Who Keeps them Up?
This is the original of an article that I sent to Radio Control Model World (RCMW); it appears in the September 2001 addition of the magazine.
Unfortunately they managed to print most of the equations (and there aren't very many!) incorrectly. They also messed up the formatting so that the mag article is badly flawed. I would be surprised if anybody could follow the arguments clearly.
However I can assure you that everything here is as I intended it to be. I don't think it is very difficult, and I have kept the maths and physics to an absolute minimum, and I hope it helps you to understand exactly what is going on when our models (and other winged aircraft for that matter) fly.
The article generated a response from John Gibson in the February 2002 edition of RCMW in which he describes in some detail how lift can be calculated using the principles of circulation. He missed the point except in the first half dozen lines where he notes that " the Bernoulli or the popular theory...is a partial adulterated version appearing only in uninformed aviation and model magazines and pilot training manuals." Exactly John, that is just the problem. He then goes onto to give a detailed history of the theory of lift with lots of numbers and technical expressions which I believe are of very little use to the average modeller trying to achieve a basic (and correct) understanding of lift.
I have also found an American Physics teaching site which continues the debate, plus a discussion board. The former can be found at http://www.amasci.com/wing/airfoil.html and the latter at http://www.amasci.com/wing/wphysl.txt. The discussion board is a bit old but and takes some wading through but it does show the range of arguments for and against the Bernoulli explanation. It also finishes up with the following quote:
"As an aside, from my experience as a pilot I can attest that what 'really' makes airplanes fly is MONEY! (There is a wonderful T-shirt for pilots that carries the slogan "If God had meant man to fly, he would have given him more money.") ".....from Roger A. Freedman of the Department of Physics and College of Creative Studies University of California, Santa Barbara.
So if you don't have a closed mind and would like to understand a bit more about lift read on. If you think you know it all, happy landings, it must be nice to be perfect.
Who Keeps Our Models Flying Bernoulli or Newton?
If you have ever asked the perfectly reasonable question " Why do aircraft fly?" you will almost certainly have found or been given an explanation based on Bernoulli’s Theorem. This says that air moves over the top of the wing faster than the bottom, faster moving air results in lower pressure, and the low pressure on the top of the wing keeps your plane up. Easily stated and sounds very reasonable doesn’t it? The trouble is it’s wrong; despite being offered as the ‘standard’ explanation for more years than you can shake a stick at, it is fundamentally incorrect. It also precludes explanation of some aspects of flight and prevents true understanding of the phenomena of lift. A correct explanation based on Newton’s three laws of physics allows both. Want to know how and why? – then read on.
In the beginning
There can be few phenomena so generally studied, and so important to modern life, which are as misunderstood as flight. In general what we have been taught as the physical explanation for lift from a wing is incorrect. As you will see later on the ‘standard’ description doesn’t even make sense; it gives incorrect values for the lift generated and can’t explain many things, for example inverted flight!
As a physicist that designs and flies model aircraft, and has a general interest in all matters aviation, I too have been fed the standard explanation. Even though it did seem to me to leave a number of questions unanswered I have never bothered to investigate why. As an undergraduate I was even exposed to something know as the "Kutta-Joukowski circulation condition" to explain lift from an aerofoil. While this explanation (which is too mind boggling to attempt here) is mathematically correct, and no doubt useful to ‘real’ aeronautical engineers it doesn’t give an intuitive explanation of lift, or one that is of much us to us amateur aeronautical types!
Recently David Anderson of Fermilab (more associated with High Energy Physics than aeronautics, but heavily involved in many subjects) has been trying to put the record straight. He and co-author Scott Eberhardt have published some papers and a book1 describing the lift of a wing using Newton’s laws of action and reaction.
The explanation of lift that David Anderson is championing, and which I will describe in this article, does give a good intuitive explanation of what is going on. It allows other flight related phenomena to be explained and understood in a way that the Bernoulli method doesn’t and can’t. At the end of the day, knowing the truth won’t make your models fly any better, and aeronautical engineers will still continue to use their highly complex mathematical methods to design aircraft. But it might make it easier to understand why aircraft do the seemingly impossible and stay airborne.
