Why is induced drag inversely proportional to airspeed SQUARED?

[Flyingwingbat1]

I learned that induced drag is inversely proportional to the amount of air a wing affects in a given amount of time. From what I've read, this can be interpreted as a cylinder of air with diameter directly proportional to wingspan, and length equal to the distance the plane covers in a given amount of time. Halve the wingspan, and the volume of the cylinder goes down 4x, hence induced drag rises 4x. Halve the airspeed, and the volume of the cylinder goes down 2x. From this, I'd be led to expect that induced drag would rise 2x, hence an inverse linear relationship with airspeed. So what's going on here? Am I misinterpreting the popular literature?

[ciurpita]

inversely proportional to airspeed squared is 1 / (v*v), which means it gets smaller as velocity increases. induced drag is proportional to lift. see
http://en.wikipedia.org/wiki/Induced_drag

induced drag is the result of downwash which makes the lift vector tilt backwards, retarding forward velocity, hence drag. the greater the lift per unit length of wing the greater the downwash. this is why induced drag is less (a smaller percentage o the lift) with higher aspect ratio wings (such as gliders, u-2)

[vintage1]

I think what the man is saying its that its profile drag that goes up as the velocity squared. 'Induced' drag is, for level flight', constant. Or certainly not v squared related.

Profile drag dominates the drag on most planes at any sorts of higher speeds anyway.

I did not think that induced drag varied with aspect ratio though.Its a function of the airfoild section surely, not its length..we quote lift to drag ratios for a section,not an aspect ratio..


Largely I had learnt that long thin wings minimize tip vortex losses. Hence they are god for minimum POWER to sustain flight at low to medium airspeeds.

At high speeds they have more profile drag, and are worse, which is why you wont see them on a Mach 2 fighter. Apart from being structurally weak.

[GeeW]

Induced drag increases with angle of attack. Typically at low speed the wing will be flying at a higher a of a and hence more drag. At high speed (but with the wing still generating lift upwards) a of a very small or even slightly negative, induced drag very small.

Gordon

[vintage1]

Yes to that..I forgot ..too much vintage brandy still in the caffeine stream.
:D

[biber]

Gentlemen, I think he means this:

http://upload.wikimedia.org/math/5/f/0/5f0051a1f9fb9acfcca7cf4811138884.png
(from this article: http://en.wikipedia.org/wiki/Induced_drag)

You can break that down into:

D_i(v) = c * 1/v²

where c is just a constant number that depends on several parameters like
air density 'rho', Lift 'L' (equal to the absolute value of TOW),
wing area, aspect ratio and an efficiency factor for the lift distribution.

Don't say FWB got it wrong, because he's perfectly right.

biber

[vintage1]

Ah. So "induced drag" DOES mean tip vortices in the wiki article.

I THOUGHT that was called something else, and that 'induced drag' was merely the vector rearwards component of lift on an 'infinite span wing'..or is it that there IS no rearwardd component with 'perfect laminar flow'?.

I actually think that is one of the WORST articles I have seen on Wiki.

[Flyingwingbat1]

The formula helps, and it illuminates a few things; if you double the span, while holding total wing area constant, the induced drag goes down 4x, because the AR term goes up 4x in the denominator. It also shows that, for a given speed, induced drag depends on SPAN loading. For example, take 2 otherwise identical planes flying at the same speed, but increase the chord on one of them by 2x. This DECREASES the aspect ratio(AR) term 2x, while increasing the wing area (S) by 2x. The net result is unchanged induced drag.

BUT, I still don't understand the PHYSICAL reason why induced drag varies as the inverse SQUARE of airspeed. I understand that it'll increase, because the wing's lift vector is "tilted back" more at the higher AOAs needed to produce the required lift (this explains why the "v" term is in the denominator). But why is the "v" term squared? :confused:

[biber]

The backward tilt is because of the finite span of a wing, the infinite wing doesn't show that tilt.

