeed increases?

Florianopolis wrote:
Don't forget your lift vector also has a backwards component, which is more backwards at higher angles of attack.

Yes, but then I ask why's that? There's no direct physical explanation why the backwards component (induced drag) is greater just because of a higher AOA. The backwards component is greater because the effective lift is higher when increasing the AOA while keeping the speed constant or because the speed has to be lower when increasing the AOA to keep the effective lift constant. Both lift (pressure difference) and speed (see explanations above) give the direct physical explanation for the resulting backwards component / induced drag - at least for me.

In other words: the lift generated by the wing doesn't tilt rearwards because of the wing tilting relative to the free stream (AOA) but because lift (pressure difference) itself results in induced drag and airspeed has an impact on the induced drag as described in the previous posts.

I'd explain it like this: on an ideal wing, you have high pressure below and low pressure above the wing, creating lift. If you tilt the wing (increase AOA), your pressure difference increases (more lift) but some of the low pressure is now behind the wing, whereas some of the high pressure is in front of the wing. This "pushes" the wing back.
I don't think that this is the reason for increased induced drag at low speeds/high AOA, though.

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

mxaxai wrote:
I'd explain it like this: on an ideal wing, you have high pressure below and low pressure above the wing, creating lift. If you tilt the wing (increase AOA), your pressure difference increases (more lift) but some of the low pressure is now behind the wing, whereas some of the high pressure is in front of the wing. This "pushes" the wing back.
I don't think that this is the reason for increased induced drag at low speeds/high AOA, though.

I don't think that's 100% accurate. That sounds like induced drag is a direct result of the tilt of the wing:
"some of the low pressure is now behind the wing, whereas some of the high pressure is in front of the wing. This "pushes" the wing back."
That would also mean, if you tilt / increase the AOA of a 2D wing, i.e. a wing with infinite aspect ratio, you would also get induced drag but that's not the case.

Induced drag is always about vortices inducing downwash and as a result influencing the pressure distribution. Greater pressure difference results in greater strength of the vortices which leads to greater downwash resulting in greater induced drag.

But yeah overall there must be a horizontal (= direction of free stream) pressure difference so that a force acting in the direction of the free stream can exist. At every point of an airfoil (a cross-section of the wing) acts a specific force (per unit span) perpendicular to the surface due to the local static pressure. Lower static pressure (relative to the ambient pressure) for example means that the force acting on the surface is lower (in comparison to the force that would result from the ambient pressure). So when you sum up the horizontal component of every force vector and you get something different than zero that means that there's drag. On the other hand, if you sum up the vertical (= perpendicular to free stream) component of every force vector you would get the lift (per unit span). In conclusion, the vortices' downwash influences the pressure distribution resulting in a horizontal pressure difference, which is the induced drag.

B737200
Posts: 211
Joined: Sun Feb 20, 2005 5:30 pm

### Re: Why does induced drag decrease as airspeed increases?

mxaxai wrote:
I don't think that's 100% accurate. That sounds like induced drag is a direct result of the tilt of the wing:
"some of the low pressure is now behind the wing, whereas some of the high pressure is in front of the wing. This "pushes" the wing back."
That would also mean, if you tilt / increase the AOA of a 2D wing, i.e. a wing with infinite aspect ratio, you would also get induced drag but that's not the case.

I believe you are right with this statement, that would be the form drag of the wing, minus drag from skin friction. It something which I would expect an infinite wing to have, though not calculable through thin aerofoil theory. As you said the induced drag is only generated by a finite wing.

The downwash as you said causes the induced drag but it is related to the C_L of the wing, that is how "hard" the wing is working to create the lift. At slower speeds the wing has to work "harder" by deflecting the air more than it would have to if the speed was higher. Hence why above it was discussed that as C_L decreases, i.e. the wing is working less "hard", the induced drag actually decreases for a specific value of lift.

As to why the downwash increases with increased C_L I honestly cannot remember. I am not sure what your background is but try to research the theory and models governing the downwash and that should give you some insight into it. Also probably worth familiarizing yourself with the term circulation which is a measure of how much the wing is deflecting the air. If you google lifting line theory you should find some information on this. It is quite mathematical but I honestly do not think you can get a more detailed explanation then delving into some equations.

One last thing, if I remember correctly, the vortices do not cause the induced drag/downwash, the downwash causes the induced drag and the downwash also causes the vortex to form. Therefore the vortex is a consequence of the phenomena which causes the induced drag, it isn't the root cause of the induced drag. I mention this because you seem to be basing your thought process on the vortices as a starting point which is in incorrect, assuming that my memory serves me well...

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

B737200 wrote:
One last thing, if I remember correctly, the vortices do not cause the induced drag/downwash, the downwash causes the induced drag and the downwash also causes the vortex to form. Therefore the vortex is a consequence of the phenomena which causes the induced drag, it isn't the root cause of the induced drag. I mention this because you seem to be basing your thought process on the vortices as a starting point which is in incorrect, assuming that my memory serves me well...

The paper Aerodynamics for Naval Aviators states on page 66:
The tip vortices are coupled with the bound vortex when circulation is induced with lift. The effect of this vortex system is to create certain vertical velocity components in the vicinity of the wing. The illustration of these vertical velocities shows that ahead of the wing the bound vortex induces an upwash. Behind the wing, the coupled action of the bound vortex and the tip vortices induces a downwash. With the action of tip and bound vortices coupled, a final vertical velocity (2w) is imparted to the airstream by the wing producing lift. This result is an inevitable consequence of a finite wing producing lift. The wing producing lift applies the equal and opposite force to the airstream and deflects it downward. One of the important factors in this system is that a downward velocity is created at the aerodynamic center (w) which is one half the final downward velocity imparted to the airstream (2w).