The Bernoulli explanation
We have all been taught in one form or another the Bernoulli description of lift. This generally presents a picture of an aerofoil similar to that in figure 1, in which streamlines are shown moving over and under the wing.
Figure 1: Typical image of wing with streamlines
It is then stated (without proof) that when the air separates at the leading edge it must move faster over the top surface in order to travel the greater distance and met again at the trailing edge. The explanation then continues that as the air is moving faster over the top surface there is a reduction in pressure in accordance with Bernoulli’s theorem. This small reduction in pressure acting over the whole wing surface generates the lift. When one stops to consider this explanation in detail there are a number of flaws in it.
In the first instance why does the air need to meet up again at the trailing edge. In fact it doesn’t always. Figure 2 shows the airflow over a wing in a wind tunnel where smoke is introduced periodically.
Figure 2: Airflow over a wing in a windtunnel with smoke pulses
One can see that the air that goes over the top of the wing gets to the trailing edge considerably before the air that goes under the wing, not at the same time at all. Also, on close inspection one sees that the air going under the wing is slowed down from the "free-stream" velocity of the air.
Secondly Bernoulli’s equation only applies to fluid under strict conditions namely the fluid must be frictionless and of constant density; the flow must be steady, and the relation holds in general only for a single streamline. A streamline is a dimensionless line in a fluid along which various conditions remain constant including pressure. Where a fluid is being accelerated, as it is with an aerofoil, we require a pressure gradient.
Thirdly (is there such a phrase as thirdly?) even if we ignore this and assume that we can apply the theorem, the numbers come out wrong. For example the difference in length over the top and bottom surface of a typical aerofoil is only about 1.5%. If this relates directly to a speed difference the numbers can be worked to calculate the lift generated. In the form most usually employed Bernoulli’s equation states that:
p + ½rv2 = C
where p is the pressure, r the density of air, v the speed of the air, and C a constant. If we take C to be constant above and below the wing we can re-write this expressing the difference in air pressure above and below as:
Dp = ½r (vT2 – vD2)
where now vT and vD are the air velocity above and below the wing, and Dp is the pressure difference.
Let us carry out this exercise for a typical model, let’s say a WOT 4. The WOT 4 has a wing area of circa 4 sq. ft or 0.37 sq. metres, weighs about 5 lbs or 2.3Kg (mine did anyway!) and probably flies at about 50mph or 22 metres per second. Putting these values into the second equation we find that at this speed the wing only generates 15% of the required lift. Alternatively the model would need to fly at 125 mph or the wing would have to provide 50% path difference top to bottom. This would require a wing as thick as its chord.
Finally the Bernoulli based explanation doesn’t answer a number of questions as follows:
How does an aeroplane with an asymmetric wing fly inverted?
Invert the wing and the airflow is now faster on the bottom of and the pressure increase is down not up.
How do aeroplanes with a symmetric wing fly at all?
Without a path difference there cannot be any pressure difference.
How does a wing adjust for a change in load at a constant speed?
A jumbo jet can be 40% fuel by weigh at takeoff and yet fly at a constant speed for the entire flight.
Of course the answer to all of these is that the aeroplane trim is adjusted so that the angle the wing makes to the airflow (the angle of attack) is changed. The problem is the Bernoulli based explanation offers no way to include the angle of attack of the wing as a factor in lift generation.
So I hope that by now I have convinced you that the Bernoulli based explanation is wrong. Not only is it qualitatively incorrect in that it cannot in simple terms explain many standard features of wing lift, but it is also quantitatively incorrect, and under estimates the lift generated by huge margins.
Lift according to Newton
With a little help from Mr. Coanda
So what does generate lift and keep all our precious models up in the air (not to mention ourselves and others when jetting off around the world)? In order to describe how lift is generated we will need to review Newton’s laws of motion. I am sure you all remember these from physics at school but just in case you are a little rusty here they are again.