The tip vortices are not only vortices art the tip, they are a vortex layer trailing the whole wing.
Or where do tips end, anyway?
If you ask me, the tip extend right towards the wing center. ;)

I'd say aero stuff is not one of the better parts of wikipedia.
Yet you rarely find really good aero stuff anywhere else either.

biber

[vintage1]

The backward tilt is because of the finite span of a wing, the infinite wing doesn't show that tilt.



Ah.. now things start to become clearer. I never normally bother with aerodynamics. If its more or less balanced, it flies,..


The tip vortices are not only vortices art the tip, they are a vortex layer trailing the whole wing.
Or where do tips end, anyway?
If you ask me, the tip extend right towards the wing center. ;)


:D


So basically ALL that drag is turbulent drag essentially. ISTR somewhere buried in some course on fluid mechanics long ago, that a frictionless surfaced object in a full laminar flow leaves no wake and has no drag as a result..its all coming back..

I suspect thats how the V squared term cones in then..the energy imparted to a vortex of a given size (hence air mass) is proportional to the velocity squared...



I'd say aero stuff is not one of the better parts of wikipedia.
Yet you rarely find really good aero stuff anywhere else either.

biber

I know, I have NEVER found a decent thesis on basic aerodynamics. Possibly because even at its most simple, its quite complicated.

Now how about profile drag..that is also presumably micro vortices spinning along in the boundary layer?

[Flyingwingbat1]

It's becoming clearer now. I thought of the energy involved in the air circulation around a wing (much as Vintage just explained). Halve the speed, and the amount of circulation doubles. KE is 1/2mv^2, so the energy in those vortices rises 4x, hence 4x the induced drag, according to the 2 brain cells bouncing around in my skull.

[biber]

Now how about profile drag..that is also presumably micro vortices spinning along in the boundary layer?Profile drag is friction only and thus clearly distinguishable
from induced drag which has nothing to do with friction.
If you look into the airflow field behing the wing you find the traces of both sorts of drag.
You'll find lack of dynamic pressure directly behind the wing,
that is the due to the friction on the wing surface
and you have just encountered the effects of profile drag.
Then there is the vortex layer behind the wing, where is energy is lost in
accelerating air perpendicular to the direction of the free stream airflow.

Examining the flow field after a wing has passed by,
the air that stood still before the wing's passing by is now moving.
Its motion vectors components in the direction of the wings traveling vector represent the friction drag,
where all vector components perpendicular to that represent the induced drag.

bibeer

[vintage1]

I am not sure that I either buy the concept that friction applies to a solid/gas interface strictly, or that if it is so applied, it is friction 'as we know it Jim'.

Any Fule know that friction is a (more or less) constant force, whereas the drag of any object moving in air is not.

Friction is a result of micro bond breaking between two surfaces and it makes the surfaces hot.

It seems to me that until you get to landing space shuttles, or transonic flight, that's not the way it works: what happens is that the energy is transferred to a wake, and a wake is vortices.

I.e. in every case the mechanism of transferring the energy from the planes momentum to the air, is the creation of vortices: splitting it into 'induced' and 'profile' drag is just a mathematical and engineering convenience.

True friction is almost zero until you get to mach 1 plus, where skin heating starts to occur.?

Leastways that's as clear as I can see it right now, but I'm willing to be persuaded otherwise..

[ciurpita]

i stand corrected (thanks), and the curve on the wikipedia page does show the induced drag getting quickly smaller as velocity increases. what explains this?

i don't understand enough to be able to calculate the loss of energy and resultant drag due to the vortices created by the wing (which i thought were just another explanation for the vertical velocity of the air).

is it because the backwards tilt of the lift vector is proportional to the AOA, so that as airspeed increases and the AOA decreases, the backward tilt (downwash) decreases as well, reducing drag?

this makes sense if you consider that the vertical velocity of the air, the amount being forced down by the wing is proportional to lift, and therefore constant. at low airspeeds the vector resulting from horizontal and vertical velocities of the air is a relatively large angle. and as the horizontal velocity (airspeed) increases, the resulting vector angle gets smaller. the backward tilt decreases and there is less drag.

i doubt this is a perfectly correct explanation. but i think i could probably calculate the vertical velocity of air, and hence the tilt. for me, this is a more quantitative description of what might be happening. i know the vortices exist, but again, i don't know how to translate vortices into a drag force.