The effect of the vertical velocities in the vicinity of the wing is best appreciated when they are added vectorially to the airstream velocity. The remote free stream well ahead of the wing is unaffected and its direction is opposite the flight path of the airplane. Aft of the wing, the vertical velocity (2w) adds to the airstream velocity to produce the downwash angle ϵ (epsilon). At the aerodynamic center of the wing, the vertical velocity (w) adds to the airstream velocity to produce a downward deflection of the airstream one-half that of the downwash angle. In other words, the wing producing lift by the deflection of an airstream incurs a downward slant to the wind in the immediate vicinity of the wing. Hence, the sections of the wing operate in an average relative wind which is inclined downward one-half the final downwash angle. This is one important feature which distinguishes the aerodynamic properties of a wing from the aerodynamic properties of an airfoil section.

The related illustration also taken from that paper:

So the tip vortices are indeed inducing the (additional) downwash?

B737200
Posts: 211
Joined: Sun Feb 20, 2005 5:30 pm

### Re: Why does induced drag decrease as airspeed increases?

As I said I may be mistaken, did this almost 10 years ago now. Could also be how you look at it, what causes what, etc. and honestly speaking these are all mathematical models of reality where the downwash and vortex exist concurrently and are intrinsically linked. I have also had my fair share of experiences when one professor somewhat contradicted what the other said.

Notwithstanding this I still think you should forget about the vortex for a bit and look at the downwash specifically since in the lifting line theory model that is what tilting your lift vector aft. There is a derivation that gives you an expression for downwash in terms of circulation.

I don't think there are any models for this apart from the lifting line theory, there was another method, I think it was called the panel method but I believe that was also based on the lifting line theory as well. For more localized detail on the specific state of the different parts of the air mass flowing around the wing and in its wake I would say that you need to go to CFD but with that I do not see how you can get a general understanding of the main parameters of wing planform, i.e. a relatively simple equation which shows you what Aspect Ratio does, etc.

Maybe someone else can chime in to prove me wrong or otherwise

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

But as far as I know the lifting line theory bases on (horseshoe) vortices inducing downwash.

[Same source as the illustration in my previous post]

Source wrote:
Near the wing the bound circulation due to lift leads to an up-wash ahead of the wing and downwash behind the wing similar to the flow produced by a two-dimensional lifting wing of infinite span.

A very important effect is generated by the flow due to the vortex pair that comprises the wake. The semi-infinite sheet of vorticity distributed in the wake produces a downward velocity component in the free-stream ahead of the wing, at the wing and far downstream [...]

The downwash by the wake leads to a reduction in the angle-of-attack of the wing relative to the free stream, reducing the lift. In addition the downwash rotates the oncoming flow vector at the wing leading to a component of drag [...]

The downwash produced by the circulation of the bound vortex of the wing is zero at the wing and therefore does not contribute to the induced drag.

I.e. the trailing vortices are the cause of the induced drag.

Topic Author
Posts: 11
Joined: Fri Jul 21, 2017 9:40 am

### Re: Why does induced drag decrease as airspeed increases?

To be more precise:
The downwash of the free trailing vortices causes the induced drag. And only the trailing vortices create a "true" downwash after the wing (see illustration above: vertical velocity: 2w) - without them there would be no downwash after the wing in the far field.

B737200
Posts: 211
Joined: Sun Feb 20, 2005 5:30 pm

### Re: Why does induced drag decrease as airspeed increases?

Hi,

Ok so I dug out my notes and yes the vortices are causing the downwash. So basically the vortices DO NOT cause the induced drag directly because the wing is dragging them around like an anchor but they ARE causing the downwash which is causing the induced drag. Piloting types I have been around spoke about the drag as being caused by the vortices because the wing is dragging them so that is where my memory of the vortices do not cause the drag came from.

So basically the higher pressure from the underside of the wing "leaks" up around the tips and causes the vortex which induces a downwash component. The downwash tilts the lift vector aft when you sum up all the velocity vectors of the airstream, this aft component is the induced drag force.

The value of this aft component is dependent on the strength of the downwash which is in turn dependent on the strength of the circulation. Recall how we said that the circulation is a measure of how much the wing is deflecting the air, how hard it is working. We said the at low speeds the wing must work harder ==> Larger Circulation, Larger Downwash, Larger Aft Component ==> Higher Induced Drag Coefficient.

In lifting line theory the downwash is used to tilt the lift vector aft so irrelevant of whether the downwash is caused by the vortex or not that is the equation you have to look at. We have however confirmed that the downwash is caused by the trailing vortex.

DougA
Posts: 1
Joined: Tue Feb 06, 2018 6:23 pm

### Re: Why does induced drag decrease as airspeed increases?

So, a little searching in Aerodynamics for Naval Aviatiors has yielded the following on page 69 -

"Such an effect would imply that a given airplane in steady flight would incur one-fourth as great an induced drag at twice as great a speed or four times as great an induced drag at half the original speed. This variation may be illustrated by assuming that an airplane in steady level flight is slowed from 300 to 150 knots. The dynamic pressure at 150 knots is one-fourth the dynamic pressure at 300 knots and the wing must deflect the airstream four times as greatly to create the same lift. The same lift force is then slanted four times as greatly and the induced drag is four times as great."

I'm not sure this provides the physical explanation you wanted but it seems the best way to explain it is that at higher airspeed, the wing has more dynamic pressure to work with, thus requiring it to not deflect the airstream as much. Since the airstream does not need to be deflected as much, the aft-pointing component of lift is also reduced, thus reducing induced drag.

Surprised no one found that explanation sooner!

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