Newton’s First law: a body at rest will remain at rest, or a body in motion will continue in straight-line motion unless subjected to an external applied force.
Rarely seen on Earth as gravity and air resistance get in the way; but can be approximated in low friction situation such as ice-skating i.e. you won’t stop until you crash into the wall around the side of the rink.
Newton’s Second law: F= ma, or force equals mass times acceleration.
This is why we all want to drive Ferraris and not buses i.e. lightweight cars with powerful engines accelerate like the clappers and are much more fun!
Newton’s Third law: for every action there is an equal and opposite reaction.
This is not the sort of reaction you get when your wife/girlfriend/partner (delete as appropriate) finds out that you have just bought another 4-stroke engine and goes out and blows a few hundred quid on clothes!
Referring to the typical Bernoulli based image of stream lines around a wing (figure 1) again it shows
Figure 1: Typical image of wing with streamlines
the air approaching from the horizontal and leaving in the same direction. From a Newtonian point of view the net change of direction of the air is zero so there is no net force acting on it (First Law). Since there is no net force on the air there can be no net force on the wing (Third Law).
Figure 3: Streamlines as they should be drawn
Figure 3 shows the streamlines, as they should be drawn. The air passes over the wing and is bent up and down. Why is it bent? Because it is sticky – or to use the scientific term it has viscosity. In fact the streamline situation shown in Figure 1 would only occur if the fluid where not viscous and lacked friction, as required by the conditions that apply to the Bernoulli equation. However we are dealing with air, which does have viscosity, albeit small it is significant.
The fact that a fluid will bend and follow a curved surface was first realised by Henri-Marie Coanda (1885-1972) a Romanian aerodynamicist, and the effect now bears his name. To demonstrate the Coanda effect touch a horizontal glass or similar smooth curved object to a stream of water from a tap. The stream of water will wrap part way around the glass as illustrated by Figure 4.
Figure 4: Demonstration of the Coanda effect
Now Newton’s first law says that there must be a force on the stream of water in the direction of the glass to cause it to bend. Newton’s third law says there must be an equal and opposite force on the glass. The glass feels a force towards the water not away from it was one might ‘guess’.
If you don’t believe this suspend a light weight smooth cylinder from some string and move it until it touches the water stream, it will be pulled further into the stream on contact with the water. The lighter the object for a given diameter the better; vitamin pill bottles work quite well! Try varying the flow rate to see the effect it has on the force produced on the cylinder.
So now we have all the elements of lift from a wing in place. As the air flows over the curved upper surface it is bent in accordance with the Coanda effect. Newton’s first law says that there must be a force on the air to bend it down (the action). Newton’s third law says that there must be an equal and opposite force (up) on the wing (the reaction).
This reaction is the lift. Case proven we can all go home – not quite. So far we have got to the same stage as the Bernoulli explanation, we have a plausible description of how lift is produced. What we now need to do is show that this explanation can produce the right amount of lift and explain aspects of lift that Bernoulli can not. I shall cover that in detail in part II of this article.
The wrong-Newtonian theory
Before we leave however I would also like to despatch another theory of lift to the dustbin, one that might be called the wrong-Newtonian theory. This description states that diverting air down produces lift, and that this lift is a reaction force. This part is true, as we have seen. Unfortunately in the wrong-Newtonian description it goes on to say that lift is produced by impact of the air with the bottom of the wing. It is interesting to note that this was the view of lift held by Sir Isaac Newton himself, so even the best people can be fooled! Although there is a little of this kind of lift, for most wings its is minimised by efficient wings. In the extreme case of the barn door (a very inefficient wing indeed) about 40% of the lift is due to air impacting the wing bottom. The rest is due to the reaction of the diverted air as for a ‘normal’ wing. As a result of generating lift in this fashion the barn door wing is very ‘draggy’, and to over come this drag you need lots of power (strap on a bigger engine!). What drag and power mean and how their relationship with lift and speed varies will also be explored in the second part of this article, again by using the correct Newtonian description of lift.