[kcaldwel]

http://www.rcgroups.com/forums/showthread.php?t=747983

[Flyingwingbat1]

I found this, which asserts that induced drag is proportional to 1/v, instead of 1/v^2 for a given aircraft. Who's right? :confused:

http://www.djaerotech.com/dj_askjd/dj_questions/induced_drag.html

[Flyingwingbat1]

Here's another assertion I found, relating to induced drag vs/ weight. Again, it somewhat contradicts the induced drag equation commonly found on the internet, (it shows that induced drag is directly proportional to weight^2) but the fellow is quite knowledgable from what I can tell. I still don't know who's right. :confused:

Don Stackhouse
Registered User


Join Date: Jun 2005
Posts: 321 Quote:
Originally Posted by canard addict
... I want it to be light and floaty. Most programs show my models stalling at 15 mph which steers me toward 25 mph level flight. The thin airfoil will reduce drag, add speed



Be careful here. Reducing drag does increase your top speed (penetrating ability), but it does not directly control your low speed performance ("float"). The way to reduce your minimum efficient flying speed (float) is by increasing lift. Lift is what controls how slow you can fly, not drag


Quote:
and call for heavier construction.



In absolute, qualitative terms, yes. In quantitative terms, the actual amount of increase might not be very significant. Also, if you go with an airfoil that's thick enough to suffer premature flow separation (which at the Re's where we work much of the time is not difficult to do), then you will need extra wing area to make up for the lost lift from that, and more wing area means more wing weight and more skin friction, both of which hurt performance.


Quote:
I will draw what looks good for strength,lightness and lift. The wing will probably be reduced to 72" and the smaller Medusa with smaller 1320 3 cell lipo used.



Which means you're carrying the same radio gear on a smaller span, so the savings in weight might not be enough to offset the induced drag penalty of the smaller span. Induced drag is linear with weight, but inversely proportional to te square of the span. OTOH, the effects on bending moments in the wing structure are pretty profound as well.

The important thing is to look at the net result of all the parameters together, and not get tunnel-visioned on any one parameter alone.

[biber]

I see, the point in Dons article, that you linked in #16,
but where in your post #17 does he talk about speed?
I see only weight (or say lift) and span.

biber

[kcaldwel]

Lift is the force the wing produces, perpendicular to the free stream velocity. The square term is real, and is there because force is proportional to the acceleration of the air, not just the final momentum ( M = m*v).

F = m*a = 1/2 *m * Delta v**2
a is proportional to the square of the velocity

This is Newtonian physics that applies to rockets throwing things out the back, or wings moving air downwards.

A very good article was referenced here on the physical description of flight the other day:

http://www.rcgroups.com/forums/showthread.php?t=794429

Hope that helps, and that I didn't make too many fundamental errors this early in the day.

Kevin

[Flyingwingbat1]

I see, the point in Dons article, that you linked in #16,
but where in your post #17 does he talk about speed?
I see only weight (or say lift) and span.

biber

I included the quote in #17 because, again, it contradicts what's shown in the induced drag equation. Now I think Don was talking about the relationship between induced drag and weight for a given AOA (i.e. same trim settings)

For instance, increase a gliders weight 9x, this makes the L^2 term in the induced drag equation go up 81x. But the "v^2" term goes up 9x. 81/9 = 9, which makes sense.

But from what I can tell, Don's explanation of induced drag linked in post 16 is partly wrong.

[Chas]

Any Fule know

Vint shouldn't that be "any fule kno"? Don't meet many Molesworth fans these days! Good on yer :D

C
(currently pinning out that wing of infinite span)

[KaneTheMediocr]

I think you are making this more complicated than it needs to be.