Part II The proof of the pudding
We have shown in part I of this article that lift on a wing is generated by a reaction force that occurs when air is diverted over the curved surface of the wing. Now let us see if we can put some numbers to this reaction and estimate the amount of air required to generate this lift. First let us look at where the wing will get the air that it diverts to generate the lift.
Some things we know:
- At the surface of the wing the air is diverted around the wing
- At some distance above the wing the air is not diverted
- At the tips of the wing the air is not diverted
From these facts we can deduce qualitatively that the volume of air affected by the wing will look something like that shown in Figure 5.
Figure 5: Volume of air affected by wing
Unfortunately the calculation of this volume is somewhat complex and beyond my limited ability to break down into simple terms. However we can easily estimate the magnitude of the air that the model must divert in order for it to create lift. We can then estimate the volume of air required, and we will see that it is not a localised surface effect, as implied by the Bernoulli explanation, but involves a considerable a mass and volume of air.
First let as look at the downward velocity of the air as it leaves a wing, again using that stalwart model the WOT 4 as an example. The degree to which air is deflected is fairly simple to estimate. If we have a wing operating at an angle of attack a as shown in Figure 6, we can simplify matters and assume that as the air leaves the trailing edge it follows a straight path at an angle a to the horizontal.
Figure 6: Downwash and angle of attack
Again if we use the WOT 4 example, at 50 mph and angle of attack of 5°, the air will be moving down at the trailing edge of the wing at circa 4 mph or 2 m/sec. Now not all of this will contribute to lift, as the air will have had to move up over the leading edge to start with. So let as assume that vertical velocity of the air contributing to the lift is only 50% of the value estimated or 1 m/sec.
We can use Newton’s second law to calculate the required rate at which air is diverted in order to lift the model. Newton’s second law states that is F= ma, or force equals mass times acceleration. Acceleration has the units of velocity per second, so that the equation can be rewritten as
Force = momentum x velocity where momentum has the units of mass per sec.
Now the weight or the mass of the model is 5lbs or 2.3kg, and the force required to hold it up will be this mass multiplied by gravity, which gives a value of 23 Newtons. (Force, weight and mass are often confused as the same thing, see end of article for further explanation if you are not sure of the difference). Hence the required mass per second to give 23N of force at 1 m/sec velocity will be 23 kg/sec. So the model is diverting ten times its own mass per second downwards to generate the necessary lift.
The fact that the wing is diverting down huge quantities of air to generate lift is shown in this image of a plane flying over fog. The hole pushed in the fog by the downwash can clearly be seen.
Downwash and wing vortices in the fog.
(Photographer Paul Bowen, courtesy of Cessna Aircraft, Co.)
Now we are in a position to estimate the size of the ‘air scoop’ above the wing described earlier. For simplicity let us assume that the scoop is a rectangle above the wing, and that the whole wing span contributes equally. Given that the density of air near sea level (where most of us fly our models!) is circa 1.2kg/cu.m. we can calculate that we need to divert a volume of 20 cu. m of air per sec. Again given that the WOT4 is lying at 50mph or 22 m/sec with a wing span of 1.3m the height of the air column above the wing will need to be 0.7m or 27 inches!.
As I said earlier this is not a localised surface effect, as implied by the Bernoulli explanation, but involves considerable a mass and volume of air. This fact helps explain why biplane designers found that staggering the wings was an advantage. Without this stagger the lower wing would be ‘starved’ of air by upper wing, and therefore generate less lift.
Lift as a function of angle of attack
We have seen that the lift of a wing is equal to the change in momentum of the air it is diverting down. An alternate form of Newton’s second law could be written as:
The lift of a wing is proportional to the amount of air diverted down times the vertical velocity of that air
It is that simple. For more lift the wing can either divert more air or increase the diverted air’s vertical velocity.
From Figure 6 we can see that the greater the angle of attack of the wing the greater the vertical velocity of the air. Likewise, for a given angle of attack, the greater the speed of the wing the greater the vertical velocity of the air. Both the increase in the speed and the increase of the angle of attack increase the length of the vertical velocity arrow. It is this vertical velocity that gives the wing lift.
So to increase lift you either increase your wings angle of attack or you increase your forward speed (or both). This is a basic fact of flight that can not be explained by the Bernoulli based explanation of lift.
There are many types of wing: conventional, symmetric, conventional in inverted flight, the early biplane wings that looked like warped boards, and even the proverbial "barn door". In all cases, the wing is diverting the air down. What each of these wings has in common is an angle of attack with respect to the oncoming air. It is the angle of attack that is the primary parameter in determining lift.
To better understand the role of the angle of attack it is useful to introduce an "effective" angle of attack, defined such that the angle of the wing to the oncoming air that gives zero lift is defined to be zero degrees. If one then changes the angle of attack both up and down one finds that the lift is proportional to the angle. Figure 7 shows the lift of a typical wing as a function of the effective angle of attack. A similar lift versus angle of attack relationship is found for all wings, independent of their design. This is true for the wing of a 747, an inverted wing, or your hand out the car window. The inverted wing can be explained by its angle of attack, despite the apparent contradiction with the popular explanation of lift. A pilot adjusts the angle of attack to adjust the lift for the speed and load. The role of the angle of attack is more important than the details of the wing’s shape in understanding lift. The shape comes into play in the understanding of stall characteristics and drag at high speed.
Fig 7 Lift versus the effective angle of attack.
Typically, the lift begins to decrease at a "critical angle" of attack of about 15 degrees. The forces necessary to bend the air to such a steep angle are greater than the viscosity of the air will support, and the air begins to separate from the wing. This separation of the airflow from the top of the wing is a stall. However up until that point the fact that lift is proportional to the angle of attack follows directly from the Newtonian description of lift outlined above.
Lift, power and drag.
Most texts on aeronautics (model or full size) seldom mention power, and introduce something-called drag, which has a complex relationship to speed. The source of this drag is generally ‘glossed over’ and such texts have a habit of making ones eyes ‘glaze over’ when reading them. However with some simple physics and more application of Newton’s laws it is relatively easy to explain the relationship between speed, power and drag, at least qualitatively.
When a plane passes overhead the still air gains a downward velocity. Thus, the air is left in motion after the plane leaves. The air has been given energy. Power is energy, or work, per time. So, lift requires power. This power is supplied by the aeroplane’s engine (or by gravity and thermals for a glider).
How much power will we need to fly?
If one fires a bullet with a mass, m, and a velocity, v, the energy given to the bullet is simply ½mv2.
Likewise, the energy given to the air by the wing is proportional to the amount of air diverted down times the vertical velocity squared of that diverted air. We have already seen that the lift of a wing is proportional to the amount of air diverted times the vertical velocity of that air. Thus, the power needed to lift the aeroplane is proportional to the weight times the vertical velocity of the air. If the speed of the plane is doubled the amount of air diverted down doubles. Thus to maintain a constant lift, the angle of attack must be reduced to give a vertical velocity that is half the original. The power required for lift has been cut in half. This shows that the power required for lift becomes less as the aeroplane’s speed increases. In fact, we have shown that this power to create lift is proportional to 1/speed of the plane.
But, we all know that to go faster in straight and level flight we must apply more power. So there must be more to power than the power required for lift. The power associated with lift is often called the "induced" power. Power is also needed to overcome what is called "parasitic" drag, which is the drag associated with moving the none lifting objects through the air e.g. wheels, undercarriage, wing walking Barbie doll, etc..
The energy the aeroplane imparts to an air molecule on impact is proportional to the speed squared (again from E = ½mv2).
The number of molecules struck per time is proportional to the speed. The faster one goes the higher the rate of impacts. Thus the parasitic power required to overcome parasitic drag increases as the speed cubed.
Figure 8 shows the "power curves" for induced power, parasitic power, and total power (the sum of induced power and parasitic power). Again, the induced power goes as 1/speed and the parasitic power goes as the speed cubed. At low speed the power requirements of flight are dominated by the induced power. The slower one flies the less air is diverted and thus the angle of attack must be increased to increase the vertical velocity of that air.