As the velocity increases, the amount of speed (relative to the aircraft) that some of the air loses as it goes by increases. Thats one factor of airspeed.

Also, as the velocity increases, the AMOUNT OF AIR that goes by the aircraft (and it slowed down) increases as well, which is the second factor of airspeed.

[floquet]

I found this, which asserts that induced drag is proportional to 1/v, instead of 1/v^2 for a given aircraft. Who's right? :confused:

You can find nice, simple (this is rare!) physical explanation of induced drag in "The simple science of flight" by Henk Tennekes, MIT Press 1997. It's a great little book.

And yes, it's inversely proportional to v^2, which was a big surprise in the beginning...

Hugh

[Don Stackhouse]

...But from what I can tell, Don's explanation of induced drag linked in post 16 is partly wrong.

Yup, I goofed. Induced drag is inversely proportional to the square of the airspeed. I was thinking about the mass flow (the size of the chunks of air that the wing makes lift from), and mentally somehow mixed that up with the induced drag itself.

The mass flow is linear with airspeed. However, as one of the other posters pointed out, when mass flow increases, the downward acceleration of the air required to make a given amount of lift, and the resulting velocity in the downwash, decreases as well. The result is a squared relationship between airspeed and induced drag.

What amazes me (besides the fact that I made such a fundamental mistake, and that I didn't catch it myself, and that nobody bothered to ask me directly about it, but instead tried to sort it out with a bunch of other folks!) was that it's taken this long for someone else to spot it!

However, the main point I was trying to make is that the induced drag is related to the square of the wing span, which is correct.

The classic equation for induced drag, which includes aspect ratio as one of the factors, is misleading. Aspect ratio only indirectly influences the induced drag, because of the effect it has on span. The actual cause of the change is the change in span, and therefore the change in mass flow.

If you hold the wing area constant, an increase in aspect ratio causes a change in span.

Conversely, if you hold span constant, a change in aspect ratio forces a change in wing area. Because of this, there is now an algebraic relationship between aspect ratio and lift coefficient. If you feed that back into the equation, you will find that the relationship simplifies into a function of span, span efficiency, airspeed, air density, and the amount of lift being made.

[Flyingwingbat1]

Great, thank you Don! I wasn't going mad after all.

[Texas Buzzard]

Flyingwingbat1 asked:
The original question was...." WHY does drag increase as the square of the increase in velocity?"

All the other subtopics should tend to confuse Flyingwingbat1. He wants to know WHY.
...............................................
I ask: If the dimensions of a 1.0 inch cube which had a mass of 100 grams were to be doubled to make it a 2.0 inch cube of the same substance - then what is the ratio of the 2.0 inch cube's mass to the mass of the 1.0 inch cube?

A. What is the volume of the 1.0 inch cube?
B. What is the volume of the 2.0 inch cube?
C. Will the increase in mas be proportional to the increase in volume?
D. Is the answer to the question 8:1? :) These are facts, this doesn't answer WHY.

Flyingwingbat1 asked WHY this is true.

A child asks, "Why is the sky blue?" Any student of physics can answer that.

Flyingwingbat1, doesn't this thing you are asking about apply to cars, planes or any object that is moving throught any liquid such as air?
I do not think anyone has answered WHY.

I didn't answer WHY either. This reminds me of college algebra for engineers. The first week we had to prove that 2-2=0 . Now that takes a whole page. Ha ha.

[Brandano]

But only for very close values of 2 :)

[macboffin]

Add to all of the foregoing, that if the plane moves fast enough the wings become merely stabilisers since the fuselage generates all the lift necessary.
Induced drag of F 105 fuselage at 1000 mph, anyone?

[Flyingwingbat1]

Flyingwingbat1 asked:
The original question was...." WHY does drag increase as the square of the increase in velocity?"

All the other subtopics should tend to confuse Flyingwingbat1. He wants to know WHY.
...............................................
I ask: If the dimensions of a 1.0 inch cube which had a mass of 100 grams were to be doubled to make it a 2.0 inch cube of the same substance - then what is the ratio of the 2.0 inch cube's mass to the mass of the 1.0 inch cube?

A. What is the volume of the 1.0 inch cube?
B. What is the volume of the 2.0 inch cube?
C. Will the increase in mas be proportional to the increase in volume?
D. Is the answer to the question 8:1? :) These are facts, this doesn't answer WHY.

Flyingwingbat1 asked WHY this is true.

A child asks, "Why is the sky blue?" Any student of physics can answer that.

Flyingwingbat1, doesn't this thing you are asking about apply to cars, planes or any object that is moving throught any liquid such as air?
I do not think anyone has answered WHY.

I didn't answer WHY either. This reminds me of college algebra for engineers. The first week we had to prove that 2-2=0 . Now that takes a whole page. Ha ha.


I think my question's been answered adequately. Thanks to all who responded. It apparently has to do with the kinetic energy (eventually dissipated via smaller and smaller eddies, down to the molecular level) imparted to each "parcel" of air being "grabbed" and shoved downwards. My understanding of the topic is still poor, but at least I can shake my head in dissapointment at the "equal transit time" theory in my COLLEGE PHYSICS book. Sigh.....

[vintage1]

I think my question's been answered adequately. Thanks to all who responded. It apparently has to do with the kinetic energy (eventually dissipated via smaller and smaller eddies, down to the molecular level) imparted to each "parcel" of air being "grabbed" and shoved downwards. My understanding of the topic is still poor, but at least I can shake my head in dissapointment at the "equal transit time" theory in my COLLEGE PHYSICS book. Sigh.....

IIRC these are true statements..

Air is viscous.
If it wasn't flight wouldn't work.
If it wasn't there would be no drag.
If it wasn't Bernouilli's would be an exact solution and flat plates wouldn't fly.
As it is, for reasonably laminar flow, Bernoulli is a very good approximation, and you just add the drag in as a separate issue.

On an aside note, as a lifetime engineer,. one realizes that there is an invisible and never explicitly stated philosophy in engineering.

"If its that hard to calculate, don't go there, unless you have to".


You will note that designers do NOT normally spend their time with supercomputers working out the absolutely optimal structure that will hold a roof up.Nor yet do they normally spend several weeks sticking modelling clay over every flush rivet in a spitfire, and removing them one by one to find out which affects airspeed the most.

Engineers have a simple yardstick of quality - "good enough".
It flies. The bridge doesn't fall down. It can be made at sensible cost.

Scientists seek exact solutions. Engineers are happy with 'better than that' or 'good enough'.

Scientists ask 'why?' Engineers simply want to know

'will it work?'
'how much will it cost?'
'Can we make it?'

The why and how of it are simply sometimes useful steps along this path, and are never pursued to the bitter end.

Somebody used the example. 'Why is the sky blue?' and posited a proposition that any student of physics could answer that one. Actually the question is unanswerable. If pursued to the bitter end it involves issues of human judgement and perception, the nature of causality itself, human language and culture, and so on.

The final answer is actually 'because it just is' or if you have the Faith, 'because God made it that way' (virtually identical statements at one level or another)

All science can do is attempt to relate an effect to what is presumed to be a cause..by that involves the start of an endless chain of causality that either ends in a statement 'thats the way it is' or something religious.

Fortunately as engineers, we try not to ask such questions. WE merely want to know 'what is the relationship between drag and airspeed and lift' and fortunately, we find that a simple set of equations allow us to be 'near enough' to calculate these things 'well enough' so that our planes are 'good enough' to fly.

Once upon a time I was briefly involved in designing military radio systems. Small ones. Tis was before the days of highly ntegrated circuits, and we had problems in packaging the stuff into a small space..it was getting unstable.