Fig 8 Power requirements versus speed.
In straight and level flight the power requirement is dominated by parasitic power. Since this goes as the speed cubed an increase in engine size gives one a faster rate of climb but does little to improve the cruise speed of the plane. Doubling the size of the engine will only increase the cruise speed by about 25%.
Since we now know how the power requirements vary with speed, we can understand drag, which is a force. Drag is simply power divided by speed. Figure 9 shows the induced, parasitic, and total drag as a function of speed. Here the induced drag varies as 1/speed squared and parasitic drag varies as the speed squared. Taking a look at these figures one can deduce a few things about how aeroplanes are designed. Slower aeroplanes, such as gliders, are designed to minimise induced power, which dominates at lower speeds. Faster propeller-driven aeroplanes are more concerned with parasite power, and jets are dominated by parasitic drag.
Fig 9 Drag versus speed.
And Finally folks!
It is also possible to extend the arguments still further and explain the features of wing efficiency, power and wing loading, wing vortices and ground effect but I think I will quit while I’m ahead (at least I hope I’m ahead!). However before I finish let me summarise what have learned and how the physical description of lift has given us a greater ability to understand flight. First what have we learned:
- The amount of air diverted by the wing is proportional to the speed of the wing.
- The vertical velocity of the diverted air is proportional to the speed of the wing and the angle of attack.
- The lift is proportional to the amount of air diverted times the vertical velocity of the air.
- The power needed for lift is proportional to the lift times the vertical velocity of the air.
Now let us look at some situations from the physical point of view and from the perspective of the popular Bernoulli explanation.
- The plane’s speed is reduced. The physical view says that the amount of air diverted is reduced so the angle of attack is increased to compensate. The power needed for lift is also increased. The popular explanation cannot address this.
- The load of the plane is increased. The physical view says that the amount of air diverted is the same but the angle of attack must be increased to give additional lift. The power needed for lift has also increased. Again, the popular explanation cannot address this.
- A plane flies upside down. The physical view has no problem with this. The plane adjusts the angle of attack of the inverted wing to give the desired lift. The popular explanation implies that inverted flight is impossible.
As one can see, the popular explanation, which fixates on the shape of the wing, may satisfy many but it does not give one the tools to really understand flight. The physical Newtonian description of lift is easy to understand and much more powerful. If you are interested in reading more, please see David Anderson and Scott Eberhardts’ book "Understanding Flight."1
Acknowledgements: I am grateful to David Anderson for supplying material, allowing me to use extracts from his publications and for his comments and corrections, and to Chris Foss for illustrations of his WOT 4.
Ref 1: "Understanding Flight", by David Anderson and Scott Eberhardt, McGraw-Hill, 2001, ISBN: 0-07-136377-7
Mass, Force and Weight (a layman’s guide)
We often confuse these three terms, and it is understandable because we live on Earth, and Earth has a permanent gravity field (and it’s a jolly good thing it does to!).
If we lived in deep space away from massive objects like stars and planets we wouldn’t have any concept of weight and so there would be no confusion. We would then know that an object’s mass, expressed in kilograms, would allow you to calculate how fast it would accelerate when acted upon by a force, expressed in Newtons.
That is a a 1kg mass acted upon by a 1N force would accelerate at a rate of 1 metre per second per second (m/s2) i.e. for every second the force was applied its speed would increase by one metre per second.
The confusion arises when we move to a gravitational field like Earth’s. Gravity has the same effect on a mass as acceleration. The gravitational field of the Earth near its surface is equivalent to an acceleration of about 9.8 m/s2.
Hence a 1kg mass at the Earth’s surface is trying to accelerate towards the Earth at 9.8 m/s2.
If we stop it by holding on to it we need to apply a force of 9.8N to it.
It is this force that we call the weight of the object. We then compound the confusion by saying it weighs 1kg! Of course if the gravitational field was less e.g. on the Moon or further away from the Earth, the 1kg mass would see less than 9.8N force, and we would say it weighs less than 1kg, where as its mass hasn’t changed at all.