"you will never get more than 40dB stable gain per inch of circuit " said a senior engineer. "WHY?" we asked. "I don't know, and I don't care: Just that no one ever has". I spent days trying to figure out the answer, and eventually realised that it was actually a function of the size of the components. The old valves (tubes) were worse even covered with screening cans, transistors were better, integrated circuits even better, simply because the size of the conductors carrying signals and the area of other parts that might pick up those signals was so much lower. Burt with the technology we had, the engineers 'rule' was as we discovered, pretty much on the button.

A further example. WE needed to fabricate a simple beam to put a hoist on to lift some loads..someone asked me how big it should be. I had forgotten all the maths..I said 'I dunno, but if you give me a day or two, I can calculate it' - then a fitter spoke up.. 'A 6x4 steel I beam will work'. 'How do you know' 'because thats what I used one to lift a car engine out of my car over that distance, and it bowed a bit, but it didn't bend, and that thing weighs about the same'

He hadn't passed a single exam in his life, apart from a diesel fitters certificate, but the boy was an engineer at heart. He was, of course, right..

In the case of this argument, I have watched as it unfolded, learnt some, had to think some, but we seem to have arrived at a conclusion that relates it all to turbulence and Newton's mechanics.

Now why does Newtons mechanics work? aha. Off we go again.. ;)

[Texas Buzzard]

vintage 1,

Your musings are enjoyable and educational. Yes, you asked why and that is good. Most of us think like engineers and empirical data suffices.

Did you get a usable answer? The word "usable" implies a tinge of engineering, eh?

I remember that when this topic first came up in H.S. Physics we were asked ," How much horsepoer will be require to make this airplane fly at 200 MPH? The maximum airspeed of this plane is 100 MPH using a 100 horsepower engine."

[Neil Stainton]

Great post Vintage, thank you.

Neil.

[Don Stackhouse]

... Most of us think like engineers and empirical data suffices...

Excuse me, but this sounds like a bunch of general public types talking about their stereotype of engineers. Speaking as an engineer with both the degree and the real-world experience, the most important question to most of the GOOD engineers I know (and I've had the privilege of working with quite a few in my career) is "Why?".

Yes, we are generally pragmatists, and can get by just fine on detail decisions with empirically based answers. However, unless we understand the "why" on at least a qualitative basis, we can have no real confidence that an empirically based answer will work on a situation that differs even slightly from the original circumstances on which the method was developed. Only by understanding the "what", "how" and especially the "why" can we reliably predict the results in a new application.

[Chas]

A: Bernouilli's would be an exact solution and flat plates wouldn't fly.


B: He hadn't passed a single exam in his life, apart from a diesel fitters certificate,


A: Causality is beautifully arraigned as the source of that argument Vint, as I suspect it is of most differences in opinion concerning empirical observation. There should be no divergence of view when the commonwealth of human senses observes the indifferent Universe, but if your antagonist believes that these same events are caused by flux variations in Capricorn, well..

B: Essential for stevadore work in Liverpool. After ramming a forklift into the crate of lingerie, the cry goes up.."Diesel fitter!". :)

C

[ciurpita]

http://selair.selkirk.bc.ca/aerodynamics1/Drag/Page7.html

[kristianb]

Add to all of the foregoing, that if the plane moves fast enough the wings become merely stabilisers since the fuselage generates all the lift necessary.
Induced drag of F 105 fuselage at 1000 mph, anyone?

At sea level a good approximation would be 0 N!
Strange question by the way!
vbr KristianB

[tommyeflight89]

IIRC these are true statements..



Scientists ask 'why?' Engineers simply want to know

'will it work?'
'how much will it cost?'
'Can we make it?'



Awesome post Vint. I'm making my way through first year Aerospace right now and I enjoyed that. The beam example was funny too. Hands on experience helps just as much as equations and calculus. 99% of the people in my Aerospace classes can't even tell you how lift works or what dihedral does. I always thought that was odd, I'm in this because I'm very very interested in airplanes, while they seem to not have a clue. I guess they are here to find out.

Relating to the quote:
Art grauduates ask:
'Would you like fries with that?'. :D